Research Article The Dynamic Programming Method of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 958920 14 pageshttpdxdoiorg1011552013958920

Research ArticleThe Dynamic Programming Method of Stochastic DifferentialGame for Functional Forward-Backward Stochastic System

Shaolin Ji1 Chuanfeng Sun2 and Qingmeng Wei2

1 Institute for Financial Studies and Institute of Mathematics Shandong University Jinan 250100 China2 Institute of Mathematics Shandong University Jinan 250100 China

Correspondence should be addressed to Shaolin Ji jslsdueducn

Received 23 October 2012 Accepted 18 December 2012

Academic Editor Guangchen Wang

Copyright copy 2013 Shaolin Ji et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is devoted to a stochastic differential game (SDG) of decoupled functional forward-backward stochastic differentialequation (FBSDE) For our SDG the associated upper and lower value functions of the SDG are defined through the solutionof controlled functional backward stochastic differential equations (BSDEs) Applying the Girsanov transformation methodintroduced by Buckdahn and Li (2008) the upper and the lower value functions are shown to be deterministic We also generalizethe Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations to the path-dependent ones By establishing the dynamic programmingprincipal (DPP) we derive that the upper and the lower value functions are the viscosity solutions of the corresponding upper andthe lower path-dependent HJBI equations respectively

1 Introduction

The theory of backward stochastic differential equations(BSDEs) has been studied widely since Pardoux and Peng[1] first introduced the nonlinear BSDEs in 1990 BSDEshave got applications in many fields such as stochasticcontrol (see Peng [2]) stochastic differential games (SDGs)(see Hamadene and Lepeltier [3] Hamadene et al [4])mathematical finance (see El Karoui et al [5]) and partialdifferential equation (PDE) theory (see Peng [6 7]) and soforth

In the aspect of finance the BSDE theory presents a sim-ple formulation of stochastic differential utilities introducedby Duffie and Epstein [8] When the generator 119892 of a BSDEdoes not depend on 119911 the solution 119884 is just the comparisontheorem for BSDEs we know

recursive utility presented in [8] From the view of BSDEby studying some important properties (such as comparisontheorem) of BSDEs El Karoui et al [5] gave the more generalclass of recursive utilities and their properties And later therecursive optimal control problems whose cost functionalsare described by the solution of BSDEare studiedwidely Peng

[7] obtained the Bellmanrsquos dynamic programming principle(DPP) for this kind of problem and proved the value functionto be a viscosity solution of one kind of quasi-linear secondorder PDE that is Hamilton-Jacobi-Bellman (HJB) equationLater for the recursive optimal control problem introducedby a BSDE under Markovian framework by introducing thenotion of backward semigroup of BSDE in Peng [9] theBellmanrsquos DPP is derived and the value function is proved tobe a viscosity solution of a generalized HJB equation

By now the DPP with related HJB equation has becomea powerful approach to solving optimal control and gameproblems (see [10ndash13]) In [10] Buckdahn and Li studieda recursive SDG problem and interpreted the relationshipbetween the controlled system and the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation A point is worthy to men-tion in order to derive the DPP they introduced a Girsanovtransformation method to prove the value functions aredeterministic which is different from the method developedin Peng [9]

There really exist some systems which are modeled onlyby stochastic systems whose evolutions depend on the pasthistory of the states Based on this phenomenon Ji and Yang

2 Mathematical Problems in Engineering

[14] investigated a controlled system governed by a functionalforward-backward stochastic differential equation (FBSDE)and proved the value function to be the viscosity solution ofthe related path-dependent HJB equation

In this paper inspired by [10 14] we will investigatethe SDG problems of the functional FBSDEs Preciselythe dynamics of our SDGs is described by the followingfunctional SDE

119889119883120574119905119906119907

(119904) = 119887 (119883120574119905119906119907

119904 119906 (119904) 119907 (119904)) 119889119904

+ 120590 (119883120574119905119906119907

119904 119906 (119904) 119907 (119904)) 119889119861 (119904) 119904 isin [119905 119879]

119883120574119905119906119907

119905= 120574119905

(1)

And the cost functional 119869(120574119905 119906 119907) is defined as119884120574119905119906119907

(119905)whichis the solution of the following functional BSDE

119889119884120574119905119906119907

(119904) = minus 119891 (119883120574119905119906119907

119904 119884

120574119905119906119907

(119904) 119885120574119905119906119907

(119904)

119906 (119904) 119907 (119904)) 119889119904 + 119885120574119905119906119907

(119904) 119889119861 (119904)

119884120574119905119906119907

(119879) = Φ (119883120574119905119906119907

119879) 119904 isin [119905 119879]

(2)

where 120574119905 is a path on [0 119905] The driver 119891 and Φ canbe interpreted as the running cost and the terminal costrespectively Also they depend on the past history of thedynamics Equations (1) and (2) compose a decoupled func-tional FBSDE The concrete conditions on 119887 120590 119891 Φ areshown in the later section

In the context we adopt the strategy against control formThe cost functional 119869(120574119905 119906 119907) is explained as a payoff forplayer I and as a cost for player II The aim of this paper isto show the following lower and upper value functions

119882(120574119905) = essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906))

119880 (120574119905) = esssup120572isinA119905119879

essinf119907isinV119905119879

119869 (120574119905 120572 (119907) 119907)

(3)

are the viscosity solutions of the following path-dependentHJBI equations respectively

119863119905119882(120574119905) + sup119906isin119880

inf119907isin119881

119867(120574119905119882119863119909119882119863119909119909119882119906 119907) = 0

119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ

119863119905119880(120574119905) + inf119907isin119881

sup119906isin119880

119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907) = 0

119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ

(4)

where

119867(120574119905 119910 119901 119883 119906 119907) =1

2tr (120590120590119879 (120574119905 119906 119907)119883)

+ 119901 sdot 119887 (120574119905 119906 119907)

+ 119891 (120574119905 119910 119901 sdot 120590 (120574119905 119906 119907) 119906 119907)

(5)

where (120574119905 119910 119901 119883) isin Λ times R times R119889times S119889 (S119889 denotes the set of

119889 times 119889 symmetric matrices)To solve the above SDG we need the functional Itorsquos

calculus and path-dependent PDEs which are recently intro-duced by Dupire [15] (for a recent account of this theorythe reader may consult [16ndash18] And under the framework offunctional Itorsquos calculus for the non-Markovian BSDEs PengandWang [19] derived a nonlinear Feynman-Kac formula forclassical solutions of path-dependent PDEs For the furtherdevelopment the readers may refer to [20 21])

In this paper we apply the Girsanov transformationmethod in Buckdahn and Li [10] to prove the determinacyof the value functions which is different from the methodintroduced by Peng [7 9] Making use of this methodand the functional Itorsquos calculus (introduced by Dupire [15]and developed by Cont and Fournie [16ndash18]) we completethe study of the zero-sum two-player SDGs in the non-Markovian case and present the lower and upper valuefunctions of our SDG are the viscosity solutions of thecorresponding path-dependent HJBI equations respectively

Different from the HJBI equations developed for stochas-tic delay systems we establish the DPP and derive the HJBIequation in the new framework of functional Itorsquos calculus

This paper is organized as follows Section 2 recalls thefunctional Itorsquos calculus and the well-known results of BSDEswe will use later In Section 3 we formulate our SDGs andget the corresponding DPP Based on the obtained DPP inSection 4 we derive the main result of the paper the lowerand upper value functions are the viscosity solutions of theassociated path-dependent HJBI equations respectively Andwe add the proof for the DPP in the appendix

2 Preliminaries

21 Functional Itorsquos Calculus We present some preliminariesfor functional Itorsquos calculus introduced firstly by Dupire [15]Here we follow the notations in [15]

Let 119879 gt 0 be fixed For each 119905 isin [0 119879] we denote Λ 119905 theset of cadlag functions from [0 119905] to R119889

For 120574 isin Λ 119879 denote 120574(119904) by the value of 120574 at time 119904 isin [0 119879]Thus 120574 = (120574(119904))0le119904le119879 is a cadlag process on [0 119879] and its valueat time 119904 is 120574(119904) 120574119905 = (120574(119904))0le119904le119905 isin Λ 119905 is the path of 120574 upto time 119905 We denote Λ = ⋃

119905isin[0119879]Λ 119905 For each 120574119905 isin Λ and

119909 isin R119889 120574119905(119904) is denoted by the value of 120574119905 at 119904 isin [0 119905] and120574119909

119905= (120574119905(119904)0le119904lt119905 120574119905(119905) + 119909) which is also an element in Λ 119905We now introduce a distance onΛ Let ⟨sdot sdot⟩ and | sdot| denote

the inner product and norm inR119889 For each 0 le 119905 119905 le 119879 and120574119905 120574119905 isin Λ we set

10038171003817100381710038171205741199051003817100381710038171003817 = sup

119904isin[0119905]

1003816100381610038161003816120574119905 (119904)1003816100381610038161003816

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817 = sup119904isin[0119905or119905]

1003816100381610038161003816120574119905 (119904 and 119905) minus 120574119905(119904 and 119905)

1003816100381610038161003816

119889infin (120574119905 120574119905) = sup119904isin[0119905or119905]

1003816100381610038161003816120574119905 (119904 and 119905) minus 120574119905(119904 and 119905)

1003816100381610038161003816 +1003816100381610038161003816119905 minus 119905

1003816100381610038161003816

(6)

It is obvious that Λ 119905 is a Banach space with respect to sdot Since Λ is not a linear space 119889infin is not a norm

Mathematical Problems in Engineering 3

Definition 1 A functional 119906 Λ 997891rarr R is Λ-continuous at120574119905 isin Λ if for any 120576 gt 0 there exists 120575 gt 0 such that for each120574119905isin Λ with 119889infin(120574119905 120574119905) lt 120575 one has |119906(120574119905) minus 119906(120574

119905)| lt 120576

119906 is said to be Λ-continuous if it is Λ-continuous at each120574119905 isin Λ

Definition 2 Let 119907 Λ 997891rarr R and 120574119905 isin Λ be given If thereexists 119901 isin R119889 such that

119907 (120574119909

119905) = 119907 (120574119905) + ⟨119901 119909⟩ + 119900 (|119909|) as 119909 997888rarr 0 119909 isin R

119889

(7)

Then we say that 119907 is (vertically) differentiable at 119905 and denotethe gradient of 119863119909119907(120574119905) = 119901 sdot 119907 is said to be verticallydifferentiable in Λ if 119863119909119907(120574119905) exists for each 120574119905 isin Λ TheHessian119863119909119909119907(120574119905) can be defined similarly It is an 119878(119889)-valuedfunction defined on Λ where 119878(119889) is the space of all 119889 times 119889

symmetric matrices

For each 120574119905 isin Λ we denote 120574119905119904(119903) = 120574119905(119903)1[0119905)(119903) +

120574119905(119905)1[119905119904](119903) 119903 isin [0 119904] It is clear that 120574119905119904 isin Λ 119904

Definition 3 For a given 120574119905 isin Λ if one has

119907 (120574119905119904) = 119907 (120574119905) + 119886 (119904 minus 119905) + 119900 (|119904 minus 119905|) as 119904 997888rarr 119905 119904 ge 119905

(8)

then we say that 119907(120574119905) is (horizontally) differentiable in 119905

at 120574119905 and denote 119863119905119907(120574119905) = 119886 119907 is said to be horizontallydifferentiable in Λ if119863119905119907(120574119905) exists for each 120574119905 isin Λ

Definition 4 Define C119895119896(Λ) as the set of function 119907 =

(119907(120574119905))120574119905isinΛ defined on Λ which are 119895 times horizontally and

119896 times vertically differentiable in Λ such that all thesederivatives are Λ-continuous

The following is about the functional Itorsquos formula whichwas firstly obtained by Dupire [15] and then developed byCont and Fournie [18] for a more general formulation

Theorem 5 (functional Itorsquos formula) Let (ΩF (F119905)119905isin[0119879]

119875) be a probability space if 119883 is a continuous semimartingaleand 119907 is in C12

(Λ) then for any 119905 isin [0 119879)

119907 (119883119905) minus 119907 (1198830) = int

119905

0

119863119904119907 (119883119904) 119889119904 + int

119905

0

119863119909119907 (119883119904) 119889119883 (119904)

+1

2int

119905

0

119863119909119909119907 (119883119904) 119889 ⟨119883⟩ (119904) 119875-as

(9)

22 BSDEs In this section we recollect some importantresults which will be used in our SDG problems

Let (ΩF 119875) be the Wiener space where Ω is the set ofcontinuous functions from [0 119879] to R119889 starting from 0 (Ω =

1198620([0 119879]R119889)) 119879 is an arbitrarily fixed real time horizonF

is the completed Borel 120590-algebra overΩ and 119875 is the Wienermeasure Let 119861 be the canonical process 119861(120596 119904) = 120596119904 119904 isin

[0 119879] 120596 isin Ω We denote by F = F119904 0 le 119904 le 119879 the natural

filtration generated by 119861(119905)119905ge0 and augmented by all 119875-nullsets that is F119904 = 120590119861(119903) 119903 le 119904 or N119875 119904 isin [0 119879] F119904

119905=

120590(119861(119903)minus119861(119905) 119905 le 119903 le 119904)orN119875 whereN119875 is the set of all119875-nullsubsets First we present two spaces of processes as follows

S2(0 119879R

119899) = (120595(119905))

0le119905le119879R

119899-valued

F-adapted continuous process

119864 [ sup0le119905le119879

1003816100381610038161003816120595 (119905)10038161003816100381610038162] lt +infin

H2(0 119879R

119899) = (120595(119905))

0le119905le119879

R119899-valued

F-progressively measurable process

100381710038171003817100381712059510038171003817100381710038172= 119864[int

119879

0

1003816100381610038161003816120595 (119905)10038161003816100381610038162119889119905] lt +infin

(10)

Consider 119892 Ω times [0 119879] times R times R119889rarr R for every (119910 119911)

in R times R119889 (119892(119905 119910 119911))119905isin[0119879] is progressively measurable andsatisfies the following conditions

(A1) there exists a constant 119862 ge 0 such that 119875-as for all119905 isin [0 119879] 119910

1 1199102 isin R 1199111 1199112 isin R119889

1003816100381610038161003816119892 (119905 1199101 1199111) minus 119892 (119905 1199102 1199112)1003816100381610038161003816 le 119862 (

10038161003816100381610038161199101 minus 11991021003816100381610038161003816 +

10038161003816100381610038161199111 minus 11991121003816100381610038161003816) (11)

(A2) 119892(sdot 0 0) isin H2(0 119879R)

In the following we suppose the driver 119892 of a BSDEsatisfies (A1) and (A2)

Lemma 6 Under the assumptions (A1) and (A2) for anyrandom variable 120585 isin 119871

2(ΩF119879 119875R) the BSDE

119910 (119905) = 120585 + int

119879

119905

119892 (119904 119910 (119904) 119911 (119904)) 119889119904 minus int

119879

119905

119911 (119904) 119889119861 (119904)

0 le 119905 le 119879

(12)

has a unique adapted solution

(119910(119905) 119911(119905))119905isin[0119879]

isin S2(0 119879R) timesH

2(0 119879R

119889) (13)

The readers may refer to Pardoux and Peng [1] forthe above well-known existence and uniqueness results onBSDEs

Lemma 7 (comparison theorem) Given two coefficients 11989211198922 satisfying (A1) (A2) and two terminal values 1205851 1205852 isin

1198712(ΩF119879 119875R) by (119910

1 119911

1) and (119910

2 119911

2) one denotes the

solution of a BSDE with the data (1205851 1198921) and (1205852 1198922)respectively Then one has the following

(i) If 1205851 ge 1205852 and 1198921 ge 1198922 P-as then 1199101(119905) ge 119910

2(119905)

P-as for all 119905 isin [0 119879]


(ii) (Strict monotonicity) If in addition to (i) one alsoassumes that 119875(1205851 gt 1205852) gt 0 then 119875(119910

1(119905) gt 119910

2(119905)) gt

0 0 le 119905 le 119879 and in particular 1199101(0) gt 119910

2(0)

With the notations in Lemma 7 we assume that for some119892 Ω times [0 119879] times R times R119889

rarr R satisfying (A1) and (A2) thedrivers 119892119894 have the following form

119892119894 (119904 119910119894(119904) 119911

119894(119904)) = 119892 (119904 119910

119894(119904) 119911

119894(119904)) + 120593119894 (119904)

119889119904119889119875-ae 119894 = 1 2

(14)

where 120593119894 isin H2(0 119879R) Then the following results hold true

for all terminal values 1205851 1205852 isin 1198712(ΩF119879 119875R)

Lemma 8 The difference of the solutions (1199101 119911

1) and (119910

2 119911

2)

of BSDE (12) with the data (1205851 1198921) and (1205852 1198922) respectivelysatisfies the following estimate

100381610038161003816100381610038161199101(119905)minus119910

2(119905)

10038161003816100381610038161003816

2

+1

2119864 [int

119879

119905

119890120573(119904minus119905)

(100381610038161003816100381610038161199101(119904)minus119910

2(119904)

10038161003816100381610038161003816

2

+100381610038161003816100381610038161199111(119904) minus 119911

2(119904)

10038161003816100381610038161003816

2

) 119889119904 | F119905]

le 119864 [119890120573(119879minus119905)10038161003816100381610038161205851 minus 1205852

10038161003816100381610038162| F119905]

+ 119864 [int

119879

119905

119890120573(119904minus119905)10038161003816100381610038161205931 (119904) minus 1205932 (119904)

10038161003816100381610038162119889119904 | F119905]

P-as forall0 le 119905 le 119879 where 120573 = 16 (1 + 1198622)

(15)

Proof The reader may refer to Proposition 21 in El Karoui etal [5] or Theorem 23 in Peng [9] for the details

3 A DPP for Stochastic Differential Games ofFunctional FBSDEs

In this section we consider the SDGs of functional FBSDEsFirst we introduce the background of SDGs Suppose

that the control state spaces 119880119881 are compact metric spacesU (resp V) is the control set of all 119880 (resp 119881)-valuedF-progressively measurable processes for the first (respsecond) player If 119906 isin U (resp 119907 isin V) we call 119906 (resp 119907)an admissible control

Let us give the following mappings

119887 Λ times 119880 times 119881 997888rarr R119899

120590 Λ times 119880 times 119881 997888rarr R119899times119889

119891 Λ timesR timesR119889times 119880 times 119881 997888rarr R

(16)

For given admissible controls 119906(sdot) isin U 119907(sdot) isin V and 119905 isin

[0 119879] 120574119905 isin Λ we consider the following functional forward-backward stochastic system

119889119883120574119905119906119907

(119904) = 119887 (119883120574119905119906119907

119904 119906 (119904) 119907 (119904)) 119889119904

+ 120590 (119883120574119905119906119907

119904 119906 (119904) 119907 (119904)) 119889119861 (119904) 119904 isin [119905 119879]

119889119884120574119905119906119907

(119904) = minus119891 (119883120574119905119906119907

119904 119884

120574119905119906119907

(119904) 119885120574119905119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

+ 119885120574119905119906119907

(119904) 119889119861 (119904)

119883120574119905119906119907

119905= 120574119905

119884120574119905119906119907

(119879) = Φ (119883120574119905119906119907

119879)

(17)

(H) (i) For all 119905 isin [0 119879] 119906 isin 119880 119907 isin 119881 119909119905 isin Λ 119910 isin

R 119911 isin R119889 119887(119909119905 119906 119907) 120590(119909119905 119906 119907) and 119891(119909119905 119910 119911 119906 119907)

areF119905-measurable(ii) There exists a constant 119862 gt 0 such that for all 119905 isin

[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091

119905 119909

2

119905isin Λ

10038161003816100381610038161003816119887 (119909

1

119905 119906 119907) minus 119887 (119909

2

119905 119906 119907)

10038161003816100381610038161003816+10038161003816100381610038161003816120590 (119909

1

119905 119906 119907) minus 120590 (119909

2

119905 119906 119907)

10038161003816100381610038161003816

le 119862100381710038171003817100381710038171199091

119905minus 119909

2

119905

10038171003817100381710038171003817

(18)1003816100381610038161003816119887 (119909119905 119906 119907)

1003816100381610038161003816 +1003816100381610038161003816120590 (119909119905 119906 119907)

1003816100381610038161003816 le 119862 (1 +1003817100381710038171003817119909119905

1003817100381710038171003817) for any 119909119905 isin Λ

(19)

(iii) There exists a constant 119862 gt 0 such that for all 119905 isin

[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091

119905 119909

2

119905isin Λ 119910

1 119910

2isin

R 1199111 119911

2isin R119889

10038161003816100381610038161003816119891 (119909

1

119905 119910

1 119911

1 119906 119907) minus 119891 (119909

2

119905 119910

2 119911

2 119906 119907)

10038161003816100381610038161003816

le 119862 (100381710038171003817100381710038171199091

119905minus 119909

2

119905

10038171003817100381710038171003817+100381610038161003816100381610038161199101minus 119910

210038161003816100381610038161003816+100381610038161003816100381610038161199111minus 119911

210038161003816100381610038161003816)

10038161003816100381610038161003816Φ (119909

1

119879) minus Φ (119909

2

119879)10038161003816100381610038161003816le 119862

100381710038171003817100381710038171199091

119879minus 119909

2

119879

10038171003817100381710038171003817

1003816100381610038161003816119891 (119909119905 0 0 119906 119907)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171199091199051003817100381710038171003817)

1003816100381610038161003816Φ (119909119879)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171199091198791003817100381710038171003817) for any 119909119905 isin Λ

(20)

Theorem 9 Under the assumption (H) there exists a uniquesolution (119883 119884 119885) isin S2

(0 119879R119899) timesS2

(0 119879R) timesH2(0 119879R119889

)

solving (17)

We recall the subspaces of admissible controls and thedefinitions of admissible strategies as follows which aresimilar to [10]

Definition 10 An admissible control process 119906 = (119906119903)119903isin[119905119904]

(resp 119907 = (119907119903)119903isin[119905119904]) for Player I (resp II) on [119905 119904] is anF119903-progressively measurable 119880 (resp 119881)-valued process Theset of all admissible controls for Player I (resp II) on [119905 119904]

is denoted byU119905119904 (respV119905119904) If 119875119906 equiv 119906 119886119890 119894119899 [119905 119904] = 1one will identify both processes 119906 and 119906 inU119905119904 Similarly oneinterprets 119907 equiv 119907 on [119905 119904] inV119905119904


Definition 11 A nonanticipative strategy for Player I on[119905 119904] (119905 lt 119904 le 119879) is a mapping 120572 V119905119904 rarr U119905119904 suchthat for any F-stopping time 119878 Ω rarr [119905 119904] and any1199071 1199072 isin V119905119904 with 1199071 equiv 1199072 on [[119905 119878]] it holds that 120572(1199071) equiv

120572(1199072) on [[119905 119878]] Nonanticipative strategies for Player II on[119905 119904] 120573 U119905119904 rarr V119905119904 are defined similarly The set ofall nonanticipative strategies 120572 V119905119904 rarr U119905119904 for PlayerI on [119905 119904] is denoted by A119905119904 The set of all nonanticipativestrategies 120573 U119905119904 rarr V119905119904 for Player II on [119905 119904] is denoted byB119905119904 (Recall that [[119905 119878]] = (119903 120596) isin [0 119879] times Ω 119905 le 119903 le 119878(120596))

For given processes 119906(sdot) isin U119905119879 119907(sdot) isin V119905119879 initial data119905 isin [0 119879] 120574119905 isin Λ the cost functional is defined as follows

119869 (120574119905 119906 119907) = 119884120574119905119906119907

(119905) 120574119905 isin Λ (21)

where the process 119884120574119905119906119907 is defined by functional FBSDE (17)

For 120574119905 isin Λ the lower and the upper value functions of ourSDGs are defined as

119882(120574119905) = essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (22)

119880 (120574119905) = esssup120572isinA119905119879

essinf119907isinV119905119879

119869 (120574119905 120572 (119907) 119907) (23)

As we know the essential infimum and essential supre-mum on a family of random variables are still random vari-ables But by applying the method introduced by Buckdahnand Li [10] we get119882(120574119905) and 119880(120574119905) are deterministic

Proposition 12 For any 119905 isin [0 119879] 120574119905 isin Λ 119882(120574119905) is a deter-ministic function in the sense that119882(120574119905) = 119864[119882(120574119905)] 119875-as

Proof Let 119867 denote the Cameron-Martin space of all abso-lutely continuous elements ℎ isin Ω whose derivative ℎ belongsto 119871

2([0 119879]R119889

)For any ℎ isin 119867 we define the mapping 120591ℎ120596 = 120596 + ℎ 120596 isin

Ω It is easy to check that 120591ℎ Ω rarr Ω is a bijectionand its law is given by 119875 ∘ [120591ℎ]

minus1= expint119879

0ℎ(119904)119889119861(119904) minus

(12) int119879

0|ℎ(119904)|

2119889119904119875 For any fixed 119905 isin [0 119879] set 119867119905 = ℎ isin

119867 | ℎ(sdot) = ℎ(sdot and 119905) The proof can be separated into thefollowing four steps

(1) For all 119906 isin U119905119879 119907 isin V119905119879 ℎ isin 119867119905 119869(120574119905 119906 119907)(120591ℎ) =

119869(120574119905 119906(120591ℎ) 119907(120591ℎ)) 119875-asFirst we make the transformation for the functional SDE

as follows

119883120574119905119906119907

(119904)∘(120591ℎ)= (120574119905 (119905)+int

119904

119905

119887 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119903

+int

119904

119905

120590 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119861 (119903)) ∘ (120591ℎ)

= 120574119905 (119905) + int

119904

119905

119887 (119883120574119905119906119907

119903(120591ℎ) 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119903

+ int

119904

119905

120590 (119883120574119905119906119907

119903(120591ℎ) 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119861 (119903)

119883120574119905119906(120591ℎ)119907(120591ℎ)(119904) = 120574119905 (119905) + int

119904

119905

119887 (119883120574119905119906(120591ℎ)119907(120591ℎ)

119903 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119903

+ int

119904

119905

120590 (119883120574119905119906(120591ℎ)119907(120591ℎ)

119903 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119861 (119903)

(24)

then from the uniqueness of the solution of the functionalSDE we get

119883120574119905119906119907

(119904) (120591ℎ) = 119883120574119905119906(120591ℎ)119907(120591ℎ)(119904)

for any 119904 isin [119905 119879] 119875-as(25)

Similarly using the transformation to the BSDE in (17) andcomparing the obtained equation with the BSDE obtainedfrom (17) by replacing the transformed control process119906(120591ℎ) 119907(120591ℎ) for 119906 119907 due to the uniqueness of the solution offunctional BSDE we obtain

119884120574119905119906119907

(119904) (120591ℎ) = 119884120574119905119906(120591ℎ)119907(120591ℎ)(119904)

for any 119904 isin [119905 119879] 119875-as

119885120574119905119906119907

(119904) (120591ℎ) = 119885120574119905119906(120591ℎ)119907(120591ℎ)(119904)

119889119904119889119875-ae on [0 119879] times Ω

(26)

Hence

119869 (120574119905 119906 119907) (120591ℎ) = 119869 (120574119905 119906 (120591ℎ) 119907 (120591ℎ)) 119875-as (27)

(2) For 120573 isin B119905119879 ℎ isin 119867119905 let 120573ℎ(119906) = 120573(119906(120591minusℎ))(120591ℎ) 119906 isin

U119905119879 Then 120573ℎisin B119905119879

Obviously 120573ℎ U119905119879 rarr V119905119879 And it is nonanticipating

In fact given an F-stopping time 119878 Ω rarr [119905 119879] and 1199061 1199062 isin

U119905119879 with 1199061 equiv 1199062 on [[119905 119878]] Accordingly 1199061(120591minusℎ) equiv 1199062(120591minusℎ)

on [[119905 119878(120591minusℎ)]] Thus

120573ℎ(1199061) = 120573 (1199061 (120591minusℎ)) (120591ℎ) = 120573 (1199062 (120591minusℎ)) (120591ℎ)

= 120573ℎ(1199062) on [[119905 119878]]

(28)

(3) For any ℎ isin 119867119905 and 120573 isin B119905119879 we have

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(29)

In fact for convenience setting 119868(120574119905 120573) = esssup119906isinU119905119879

119869(120574119905 119906

120573(119906)) 120573 isin B119905119879 we know 119868(120574119905 120573) ge 119869(120574119905 119906 120573(119906)) Then


119868(120574119905 120573)(120591ℎ) ge 119869(120574119905 119906 120573(119906))(120591ℎ) 119875-as for all 119906 isin U119905119879Therefore

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

ge esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(30)

From the definition of essential supremum for anyrandom variable 120585 which satisfies 120585 ge 119869(120574119905 119906 120573(119906))(120591ℎ) wehave 120585(120591minusℎ) ge 119869(120574119905 119906 120573(119906)) 119875-as for all 119906 isin U119905119879 So120585(120591minusℎ) ge 119868(120574119905 120573) 119875-as that is 120585 ge 119868(120574119905 120573)(120591ℎ) 119875-as Thus

119869 (120574119905 119906 120573 (119906)) (120591ℎ) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

119875-as for any 119906 isin U119905119879

(31)

Therefore

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(32)

From above we get

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(33)

(4) Under the Girsanov transformation 120591ℎ 119882(120574119905) isinvariant that is

119882(120574119905) (120591ℎ) = 119882(120574119905) 119875-as for any ℎ isin 119867 (34)

In fact we can prove essinf120573isinB119905119879

119868(120574119905 120573)(120591ℎ) =

essinf120573isinB119905119879

119868(120574119905 120573)(120591ℎ) 119875-as for all ℎ isin 119867119905 which issimilar to the above step From the above three steps for allℎ isin 119867119905 we get

119882(120574119905) (120591ℎ) = essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 (120591ℎ) 120573ℎ(119906 (120591ℎ)))

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906))

= 119882 (120574119905) 119875-as(35)

Note that 119906(120591ℎ) | 119906(sdot) isin U119905119879 = U119905119879 and 120573ℎ

| 120573 isin

B119905119879 = B119905119879 have been used in the above equalities So forany ℎ isin 119867119905119882(120574119905)(120591ℎ) = 119882(120574119905) 119875-as Thanks to 119882(120574119905) isF119905-measurable this relation holds true for all ℎ isin 119867

To finish the proof we also need the auxiliary lemma asfollows

Lemma 13 Let 120577 be a random variable defined over theclassical Wiener space (ΩF119879 119875) such that 120577(120591ℎ) = 120577 119875-asfor any ℎ isin 119867 Then 120577 = 119864120577 119875-as

Proof From Lemma 13 in Buckdahn and Li [10] we know forany 119860 isin B(R) 120593 isin 119871

2([0 119879]R119889

)

119864[1120577isin119860 expint

119879

0

120593 (119904) 119889119861 (119904)]

= 119864 [1120577isin119860] 119864 [expint

119879

0

120593 (119904) 119889119861 (119904)]

(36)

For any 120593 = sum119873

119894=11205931198941(119905

119894minus1119905119894] where 120593119894 isin R119889 for 0 le 119894 le 119873 and

119905119894119873

119894=0is a finite partition of [0 119879] from (36)

119864[1120577isin119860 exp(

119873

sum

119894=1

120593119894119861 (119905119894) minus 119861 (119905119894minus1))]

= 119864 [1120577isin119860] 119864[exp119873

sum

119894=1

120593119894 (119861 (119905119894) minus 119861 (119905119894minus1))]

(37)

Therefore for any nonnegative integer 119896119894

119864[1120577isin119860

119873

prod

119894=1

(119861 (119905119894) minus 119861 (119905119894minus1))119896119894

]

= 119864 [1120577isin119860] 119864[

119873

prod

119894=1

(119861 (119905119894) minus 119861 (119905119894minus1))119896119894

]

(38)

So for any polynomial function 119876 we have

119864 [1120577isin119860119876 (119861 (1199051) minus 119861 (1199050) 119861 (119905119899) minus 119861 (119905119899minus1))]

= 119864 [1120577isin119860] 119864 [119876 (119861 (1199051) minus 119861 (1199050) 119861 (119905119899) minus 119861 (119905119899minus1))]

(39)

Furthermore for any 119876 isin 119862119887(R119899) we still have (39)

Combining the arbitrariness of 119860 isin B(R) we obtain 120577 isindependent of (119861(1199051) minus 119861(1199050) 119861(119905119899) minus 119861(119905119899minus1)) for allpartition of [0 119879] Therefore 120577 is independent of F119879 whichimplies 120577 is independent of itself that is 119864120577 = 120577 119875-as

In Ji and Yang [14] they proved the following estimates


Lemma 14 Under the assumption (H) there exists someconstant 119862 gt 0 such that for any 119905 isin [0 119879] 120574119905 120574119905 isin Λ 119906(sdot) isin

U 119907(sdot) isin V

119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905]le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172)

119864 [ sup119904isin[119905119879]

10038161003816100381610038161003816119883

120574119905119906119907

(119904) minus 119883120574119905119906119907

(119904)10038161003816100381610038161003816

2

| F119905]le 1198621003817100381710038171003817120574119905 minus 120574

119905

10038171003817100381710038172

119864 [ sup119904isin[119905119879]

1003816100381610038161003816119884120574119905119906119907

(119904)10038161003816100381610038162+int

119879

119905

1003816100381610038161003816119885120574119905119906119907

(119904)10038161003816100381610038162119889119904 | F119905]le 119862 (1+

100381710038171003817100381712057411990510038171003817100381710038172)

119864 [ sup119904isin[119905119879]

10038161003816100381610038161003816119884120574119905119906119907

(119904) minus 119884120574119905119906119907

(119904)10038161003816100381610038161003816

2

+int

119879

119905

10038161003816100381610038161003816119885120574119905119906119907

(119904) minus 119885120574119905119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]le 1198621003817100381710038171003817120574119905 minus 120574

119905

10038171003817100381710038172

(40)From the definition of119882(120574119905) and Lemma 14 we have the

following property

Lemma 15 There exists some constant 119862 gt 0 such that for all0 le 119905 le 119879 120574119905 120574119905 isin Λ

(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817

(ii) 1003816100381610038161003816119882 (120574119905)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171205741199051003817100381710038171003817)

(41)

Nowwe adopt Pengrsquos notion of stochastic backward semi-group (which was first introduced by Peng [9] to prove theDPP for stochastic control problems) to discuss a generalizedDPP for our SDG (17) (22) First we define the family ofbackward semigroups associated with FBSDE (17)

For given 119905 isin [0 119879] 120574119905 isin Λ a number 120575 isin (0 119879 minus 119905]admissible control processes 119906(sdot) isin U119905119905+120575 119907(sdot) isin V119905119905+120575 weset

119866120574119905119906119907

119904119905+120575[120578] =

120574119905119906119907

(119904) 119904 isin [119905 119905 + 120575] (42)

where 120578 isin 1198712(ΩF119905+120575

119905 119875R) (

120574119905119906119907

(119904) 119885120574119905119906119907

(119904))119905le119904le119905+120575

solves the following functional FBSDE on [119905 119905 + 120575]

119889120574119905119906119907

(119904)

= minus119891 (119904 119883120574119905119906119907

119904

120574119905119906119907

(119904) 119885120574119905119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

+ 119885120574119905119906119907

(119904) 119889119861 (119904)

120574119905119906119907

(119905 + 120575) = 120578 119904 isin [119905 119905 + 120575]

(43)Also we have

119866120574119905119906119907

119905119879[Φ (119883

120574119905119906119907

119879)] = 119866

120574119905119906119907

119905119905+120575[119884

120574119905119906119907

(119905 + 120575)] (44)Theorem 16 Suppose (H) holds true the lower value function119882(120574119905) satisfies the followingDPP for any 119905 isin [0 119879] 120574119905 isin Λ 120575 gt

0

119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (45)

The proof is given in the appendix

4 Viscosity Solutions of Path-DependentHJBI Equation

Now we study the following path-dependent PDEs

119863119905119882(120574119905) + 119867minus(120574119905119882119863119909119882119863119909119909119882) = 0

119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ

(46)

119863119905119880 (120574119905) + 119867+(120574119905 119880119863119909119880119863119909119909119880) = 0

119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ

(47)

where

119867minus(120574119905119882119863119909119882119863119909119909119882)

= sup119906isin119880

inf119907isin119881

119867(120574119905119882119863119909119882119863119909119909119882119906 119907)

119867+(120574119905 119880119863119909119880119863119909119909119880)

= inf119907isin119881

sup119906isin119880

119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907)

119867 (120574119905 119910 119901 119883 119906 119907)

=1

2tr (120590120590119879 (120574119905 119906 119907)119883) + 119901 sdot 119887 (120574119905 119906 119907)

+ 119891 (120574119905 119910 119901120590 (120574119905 119906 119907) 119906 119907)

(48)


119889 times 119889 symmetric matrices)We will show that the value function119882(120574119905) (resp 119880(120574119905))

defined in (22) (resp (23)) is a viscosity solution of thecorresponding equation (46) (resp (47)) First we give thedefinition of viscosity solution for this kind of PDEs Formore information on viscosity solution the reader is referredto Crandall et al [22]

Definition 17 A real-valued Λ-continuous function 119882 isin

C(Λ) is called

(i) a viscosity subsolution of (46) if for any 120575 gt 0 Γ isin

C12

119897119887(Λ) 120574119905 isin Λ satisfying Γ ge 119882 on ⋃

0le119904le120575Λ 119905+119904 and

Γ(120574119905) = 119882(120574119905) one has

119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) ge 0 (49)

(ii) a viscosity supersolution of (46) if for any 120575 gt 0 Γ isin

C12

119897119887(Λ) 120574119905 isin Λ satisfying Γ le 119882 on ⋃

0le119904le120575Λ 119905+119904 and

Γ(120574119905) = 119882(120574119905) one has

119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) le 0 (50)

(iii) a viscosity solution of (46) if it is both a viscosity sub-and supersolution of (46)

Theorem 18 Assume (H) holds the lower value function119882 isa viscosity solution of path-dependent HJBI Equation (46) onΛ the upper value function 119880 is a viscosity solution of (47)


First we prove some helpful lemmas For some fixed Γ isin

C12

119897119887(Λ) denote

119865 (120574119904 119910 119911 119906 119907)

= 119863119904Γ (120574119904) +1

2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))

+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)

+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)

(51)

where (120574119904 119910 119911 119906 119907) isin Λ timesR timesR119889times 119880 times 119881

Consider the following BSDE

minus 1198891198841119906119907

(119904)

= 119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198851119906119907

(119904) 119889119861 (119904)

1198841119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]

(52)

Lemma 19 For every 119904 isin [119905 119905 + 120575] one has the following

1198841119906119907

(119904) = 119866120574119905119906119907

119904119905+120575[Γ (119883

120574119905119906119907

119905+120575)] minus Γ (119883

120574119905119906119907

119904) 119875-as (53)

Proof Note119866120574119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] is defined as119866120574

119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] =

119884119906119907

(119904) through the following BSDE

minus 119889119884119906119907

(119904) = 119891 (119883120574119905119906119907

119904 119884

119906119907(119904) 119885

119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 119885119906119907

(119904) 119889119861 (119904)

119884119906119907

(119905 + 120575) = Γ (119883120574119905119906119907

119905+120575) 119904 isin [119905 119905 + 120575]

(54)

Using Itorsquos formula to Γ(119883120574119905119906119907

119904) we have

119889 (119884119906119907

(119904) minus Γ (119883120574119905119906119907

119904)) = 119889119884

1119906119907(119904) (55)

Combined with 119884119906119907

(119905 + 120575) minus Γ(119883120574119905119906119907

119905+120575) = 0 = 119884

1119906119907(119905 + 120575) we

get the desired result

Now consider the following BSDE

minus 1198891198842119906119907

(119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198852119906119907

(119904) 119889119861 (119904)

(56)

1198842119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)

Then we have the following lemma

Lemma 20 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has100381610038161003816100381610038161198841119906119907

(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816le 119862120575

32 119875-as (58)

where 119862 is independent of the control processes 119906 119907

Proof From Lemma 14 we know there exists some constant119862 gt 0 such that

119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905] le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172) (59)

combined with

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]

le 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

119887 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119903

10038161003816100381610038161003816100381610038161003816

2

| F119905]

+ 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

120590 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119861 (119903)

10038161003816100381610038161003816100381610038161003816

2

| F119905]

(60)

we have

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)

From (52) and (54) using Lemma 8 set

1205851 = 1205852 = 0

119892 (119904 119910 119911) = 119865 (119883120574119905119906119907

119904 119910 119911 119906119904 119907119904)

1205931 (119904) = 0

1205932 (119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906119904 119907119904)

minus 119865 (119883120574119905119906119907

119904 119884

2119906119907(119904) 119885

2119906119907(119904) 119906119904 119907119904)

(62)

Denote 1205880(119903) = (1 + |119909|2)119903 due to 119887 120590 119891 are Lipishtiz and

that they are of linear growth Γ isin C12

119887 |1205932(119904)| le 1205880(|119883

120574119905119906119907

119904minus

120574119905(119905)|) we have

119864[int

119905+120575

119905

(100381610038161003816100381610038161198841119906119907

(119904) minus1198842119906119907

(119904)10038161003816100381610038161003816

2

+100381610038161003816100381610038161198851119906119907

(119904) minus1198852119906119907

(119904)10038161003816100381610038161003816

2

) 119889119904 | F119905]

le 119864[int

119905+120575

119905

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) 119889119904 | F119905]

le 119862120575119864[ sup119904isin[119905119905+120575]

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) | F119905]

le 1198621205752

(63)


So100381610038161003816100381610038161198841119906119907

(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816

=10038161003816100381610038161003816119864 [(119884

1119906119907(119904) minus 119884

2119906119907(119904)) | F119905]

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

119864 [int

119905+120575

119905

(119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904))

minus119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]

100381610038161003816100381610038161003816100381610038161003816

le 119862119864[int

119905+120575

119905

[1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

+100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816] 119889119904 | F119905]

le 119862119864[int

119905+120575

119905

1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) 119889119904 | F119905]

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

le 11986212057532

(64)

Lemma 21 Denote 1198840(sdot) by the solution of the followingordinary differential equation

minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]

1198840 (119905 + 120575) = 0

(65)

where 1198650(120574119905 119910 119911) = sup119906isin119880

inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as

1198840 (119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) (66)

Proof First we define a function as follows

1198651 (120574119905 119910 119911 119906) = inf119907isin119881

119865 (120574119905 119910 119911 119906 119907)

(120574119905 119910 119911 119906) isin Λ timesR timesR119889times 119880

(67)

Consider the following equation

minus1198891198843119906

(119904) = 1198651 (120574119905 1198843119906

(119904) 1198853119906

(119904) 119906 (119904)) 119889119904

minus1198853119906

(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]

1198843119906

(119905 + 120575) = 0

(68)

according to Lemma 6 for every 119906 isin U119905119905+120575 there exists aunique (1198843119906

1198853119906

) solving (68) Also

1198843119906

(119905) = essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-as for every 119906 isin U119905119905+120575

(69)

In fact according to the definition of 1198651 and Lemma 7 wehave

1198843119906

(119905) le 1198842119906119907

(119905) 119875-as for any 119907 isin V119905119905+120575

for every 119906 isin U119905119905+120575

(70)

On the other side we have the existence of a measurablefunction 119907

4 R timesR119889

times 119880 rarr 119881 such that

1198651 (120574119905 119910 119911 119906)

= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906

(71)

Set 1199074(119904) = 1199074(119884

3119906(119904) 119885

3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know

1199074isin V119905119905+120575 Therefore (1198843119906

1198853119906

) = (11988421199061199074

11988521199061199074

) which isdue to the uniqueness of the solution of (68) In particular1198843119906

(119905) = 11988421199061199074

(119905) 119875-as for every 119906 isin U119905119905+120575 Hence1198843119906

(119905) = essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-119886119904 for every 119906 isin U119905119905+120575

Similarly from 1198650(120574119905 119910 119911) = sup119906isin119880

1198651(120574119905 119910 119911 119906) wealso derive

1198840 (119905) = esssup119906isinU119905119905+120575

1198843119906

(119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-as

(72)

Lemma 22 For every 119906 isin U119905119905+120575 119907 isin V119905119905+120575 one has

119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575

32 119875-as

(73)

where the constant 119862 is independent of the control processes119906 119907

Proof Due to 119865(120574119905 sdot sdot 119906 119907) is linear growth in (119910 119911) uni-formly in (119906 119907) from Lemma 8 we know there exists aconstant 119862 gt 0 which does not depend on 120575 nor on thecontrols 119906 and 119907 such that 119875-as

100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816

2

le 119862120575

119864 [int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904] le 119862120575

(74)


Moreover from (54) we have100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816

le 119864 [int

119905+120575

119904

119865 (120574119905 1198842119906119907

(119903) 1198852119906119907

(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]

le 119862119864[int

119905+120575

119904

(1 +1003817100381710038171003817120574119905

10038171003817100381710038172+100381610038161003816100381610038161198842119906119907

(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816) 119889119903 | F119904]

le 119862120575 + 11986212057512

119864[int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904]

le 119862120575 119904 isin [119905 119905 + 120575]

(75)

and applying Itorsquos formula we get 119864[int119905+120575

119905|119885

2119906119907(119904)|

2

119889119904 |

F119905] le 1198621205752 119875-as Therefore

119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905]

le 1198621205752+ 119862120575

12(119864[int

119905+120575

119905

100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816

2

119889119903 | F119905])

12

le 11986212057532

119875-as

(76)

Now we continue the proof of Theorem 18

Proof (1) First we will prove 119882(120574119905) is a viscosity supersolu-tion Given 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and119882 ge Γ on⋃

0le119904le120575Λ 119905+119904

According to the DPP (Theorem 16) we have

Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(77)

From 119882 ge Γ on ⋃0le119904le120575

Λ 119905+119904 as well as the monotonicityproperty of 119866120574

119905119906120573(119906)

119905119905+120575[sdot] (Lemma 7) we have

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

120574119905119906120573(119906)

119905+120575)] minus Γ (120574119905) le 0 (78)

By Lemma 19 we have

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) le 0 119875-as (79)

Thus from Lemma 20

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as (80)

Therefore from essinf119907isinV119905119905+120575

1198842119906119907

(119905) le 1198842119906120573(119906)

(119905) 120573 isin

B119905119905+120575 we have

esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905)

le essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as(81)

Thus by Lemma 21 1198840(119905) le 11986212057532

119875-as where 1198840(sdot) solvesthe ODE (65) Consequently

11986212057512

ge1

1205751198840 (119905) =

1

120575int

119905+120575

119905

1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)

sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)

Obviously according to the definition of 119865 119882 is a viscositysupersolution of (46)

(2) Now we prove119882 is a viscosity subsolutionGiven 120574119905 isin Λ for any 120575 gt 0 suppose Γ(120574119905) = 119882(120574119905) and

Γ ge 119882 on⋃0le119904le120575

Λ 119905+119904We need to prove

sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)

Suppose it does not hold true So we have some 120579 gt 0 suchthat

1198650 (120574119905 0 0) = sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)

and there exists a measurable function 120595 119880 rarr 119881 such that

119865 (120574119905 0 0 119906 120595 (119906)) le minus3

4120579 forall119906 isin 119880 (86)

Moreover from the DPP (Theorem 16)

Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(87)

Again from119882 le Γ on⋃0le119904le120575

Λ 119905+119904 as well as themonotonicityproperty of 119866120574

119905119906120573(119906)

119905119905+120575[sdot] (Lemma 7) we get

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

119906120573(119906)

119905+120575)] minus Γ (120574119905) ge 0 119875-as

(88)

Then similar to (1) from the definition of backward semi-group we have

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) ge 0 119875-as (89)

in particular

esssup119906isinU119905119905+120575

1198841119906120595(119906)

(119905) ge 0 119875-as (90)

Setting 120595119904(119906)(120596) = 120595(119906(119904)(120596)) (119904 120596) isin [119905 119879] times Ω 120595 can beregarded as an element of B119905119905+120575 Given any 120576 gt 0 we canselect 119906120576 isin U119905119905+120575 satisfying 119884

1119906120576120595(119906120576)(119905) ge minus120576120575

From Lemma 20 we have

1198842119906120576120595(119906120576)(119905) ge minus119862120575

32minus 120576120575 119875-as (91)


Moreover from (54) we have

1198842119906120576120595(119906120576)(119905)

= 119864 [int

119905+120575

119905

119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885

2119906120576120595(119906120576)(119904)

119906120576(119904) 120595 (119906

120576(119904))) 119889119904 | F119905]

le 119864[int

119905+120575

119905

(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816+ 119862

1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816

+119865 (120574119905 0 0 119906120576(119904) 120595 (119906

120576(119904)))) 119889119904 | F119905]

le 11986212057532

minus3

4120579120575 119875-as

(92)

From (91) and (92) we have minus11986212057512

minus 120576 le 11986212057512

minus

(34)120579 119875-as Letting 120575 darr 0 and then 120576 darr 0 we get 120579 le 0which produces a contradiction Consequently

sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)

According to the definition of 119865 119882 is a viscosity supersolu-tion of (46) At last from the above two steps we derive that119882 is a viscosity solution of (46)

The similar argument for 119880 we also get that 119880 is aviscosity solution of (47)

Appendix

Proof of Theorem 16 (DPP)

Proof For convenience we set

119882120575 (120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (A1)

We want to prove 119882120575(120574119905) and 119882(120574119905) coincide For it we onlyneed to prove the following three lemmas

Lemma A1 119882120575(120574119905) is deterministic

The proof of this lemma is similar to the proof of Proposi-tion 12 so we omit it here

Lemma A2 119882120575(120574119905) le 119882(120574119905)

Proof For any fixed 120573 isin B119905119879 given a 1199062(sdot) isin U119905+120575119879 therestriction 1205731 of 120573 toU119905119905+120575 can be defined as follows

1205731(1199061) = 120573 (1199061 oplus 1199062) |[119905119905+120575] 1199061(sdot) isin U119905119905+120575 (A2)

where 1199061 oplus 1199062 = 11990611[119905119905+120575] + 11990621(119905+120575119879] generalizes 1199061(sdot) toan element of U119905119879 Clearly 1205731 isin B119905119905+120575 Also due to thenonanticipativity property of 120573 we know 1205731 is free of the

choice of 1199062(sdot) isin U119905+120575119879 Therefore due to the definition of119882120575(120574119905)

119882120575 (120574119905) le esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)] 119875-as

(A3)

We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907

119905119905+120575[119882(119883

120574119905119906119907

119905+120575)] Then there exists a

sequence 1199061119894 119894 ge 1 sub U119905119905+120575 such that

119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575

119868120575 (120574119905 1199061 1205731 (1199061))

= sup119894ge1

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) 119875-as

(A4)

For any 120576 gt 0 we set Γ119894 = 119868120575(120574119905 1205731) le 119868120575(120574119905 1199061

119894 1205731(119906

1

119894)) +

120576 isin F119905 119894 ge 1 Then the sets Γ1 = Γ1 Γ119894 = Γ119894

(⋃119894minus1

119897=1Γ119897) isin F119905 119894 ge 2 form an (ΩF119905)-partition Thus

119906120576

1= sum

119894ge11Γ119894

1199061

119894isin U119905119905+120575 Also from the nonanticipativity

of 1205731 we have 1205731(119906120576

1) = sum

119894ge11Γ119894

1205731(1199061

119894) and due to the

uniqueness of the solution of the functional FBSDE we have119868120575(120574119905 119906

120576

1 1205731(119906

120576

1)) = sum

119894ge11Γ119894

119868120575(120574119905 1199061

119894 1205731(119906

1

119894)) 119875-as So

119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum

119894ge1

1Γ119894

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) + 120576

= 119868120575 (120574119905 119906120576

1 1205731 (119906

120576

1)) + 120576

= 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119882(119883

120574119905119906120576

11205731(119906120576

1)

119905+120575)] + 120576

(A5)

On the other hand since 1205731(sdot) = 120573(sdot oplus 1199062) isin B119905119905+120575 isindependent of 1199062(sdot) isin U119905+120575119879 we define 1205732(1199062) = 120573(119906

120576

1oplus

1199062)|[119905+120575119879] for all 1199062(sdot) isin U119905+120575119879 According to the definitionof119882(120574119905) we get for any 119910 isin R119899

119882(120574119905) le esssup1199062isinU119905+120575119879

119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)

Recalling that there exists a constant 119862 isin R such that

(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817 for any 120574119905 120574119905 isin Λ

(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))

1003816100381610038161003816

le 1198621003817100381710038171003817120574119905 minus 120574

119905

1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879

(A7)

We can show by approximating119883120574119905119906120576

11205731(119906120576

1)

119905+120575that

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062)) 119875-as

(A8)


Now we estimate the right side of the latter inequality thereexists some sequence 1199062

119895 119895 ge 1 sub U119905+120575119879 such that

esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

= sup119895ge1

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A9)

Then setting Δ 119895 = esssup1199062isinU119905+120575119879

119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732(1199062)) le

119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732(119906

2

119895)) + 120576 isin F119905+120575 119895 ge 1 we have the sets

Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1

119897=1Δ 119897) isin F119905+120575 119895 ge 2 forms an

(ΩF119905+120575)-partition and 119906120576

2= sum

119895ge11Δ119895

1199062

119895isin U119905+120575119879 Due to

the nonanticipativity of 1205732 we get 1205732(119906120576

2) = sum

119895ge11Δ119895

1205732(1199062

119895)

also from the definitions of 1205731 1205732 we have 120573(119906120576

1oplus 119906

120576

2) =

1205731(119906120576

1) oplus 1205732(119906

120576

2) Therefore due to the uniqueness of the

solution of functional FBSDE we get

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

2 1205732 (119906

120576

2))

= 119884119883120574119905119906120576

11205731(119906120576

1)

119905+120575119906120576

21205732(119906120576

2)(119905 + 120575)

= sum

119895ge1

1Δ119895

119884119883120574119905119906120576

11205731(119906120576

1)

119905+1205751199062

1198951205732(1199062

119895)(119905 + 120575)

= sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A10)

So

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

le sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

2

119895 120573 (119906

120576

1oplus 119906

2

119895)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

120576

2 120573 (119906

120576

1oplus 119906

120576

2)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576 119875-as

(A11)

where 119906120576

= 119906120576

1oplus 119906

120576

2isin U119905119879 From (A5) (A11) and the

comparison theorem for BSDEs we know

119882120575 (120574119905)

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576] + 120576

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119866120574119905119906120576120573(119906120576)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119869 (120574119905 119906120576 120573 (119906

120576)) + (119862 + 1) 120576

= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576

le esssup119906isinU119905119879

119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576 119875-as

(A12)

Because of the arbitrariness of 120573 isin B119905119879 (A12) holds true forall 120573 isin B119905119879 Thus

119882120575 (120574119905) le essinf120573isinB119905119879

esssup119906isinU119905119879

119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576

= 119882(120574119905) + (119862 + 1) 120576

(A13)

Therefore letting 120576 darr 0 we have119882120575(120574119905) le 119882(120574119905)

Lemma A3 119882(120574119905) le 119882120575(120574119905)

Proof We keep the notations in the above lemma From thedefinition of119882120575(120574119905) we know

119882120575 (120574119905) = essinf1205731isinB119905119905+120575

esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)]

= essinf1205731isinB119905119905+120575

119868120575 (120574119905 1205731)

(A14)

and the existence of some sequence 1205731

119894 119894 ge 1 sub B119905119905+120575 such

that

119882120575 (120574119905) = inf119894ge1

119868120575 (120574119905 1205731

119894) 119875-as (A15)

For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731

119894) minus 120576 le 119882120575(120574119905) isin

F119905 119894 ge 1 Π1 = Π1 and Π119894 = Π119894 (⋃119894minus1

119897=1Π119897) isin F119905 119894 ge

2 Then Π119894 119894 ge 1 forms an (Ω F119905)-partition 120573120576

1=

sum119894ge1

1Λ119894

1205731

119894isin B119905119905+120575 combining the uniqueness of the solu-

tion of the functional FBSDE we derive 119868120575(120574119905 1199061 120573120576

1(1199061)) =

sum119894ge1

1Λ119894

119868120575(120574119905 1199061 1205731

119894(1199061)) 119875-as for all 1199061(sdot) isin U119905119905+120575

119882120575 (120574119905) ge sum

119894ge1

1Π119894

119868120575 (120574119905 1205731

119894) minus 120576

ge sum

119894ge1

1Π119894

119868120575 (120574119905 1199061 1205731

119894(1199061)) minus 120576

= 119868120575 (120574119905 1199061 120573120576

1(1199061)) minus 120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

119875-as forall1199061 isin U119905119905+120575

(A16)


On the other hand according to the definition of 119882(120574119905) wededuce that for any 119909 isin 119862([0 119905 + 120575]R119899

) there exists 120573119909120576isin

B119905+120575119879 such that

119882(119909119905+120575) ge esssup1199062isinU119905+120575119879

119869 (119909119905+120575 1199062 120573119909120576

(1199062)) minus 120576 119875-as

(A17)

Let 119874119894119894ge1 be a Borel partition of 119862([0 119905 + 120575]R119899) and

diam(119874119894) le 120576 119894 ge 1 Let 120574119894 be an arbitrarily fixed element of119874119894 119894 ge 1 Defining [119883

1205741199051199061120573120576

1(1199061)

119905+120575] = sum

119894ge1120574119894

119905+12057511198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894

we have

100381610038161003816100381610038161003816119883

1205741199051199061120573120576

1(1199061)

119905+120575minus [119883

1205741199051199061120573120576

1(1199061)

119905+120575]100381610038161003816100381610038161003816le 120576

everywhere on Ω forall1199061 isin U119905119905+120575

(A18)

And for every 120574119894 there exists some 120573

120574119894120576isin B119905+120575119879 satisfying

(A17) and 120573120576

1199061

= sum119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894120573120574119894120576isin B119905+120575119879

Next we define the new 120573120576(119906) = 120573

120576

1(1199061) oplus 120573

120576

1199061

(1199062) 119906 isin

U119905119879 where 1199061 = 119906|[119905119905+120575] 1199062 = 119906|(119905+120575119879] (which are therestrictions of 119906 to [119905 119905 + 120575] times Ω and (119905 + 120575 119879] times Ω resp)Clearly 120573120576

U119905119879 rarr V119905119879 And 120573120576 is nonanticipating In fact

let 119878 Ω rarr [119905 119879] be an F-stopping time and 119906 1199061015840isin U119905119879 be

such that 119906 equiv 1199061015840 on [[119905 119878]] Decomposing 119906 119906

1015840 into 1199061 1199061015840

1isin

U119905119905+120575 1199062 1199061015840

2isin U119905+120575119879 such that 119906 = 1199061 oplus 119906

1015840

1and 119906 = 1199062 oplus 119906

1015840

2

From 1199061 equiv 1199061015840

1on [[119905 119878 and (119905 + 120575)]] we get 120573120576

1(1199061) = 120573

120576

1(119906

1015840

1) on

[[119905 119878and(119905+120575)]] (note that 120573120576

1is nonanticipating) On the other

side from 1199062 equiv 1199061015840

2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt

119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576

1(1199061)

119905+120575= 119883

1205741199051199061015840

1120573120576

1(1199061015840

1)

119905+120575

Thus from the definition 120573120576

1199061

= 120573120576

11990610158401

on 119878 gt 119905 + 120575 and120573120576

1199061

(1199062) = 120573120576

11990610158401

(1199061015840

2) on]]119905 + 120575 119878 or (119905 + 120575)]] This produces

120573120576(119906) = 120573

120576

1(1199061) oplus 120573

120576

1199061

(1199062) = 120573120576

1(119906

1015840

1) oplus 120573

120576

11990610158401

(1199061015840

2) on [[119905 119878]] from

which we know 120573120576isin B119905119879

For an arbitrarily chosen 119906 isin U119905119879 decompose it into 1199061 =

119906|(119905119905+120575] isin U119905119905+120575 and 1199062 = 119906|[119905+120575119879] isin U119905+120575119879 From (A16)(A7)-(i) (A18) and the comparison theorem we get

119882120575 (120574119905) ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575]) minus 119862120576] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575])] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119882(120574

119894

119905+120575)] minus 119862120576

119875-as

(A19)

Moreover from (A19) (A7)-(ii) (A17) and the comparisontheorem we have

119882120575 (120574119905)

ge1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))minus 120576]minus119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 ([119883

1205741199051199061120573120576

1(1199061)

119905+120575] 1199062 120573

120576

1199061

(1199062))] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062)) minus 119862120576] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062))] minus 119862120576

= 119866120574119905119906120573120576(119906)

119905119905+120575[119884

120574119905119906120573120576(119906)

(119905 + 120575)] minus 119862120576

= 119884120574119905119906120573120576(119906)

(119905) minus 119862120576 119875-as for any 119906 isin U119905119879

(A20)

Therefore

119882120575 (120574119905) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573120576(119906)) minus 119862120576

ge essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) minus 119862120576

= 119882(120574119905) minus 119862120576 119875-as

(A21)

Letting 120576 darr 0 we derive119882120575(120574119905) ge 119882(120574119905)

Remark A4 (a) (i) For each 120573 isin B119905119905+120575 there exists some119906120576(sdot) isin 119880119905119905+120575 such that

119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)

119905119905+120575[119882(119883

120574119905119906120576120573(119906120576)

119905+120575)] + 120576

119875-as(A22)

(ii)There exists some 120573120576(sdot) isin 119861119905119905+120575 such that for all 119906(sdot) isin

119880119905119905+120575

119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)

119905119905+120575[119882(119883

120574119905119906120573120576(119906)

119905+120575)] minus 120576

119875-as(A23)

(b) From Proposition 12 we know the lower value function119882 is deterministic So by choosing 120575 = 119879 minus 119905 and takingexpectation on both sides of (A22) (A23) we get 119882(120574119905) =

inf120573isinB119905119879

sup119906isinU119905119879

119864[119869(120574119905 119906 120573(119906))]

Acknowledgments

This work was supported by National Natural Science Foun-dation of China (no 11171187 no 10871118 and no 10921101)


the Programme of Introducing Talents of Discipline toUniversities of China (no B12023) and Program for NewCentury Excellent Talents in University of China

References

[1] E Pardoux and S G Peng ldquoAdapted solution of a backwardstochastic differential equationrdquo Systems amp Control Letters vol14 no 1 pp 55ndash61 1990

[2] SG Peng ldquoA general stochasticmaximumprinciple for optimalcontrol problemsrdquo SIAM Journal on Control and Optimizationvol 28 no 4 pp 966ndash979 1990

[3] S Hamadene and J-P Lepeltier ldquoZero-sum stochastic differen-tial games and backward equationsrdquo Systems amp Control Lettersvol 24 no 4 pp 259ndash263 1995

[4] S Hamadene J-P Lepeltier and S G Peng ldquoBSDEs withcontinuous coefficients and stochastic differential gamesrdquo inBackward Stochastic Differential Equations vol 364 of PitmanResearch Notes in Math Series pp 115ndash128 El Karoui Mazliak1997

[5] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997

[6] S G Peng ldquoProbabilistic interpretation for systems of quasi-linear parabolic partial differential equationsrdquo Stochastics andStochastics Reports vol 37 no 1-2 pp 61ndash74 1991

[7] S G Peng ldquoA generalized dynamic programming principle andHamilton-Jacobi-Bellman equationrdquo Stochastics and StochasticsReports vol 38 no 2 pp 119ndash134 1992

[8] D Duffie and L G Epstein ldquoStochastic differential utilityrdquoEconometrica vol 60 no 2 pp 353ndash394 1992

[9] S G Peng ldquoBackward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJBequationsrdquo in Topics on Stochastic Analysis J A Yan S GPeng S Z Fang and L M Wu Eds pp 85ndash138 Science PressBeijing China 1997

[10] R Buckdahn and J Li ldquoStochastic differential games and vis-cosity solutions of Hamilton-Jacobi-Bellman-Isaacs equationsrdquoSIAM Journal on Control and Optimization vol 47 no 1 pp444ndash475 2008

[11] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Springer New York NY USASecond edition 2006

[12] Z Wu and Z Y Yu ldquoDynamic programming principle forone kind of stochastic recursive optimal control problem andHamilton-Jacobi-Bellman equationrdquo SIAM Journal on Controland Optimization vol 47 no 5 pp 2616ndash2641 2008

[13] J Yong andX Y Zhou Stochastic Controls Hamiltonian Systemsand HJB Equations vol 43 Springer New York NY USA 1999

[14] S L Ji and S Z Yang ldquoAn optimal control problem forfunctional forward-backward stochastic systems and relatedPath-dependent HJB equationsrdquo httparxivorgabs12046543

[15] B Dupire ldquoFunctional Itorsquos Calculusrdquo Portfolio Research PaperBloomberg 2009

[16] R Cont and D A Fournie ldquoA functional extension of the Itorsquosformulardquo Comptes Rendus Mathematique vol 348 no 1-2 pp57ndash61 2010

[17] R Cont and D A Fournie ldquoChange of variable formulasfor non-anticipative functionals on path spacerdquo Journal ofFunctional Analysis vol 259 no 4 pp 1043ndash1072 2010

[18] R Cont and D A Fournie ldquoFunctional Itorsquos calculus andstochastic integral representation of martingalesrdquoTheAnnals ofProbability vol 41 no 1 pp 109ndash133 2013

[19] S G Peng and F L Wang ldquoBSDE path-dependent PDE andnonlinear Feynman-Kac formulardquo In press httparxivorgabs11084317

[20] I Ekren C Keller N Touzi and J Zhang ldquoOn viscositysolutions of path dependent PDEsrdquo Annals of Probability Inpress httparxivorgabs11095971

[21] S G Peng ldquoNote on viscosity solution of path-dependentPDE and G-Martingalesrdquo Probability In press httparxivorgabs11061144

[22] M G Crandall H Ishii and P-L Lions ldquoUserrsquos guide to vis-cosity solutions of second order partial differential equationsrdquoBulletin of the American Mathematical Society vol 27 no 1 pp1ndash67 1992

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


[14] investigated a controlled system governed by a functionalforward-backward stochastic differential equation (FBSDE)and proved the value function to be the viscosity solution ofthe related path-dependent HJB equation

In this paper inspired by [10 14] we will investigatethe SDG problems of the functional FBSDEs Preciselythe dynamics of our SDGs is described by the followingfunctional SDE

119889119883120574119905119906119907

(119904) = 119887 (119883120574119905119906119907

119904 119906 (119904) 119907 (119904)) 119889119904

+ 120590 (119883120574119905119906119907

119904 119906 (119904) 119907 (119904)) 119889119861 (119904) 119904 isin [119905 119879]

119883120574119905119906119907

119905= 120574119905

(1)

And the cost functional 119869(120574119905 119906 119907) is defined as119884120574119905119906119907

(119905)whichis the solution of the following functional BSDE

119889119884120574119905119906119907

(119904) = minus 119891 (119883120574119905119906119907

119904 119884

120574119905119906119907

(119904) 119885120574119905119906119907

(119904)

119906 (119904) 119907 (119904)) 119889119904 + 119885120574119905119906119907

(119904) 119889119861 (119904)

119884120574119905119906119907

(119879) = Φ (119883120574119905119906119907

119879) 119904 isin [119905 119879]

(2)

where 120574119905 is a path on [0 119905] The driver 119891 and Φ canbe interpreted as the running cost and the terminal costrespectively Also they depend on the past history of thedynamics Equations (1) and (2) compose a decoupled func-tional FBSDE The concrete conditions on 119887 120590 119891 Φ areshown in the later section

In the context we adopt the strategy against control formThe cost functional 119869(120574119905 119906 119907) is explained as a payoff forplayer I and as a cost for player II The aim of this paper isto show the following lower and upper value functions

119882(120574119905) = essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906))

119880 (120574119905) = esssup120572isinA119905119879

essinf119907isinV119905119879

119869 (120574119905 120572 (119907) 119907)

(3)

are the viscosity solutions of the following path-dependentHJBI equations respectively

119863119905119882(120574119905) + sup119906isin119880

inf119907isin119881

119867(120574119905119882119863119909119882119863119909119909119882119906 119907) = 0

119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ

119863119905119880(120574119905) + inf119907isin119881

sup119906isin119880

119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907) = 0

119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ

(4)

where

119867(120574119905 119910 119901 119883 119906 119907) =1

2tr (120590120590119879 (120574119905 119906 119907)119883)

+ 119901 sdot 119887 (120574119905 119906 119907)

+ 119891 (120574119905 119910 119901 sdot 120590 (120574119905 119906 119907) 119906 119907)

(5)


119889 times 119889 symmetric matrices)To solve the above SDG we need the functional Itorsquos

calculus and path-dependent PDEs which are recently intro-duced by Dupire [15] (for a recent account of this theorythe reader may consult [16ndash18] And under the framework offunctional Itorsquos calculus for the non-Markovian BSDEs PengandWang [19] derived a nonlinear Feynman-Kac formula forclassical solutions of path-dependent PDEs For the furtherdevelopment the readers may refer to [20 21])

In this paper we apply the Girsanov transformationmethod in Buckdahn and Li [10] to prove the determinacyof the value functions which is different from the methodintroduced by Peng [7 9] Making use of this methodand the functional Itorsquos calculus (introduced by Dupire [15]and developed by Cont and Fournie [16ndash18]) we completethe study of the zero-sum two-player SDGs in the non-Markovian case and present the lower and upper valuefunctions of our SDG are the viscosity solutions of thecorresponding path-dependent HJBI equations respectively

Different from the HJBI equations developed for stochas-tic delay systems we establish the DPP and derive the HJBIequation in the new framework of functional Itorsquos calculus

This paper is organized as follows Section 2 recalls thefunctional Itorsquos calculus and the well-known results of BSDEswe will use later In Section 3 we formulate our SDGs andget the corresponding DPP Based on the obtained DPP inSection 4 we derive the main result of the paper the lowerand upper value functions are the viscosity solutions of theassociated path-dependent HJBI equations respectively Andwe add the proof for the DPP in the appendix

2 Preliminaries

21 Functional Itorsquos Calculus We present some preliminariesfor functional Itorsquos calculus introduced firstly by Dupire [15]Here we follow the notations in [15]

Let 119879 gt 0 be fixed For each 119905 isin [0 119879] we denote Λ 119905 theset of cadlag functions from [0 119905] to R119889

For 120574 isin Λ 119879 denote 120574(119904) by the value of 120574 at time 119904 isin [0 119879]Thus 120574 = (120574(119904))0le119904le119879 is a cadlag process on [0 119879] and its valueat time 119904 is 120574(119904) 120574119905 = (120574(119904))0le119904le119905 isin Λ 119905 is the path of 120574 upto time 119905 We denote Λ = ⋃

119905isin[0119879]Λ 119905 For each 120574119905 isin Λ and

119909 isin R119889 120574119905(119904) is denoted by the value of 120574119905 at 119904 isin [0 119905] and120574119909

119905= (120574119905(119904)0le119904lt119905 120574119905(119905) + 119909) which is also an element in Λ 119905We now introduce a distance onΛ Let ⟨sdot sdot⟩ and | sdot| denote

the inner product and norm inR119889 For each 0 le 119905 119905 le 119879 and120574119905 120574119905 isin Λ we set

10038171003817100381710038171205741199051003817100381710038171003817 = sup

119904isin[0119905]

1003816100381610038161003816120574119905 (119904)1003816100381610038161003816

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817 = sup119904isin[0119905or119905]

1003816100381610038161003816120574119905 (119904 and 119905) minus 120574119905(119904 and 119905)

1003816100381610038161003816

119889infin (120574119905 120574119905) = sup119904isin[0119905or119905]

1003816100381610038161003816120574119905 (119904 and 119905) minus 120574119905(119904 and 119905)

1003816100381610038161003816 +1003816100381610038161003816119905 minus 119905

1003816100381610038161003816

(6)

It is obvious that Λ 119905 is a Banach space with respect to sdot Since Λ is not a linear space 119889infin is not a norm



119905)| lt 120576



119907 (120574119909

119905) = 119907 (120574119905) + ⟨119901 119909⟩ + 119900 (|119909|) as 119909 997888rarr 0 119909 isin R

119889

(7)


symmetric matrices





(8)










119907 (119883119905) minus 119907 (1198830) = int

119905

0

119863119904119907 (119883119904) 119889119904 + int

119905

0

119863119909119907 (119883119904) 119889119883 (119904)

+1

2int

119905

0

119863119909119909119907 (119883119904) 119889 ⟨119883⟩ (119904) 119875-as

(9)







119905=


S2(0 119879R

119899) = (120595(119905))

0le119905le119879R

119899-valued


119864 [ sup0le119905le119879

1003816100381610038161003816120595 (119905)10038161003816100381610038162] lt +infin

H2(0 119879R

119899) = (120595(119905))

0le119905le119879

R119899-valued


100381710038171003817100381712059510038171003817100381710038172= 119864[int

119879

0

1003816100381610038161003816120595 (119905)10038161003816100381610038162119889119905] lt +infin

(10)




1 1199102 isin R 1199111 1199112 isin R119889

1003816100381610038161003816119892 (119905 1199101 1199111) minus 119892 (119905 1199102 1199112)1003816100381610038161003816 le 119862 (

10038161003816100381610038161199101 minus 11991021003816100381610038161003816 +

10038161003816100381610038161199111 minus 11991121003816100381610038161003816) (11)

(A2) 119892(sdot 0 0) isin H2(0 119879R)



2(ΩF119879 119875R) the BSDE

119910 (119905) = 120585 + int

119879

119905

119892 (119904 119910 (119904) 119911 (119904)) 119889119904 minus int

119879

119905

119911 (119904) 119889119861 (119904)

0 le 119905 le 119879

(12)


(119910(119905) 119911(119905))119905isin[0119879]


2(0 119879R

119889) (13)



1198712(ΩF119879 119875R) by (119910

1 119911

1) and (119910

2 119911

2) one denotes the



2(119905)

P-as for all 119905 isin [0 119879]



1(119905) gt 119910

2(119905)) gt


2(0)



119892119894 (119904 119910119894(119904) 119911

119894(119904)) = 119892 (119904 119910

119894(119904) 119911

119894(119904)) + 120593119894 (119904)

119889119904119889119875-ae 119894 = 1 2

(14)




1) and (119910

2 119911

2)


100381610038161003816100381610038161199101(119905)minus119910

2(119905)

10038161003816100381610038161003816

2

+1

2119864 [int

119879

119905

119890120573(119904minus119905)

(100381610038161003816100381610038161199101(119904)minus119910

2(119904)

10038161003816100381610038161003816

2

+100381610038161003816100381610038161199111(119904) minus 119911

2(119904)

10038161003816100381610038161003816

2

) 119889119904 | F119905]

le 119864 [119890120573(119879minus119905)10038161003816100381610038161205851 minus 1205852

10038161003816100381610038162| F119905]

+ 119864 [int

119879

119905

119890120573(119904minus119905)10038161003816100381610038161205931 (119904) minus 1205932 (119904)

10038161003816100381610038162119889119904 | F119905]


(15)









(16)



119889119883120574119905119906119907

(119904) = 119887 (119883120574119905119906119907

119904 119906 (119904) 119907 (119904)) 119889119904

+ 120590 (119883120574119905119906119907

119904 119906 (119904) 119907 (119904)) 119889119861 (119904) 119904 isin [119905 119879]

119889119884120574119905119906119907

(119904) = minus119891 (119883120574119905119906119907

119904 119884

120574119905119906119907

(119904) 119885120574119905119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

+ 119885120574119905119906119907

(119904) 119889119861 (119904)

119883120574119905119906119907

119905= 120574119905

119884120574119905119906119907

(119879) = Φ (119883120574119905119906119907

119879)

(17)


R 119911 isin R119889 119887(119909119905 119906 119907) 120590(119909119905 119906 119907) and 119891(119909119905 119910 119911 119906 119907)


[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091

119905 119909

2

119905isin Λ

10038161003816100381610038161003816119887 (119909

1

119905 119906 119907) minus 119887 (119909

2

119905 119906 119907)

10038161003816100381610038161003816+10038161003816100381610038161003816120590 (119909

1

119905 119906 119907) minus 120590 (119909

2

119905 119906 119907)

10038161003816100381610038161003816

le 119862100381710038171003817100381710038171199091

119905minus 119909

2

119905

10038171003817100381710038171003817

(18)1003816100381610038161003816119887 (119909119905 119906 119907)

1003816100381610038161003816 +1003816100381610038161003816120590 (119909119905 119906 119907)

1003816100381610038161003816 le 119862 (1 +1003817100381710038171003817119909119905

1003817100381710038171003817) for any 119909119905 isin Λ

(19)


[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091

119905 119909

2

119905isin Λ 119910

1 119910

2isin

R 1199111 119911

2isin R119889

10038161003816100381610038161003816119891 (119909

1

119905 119910

1 119911

1 119906 119907) minus 119891 (119909

2

119905 119910

2 119911

2 119906 119907)

10038161003816100381610038161003816

le 119862 (100381710038171003817100381710038171199091

119905minus 119909

2

119905

10038171003817100381710038171003817+100381610038161003816100381610038161199101minus 119910

210038161003816100381610038161003816+100381610038161003816100381610038161199111minus 119911

210038161003816100381610038161003816)

10038161003816100381610038161003816Φ (119909

1

119879) minus Φ (119909

2

119879)10038161003816100381610038161003816le 119862

100381710038171003817100381710038171199091

119879minus 119909

2

119879

10038171003817100381710038171003817

1003816100381610038161003816119891 (119909119905 0 0 119906 119907)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171199091199051003817100381710038171003817)

1003816100381610038161003816Φ (119909119879)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171199091198791003817100381710038171003817) for any 119909119905 isin Λ

(20)


(0 119879R119899) timesS2

(0 119879R) timesH2(0 119879R119889

)

solving (17)









119869 (120574119905 119906 119907) = 119884120574119905119906119907

(119905) 120574119905 isin Λ (21)



119882(120574119905) = essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (22)

119880 (120574119905) = esssup120572isinA119905119879

essinf119907isinV119905119879

119869 (120574119905 120572 (119907) 119907) (23)




2([0 119879]R119889




0ℎ(119904)119889119861(119904) minus

(12) int119879

0|ℎ(119904)|





as follows

119883120574119905119906119907

(119904)∘(120591ℎ)= (120574119905 (119905)+int

119904

119905

119887 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119903

+int

119904

119905

120590 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119861 (119903)) ∘ (120591ℎ)

= 120574119905 (119905) + int

119904

119905

119887 (119883120574119905119906119907

119903(120591ℎ) 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119903

+ int

119904

119905

120590 (119883120574119905119906119907

119903(120591ℎ) 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119861 (119903)

119883120574119905119906(120591ℎ)119907(120591ℎ)(119904) = 120574119905 (119905) + int

119904

119905

119887 (119883120574119905119906(120591ℎ)119907(120591ℎ)

119903 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119903

+ int

119904

119905

120590 (119883120574119905119906(120591ℎ)119907(120591ℎ)

119903 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119861 (119903)

(24)


119883120574119905119906119907

(119904) (120591ℎ) = 119883120574119905119906(120591ℎ)119907(120591ℎ)(119904)

for any 119904 isin [119905 119879] 119875-as(25)


119884120574119905119906119907

(119904) (120591ℎ) = 119884120574119905119906(120591ℎ)119907(120591ℎ)(119904)

for any 119904 isin [119905 119879] 119875-as

119885120574119905119906119907

(119904) (120591ℎ) = 119885120574119905119906(120591ℎ)119907(120591ℎ)(119904)

119889119904119889119875-ae on [0 119879] times Ω

(26)

Hence

119869 (120574119905 119906 119907) (120591ℎ) = 119869 (120574119905 119906 (120591ℎ) 119907 (120591ℎ)) 119875-as (27)


U119905119879 Then 120573ℎisin B119905119879




on [[119905 119878(120591minusℎ)]] Thus


= 120573ℎ(1199062) on [[119905 119878]]

(28)


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(29)


119869(120574119905 119906




esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(30)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

119875-as for any 119906 isin U119905119879

(31)

Therefore

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(32)

From above we get

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(33)


119882(120574119905) (120591ℎ) = 119882(120574119905) 119875-as for any ℎ isin 119867 (34)


119868(120574119905 120573)(120591ℎ) =

essinf120573isinB119905119879


119882(120574119905) (120591ℎ) = essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 (120591ℎ) 120573ℎ(119906 (120591ℎ)))

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906))

= 119882 (120574119905) 119875-as(35)


| 120573 isin





2([0 119879]R119889

)

119864[1120577isin119860 expint

119879

0

120593 (119904) 119889119861 (119904)]

= 119864 [1120577isin119860] 119864 [expint

119879

0

120593 (119904) 119889119861 (119904)]

(36)

For any 120593 = sum119873

119894=11205931198941(119905


119905119894119873


119864[1120577isin119860 exp(

119873

sum

119894=1

120593119894119861 (119905119894) minus 119861 (119905119894minus1))]

= 119864 [1120577isin119860] 119864[exp119873

sum

119894=1

120593119894 (119861 (119905119894) minus 119861 (119905119894minus1))]

(37)


119864[1120577isin119860

119873

prod

119894=1

(119861 (119905119894) minus 119861 (119905119894minus1))119896119894

]

= 119864 [1120577isin119860] 119864[

119873

prod

119894=1

(119861 (119905119894) minus 119861 (119905119894minus1))119896119894

]

(38)




(39)







119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905]le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172)

119864 [ sup119904isin[119905119879]

10038161003816100381610038161003816119883

120574119905119906119907

(119904) minus 119883120574119905119906119907

(119904)10038161003816100381610038161003816

2

| F119905]le 1198621003817100381710038171003817120574119905 minus 120574

119905

10038171003817100381710038172

119864 [ sup119904isin[119905119879]

1003816100381610038161003816119884120574119905119906119907

(119904)10038161003816100381610038162+int

119879

119905

1003816100381610038161003816119885120574119905119906119907

(119904)10038161003816100381610038162119889119904 | F119905]le 119862 (1+

100381710038171003817100381712057411990510038171003817100381710038172)

119864 [ sup119904isin[119905119879]

10038161003816100381610038161003816119884120574119905119906119907

(119904) minus 119884120574119905119906119907

(119904)10038161003816100381610038161003816

2

+int

119879

119905

10038161003816100381610038161003816119885120574119905119906119907

(119904) minus 119885120574119905119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]le 1198621003817100381710038171003817120574119905 minus 120574

119905

10038171003817100381710038172


following property


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817

(ii) 1003816100381610038161003816119882 (120574119905)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171205741199051003817100381710038171003817)

(41)



119866120574119905119906119907

119904119905+120575[120578] =

120574119905119906119907

(119904) 119904 isin [119905 119905 + 120575] (42)

where 120578 isin 1198712(ΩF119905+120575

119905 119875R) (

120574119905119906119907

(119904) 119885120574119905119906119907

(119904))119905le119904le119905+120575


119889120574119905119906119907

(119904)

= minus119891 (119904 119883120574119905119906119907

119904

120574119905119906119907

(119904) 119885120574119905119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

+ 119885120574119905119906119907

(119904) 119889119861 (119904)

120574119905119906119907

(119905 + 120575) = 120578 119904 isin [119905 119905 + 120575]

(43)Also we have

119866120574119905119906119907

119905119879[Φ (119883

120574119905119906119907

119879)] = 119866

120574119905119906119907

119905119905+120575[119884

120574119905119906119907


0

119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (45)




119863119905119882(120574119905) + 119867minus(120574119905119882119863119909119882119863119909119909119882) = 0

119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ

(46)

119863119905119880 (120574119905) + 119867+(120574119905 119880119863119909119880119863119909119909119880) = 0

119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ

(47)

where

119867minus(120574119905119882119863119909119882119863119909119909119882)

= sup119906isin119880

inf119907isin119881

119867(120574119905119882119863119909119882119863119909119909119882119906 119907)

119867+(120574119905 119880119863119909119880119863119909119909119880)

= inf119907isin119881

sup119906isin119880

119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907)

119867 (120574119905 119910 119901 119883 119906 119907)

=1

2tr (120590120590119879 (120574119905 119906 119907)119883) + 119901 sdot 119887 (120574119905 119906 119907)

+ 119891 (120574119905 119910 119901120590 (120574119905 119906 119907) 119906 119907)

(48)





C(Λ) is called


C12


0le119904le120575Λ 119905+119904 and

Γ(120574119905) = 119882(120574119905) one has

119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) ge 0 (49)


C12


0le119904le120575Λ 119905+119904 and

Γ(120574119905) = 119882(120574119905) one has

119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) le 0 (50)





C12

119897119887(Λ) denote

119865 (120574119904 119910 119911 119906 119907)

= 119863119904Γ (120574119904) +1

2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))

+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)

+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)

(51)



minus 1198891198841119906119907

(119904)

= 119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198851119906119907

(119904) 119889119861 (119904)

1198841119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]

(52)


1198841119906119907

(119904) = 119866120574119905119906119907

119904119905+120575[Γ (119883

120574119905119906119907

119905+120575)] minus Γ (119883

120574119905119906119907

119904) 119875-as (53)

Proof Note119866120574119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] is defined as119866120574

119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] =

119884119906119907


minus 119889119884119906119907

(119904) = 119891 (119883120574119905119906119907

119904 119884

119906119907(119904) 119885

119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 119885119906119907

(119904) 119889119861 (119904)

119884119906119907

(119905 + 120575) = Γ (119883120574119905119906119907

119905+120575) 119904 isin [119905 119905 + 120575]

(54)


119904) we have

119889 (119884119906119907

(119904) minus Γ (119883120574119905119906119907

119904)) = 119889119884

1119906119907(119904) (55)

Combined with 119884119906119907

(119905 + 120575) minus Γ(119883120574119905119906119907

119905+120575) = 0 = 119884

1119906119907(119905 + 120575) we



minus 1198891198842119906119907

(119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198852119906119907

(119904) 119889119861 (119904)

(56)

1198842119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)



(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816le 119862120575

32 119875-as (58)



119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905] le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172) (59)

combined with

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]

le 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

119887 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119903

10038161003816100381610038161003816100381610038161003816

2

| F119905]

+ 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

120590 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119861 (119903)

10038161003816100381610038161003816100381610038161003816

2

| F119905]

(60)

we have

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)


1205851 = 1205852 = 0

119892 (119904 119910 119911) = 119865 (119883120574119905119906119907

119904 119910 119911 119906119904 119907119904)

1205931 (119904) = 0

1205932 (119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906119904 119907119904)

minus 119865 (119883120574119905119906119907

119904 119884

2119906119907(119904) 119885

2119906119907(119904) 119906119904 119907119904)

(62)



119887 |1205932(119904)| le 1205880(|119883

120574119905119906119907

119904minus

120574119905(119905)|) we have

119864[int

119905+120575

119905

(100381610038161003816100381610038161198841119906119907

(119904) minus1198842119906119907

(119904)10038161003816100381610038161003816

2

+100381610038161003816100381610038161198851119906119907

(119904) minus1198852119906119907

(119904)10038161003816100381610038161003816

2

) 119889119904 | F119905]

le 119864[int

119905+120575

119905

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) 119889119904 | F119905]

le 119862120575119864[ sup119904isin[119905119905+120575]

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) | F119905]

le 1198621205752

(63)


So100381610038161003816100381610038161198841119906119907

(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816

=10038161003816100381610038161003816119864 [(119884

1119906119907(119904) minus 119884

2119906119907(119904)) | F119905]

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

119864 [int

119905+120575

119905

(119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904))

minus119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]

100381610038161003816100381610038161003816100381610038161003816

le 119862119864[int

119905+120575

119905

[1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

+100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816] 119889119904 | F119905]

le 119862119864[int

119905+120575

119905

1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) 119889119904 | F119905]

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

le 11986212057532

(64)


minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]

1198840 (119905 + 120575) = 0

(65)

where 1198650(120574119905 119910 119911) = sup119906isin119880

inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as

1198840 (119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) (66)


1198651 (120574119905 119910 119911 119906) = inf119907isin119881

119865 (120574119905 119910 119911 119906 119907)


(67)


minus1198891198843119906

(119904) = 1198651 (120574119905 1198843119906

(119904) 1198853119906

(119904) 119906 (119904)) 119889119904

minus1198853119906

(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]

1198843119906

(119905 + 120575) = 0

(68)


1198853119906

) solving (68) Also

1198843119906

(119905) = essinf119907isinV119905119905+120575

1198842119906119907


(69)


1198843119906

(119905) le 1198842119906119907

(119905) 119875-as for any 119907 isin V119905119905+120575

for every 119906 isin U119905119905+120575

(70)


4 R timesR119889


1198651 (120574119905 119910 119911 119906)

= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906

(71)

Set 1199074(119904) = 1199074(119884

3119906(119904) 119885

3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know

1199074isin V119905119905+120575 Therefore (1198843119906

1198853119906

) = (11988421199061199074

11988521199061199074


(119905) = 11988421199061199074


(119905) = essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-119886119904 for every 119906 isin U119905119905+120575


1198651(120574119905 119910 119911 119906) wealso derive

1198840 (119905) = esssup119906isinU119905119905+120575

1198843119906

(119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-as

(72)


119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575

32 119875-as

(73)



100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816

2

le 119862120575

119864 [int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904] le 119862120575

(74)



(119904)10038161003816100381610038161003816

le 119864 [int

119905+120575

119904

119865 (120574119905 1198842119906119907

(119903) 1198852119906119907

(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]

le 119862119864[int

119905+120575

119904

(1 +1003817100381710038171003817120574119905

10038171003817100381710038172+100381610038161003816100381610038161198842119906119907

(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816) 119889119903 | F119904]

le 119862120575 + 11986212057512

119864[int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904]

le 119862120575 119904 isin [119905 119905 + 120575]

(75)


119905|119885

2119906119907(119904)|

2

119889119904 |

F119905] le 1198621205752 119875-as Therefore

119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905]

le 1198621205752+ 119862120575

12(119864[int

119905+120575

119905

100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816

2

119889119903 | F119905])

12

le 11986212057532

119875-as

(76)



0le119904le120575Λ 119905+119904


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(77)

From 119882 ge Γ on ⋃0le119904le120575


119905119906120573(119906)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

120574119905119906120573(119906)

119905+120575)] minus Γ (120574119905) le 0 (78)

By Lemma 19 we have

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) le 0 119875-as (79)

Thus from Lemma 20

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as (80)


1198842119906119907

(119905) le 1198842119906120573(119906)

(119905) 120573 isin

B119905119905+120575 we have

esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905)

le essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as(81)

Thus by Lemma 21 1198840(119905) le 11986212057532


11986212057512

ge1

1205751198840 (119905) =

1

120575int

119905+120575

119905

1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)

sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)



Γ ge 119882 on⋃0le119904le120575


sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)


1198650 (120574119905 0 0) = sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)


119865 (120574119905 0 0 119906 120595 (119906)) le minus3

4120579 forall119906 isin 119880 (86)


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(87)



119905119906120573(119906)

119905119905+120575[sdot] (Lemma 7) we get

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

119906120573(119906)

119905+120575)] minus Γ (120574119905) ge 0 119875-as

(88)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) ge 0 119875-as (89)

in particular

esssup119906isinU119905119905+120575

1198841119906120595(119906)

(119905) ge 0 119875-as (90)


1119906120576120595(119906120576)(119905) ge minus120576120575


1198842119906120576120595(119906120576)(119905) ge minus119862120575

32minus 120576120575 119875-as (91)



1198842119906120576120595(119906120576)(119905)

= 119864 [int

119905+120575

119905

119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885

2119906120576120595(119906120576)(119904)

119906120576(119904) 120595 (119906

120576(119904))) 119889119904 | F119905]

le 119864[int

119905+120575

119905

(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816+ 119862

1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816

+119865 (120574119905 0 0 119906120576(119904) 120595 (119906

120576(119904)))) 119889119904 | F119905]

le 11986212057532

minus3

4120579120575 119875-as

(92)


minus 120576 le 11986212057512

minus


sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)



Appendix



119882120575 (120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (A1)




Lemma A2 119882120575(120574119905) le 119882(120574119905)





119882120575 (120574119905) le esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)] 119875-as

(A3)

We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907

119905119905+120575[119882(119883

120574119905119906119907



119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575

119868120575 (120574119905 1199061 1205731 (1199061))

= sup119894ge1

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) 119875-as

(A4)


119894 1205731(119906

1

119894)) +


(⋃119894minus1


119906120576

1= sum

119894ge11Γ119894

1199061


of 1205731 we have 1205731(119906120576

1) = sum

119894ge11Γ119894

1205731(1199061



120576

1 1205731(119906

120576

1)) = sum

119894ge11Γ119894

119868120575(120574119905 1199061

119894 1205731(119906

1

119894)) 119875-as So

119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum

119894ge1

1Γ119894

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) + 120576

= 119868120575 (120574119905 119906120576

1 1205731 (119906

120576

1)) + 120576

= 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119882(119883

120574119905119906120576

11205731(119906120576

1)

119905+120575)] + 120576

(A5)


120576

1oplus


119882(120574119905) le esssup1199062isinU119905+120575119879

119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817 for any 120574119905 120574119905 isin Λ

(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))

1003816100381610038161003816

le 1198621003817100381710038171003817120574119905 minus 120574

119905

1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879

(A7)


11205731(119906120576

1)

119905+120575that

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062)) 119875-as

(A8)



119895 119895 ge 1 sub U119905+120575119879 such that

esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

= sup119895ge1

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A9)


119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732(1199062)) le

119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732(119906

2


Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1


(ΩF119905+120575)-partition and 119906120576

2= sum

119895ge11Δ119895

1199062

119895isin U119905+120575119879 Due to


2) = sum

119895ge11Δ119895

1205732(1199062

119895)


1oplus 119906

120576

2) =

1205731(119906120576

1) oplus 1205732(119906

120576



119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

2 1205732 (119906

120576

2))

= 119884119883120574119905119906120576

11205731(119906120576

1)

119905+120575119906120576

21205732(119906120576

2)(119905 + 120575)

= sum

119895ge1

1Δ119895

119884119883120574119905119906120576

11205731(119906120576

1)

119905+1205751199062

1198951205732(1199062

119895)(119905 + 120575)

= sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A10)

So

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

le sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

2

119895 120573 (119906

120576

1oplus 119906

2

119895)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

120576

2 120573 (119906

120576

1oplus 119906

120576

2)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576 119875-as

(A11)

where 119906120576

= 119906120576

1oplus 119906

120576



119882120575 (120574119905)

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576] + 120576

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119866120574119905119906120576120573(119906120576)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119869 (120574119905 119906120576 120573 (119906

120576)) + (119862 + 1) 120576

= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576


119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576 119875-as

(A12)


119882120575 (120574119905) le essinf120573isinB119905119879

esssup119906isinU119905119879

119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576

= 119882(120574119905) + (119862 + 1) 120576

(A13)


Lemma A3 119882(120574119905) le 119882120575(120574119905)


119882120575 (120574119905) = essinf1205731isinB119905119905+120575

esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)]

= essinf1205731isinB119905119905+120575

119868120575 (120574119905 1205731)

(A14)


119894 119894 ge 1 sub B119905119905+120575 such

that

119882120575 (120574119905) = inf119894ge1

119868120575 (120574119905 1205731

119894) 119875-as (A15)

For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731

119894) minus 120576 le 119882120575(120574119905) isin


119897=1Π119897) isin F119905 119894 ge


1=

sum119894ge1

1Λ119894

1205731



1(1199061)) =

sum119894ge1

1Λ119894

119868120575(120574119905 1199061 1205731


119882120575 (120574119905) ge sum

119894ge1

1Π119894

119868120575 (120574119905 1205731

119894) minus 120576

ge sum

119894ge1

1Π119894

119868120575 (120574119905 1199061 1205731

119894(1199061)) minus 120576

= 119868120575 (120574119905 1199061 120573120576

1(1199061)) minus 120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

119875-as forall1199061 isin U119905119905+120575

(A16)




B119905+120575119879 such that

119882(119909119905+120575) ge esssup1199062isinU119905+120575119879

119869 (119909119905+120575 1199062 120573119909120576

(1199062)) minus 120576 119875-as

(A17)



1205741199051199061120573120576

1(1199061)

119905+120575] = sum

119894ge1120574119894

119905+12057511198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894

we have

100381610038161003816100381610038161003816119883

1205741199051199061120573120576

1(1199061)

119905+120575minus [119883

1205741199051199061120573120576

1(1199061)

119905+120575]100381610038161003816100381610038161003816le 120576


(A18)



(A17) and 120573120576

1199061

= sum119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894120573120574119894120576isin B119905+120575119879


120576

1(1199061) oplus 120573

120576

1199061

(1199062) 119906 isin





1015840 into 1199061 1199061015840

1isin

U119905119905+120575 1199062 1199061015840


1015840

1and 119906 = 1199062 oplus 119906

1015840

2

From 1199061 equiv 1199061015840

1on [[119905 119878 and (119905 + 120575)]] we get 120573120576

1(1199061) = 120573

120576

1(119906

1015840

1) on

[[119905 119878and(119905+120575)]] (note that 120573120576


side from 1199062 equiv 1199061015840

2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt

119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576

1(1199061)

119905+120575= 119883

1205741199051199061015840

1120573120576

1(1199061015840

1)

119905+120575


1199061

= 120573120576

11990610158401

on 119878 gt 119905 + 120575 and120573120576

1199061

(1199062) = 120573120576

11990610158401

(1199061015840


120573120576(119906) = 120573

120576

1(1199061) oplus 120573

120576

1199061

(1199062) = 120573120576

1(119906

1015840

1) oplus 120573

120576

11990610158401

(1199061015840

2) on [[119905 119878]] from




119882120575 (120574119905) ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575]) minus 119862120576] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575])] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119882(120574

119894

119905+120575)] minus 119862120576

119875-as

(A19)


119882120575 (120574119905)

ge1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))minus 120576]minus119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 ([119883

1205741199051199061120573120576

1(1199061)

119905+120575] 1199062 120573

120576

1199061

(1199062))] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062)) minus 119862120576] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062))] minus 119862120576

= 119866120574119905119906120573120576(119906)

119905119905+120575[119884

120574119905119906120573120576(119906)

(119905 + 120575)] minus 119862120576

= 119884120574119905119906120573120576(119906)


(A20)

Therefore

119882120575 (120574119905) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573120576(119906)) minus 119862120576


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) minus 119862120576

= 119882(120574119905) minus 119862120576 119875-as

(A21)



119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)

119905119905+120575[119882(119883

120574119905119906120576120573(119906120576)

119905+120575)] + 120576

119875-as(A22)


119880119905119905+120575

119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)

119905119905+120575[119882(119883

120574119905119906120573120576(119906)

119905+120575)] minus 120576

119875-as(A23)


inf120573isinB119905119879

sup119906isinU119905119879

119864[119869(120574119905 119906 120573(119906))]

Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119905)| lt 120576



119907 (120574119909

119905) = 119907 (120574119905) + ⟨119901 119909⟩ + 119900 (|119909|) as 119909 997888rarr 0 119909 isin R

119889

(7)


symmetric matrices





(8)










119907 (119883119905) minus 119907 (1198830) = int

119905

0

119863119904119907 (119883119904) 119889119904 + int

119905

0

119863119909119907 (119883119904) 119889119883 (119904)

+1

2int

119905

0

119863119909119909119907 (119883119904) 119889 ⟨119883⟩ (119904) 119875-as

(9)







119905=


S2(0 119879R

119899) = (120595(119905))

0le119905le119879R

119899-valued


119864 [ sup0le119905le119879

1003816100381610038161003816120595 (119905)10038161003816100381610038162] lt +infin

H2(0 119879R

119899) = (120595(119905))

0le119905le119879

R119899-valued


100381710038171003817100381712059510038171003817100381710038172= 119864[int

119879

0

1003816100381610038161003816120595 (119905)10038161003816100381610038162119889119905] lt +infin

(10)




1 1199102 isin R 1199111 1199112 isin R119889

1003816100381610038161003816119892 (119905 1199101 1199111) minus 119892 (119905 1199102 1199112)1003816100381610038161003816 le 119862 (

10038161003816100381610038161199101 minus 11991021003816100381610038161003816 +

10038161003816100381610038161199111 minus 11991121003816100381610038161003816) (11)

(A2) 119892(sdot 0 0) isin H2(0 119879R)



2(ΩF119879 119875R) the BSDE

119910 (119905) = 120585 + int

119879

119905

119892 (119904 119910 (119904) 119911 (119904)) 119889119904 minus int

119879

119905

119911 (119904) 119889119861 (119904)

0 le 119905 le 119879

(12)


(119910(119905) 119911(119905))119905isin[0119879]


2(0 119879R

119889) (13)



1198712(ΩF119879 119875R) by (119910

1 119911

1) and (119910

2 119911

2) one denotes the



2(119905)

P-as for all 119905 isin [0 119879]



1(119905) gt 119910

2(119905)) gt


2(0)



119892119894 (119904 119910119894(119904) 119911

119894(119904)) = 119892 (119904 119910

119894(119904) 119911

119894(119904)) + 120593119894 (119904)

119889119904119889119875-ae 119894 = 1 2

(14)




1) and (119910

2 119911

2)


100381610038161003816100381610038161199101(119905)minus119910

2(119905)

10038161003816100381610038161003816

2

+1

2119864 [int

119879

119905

119890120573(119904minus119905)

(100381610038161003816100381610038161199101(119904)minus119910

2(119904)

10038161003816100381610038161003816

2

+100381610038161003816100381610038161199111(119904) minus 119911

2(119904)

10038161003816100381610038161003816

2

) 119889119904 | F119905]

le 119864 [119890120573(119879minus119905)10038161003816100381610038161205851 minus 1205852

10038161003816100381610038162| F119905]

+ 119864 [int

119879

119905

119890120573(119904minus119905)10038161003816100381610038161205931 (119904) minus 1205932 (119904)

10038161003816100381610038162119889119904 | F119905]


(15)









(16)



119889119883120574119905119906119907

(119904) = 119887 (119883120574119905119906119907

119904 119906 (119904) 119907 (119904)) 119889119904

+ 120590 (119883120574119905119906119907

119904 119906 (119904) 119907 (119904)) 119889119861 (119904) 119904 isin [119905 119879]

119889119884120574119905119906119907

(119904) = minus119891 (119883120574119905119906119907

119904 119884

120574119905119906119907

(119904) 119885120574119905119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

+ 119885120574119905119906119907

(119904) 119889119861 (119904)

119883120574119905119906119907

119905= 120574119905

119884120574119905119906119907

(119879) = Φ (119883120574119905119906119907

119879)

(17)


R 119911 isin R119889 119887(119909119905 119906 119907) 120590(119909119905 119906 119907) and 119891(119909119905 119910 119911 119906 119907)


[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091

119905 119909

2

119905isin Λ

10038161003816100381610038161003816119887 (119909

1

119905 119906 119907) minus 119887 (119909

2

119905 119906 119907)

10038161003816100381610038161003816+10038161003816100381610038161003816120590 (119909

1

119905 119906 119907) minus 120590 (119909

2

119905 119906 119907)

10038161003816100381610038161003816

le 119862100381710038171003817100381710038171199091

119905minus 119909

2

119905

10038171003817100381710038171003817

(18)1003816100381610038161003816119887 (119909119905 119906 119907)

1003816100381610038161003816 +1003816100381610038161003816120590 (119909119905 119906 119907)

1003816100381610038161003816 le 119862 (1 +1003817100381710038171003817119909119905

1003817100381710038171003817) for any 119909119905 isin Λ

(19)


[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091

119905 119909

2

119905isin Λ 119910

1 119910

2isin

R 1199111 119911

2isin R119889

10038161003816100381610038161003816119891 (119909

1

119905 119910

1 119911

1 119906 119907) minus 119891 (119909

2

119905 119910

2 119911

2 119906 119907)

10038161003816100381610038161003816

le 119862 (100381710038171003817100381710038171199091

119905minus 119909

2

119905

10038171003817100381710038171003817+100381610038161003816100381610038161199101minus 119910

210038161003816100381610038161003816+100381610038161003816100381610038161199111minus 119911

210038161003816100381610038161003816)

10038161003816100381610038161003816Φ (119909

1

119879) minus Φ (119909

2

119879)10038161003816100381610038161003816le 119862

100381710038171003817100381710038171199091

119879minus 119909

2

119879

10038171003817100381710038171003817

1003816100381610038161003816119891 (119909119905 0 0 119906 119907)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171199091199051003817100381710038171003817)

1003816100381610038161003816Φ (119909119879)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171199091198791003817100381710038171003817) for any 119909119905 isin Λ

(20)


(0 119879R119899) timesS2

(0 119879R) timesH2(0 119879R119889

)

solving (17)









119869 (120574119905 119906 119907) = 119884120574119905119906119907

(119905) 120574119905 isin Λ (21)



119882(120574119905) = essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (22)

119880 (120574119905) = esssup120572isinA119905119879

essinf119907isinV119905119879

119869 (120574119905 120572 (119907) 119907) (23)




2([0 119879]R119889




0ℎ(119904)119889119861(119904) minus

(12) int119879

0|ℎ(119904)|





as follows

119883120574119905119906119907

(119904)∘(120591ℎ)= (120574119905 (119905)+int

119904

119905

119887 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119903

+int

119904

119905

120590 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119861 (119903)) ∘ (120591ℎ)

= 120574119905 (119905) + int

119904

119905

119887 (119883120574119905119906119907

119903(120591ℎ) 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119903

+ int

119904

119905

120590 (119883120574119905119906119907

119903(120591ℎ) 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119861 (119903)

119883120574119905119906(120591ℎ)119907(120591ℎ)(119904) = 120574119905 (119905) + int

119904

119905

119887 (119883120574119905119906(120591ℎ)119907(120591ℎ)

119903 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119903

+ int

119904

119905

120590 (119883120574119905119906(120591ℎ)119907(120591ℎ)

119903 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119861 (119903)

(24)


119883120574119905119906119907

(119904) (120591ℎ) = 119883120574119905119906(120591ℎ)119907(120591ℎ)(119904)

for any 119904 isin [119905 119879] 119875-as(25)


119884120574119905119906119907

(119904) (120591ℎ) = 119884120574119905119906(120591ℎ)119907(120591ℎ)(119904)

for any 119904 isin [119905 119879] 119875-as

119885120574119905119906119907

(119904) (120591ℎ) = 119885120574119905119906(120591ℎ)119907(120591ℎ)(119904)

119889119904119889119875-ae on [0 119879] times Ω

(26)

Hence

119869 (120574119905 119906 119907) (120591ℎ) = 119869 (120574119905 119906 (120591ℎ) 119907 (120591ℎ)) 119875-as (27)


U119905119879 Then 120573ℎisin B119905119879




on [[119905 119878(120591minusℎ)]] Thus


= 120573ℎ(1199062) on [[119905 119878]]

(28)


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(29)


119869(120574119905 119906




esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(30)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

119875-as for any 119906 isin U119905119879

(31)

Therefore

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(32)

From above we get

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(33)


119882(120574119905) (120591ℎ) = 119882(120574119905) 119875-as for any ℎ isin 119867 (34)


119868(120574119905 120573)(120591ℎ) =

essinf120573isinB119905119879


119882(120574119905) (120591ℎ) = essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 (120591ℎ) 120573ℎ(119906 (120591ℎ)))

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906))

= 119882 (120574119905) 119875-as(35)


| 120573 isin





2([0 119879]R119889

)

119864[1120577isin119860 expint

119879

0

120593 (119904) 119889119861 (119904)]

= 119864 [1120577isin119860] 119864 [expint

119879

0

120593 (119904) 119889119861 (119904)]

(36)

For any 120593 = sum119873

119894=11205931198941(119905


119905119894119873


119864[1120577isin119860 exp(

119873

sum

119894=1

120593119894119861 (119905119894) minus 119861 (119905119894minus1))]

= 119864 [1120577isin119860] 119864[exp119873

sum

119894=1

120593119894 (119861 (119905119894) minus 119861 (119905119894minus1))]

(37)


119864[1120577isin119860

119873

prod

119894=1

(119861 (119905119894) minus 119861 (119905119894minus1))119896119894

]

= 119864 [1120577isin119860] 119864[

119873

prod

119894=1

(119861 (119905119894) minus 119861 (119905119894minus1))119896119894

]

(38)




(39)







119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905]le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172)

119864 [ sup119904isin[119905119879]

10038161003816100381610038161003816119883

120574119905119906119907

(119904) minus 119883120574119905119906119907

(119904)10038161003816100381610038161003816

2

| F119905]le 1198621003817100381710038171003817120574119905 minus 120574

119905

10038171003817100381710038172

119864 [ sup119904isin[119905119879]

1003816100381610038161003816119884120574119905119906119907

(119904)10038161003816100381610038162+int

119879

119905

1003816100381610038161003816119885120574119905119906119907

(119904)10038161003816100381610038162119889119904 | F119905]le 119862 (1+

100381710038171003817100381712057411990510038171003817100381710038172)

119864 [ sup119904isin[119905119879]

10038161003816100381610038161003816119884120574119905119906119907

(119904) minus 119884120574119905119906119907

(119904)10038161003816100381610038161003816

2

+int

119879

119905

10038161003816100381610038161003816119885120574119905119906119907

(119904) minus 119885120574119905119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]le 1198621003817100381710038171003817120574119905 minus 120574

119905

10038171003817100381710038172


following property


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817

(ii) 1003816100381610038161003816119882 (120574119905)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171205741199051003817100381710038171003817)

(41)



119866120574119905119906119907

119904119905+120575[120578] =

120574119905119906119907

(119904) 119904 isin [119905 119905 + 120575] (42)

where 120578 isin 1198712(ΩF119905+120575

119905 119875R) (

120574119905119906119907

(119904) 119885120574119905119906119907

(119904))119905le119904le119905+120575


119889120574119905119906119907

(119904)

= minus119891 (119904 119883120574119905119906119907

119904

120574119905119906119907

(119904) 119885120574119905119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

+ 119885120574119905119906119907

(119904) 119889119861 (119904)

120574119905119906119907

(119905 + 120575) = 120578 119904 isin [119905 119905 + 120575]

(43)Also we have

119866120574119905119906119907

119905119879[Φ (119883

120574119905119906119907

119879)] = 119866

120574119905119906119907

119905119905+120575[119884

120574119905119906119907


0

119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (45)




119863119905119882(120574119905) + 119867minus(120574119905119882119863119909119882119863119909119909119882) = 0

119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ

(46)

119863119905119880 (120574119905) + 119867+(120574119905 119880119863119909119880119863119909119909119880) = 0

119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ

(47)

where

119867minus(120574119905119882119863119909119882119863119909119909119882)

= sup119906isin119880

inf119907isin119881

119867(120574119905119882119863119909119882119863119909119909119882119906 119907)

119867+(120574119905 119880119863119909119880119863119909119909119880)

= inf119907isin119881

sup119906isin119880

119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907)

119867 (120574119905 119910 119901 119883 119906 119907)

=1

2tr (120590120590119879 (120574119905 119906 119907)119883) + 119901 sdot 119887 (120574119905 119906 119907)

+ 119891 (120574119905 119910 119901120590 (120574119905 119906 119907) 119906 119907)

(48)





C(Λ) is called


C12


0le119904le120575Λ 119905+119904 and

Γ(120574119905) = 119882(120574119905) one has

119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) ge 0 (49)


C12


0le119904le120575Λ 119905+119904 and

Γ(120574119905) = 119882(120574119905) one has

119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) le 0 (50)





C12

119897119887(Λ) denote

119865 (120574119904 119910 119911 119906 119907)

= 119863119904Γ (120574119904) +1

2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))

+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)

+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)

(51)



minus 1198891198841119906119907

(119904)

= 119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198851119906119907

(119904) 119889119861 (119904)

1198841119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]

(52)


1198841119906119907

(119904) = 119866120574119905119906119907

119904119905+120575[Γ (119883

120574119905119906119907

119905+120575)] minus Γ (119883

120574119905119906119907

119904) 119875-as (53)

Proof Note119866120574119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] is defined as119866120574

119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] =

119884119906119907


minus 119889119884119906119907

(119904) = 119891 (119883120574119905119906119907

119904 119884

119906119907(119904) 119885

119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 119885119906119907

(119904) 119889119861 (119904)

119884119906119907

(119905 + 120575) = Γ (119883120574119905119906119907

119905+120575) 119904 isin [119905 119905 + 120575]

(54)


119904) we have

119889 (119884119906119907

(119904) minus Γ (119883120574119905119906119907

119904)) = 119889119884

1119906119907(119904) (55)

Combined with 119884119906119907

(119905 + 120575) minus Γ(119883120574119905119906119907

119905+120575) = 0 = 119884

1119906119907(119905 + 120575) we



minus 1198891198842119906119907

(119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198852119906119907

(119904) 119889119861 (119904)

(56)

1198842119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)



(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816le 119862120575

32 119875-as (58)



119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905] le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172) (59)

combined with

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]

le 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

119887 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119903

10038161003816100381610038161003816100381610038161003816

2

| F119905]

+ 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

120590 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119861 (119903)

10038161003816100381610038161003816100381610038161003816

2

| F119905]

(60)

we have

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)


1205851 = 1205852 = 0

119892 (119904 119910 119911) = 119865 (119883120574119905119906119907

119904 119910 119911 119906119904 119907119904)

1205931 (119904) = 0

1205932 (119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906119904 119907119904)

minus 119865 (119883120574119905119906119907

119904 119884

2119906119907(119904) 119885

2119906119907(119904) 119906119904 119907119904)

(62)



119887 |1205932(119904)| le 1205880(|119883

120574119905119906119907

119904minus

120574119905(119905)|) we have

119864[int

119905+120575

119905

(100381610038161003816100381610038161198841119906119907

(119904) minus1198842119906119907

(119904)10038161003816100381610038161003816

2

+100381610038161003816100381610038161198851119906119907

(119904) minus1198852119906119907

(119904)10038161003816100381610038161003816

2

) 119889119904 | F119905]

le 119864[int

119905+120575

119905

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) 119889119904 | F119905]

le 119862120575119864[ sup119904isin[119905119905+120575]

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) | F119905]

le 1198621205752

(63)


So100381610038161003816100381610038161198841119906119907

(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816

=10038161003816100381610038161003816119864 [(119884

1119906119907(119904) minus 119884

2119906119907(119904)) | F119905]

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

119864 [int

119905+120575

119905

(119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904))

minus119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]

100381610038161003816100381610038161003816100381610038161003816

le 119862119864[int

119905+120575

119905

[1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

+100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816] 119889119904 | F119905]

le 119862119864[int

119905+120575

119905

1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) 119889119904 | F119905]

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

le 11986212057532

(64)


minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]

1198840 (119905 + 120575) = 0

(65)

where 1198650(120574119905 119910 119911) = sup119906isin119880

inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as

1198840 (119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) (66)


1198651 (120574119905 119910 119911 119906) = inf119907isin119881

119865 (120574119905 119910 119911 119906 119907)


(67)


minus1198891198843119906

(119904) = 1198651 (120574119905 1198843119906

(119904) 1198853119906

(119904) 119906 (119904)) 119889119904

minus1198853119906

(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]

1198843119906

(119905 + 120575) = 0

(68)


1198853119906

) solving (68) Also

1198843119906

(119905) = essinf119907isinV119905119905+120575

1198842119906119907


(69)


1198843119906

(119905) le 1198842119906119907

(119905) 119875-as for any 119907 isin V119905119905+120575

for every 119906 isin U119905119905+120575

(70)


4 R timesR119889


1198651 (120574119905 119910 119911 119906)

= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906

(71)

Set 1199074(119904) = 1199074(119884

3119906(119904) 119885

3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know

1199074isin V119905119905+120575 Therefore (1198843119906

1198853119906

) = (11988421199061199074

11988521199061199074


(119905) = 11988421199061199074


(119905) = essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-119886119904 for every 119906 isin U119905119905+120575


1198651(120574119905 119910 119911 119906) wealso derive

1198840 (119905) = esssup119906isinU119905119905+120575

1198843119906

(119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-as

(72)


119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575

32 119875-as

(73)



100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816

2

le 119862120575

119864 [int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904] le 119862120575

(74)



(119904)10038161003816100381610038161003816

le 119864 [int

119905+120575

119904

119865 (120574119905 1198842119906119907

(119903) 1198852119906119907

(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]

le 119862119864[int

119905+120575

119904

(1 +1003817100381710038171003817120574119905

10038171003817100381710038172+100381610038161003816100381610038161198842119906119907

(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816) 119889119903 | F119904]

le 119862120575 + 11986212057512

119864[int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904]

le 119862120575 119904 isin [119905 119905 + 120575]

(75)


119905|119885

2119906119907(119904)|

2

119889119904 |

F119905] le 1198621205752 119875-as Therefore

119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905]

le 1198621205752+ 119862120575

12(119864[int

119905+120575

119905

100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816

2

119889119903 | F119905])

12

le 11986212057532

119875-as

(76)



0le119904le120575Λ 119905+119904


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(77)

From 119882 ge Γ on ⋃0le119904le120575


119905119906120573(119906)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

120574119905119906120573(119906)

119905+120575)] minus Γ (120574119905) le 0 (78)

By Lemma 19 we have

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) le 0 119875-as (79)

Thus from Lemma 20

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as (80)


1198842119906119907

(119905) le 1198842119906120573(119906)

(119905) 120573 isin

B119905119905+120575 we have

esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905)

le essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as(81)

Thus by Lemma 21 1198840(119905) le 11986212057532


11986212057512

ge1

1205751198840 (119905) =

1

120575int

119905+120575

119905

1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)

sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)



Γ ge 119882 on⋃0le119904le120575


sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)


1198650 (120574119905 0 0) = sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)


119865 (120574119905 0 0 119906 120595 (119906)) le minus3

4120579 forall119906 isin 119880 (86)


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(87)



119905119906120573(119906)

119905119905+120575[sdot] (Lemma 7) we get

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

119906120573(119906)

119905+120575)] minus Γ (120574119905) ge 0 119875-as

(88)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) ge 0 119875-as (89)

in particular

esssup119906isinU119905119905+120575

1198841119906120595(119906)

(119905) ge 0 119875-as (90)


1119906120576120595(119906120576)(119905) ge minus120576120575


1198842119906120576120595(119906120576)(119905) ge minus119862120575

32minus 120576120575 119875-as (91)



1198842119906120576120595(119906120576)(119905)

= 119864 [int

119905+120575

119905

119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885

2119906120576120595(119906120576)(119904)

119906120576(119904) 120595 (119906

120576(119904))) 119889119904 | F119905]

le 119864[int

119905+120575

119905

(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816+ 119862

1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816

+119865 (120574119905 0 0 119906120576(119904) 120595 (119906

120576(119904)))) 119889119904 | F119905]

le 11986212057532

minus3

4120579120575 119875-as

(92)


minus 120576 le 11986212057512

minus


sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)



Appendix



119882120575 (120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (A1)




Lemma A2 119882120575(120574119905) le 119882(120574119905)





119882120575 (120574119905) le esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)] 119875-as

(A3)

We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907

119905119905+120575[119882(119883

120574119905119906119907



119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575

119868120575 (120574119905 1199061 1205731 (1199061))

= sup119894ge1

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) 119875-as

(A4)


119894 1205731(119906

1

119894)) +


(⋃119894minus1


119906120576

1= sum

119894ge11Γ119894

1199061


of 1205731 we have 1205731(119906120576

1) = sum

119894ge11Γ119894

1205731(1199061



120576

1 1205731(119906

120576

1)) = sum

119894ge11Γ119894

119868120575(120574119905 1199061

119894 1205731(119906

1

119894)) 119875-as So

119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum

119894ge1

1Γ119894

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) + 120576

= 119868120575 (120574119905 119906120576

1 1205731 (119906

120576

1)) + 120576

= 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119882(119883

120574119905119906120576

11205731(119906120576

1)

119905+120575)] + 120576

(A5)


120576

1oplus


119882(120574119905) le esssup1199062isinU119905+120575119879

119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817 for any 120574119905 120574119905 isin Λ

(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))

1003816100381610038161003816

le 1198621003817100381710038171003817120574119905 minus 120574

119905

1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879

(A7)


11205731(119906120576

1)

119905+120575that

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062)) 119875-as

(A8)



119895 119895 ge 1 sub U119905+120575119879 such that

esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

= sup119895ge1

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A9)


119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732(1199062)) le

119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732(119906

2


Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1


(ΩF119905+120575)-partition and 119906120576

2= sum

119895ge11Δ119895

1199062

119895isin U119905+120575119879 Due to


2) = sum

119895ge11Δ119895

1205732(1199062

119895)


1oplus 119906

120576

2) =

1205731(119906120576

1) oplus 1205732(119906

120576



119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

2 1205732 (119906

120576

2))

= 119884119883120574119905119906120576

11205731(119906120576

1)

119905+120575119906120576

21205732(119906120576

2)(119905 + 120575)

= sum

119895ge1

1Δ119895

119884119883120574119905119906120576

11205731(119906120576

1)

119905+1205751199062

1198951205732(1199062

119895)(119905 + 120575)

= sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A10)

So

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

le sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

2

119895 120573 (119906

120576

1oplus 119906

2

119895)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

120576

2 120573 (119906

120576

1oplus 119906

120576

2)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576 119875-as

(A11)

where 119906120576

= 119906120576

1oplus 119906

120576



119882120575 (120574119905)

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576] + 120576

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119866120574119905119906120576120573(119906120576)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119869 (120574119905 119906120576 120573 (119906

120576)) + (119862 + 1) 120576

= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576


119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576 119875-as

(A12)


119882120575 (120574119905) le essinf120573isinB119905119879

esssup119906isinU119905119879

119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576

= 119882(120574119905) + (119862 + 1) 120576

(A13)


Lemma A3 119882(120574119905) le 119882120575(120574119905)


119882120575 (120574119905) = essinf1205731isinB119905119905+120575

esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)]

= essinf1205731isinB119905119905+120575

119868120575 (120574119905 1205731)

(A14)


119894 119894 ge 1 sub B119905119905+120575 such

that

119882120575 (120574119905) = inf119894ge1

119868120575 (120574119905 1205731

119894) 119875-as (A15)

For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731

119894) minus 120576 le 119882120575(120574119905) isin


119897=1Π119897) isin F119905 119894 ge


1=

sum119894ge1

1Λ119894

1205731



1(1199061)) =

sum119894ge1

1Λ119894

119868120575(120574119905 1199061 1205731


119882120575 (120574119905) ge sum

119894ge1

1Π119894

119868120575 (120574119905 1205731

119894) minus 120576

ge sum

119894ge1

1Π119894

119868120575 (120574119905 1199061 1205731

119894(1199061)) minus 120576

= 119868120575 (120574119905 1199061 120573120576

1(1199061)) minus 120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

119875-as forall1199061 isin U119905119905+120575

(A16)




B119905+120575119879 such that

119882(119909119905+120575) ge esssup1199062isinU119905+120575119879

119869 (119909119905+120575 1199062 120573119909120576

(1199062)) minus 120576 119875-as

(A17)



1205741199051199061120573120576

1(1199061)

119905+120575] = sum

119894ge1120574119894

119905+12057511198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894

we have

100381610038161003816100381610038161003816119883

1205741199051199061120573120576

1(1199061)

119905+120575minus [119883

1205741199051199061120573120576

1(1199061)

119905+120575]100381610038161003816100381610038161003816le 120576


(A18)



(A17) and 120573120576

1199061

= sum119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894120573120574119894120576isin B119905+120575119879


120576

1(1199061) oplus 120573

120576

1199061

(1199062) 119906 isin





1015840 into 1199061 1199061015840

1isin

U119905119905+120575 1199062 1199061015840


1015840

1and 119906 = 1199062 oplus 119906

1015840

2

From 1199061 equiv 1199061015840

1on [[119905 119878 and (119905 + 120575)]] we get 120573120576

1(1199061) = 120573

120576

1(119906

1015840

1) on

[[119905 119878and(119905+120575)]] (note that 120573120576


side from 1199062 equiv 1199061015840

2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt

119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576

1(1199061)

119905+120575= 119883

1205741199051199061015840

1120573120576

1(1199061015840

1)

119905+120575


1199061

= 120573120576

11990610158401

on 119878 gt 119905 + 120575 and120573120576

1199061

(1199062) = 120573120576

11990610158401

(1199061015840


120573120576(119906) = 120573

120576

1(1199061) oplus 120573

120576

1199061

(1199062) = 120573120576

1(119906

1015840

1) oplus 120573

120576

11990610158401

(1199061015840

2) on [[119905 119878]] from




119882120575 (120574119905) ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575]) minus 119862120576] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575])] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119882(120574

119894

119905+120575)] minus 119862120576

119875-as

(A19)


119882120575 (120574119905)

ge1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))minus 120576]minus119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 ([119883

1205741199051199061120573120576

1(1199061)

119905+120575] 1199062 120573

120576

1199061

(1199062))] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062)) minus 119862120576] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062))] minus 119862120576

= 119866120574119905119906120573120576(119906)

119905119905+120575[119884

120574119905119906120573120576(119906)

(119905 + 120575)] minus 119862120576

= 119884120574119905119906120573120576(119906)


(A20)

Therefore

119882120575 (120574119905) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573120576(119906)) minus 119862120576


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) minus 119862120576

= 119882(120574119905) minus 119862120576 119875-as

(A21)



119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)

119905119905+120575[119882(119883

120574119905119906120576120573(119906120576)

119905+120575)] + 120576

119875-as(A22)


119880119905119905+120575

119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)

119905119905+120575[119882(119883

120574119905119906120573120576(119906)

119905+120575)] minus 120576

119875-as(A23)


inf120573isinB119905119879

sup119906isinU119905119879

119864[119869(120574119905 119906 120573(119906))]

Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1(119905) gt 119910

2(119905)) gt


2(0)



119892119894 (119904 119910119894(119904) 119911

119894(119904)) = 119892 (119904 119910

119894(119904) 119911

119894(119904)) + 120593119894 (119904)

119889119904119889119875-ae 119894 = 1 2

(14)




1) and (119910

2 119911

2)


100381610038161003816100381610038161199101(119905)minus119910

2(119905)

10038161003816100381610038161003816

2

+1

2119864 [int

119879

119905

119890120573(119904minus119905)

(100381610038161003816100381610038161199101(119904)minus119910

2(119904)

10038161003816100381610038161003816

2

+100381610038161003816100381610038161199111(119904) minus 119911

2(119904)

10038161003816100381610038161003816

2

) 119889119904 | F119905]

le 119864 [119890120573(119879minus119905)10038161003816100381610038161205851 minus 1205852

10038161003816100381610038162| F119905]

+ 119864 [int

119879

119905

119890120573(119904minus119905)10038161003816100381610038161205931 (119904) minus 1205932 (119904)

10038161003816100381610038162119889119904 | F119905]


(15)









(16)



119889119883120574119905119906119907

(119904) = 119887 (119883120574119905119906119907

119904 119906 (119904) 119907 (119904)) 119889119904

+ 120590 (119883120574119905119906119907

119904 119906 (119904) 119907 (119904)) 119889119861 (119904) 119904 isin [119905 119879]

119889119884120574119905119906119907

(119904) = minus119891 (119883120574119905119906119907

119904 119884

120574119905119906119907

(119904) 119885120574119905119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

+ 119885120574119905119906119907

(119904) 119889119861 (119904)

119883120574119905119906119907

119905= 120574119905

119884120574119905119906119907

(119879) = Φ (119883120574119905119906119907

119879)

(17)


R 119911 isin R119889 119887(119909119905 119906 119907) 120590(119909119905 119906 119907) and 119891(119909119905 119910 119911 119906 119907)


[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091

119905 119909

2

119905isin Λ

10038161003816100381610038161003816119887 (119909

1

119905 119906 119907) minus 119887 (119909

2

119905 119906 119907)

10038161003816100381610038161003816+10038161003816100381610038161003816120590 (119909

1

119905 119906 119907) minus 120590 (119909

2

119905 119906 119907)

10038161003816100381610038161003816

le 119862100381710038171003817100381710038171199091

119905minus 119909

2

119905

10038171003817100381710038171003817

(18)1003816100381610038161003816119887 (119909119905 119906 119907)

1003816100381610038161003816 +1003816100381610038161003816120590 (119909119905 119906 119907)

1003816100381610038161003816 le 119862 (1 +1003817100381710038171003817119909119905

1003817100381710038171003817) for any 119909119905 isin Λ

(19)


[0 119879] 119906 isin 119880 119907 isin 119881 for any 1199091

119905 119909

2

119905isin Λ 119910

1 119910

2isin

R 1199111 119911

2isin R119889

10038161003816100381610038161003816119891 (119909

1

119905 119910

1 119911

1 119906 119907) minus 119891 (119909

2

119905 119910

2 119911

2 119906 119907)

10038161003816100381610038161003816

le 119862 (100381710038171003817100381710038171199091

119905minus 119909

2

119905

10038171003817100381710038171003817+100381610038161003816100381610038161199101minus 119910

210038161003816100381610038161003816+100381610038161003816100381610038161199111minus 119911

210038161003816100381610038161003816)

10038161003816100381610038161003816Φ (119909

1

119879) minus Φ (119909

2

119879)10038161003816100381610038161003816le 119862

100381710038171003817100381710038171199091

119879minus 119909

2

119879

10038171003817100381710038171003817

1003816100381610038161003816119891 (119909119905 0 0 119906 119907)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171199091199051003817100381710038171003817)

1003816100381610038161003816Φ (119909119879)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171199091198791003817100381710038171003817) for any 119909119905 isin Λ

(20)


(0 119879R119899) timesS2

(0 119879R) timesH2(0 119879R119889

)

solving (17)









119869 (120574119905 119906 119907) = 119884120574119905119906119907

(119905) 120574119905 isin Λ (21)



119882(120574119905) = essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (22)

119880 (120574119905) = esssup120572isinA119905119879

essinf119907isinV119905119879

119869 (120574119905 120572 (119907) 119907) (23)




2([0 119879]R119889




0ℎ(119904)119889119861(119904) minus

(12) int119879

0|ℎ(119904)|





as follows

119883120574119905119906119907

(119904)∘(120591ℎ)= (120574119905 (119905)+int

119904

119905

119887 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119903

+int

119904

119905

120590 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119861 (119903)) ∘ (120591ℎ)

= 120574119905 (119905) + int

119904

119905

119887 (119883120574119905119906119907

119903(120591ℎ) 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119903

+ int

119904

119905

120590 (119883120574119905119906119907

119903(120591ℎ) 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119861 (119903)

119883120574119905119906(120591ℎ)119907(120591ℎ)(119904) = 120574119905 (119905) + int

119904

119905

119887 (119883120574119905119906(120591ℎ)119907(120591ℎ)

119903 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119903

+ int

119904

119905

120590 (119883120574119905119906(120591ℎ)119907(120591ℎ)

119903 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119861 (119903)

(24)


119883120574119905119906119907

(119904) (120591ℎ) = 119883120574119905119906(120591ℎ)119907(120591ℎ)(119904)

for any 119904 isin [119905 119879] 119875-as(25)


119884120574119905119906119907

(119904) (120591ℎ) = 119884120574119905119906(120591ℎ)119907(120591ℎ)(119904)

for any 119904 isin [119905 119879] 119875-as

119885120574119905119906119907

(119904) (120591ℎ) = 119885120574119905119906(120591ℎ)119907(120591ℎ)(119904)

119889119904119889119875-ae on [0 119879] times Ω

(26)

Hence

119869 (120574119905 119906 119907) (120591ℎ) = 119869 (120574119905 119906 (120591ℎ) 119907 (120591ℎ)) 119875-as (27)


U119905119879 Then 120573ℎisin B119905119879




on [[119905 119878(120591minusℎ)]] Thus


= 120573ℎ(1199062) on [[119905 119878]]

(28)


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(29)


119869(120574119905 119906




esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(30)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

119875-as for any 119906 isin U119905119879

(31)

Therefore

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(32)

From above we get

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(33)


119882(120574119905) (120591ℎ) = 119882(120574119905) 119875-as for any ℎ isin 119867 (34)


119868(120574119905 120573)(120591ℎ) =

essinf120573isinB119905119879


119882(120574119905) (120591ℎ) = essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 (120591ℎ) 120573ℎ(119906 (120591ℎ)))

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906))

= 119882 (120574119905) 119875-as(35)


| 120573 isin





2([0 119879]R119889

)

119864[1120577isin119860 expint

119879

0

120593 (119904) 119889119861 (119904)]

= 119864 [1120577isin119860] 119864 [expint

119879

0

120593 (119904) 119889119861 (119904)]

(36)

For any 120593 = sum119873

119894=11205931198941(119905


119905119894119873


119864[1120577isin119860 exp(

119873

sum

119894=1

120593119894119861 (119905119894) minus 119861 (119905119894minus1))]

= 119864 [1120577isin119860] 119864[exp119873

sum

119894=1

120593119894 (119861 (119905119894) minus 119861 (119905119894minus1))]

(37)


119864[1120577isin119860

119873

prod

119894=1

(119861 (119905119894) minus 119861 (119905119894minus1))119896119894

]

= 119864 [1120577isin119860] 119864[

119873

prod

119894=1

(119861 (119905119894) minus 119861 (119905119894minus1))119896119894

]

(38)




(39)







119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905]le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172)

119864 [ sup119904isin[119905119879]

10038161003816100381610038161003816119883

120574119905119906119907

(119904) minus 119883120574119905119906119907

(119904)10038161003816100381610038161003816

2

| F119905]le 1198621003817100381710038171003817120574119905 minus 120574

119905

10038171003817100381710038172

119864 [ sup119904isin[119905119879]

1003816100381610038161003816119884120574119905119906119907

(119904)10038161003816100381610038162+int

119879

119905

1003816100381610038161003816119885120574119905119906119907

(119904)10038161003816100381610038162119889119904 | F119905]le 119862 (1+

100381710038171003817100381712057411990510038171003817100381710038172)

119864 [ sup119904isin[119905119879]

10038161003816100381610038161003816119884120574119905119906119907

(119904) minus 119884120574119905119906119907

(119904)10038161003816100381610038161003816

2

+int

119879

119905

10038161003816100381610038161003816119885120574119905119906119907

(119904) minus 119885120574119905119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]le 1198621003817100381710038171003817120574119905 minus 120574

119905

10038171003817100381710038172


following property


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817

(ii) 1003816100381610038161003816119882 (120574119905)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171205741199051003817100381710038171003817)

(41)



119866120574119905119906119907

119904119905+120575[120578] =

120574119905119906119907

(119904) 119904 isin [119905 119905 + 120575] (42)

where 120578 isin 1198712(ΩF119905+120575

119905 119875R) (

120574119905119906119907

(119904) 119885120574119905119906119907

(119904))119905le119904le119905+120575


119889120574119905119906119907

(119904)

= minus119891 (119904 119883120574119905119906119907

119904

120574119905119906119907

(119904) 119885120574119905119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

+ 119885120574119905119906119907

(119904) 119889119861 (119904)

120574119905119906119907

(119905 + 120575) = 120578 119904 isin [119905 119905 + 120575]

(43)Also we have

119866120574119905119906119907

119905119879[Φ (119883

120574119905119906119907

119879)] = 119866

120574119905119906119907

119905119905+120575[119884

120574119905119906119907


0

119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (45)




119863119905119882(120574119905) + 119867minus(120574119905119882119863119909119882119863119909119909119882) = 0

119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ

(46)

119863119905119880 (120574119905) + 119867+(120574119905 119880119863119909119880119863119909119909119880) = 0

119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ

(47)

where

119867minus(120574119905119882119863119909119882119863119909119909119882)

= sup119906isin119880

inf119907isin119881

119867(120574119905119882119863119909119882119863119909119909119882119906 119907)

119867+(120574119905 119880119863119909119880119863119909119909119880)

= inf119907isin119881

sup119906isin119880

119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907)

119867 (120574119905 119910 119901 119883 119906 119907)

=1

2tr (120590120590119879 (120574119905 119906 119907)119883) + 119901 sdot 119887 (120574119905 119906 119907)

+ 119891 (120574119905 119910 119901120590 (120574119905 119906 119907) 119906 119907)

(48)





C(Λ) is called


C12


0le119904le120575Λ 119905+119904 and

Γ(120574119905) = 119882(120574119905) one has

119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) ge 0 (49)


C12


0le119904le120575Λ 119905+119904 and

Γ(120574119905) = 119882(120574119905) one has

119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) le 0 (50)





C12

119897119887(Λ) denote

119865 (120574119904 119910 119911 119906 119907)

= 119863119904Γ (120574119904) +1

2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))

+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)

+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)

(51)



minus 1198891198841119906119907

(119904)

= 119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198851119906119907

(119904) 119889119861 (119904)

1198841119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]

(52)


1198841119906119907

(119904) = 119866120574119905119906119907

119904119905+120575[Γ (119883

120574119905119906119907

119905+120575)] minus Γ (119883

120574119905119906119907

119904) 119875-as (53)

Proof Note119866120574119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] is defined as119866120574

119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] =

119884119906119907


minus 119889119884119906119907

(119904) = 119891 (119883120574119905119906119907

119904 119884

119906119907(119904) 119885

119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 119885119906119907

(119904) 119889119861 (119904)

119884119906119907

(119905 + 120575) = Γ (119883120574119905119906119907

119905+120575) 119904 isin [119905 119905 + 120575]

(54)


119904) we have

119889 (119884119906119907

(119904) minus Γ (119883120574119905119906119907

119904)) = 119889119884

1119906119907(119904) (55)

Combined with 119884119906119907

(119905 + 120575) minus Γ(119883120574119905119906119907

119905+120575) = 0 = 119884

1119906119907(119905 + 120575) we



minus 1198891198842119906119907

(119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198852119906119907

(119904) 119889119861 (119904)

(56)

1198842119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)



(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816le 119862120575

32 119875-as (58)



119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905] le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172) (59)

combined with

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]

le 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

119887 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119903

10038161003816100381610038161003816100381610038161003816

2

| F119905]

+ 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

120590 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119861 (119903)

10038161003816100381610038161003816100381610038161003816

2

| F119905]

(60)

we have

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)


1205851 = 1205852 = 0

119892 (119904 119910 119911) = 119865 (119883120574119905119906119907

119904 119910 119911 119906119904 119907119904)

1205931 (119904) = 0

1205932 (119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906119904 119907119904)

minus 119865 (119883120574119905119906119907

119904 119884

2119906119907(119904) 119885

2119906119907(119904) 119906119904 119907119904)

(62)



119887 |1205932(119904)| le 1205880(|119883

120574119905119906119907

119904minus

120574119905(119905)|) we have

119864[int

119905+120575

119905

(100381610038161003816100381610038161198841119906119907

(119904) minus1198842119906119907

(119904)10038161003816100381610038161003816

2

+100381610038161003816100381610038161198851119906119907

(119904) minus1198852119906119907

(119904)10038161003816100381610038161003816

2

) 119889119904 | F119905]

le 119864[int

119905+120575

119905

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) 119889119904 | F119905]

le 119862120575119864[ sup119904isin[119905119905+120575]

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) | F119905]

le 1198621205752

(63)


So100381610038161003816100381610038161198841119906119907

(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816

=10038161003816100381610038161003816119864 [(119884

1119906119907(119904) minus 119884

2119906119907(119904)) | F119905]

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

119864 [int

119905+120575

119905

(119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904))

minus119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]

100381610038161003816100381610038161003816100381610038161003816

le 119862119864[int

119905+120575

119905

[1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

+100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816] 119889119904 | F119905]

le 119862119864[int

119905+120575

119905

1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) 119889119904 | F119905]

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

le 11986212057532

(64)


minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]

1198840 (119905 + 120575) = 0

(65)

where 1198650(120574119905 119910 119911) = sup119906isin119880

inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as

1198840 (119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) (66)


1198651 (120574119905 119910 119911 119906) = inf119907isin119881

119865 (120574119905 119910 119911 119906 119907)


(67)


minus1198891198843119906

(119904) = 1198651 (120574119905 1198843119906

(119904) 1198853119906

(119904) 119906 (119904)) 119889119904

minus1198853119906

(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]

1198843119906

(119905 + 120575) = 0

(68)


1198853119906

) solving (68) Also

1198843119906

(119905) = essinf119907isinV119905119905+120575

1198842119906119907


(69)


1198843119906

(119905) le 1198842119906119907

(119905) 119875-as for any 119907 isin V119905119905+120575

for every 119906 isin U119905119905+120575

(70)


4 R timesR119889


1198651 (120574119905 119910 119911 119906)

= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906

(71)

Set 1199074(119904) = 1199074(119884

3119906(119904) 119885

3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know

1199074isin V119905119905+120575 Therefore (1198843119906

1198853119906

) = (11988421199061199074

11988521199061199074


(119905) = 11988421199061199074


(119905) = essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-119886119904 for every 119906 isin U119905119905+120575


1198651(120574119905 119910 119911 119906) wealso derive

1198840 (119905) = esssup119906isinU119905119905+120575

1198843119906

(119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-as

(72)


119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575

32 119875-as

(73)



100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816

2

le 119862120575

119864 [int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904] le 119862120575

(74)



(119904)10038161003816100381610038161003816

le 119864 [int

119905+120575

119904

119865 (120574119905 1198842119906119907

(119903) 1198852119906119907

(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]

le 119862119864[int

119905+120575

119904

(1 +1003817100381710038171003817120574119905

10038171003817100381710038172+100381610038161003816100381610038161198842119906119907

(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816) 119889119903 | F119904]

le 119862120575 + 11986212057512

119864[int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904]

le 119862120575 119904 isin [119905 119905 + 120575]

(75)


119905|119885

2119906119907(119904)|

2

119889119904 |

F119905] le 1198621205752 119875-as Therefore

119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905]

le 1198621205752+ 119862120575

12(119864[int

119905+120575

119905

100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816

2

119889119903 | F119905])

12

le 11986212057532

119875-as

(76)



0le119904le120575Λ 119905+119904


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(77)

From 119882 ge Γ on ⋃0le119904le120575


119905119906120573(119906)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

120574119905119906120573(119906)

119905+120575)] minus Γ (120574119905) le 0 (78)

By Lemma 19 we have

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) le 0 119875-as (79)

Thus from Lemma 20

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as (80)


1198842119906119907

(119905) le 1198842119906120573(119906)

(119905) 120573 isin

B119905119905+120575 we have

esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905)

le essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as(81)

Thus by Lemma 21 1198840(119905) le 11986212057532


11986212057512

ge1

1205751198840 (119905) =

1

120575int

119905+120575

119905

1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)

sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)



Γ ge 119882 on⋃0le119904le120575


sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)


1198650 (120574119905 0 0) = sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)


119865 (120574119905 0 0 119906 120595 (119906)) le minus3

4120579 forall119906 isin 119880 (86)


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(87)



119905119906120573(119906)

119905119905+120575[sdot] (Lemma 7) we get

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

119906120573(119906)

119905+120575)] minus Γ (120574119905) ge 0 119875-as

(88)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) ge 0 119875-as (89)

in particular

esssup119906isinU119905119905+120575

1198841119906120595(119906)

(119905) ge 0 119875-as (90)


1119906120576120595(119906120576)(119905) ge minus120576120575


1198842119906120576120595(119906120576)(119905) ge minus119862120575

32minus 120576120575 119875-as (91)



1198842119906120576120595(119906120576)(119905)

= 119864 [int

119905+120575

119905

119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885

2119906120576120595(119906120576)(119904)

119906120576(119904) 120595 (119906

120576(119904))) 119889119904 | F119905]

le 119864[int

119905+120575

119905

(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816+ 119862

1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816

+119865 (120574119905 0 0 119906120576(119904) 120595 (119906

120576(119904)))) 119889119904 | F119905]

le 11986212057532

minus3

4120579120575 119875-as

(92)


minus 120576 le 11986212057512

minus


sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)



Appendix



119882120575 (120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (A1)




Lemma A2 119882120575(120574119905) le 119882(120574119905)





119882120575 (120574119905) le esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)] 119875-as

(A3)

We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907

119905119905+120575[119882(119883

120574119905119906119907



119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575

119868120575 (120574119905 1199061 1205731 (1199061))

= sup119894ge1

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) 119875-as

(A4)


119894 1205731(119906

1

119894)) +


(⋃119894minus1


119906120576

1= sum

119894ge11Γ119894

1199061


of 1205731 we have 1205731(119906120576

1) = sum

119894ge11Γ119894

1205731(1199061



120576

1 1205731(119906

120576

1)) = sum

119894ge11Γ119894

119868120575(120574119905 1199061

119894 1205731(119906

1

119894)) 119875-as So

119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum

119894ge1

1Γ119894

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) + 120576

= 119868120575 (120574119905 119906120576

1 1205731 (119906

120576

1)) + 120576

= 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119882(119883

120574119905119906120576

11205731(119906120576

1)

119905+120575)] + 120576

(A5)


120576

1oplus


119882(120574119905) le esssup1199062isinU119905+120575119879

119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817 for any 120574119905 120574119905 isin Λ

(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))

1003816100381610038161003816

le 1198621003817100381710038171003817120574119905 minus 120574

119905

1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879

(A7)


11205731(119906120576

1)

119905+120575that

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062)) 119875-as

(A8)



119895 119895 ge 1 sub U119905+120575119879 such that

esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

= sup119895ge1

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A9)


119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732(1199062)) le

119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732(119906

2


Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1


(ΩF119905+120575)-partition and 119906120576

2= sum

119895ge11Δ119895

1199062

119895isin U119905+120575119879 Due to


2) = sum

119895ge11Δ119895

1205732(1199062

119895)


1oplus 119906

120576

2) =

1205731(119906120576

1) oplus 1205732(119906

120576



119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

2 1205732 (119906

120576

2))

= 119884119883120574119905119906120576

11205731(119906120576

1)

119905+120575119906120576

21205732(119906120576

2)(119905 + 120575)

= sum

119895ge1

1Δ119895

119884119883120574119905119906120576

11205731(119906120576

1)

119905+1205751199062

1198951205732(1199062

119895)(119905 + 120575)

= sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A10)

So

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

le sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

2

119895 120573 (119906

120576

1oplus 119906

2

119895)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

120576

2 120573 (119906

120576

1oplus 119906

120576

2)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576 119875-as

(A11)

where 119906120576

= 119906120576

1oplus 119906

120576



119882120575 (120574119905)

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576] + 120576

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119866120574119905119906120576120573(119906120576)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119869 (120574119905 119906120576 120573 (119906

120576)) + (119862 + 1) 120576

= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576


119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576 119875-as

(A12)


119882120575 (120574119905) le essinf120573isinB119905119879

esssup119906isinU119905119879

119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576

= 119882(120574119905) + (119862 + 1) 120576

(A13)


Lemma A3 119882(120574119905) le 119882120575(120574119905)


119882120575 (120574119905) = essinf1205731isinB119905119905+120575

esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)]

= essinf1205731isinB119905119905+120575

119868120575 (120574119905 1205731)

(A14)


119894 119894 ge 1 sub B119905119905+120575 such

that

119882120575 (120574119905) = inf119894ge1

119868120575 (120574119905 1205731

119894) 119875-as (A15)

For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731

119894) minus 120576 le 119882120575(120574119905) isin


119897=1Π119897) isin F119905 119894 ge


1=

sum119894ge1

1Λ119894

1205731



1(1199061)) =

sum119894ge1

1Λ119894

119868120575(120574119905 1199061 1205731


119882120575 (120574119905) ge sum

119894ge1

1Π119894

119868120575 (120574119905 1205731

119894) minus 120576

ge sum

119894ge1

1Π119894

119868120575 (120574119905 1199061 1205731

119894(1199061)) minus 120576

= 119868120575 (120574119905 1199061 120573120576

1(1199061)) minus 120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

119875-as forall1199061 isin U119905119905+120575

(A16)




B119905+120575119879 such that

119882(119909119905+120575) ge esssup1199062isinU119905+120575119879

119869 (119909119905+120575 1199062 120573119909120576

(1199062)) minus 120576 119875-as

(A17)



1205741199051199061120573120576

1(1199061)

119905+120575] = sum

119894ge1120574119894

119905+12057511198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894

we have

100381610038161003816100381610038161003816119883

1205741199051199061120573120576

1(1199061)

119905+120575minus [119883

1205741199051199061120573120576

1(1199061)

119905+120575]100381610038161003816100381610038161003816le 120576


(A18)



(A17) and 120573120576

1199061

= sum119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894120573120574119894120576isin B119905+120575119879


120576

1(1199061) oplus 120573

120576

1199061

(1199062) 119906 isin





1015840 into 1199061 1199061015840

1isin

U119905119905+120575 1199062 1199061015840


1015840

1and 119906 = 1199062 oplus 119906

1015840

2

From 1199061 equiv 1199061015840

1on [[119905 119878 and (119905 + 120575)]] we get 120573120576

1(1199061) = 120573

120576

1(119906

1015840

1) on

[[119905 119878and(119905+120575)]] (note that 120573120576


side from 1199062 equiv 1199061015840

2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt

119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576

1(1199061)

119905+120575= 119883

1205741199051199061015840

1120573120576

1(1199061015840

1)

119905+120575


1199061

= 120573120576

11990610158401

on 119878 gt 119905 + 120575 and120573120576

1199061

(1199062) = 120573120576

11990610158401

(1199061015840


120573120576(119906) = 120573

120576

1(1199061) oplus 120573

120576

1199061

(1199062) = 120573120576

1(119906

1015840

1) oplus 120573

120576

11990610158401

(1199061015840

2) on [[119905 119878]] from




119882120575 (120574119905) ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575]) minus 119862120576] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575])] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119882(120574

119894

119905+120575)] minus 119862120576

119875-as

(A19)


119882120575 (120574119905)

ge1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))minus 120576]minus119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 ([119883

1205741199051199061120573120576

1(1199061)

119905+120575] 1199062 120573

120576

1199061

(1199062))] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062)) minus 119862120576] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062))] minus 119862120576

= 119866120574119905119906120573120576(119906)

119905119905+120575[119884

120574119905119906120573120576(119906)

(119905 + 120575)] minus 119862120576

= 119884120574119905119906120573120576(119906)


(A20)

Therefore

119882120575 (120574119905) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573120576(119906)) minus 119862120576


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) minus 119862120576

= 119882(120574119905) minus 119862120576 119875-as

(A21)



119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)

119905119905+120575[119882(119883

120574119905119906120576120573(119906120576)

119905+120575)] + 120576

119875-as(A22)


119880119905119905+120575

119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)

119905119905+120575[119882(119883

120574119905119906120573120576(119906)

119905+120575)] minus 120576

119875-as(A23)


inf120573isinB119905119879

sup119906isinU119905119879

119864[119869(120574119905 119906 120573(119906))]

Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119869 (120574119905 119906 119907) = 119884120574119905119906119907

(119905) 120574119905 isin Λ (21)



119882(120574119905) = essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (22)

119880 (120574119905) = esssup120572isinA119905119879

essinf119907isinV119905119879

119869 (120574119905 120572 (119907) 119907) (23)




2([0 119879]R119889




0ℎ(119904)119889119861(119904) minus

(12) int119879

0|ℎ(119904)|





as follows

119883120574119905119906119907

(119904)∘(120591ℎ)= (120574119905 (119905)+int

119904

119905

119887 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119903

+int

119904

119905

120590 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119861 (119903)) ∘ (120591ℎ)

= 120574119905 (119905) + int

119904

119905

119887 (119883120574119905119906119907

119903(120591ℎ) 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119903

+ int

119904

119905

120590 (119883120574119905119906119907

119903(120591ℎ) 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119861 (119903)

119883120574119905119906(120591ℎ)119907(120591ℎ)(119904) = 120574119905 (119905) + int

119904

119905

119887 (119883120574119905119906(120591ℎ)119907(120591ℎ)

119903 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119903

+ int

119904

119905

120590 (119883120574119905119906(120591ℎ)119907(120591ℎ)

119903 119906 (120591ℎ) (119903)

119907 (120591ℎ) (119903)) 119889119861 (119903)

(24)


119883120574119905119906119907

(119904) (120591ℎ) = 119883120574119905119906(120591ℎ)119907(120591ℎ)(119904)

for any 119904 isin [119905 119879] 119875-as(25)


119884120574119905119906119907

(119904) (120591ℎ) = 119884120574119905119906(120591ℎ)119907(120591ℎ)(119904)

for any 119904 isin [119905 119879] 119875-as

119885120574119905119906119907

(119904) (120591ℎ) = 119885120574119905119906(120591ℎ)119907(120591ℎ)(119904)

119889119904119889119875-ae on [0 119879] times Ω

(26)

Hence

119869 (120574119905 119906 119907) (120591ℎ) = 119869 (120574119905 119906 (120591ℎ) 119907 (120591ℎ)) 119875-as (27)


U119905119879 Then 120573ℎisin B119905119879




on [[119905 119878(120591minusℎ)]] Thus


= 120573ℎ(1199062) on [[119905 119878]]

(28)


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(29)


119869(120574119905 119906




esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(30)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

119875-as for any 119906 isin U119905119879

(31)

Therefore

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(32)

From above we get

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(33)


119882(120574119905) (120591ℎ) = 119882(120574119905) 119875-as for any ℎ isin 119867 (34)


119868(120574119905 120573)(120591ℎ) =

essinf120573isinB119905119879


119882(120574119905) (120591ℎ) = essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 (120591ℎ) 120573ℎ(119906 (120591ℎ)))

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906))

= 119882 (120574119905) 119875-as(35)


| 120573 isin





2([0 119879]R119889

)

119864[1120577isin119860 expint

119879

0

120593 (119904) 119889119861 (119904)]

= 119864 [1120577isin119860] 119864 [expint

119879

0

120593 (119904) 119889119861 (119904)]

(36)

For any 120593 = sum119873

119894=11205931198941(119905


119905119894119873


119864[1120577isin119860 exp(

119873

sum

119894=1

120593119894119861 (119905119894) minus 119861 (119905119894minus1))]

= 119864 [1120577isin119860] 119864[exp119873

sum

119894=1

120593119894 (119861 (119905119894) minus 119861 (119905119894minus1))]

(37)


119864[1120577isin119860

119873

prod

119894=1

(119861 (119905119894) minus 119861 (119905119894minus1))119896119894

]

= 119864 [1120577isin119860] 119864[

119873

prod

119894=1

(119861 (119905119894) minus 119861 (119905119894minus1))119896119894

]

(38)




(39)







119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905]le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172)

119864 [ sup119904isin[119905119879]

10038161003816100381610038161003816119883

120574119905119906119907

(119904) minus 119883120574119905119906119907

(119904)10038161003816100381610038161003816

2

| F119905]le 1198621003817100381710038171003817120574119905 minus 120574

119905

10038171003817100381710038172

119864 [ sup119904isin[119905119879]

1003816100381610038161003816119884120574119905119906119907

(119904)10038161003816100381610038162+int

119879

119905

1003816100381610038161003816119885120574119905119906119907

(119904)10038161003816100381610038162119889119904 | F119905]le 119862 (1+

100381710038171003817100381712057411990510038171003817100381710038172)

119864 [ sup119904isin[119905119879]

10038161003816100381610038161003816119884120574119905119906119907

(119904) minus 119884120574119905119906119907

(119904)10038161003816100381610038161003816

2

+int

119879

119905

10038161003816100381610038161003816119885120574119905119906119907

(119904) minus 119885120574119905119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]le 1198621003817100381710038171003817120574119905 minus 120574

119905

10038171003817100381710038172


following property


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817

(ii) 1003816100381610038161003816119882 (120574119905)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171205741199051003817100381710038171003817)

(41)



119866120574119905119906119907

119904119905+120575[120578] =

120574119905119906119907

(119904) 119904 isin [119905 119905 + 120575] (42)

where 120578 isin 1198712(ΩF119905+120575

119905 119875R) (

120574119905119906119907

(119904) 119885120574119905119906119907

(119904))119905le119904le119905+120575


119889120574119905119906119907

(119904)

= minus119891 (119904 119883120574119905119906119907

119904

120574119905119906119907

(119904) 119885120574119905119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

+ 119885120574119905119906119907

(119904) 119889119861 (119904)

120574119905119906119907

(119905 + 120575) = 120578 119904 isin [119905 119905 + 120575]

(43)Also we have

119866120574119905119906119907

119905119879[Φ (119883

120574119905119906119907

119879)] = 119866

120574119905119906119907

119905119905+120575[119884

120574119905119906119907


0

119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (45)




119863119905119882(120574119905) + 119867minus(120574119905119882119863119909119882119863119909119909119882) = 0

119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ

(46)

119863119905119880 (120574119905) + 119867+(120574119905 119880119863119909119880119863119909119909119880) = 0

119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ

(47)

where

119867minus(120574119905119882119863119909119882119863119909119909119882)

= sup119906isin119880

inf119907isin119881

119867(120574119905119882119863119909119882119863119909119909119882119906 119907)

119867+(120574119905 119880119863119909119880119863119909119909119880)

= inf119907isin119881

sup119906isin119880

119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907)

119867 (120574119905 119910 119901 119883 119906 119907)

=1

2tr (120590120590119879 (120574119905 119906 119907)119883) + 119901 sdot 119887 (120574119905 119906 119907)

+ 119891 (120574119905 119910 119901120590 (120574119905 119906 119907) 119906 119907)

(48)





C(Λ) is called


C12


0le119904le120575Λ 119905+119904 and

Γ(120574119905) = 119882(120574119905) one has

119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) ge 0 (49)


C12


0le119904le120575Λ 119905+119904 and

Γ(120574119905) = 119882(120574119905) one has

119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) le 0 (50)





C12

119897119887(Λ) denote

119865 (120574119904 119910 119911 119906 119907)

= 119863119904Γ (120574119904) +1

2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))

+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)

+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)

(51)



minus 1198891198841119906119907

(119904)

= 119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198851119906119907

(119904) 119889119861 (119904)

1198841119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]

(52)


1198841119906119907

(119904) = 119866120574119905119906119907

119904119905+120575[Γ (119883

120574119905119906119907

119905+120575)] minus Γ (119883

120574119905119906119907

119904) 119875-as (53)

Proof Note119866120574119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] is defined as119866120574

119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] =

119884119906119907


minus 119889119884119906119907

(119904) = 119891 (119883120574119905119906119907

119904 119884

119906119907(119904) 119885

119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 119885119906119907

(119904) 119889119861 (119904)

119884119906119907

(119905 + 120575) = Γ (119883120574119905119906119907

119905+120575) 119904 isin [119905 119905 + 120575]

(54)


119904) we have

119889 (119884119906119907

(119904) minus Γ (119883120574119905119906119907

119904)) = 119889119884

1119906119907(119904) (55)

Combined with 119884119906119907

(119905 + 120575) minus Γ(119883120574119905119906119907

119905+120575) = 0 = 119884

1119906119907(119905 + 120575) we



minus 1198891198842119906119907

(119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198852119906119907

(119904) 119889119861 (119904)

(56)

1198842119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)



(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816le 119862120575

32 119875-as (58)



119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905] le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172) (59)

combined with

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]

le 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

119887 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119903

10038161003816100381610038161003816100381610038161003816

2

| F119905]

+ 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

120590 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119861 (119903)

10038161003816100381610038161003816100381610038161003816

2

| F119905]

(60)

we have

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)


1205851 = 1205852 = 0

119892 (119904 119910 119911) = 119865 (119883120574119905119906119907

119904 119910 119911 119906119904 119907119904)

1205931 (119904) = 0

1205932 (119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906119904 119907119904)

minus 119865 (119883120574119905119906119907

119904 119884

2119906119907(119904) 119885

2119906119907(119904) 119906119904 119907119904)

(62)



119887 |1205932(119904)| le 1205880(|119883

120574119905119906119907

119904minus

120574119905(119905)|) we have

119864[int

119905+120575

119905

(100381610038161003816100381610038161198841119906119907

(119904) minus1198842119906119907

(119904)10038161003816100381610038161003816

2

+100381610038161003816100381610038161198851119906119907

(119904) minus1198852119906119907

(119904)10038161003816100381610038161003816

2

) 119889119904 | F119905]

le 119864[int

119905+120575

119905

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) 119889119904 | F119905]

le 119862120575119864[ sup119904isin[119905119905+120575]

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) | F119905]

le 1198621205752

(63)


So100381610038161003816100381610038161198841119906119907

(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816

=10038161003816100381610038161003816119864 [(119884

1119906119907(119904) minus 119884

2119906119907(119904)) | F119905]

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

119864 [int

119905+120575

119905

(119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904))

minus119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]

100381610038161003816100381610038161003816100381610038161003816

le 119862119864[int

119905+120575

119905

[1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

+100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816] 119889119904 | F119905]

le 119862119864[int

119905+120575

119905

1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) 119889119904 | F119905]

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

le 11986212057532

(64)


minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]

1198840 (119905 + 120575) = 0

(65)

where 1198650(120574119905 119910 119911) = sup119906isin119880

inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as

1198840 (119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) (66)


1198651 (120574119905 119910 119911 119906) = inf119907isin119881

119865 (120574119905 119910 119911 119906 119907)


(67)


minus1198891198843119906

(119904) = 1198651 (120574119905 1198843119906

(119904) 1198853119906

(119904) 119906 (119904)) 119889119904

minus1198853119906

(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]

1198843119906

(119905 + 120575) = 0

(68)


1198853119906

) solving (68) Also

1198843119906

(119905) = essinf119907isinV119905119905+120575

1198842119906119907


(69)


1198843119906

(119905) le 1198842119906119907

(119905) 119875-as for any 119907 isin V119905119905+120575

for every 119906 isin U119905119905+120575

(70)


4 R timesR119889


1198651 (120574119905 119910 119911 119906)

= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906

(71)

Set 1199074(119904) = 1199074(119884

3119906(119904) 119885

3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know

1199074isin V119905119905+120575 Therefore (1198843119906

1198853119906

) = (11988421199061199074

11988521199061199074


(119905) = 11988421199061199074


(119905) = essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-119886119904 for every 119906 isin U119905119905+120575


1198651(120574119905 119910 119911 119906) wealso derive

1198840 (119905) = esssup119906isinU119905119905+120575

1198843119906

(119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-as

(72)


119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575

32 119875-as

(73)



100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816

2

le 119862120575

119864 [int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904] le 119862120575

(74)



(119904)10038161003816100381610038161003816

le 119864 [int

119905+120575

119904

119865 (120574119905 1198842119906119907

(119903) 1198852119906119907

(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]

le 119862119864[int

119905+120575

119904

(1 +1003817100381710038171003817120574119905

10038171003817100381710038172+100381610038161003816100381610038161198842119906119907

(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816) 119889119903 | F119904]

le 119862120575 + 11986212057512

119864[int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904]

le 119862120575 119904 isin [119905 119905 + 120575]

(75)


119905|119885

2119906119907(119904)|

2

119889119904 |

F119905] le 1198621205752 119875-as Therefore

119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905]

le 1198621205752+ 119862120575

12(119864[int

119905+120575

119905

100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816

2

119889119903 | F119905])

12

le 11986212057532

119875-as

(76)



0le119904le120575Λ 119905+119904


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(77)

From 119882 ge Γ on ⋃0le119904le120575


119905119906120573(119906)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

120574119905119906120573(119906)

119905+120575)] minus Γ (120574119905) le 0 (78)

By Lemma 19 we have

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) le 0 119875-as (79)

Thus from Lemma 20

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as (80)


1198842119906119907

(119905) le 1198842119906120573(119906)

(119905) 120573 isin

B119905119905+120575 we have

esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905)

le essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as(81)

Thus by Lemma 21 1198840(119905) le 11986212057532


11986212057512

ge1

1205751198840 (119905) =

1

120575int

119905+120575

119905

1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)

sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)



Γ ge 119882 on⋃0le119904le120575


sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)


1198650 (120574119905 0 0) = sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)


119865 (120574119905 0 0 119906 120595 (119906)) le minus3

4120579 forall119906 isin 119880 (86)


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(87)



119905119906120573(119906)

119905119905+120575[sdot] (Lemma 7) we get

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

119906120573(119906)

119905+120575)] minus Γ (120574119905) ge 0 119875-as

(88)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) ge 0 119875-as (89)

in particular

esssup119906isinU119905119905+120575

1198841119906120595(119906)

(119905) ge 0 119875-as (90)


1119906120576120595(119906120576)(119905) ge minus120576120575


1198842119906120576120595(119906120576)(119905) ge minus119862120575

32minus 120576120575 119875-as (91)



1198842119906120576120595(119906120576)(119905)

= 119864 [int

119905+120575

119905

119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885

2119906120576120595(119906120576)(119904)

119906120576(119904) 120595 (119906

120576(119904))) 119889119904 | F119905]

le 119864[int

119905+120575

119905

(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816+ 119862

1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816

+119865 (120574119905 0 0 119906120576(119904) 120595 (119906

120576(119904)))) 119889119904 | F119905]

le 11986212057532

minus3

4120579120575 119875-as

(92)


minus 120576 le 11986212057512

minus


sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)



Appendix



119882120575 (120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (A1)




Lemma A2 119882120575(120574119905) le 119882(120574119905)





119882120575 (120574119905) le esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)] 119875-as

(A3)

We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907

119905119905+120575[119882(119883

120574119905119906119907



119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575

119868120575 (120574119905 1199061 1205731 (1199061))

= sup119894ge1

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) 119875-as

(A4)


119894 1205731(119906

1

119894)) +


(⋃119894minus1


119906120576

1= sum

119894ge11Γ119894

1199061


of 1205731 we have 1205731(119906120576

1) = sum

119894ge11Γ119894

1205731(1199061



120576

1 1205731(119906

120576

1)) = sum

119894ge11Γ119894

119868120575(120574119905 1199061

119894 1205731(119906

1

119894)) 119875-as So

119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum

119894ge1

1Γ119894

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) + 120576

= 119868120575 (120574119905 119906120576

1 1205731 (119906

120576

1)) + 120576

= 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119882(119883

120574119905119906120576

11205731(119906120576

1)

119905+120575)] + 120576

(A5)


120576

1oplus


119882(120574119905) le esssup1199062isinU119905+120575119879

119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817 for any 120574119905 120574119905 isin Λ

(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))

1003816100381610038161003816

le 1198621003817100381710038171003817120574119905 minus 120574

119905

1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879

(A7)


11205731(119906120576

1)

119905+120575that

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062)) 119875-as

(A8)



119895 119895 ge 1 sub U119905+120575119879 such that

esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

= sup119895ge1

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A9)


119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732(1199062)) le

119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732(119906

2


Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1


(ΩF119905+120575)-partition and 119906120576

2= sum

119895ge11Δ119895

1199062

119895isin U119905+120575119879 Due to


2) = sum

119895ge11Δ119895

1205732(1199062

119895)


1oplus 119906

120576

2) =

1205731(119906120576

1) oplus 1205732(119906

120576



119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

2 1205732 (119906

120576

2))

= 119884119883120574119905119906120576

11205731(119906120576

1)

119905+120575119906120576

21205732(119906120576

2)(119905 + 120575)

= sum

119895ge1

1Δ119895

119884119883120574119905119906120576

11205731(119906120576

1)

119905+1205751199062

1198951205732(1199062

119895)(119905 + 120575)

= sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A10)

So

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

le sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

2

119895 120573 (119906

120576

1oplus 119906

2

119895)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

120576

2 120573 (119906

120576

1oplus 119906

120576

2)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576 119875-as

(A11)

where 119906120576

= 119906120576

1oplus 119906

120576



119882120575 (120574119905)

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576] + 120576

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119866120574119905119906120576120573(119906120576)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119869 (120574119905 119906120576 120573 (119906

120576)) + (119862 + 1) 120576

= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576


119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576 119875-as

(A12)


119882120575 (120574119905) le essinf120573isinB119905119879

esssup119906isinU119905119879

119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576

= 119882(120574119905) + (119862 + 1) 120576

(A13)


Lemma A3 119882(120574119905) le 119882120575(120574119905)


119882120575 (120574119905) = essinf1205731isinB119905119905+120575

esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)]

= essinf1205731isinB119905119905+120575

119868120575 (120574119905 1205731)

(A14)


119894 119894 ge 1 sub B119905119905+120575 such

that

119882120575 (120574119905) = inf119894ge1

119868120575 (120574119905 1205731

119894) 119875-as (A15)

For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731

119894) minus 120576 le 119882120575(120574119905) isin


119897=1Π119897) isin F119905 119894 ge


1=

sum119894ge1

1Λ119894

1205731



1(1199061)) =

sum119894ge1

1Λ119894

119868120575(120574119905 1199061 1205731


119882120575 (120574119905) ge sum

119894ge1

1Π119894

119868120575 (120574119905 1205731

119894) minus 120576

ge sum

119894ge1

1Π119894

119868120575 (120574119905 1199061 1205731

119894(1199061)) minus 120576

= 119868120575 (120574119905 1199061 120573120576

1(1199061)) minus 120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

119875-as forall1199061 isin U119905119905+120575

(A16)




B119905+120575119879 such that

119882(119909119905+120575) ge esssup1199062isinU119905+120575119879

119869 (119909119905+120575 1199062 120573119909120576

(1199062)) minus 120576 119875-as

(A17)



1205741199051199061120573120576

1(1199061)

119905+120575] = sum

119894ge1120574119894

119905+12057511198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894

we have

100381610038161003816100381610038161003816119883

1205741199051199061120573120576

1(1199061)

119905+120575minus [119883

1205741199051199061120573120576

1(1199061)

119905+120575]100381610038161003816100381610038161003816le 120576


(A18)



(A17) and 120573120576

1199061

= sum119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894120573120574119894120576isin B119905+120575119879


120576

1(1199061) oplus 120573

120576

1199061

(1199062) 119906 isin





1015840 into 1199061 1199061015840

1isin

U119905119905+120575 1199062 1199061015840


1015840

1and 119906 = 1199062 oplus 119906

1015840

2

From 1199061 equiv 1199061015840

1on [[119905 119878 and (119905 + 120575)]] we get 120573120576

1(1199061) = 120573

120576

1(119906

1015840

1) on

[[119905 119878and(119905+120575)]] (note that 120573120576


side from 1199062 equiv 1199061015840

2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt

119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576

1(1199061)

119905+120575= 119883

1205741199051199061015840

1120573120576

1(1199061015840

1)

119905+120575


1199061

= 120573120576

11990610158401

on 119878 gt 119905 + 120575 and120573120576

1199061

(1199062) = 120573120576

11990610158401

(1199061015840


120573120576(119906) = 120573

120576

1(1199061) oplus 120573

120576

1199061

(1199062) = 120573120576

1(119906

1015840

1) oplus 120573

120576

11990610158401

(1199061015840

2) on [[119905 119878]] from




119882120575 (120574119905) ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575]) minus 119862120576] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575])] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119882(120574

119894

119905+120575)] minus 119862120576

119875-as

(A19)


119882120575 (120574119905)

ge1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))minus 120576]minus119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 ([119883

1205741199051199061120573120576

1(1199061)

119905+120575] 1199062 120573

120576

1199061

(1199062))] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062)) minus 119862120576] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062))] minus 119862120576

= 119866120574119905119906120573120576(119906)

119905119905+120575[119884

120574119905119906120573120576(119906)

(119905 + 120575)] minus 119862120576

= 119884120574119905119906120573120576(119906)


(A20)

Therefore

119882120575 (120574119905) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573120576(119906)) minus 119862120576


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) minus 119862120576

= 119882(120574119905) minus 119862120576 119875-as

(A21)



119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)

119905119905+120575[119882(119883

120574119905119906120576120573(119906120576)

119905+120575)] + 120576

119875-as(A22)


119880119905119905+120575

119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)

119905119905+120575[119882(119883

120574119905119906120573120576(119906)

119905+120575)] minus 120576

119875-as(A23)


inf120573isinB119905119879

sup119906isinU119905119879

119864[119869(120574119905 119906 120573(119906))]

Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(30)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

119875-as for any 119906 isin U119905119879

(31)

Therefore

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)


119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(32)

From above we get

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ) 119875-as(33)


119882(120574119905) (120591ℎ) = 119882(120574119905) 119875-as for any ℎ isin 119867 (34)


119868(120574119905 120573)(120591ℎ) =

essinf120573isinB119905119879


119882(120574119905) (120591ℎ) = essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) (120591ℎ)

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 (120591ℎ) 120573ℎ(119906 (120591ℎ)))

= essinf120573isinB119905119879

esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906))

= 119882 (120574119905) 119875-as(35)


| 120573 isin





2([0 119879]R119889

)

119864[1120577isin119860 expint

119879

0

120593 (119904) 119889119861 (119904)]

= 119864 [1120577isin119860] 119864 [expint

119879

0

120593 (119904) 119889119861 (119904)]

(36)

For any 120593 = sum119873

119894=11205931198941(119905


119905119894119873


119864[1120577isin119860 exp(

119873

sum

119894=1

120593119894119861 (119905119894) minus 119861 (119905119894minus1))]

= 119864 [1120577isin119860] 119864[exp119873

sum

119894=1

120593119894 (119861 (119905119894) minus 119861 (119905119894minus1))]

(37)


119864[1120577isin119860

119873

prod

119894=1

(119861 (119905119894) minus 119861 (119905119894minus1))119896119894

]

= 119864 [1120577isin119860] 119864[

119873

prod

119894=1

(119861 (119905119894) minus 119861 (119905119894minus1))119896119894

]

(38)




(39)







119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905]le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172)

119864 [ sup119904isin[119905119879]

10038161003816100381610038161003816119883

120574119905119906119907

(119904) minus 119883120574119905119906119907

(119904)10038161003816100381610038161003816

2

| F119905]le 1198621003817100381710038171003817120574119905 minus 120574

119905

10038171003817100381710038172

119864 [ sup119904isin[119905119879]

1003816100381610038161003816119884120574119905119906119907

(119904)10038161003816100381610038162+int

119879

119905

1003816100381610038161003816119885120574119905119906119907

(119904)10038161003816100381610038162119889119904 | F119905]le 119862 (1+

100381710038171003817100381712057411990510038171003817100381710038172)

119864 [ sup119904isin[119905119879]

10038161003816100381610038161003816119884120574119905119906119907

(119904) minus 119884120574119905119906119907

(119904)10038161003816100381610038161003816

2

+int

119879

119905

10038161003816100381610038161003816119885120574119905119906119907

(119904) minus 119885120574119905119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]le 1198621003817100381710038171003817120574119905 minus 120574

119905

10038171003817100381710038172


following property


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817

(ii) 1003816100381610038161003816119882 (120574119905)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171205741199051003817100381710038171003817)

(41)



119866120574119905119906119907

119904119905+120575[120578] =

120574119905119906119907

(119904) 119904 isin [119905 119905 + 120575] (42)

where 120578 isin 1198712(ΩF119905+120575

119905 119875R) (

120574119905119906119907

(119904) 119885120574119905119906119907

(119904))119905le119904le119905+120575


119889120574119905119906119907

(119904)

= minus119891 (119904 119883120574119905119906119907

119904

120574119905119906119907

(119904) 119885120574119905119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

+ 119885120574119905119906119907

(119904) 119889119861 (119904)

120574119905119906119907

(119905 + 120575) = 120578 119904 isin [119905 119905 + 120575]

(43)Also we have

119866120574119905119906119907

119905119879[Φ (119883

120574119905119906119907

119879)] = 119866

120574119905119906119907

119905119905+120575[119884

120574119905119906119907


0

119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (45)




119863119905119882(120574119905) + 119867minus(120574119905119882119863119909119882119863119909119909119882) = 0

119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ

(46)

119863119905119880 (120574119905) + 119867+(120574119905 119880119863119909119880119863119909119909119880) = 0

119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ

(47)

where

119867minus(120574119905119882119863119909119882119863119909119909119882)

= sup119906isin119880

inf119907isin119881

119867(120574119905119882119863119909119882119863119909119909119882119906 119907)

119867+(120574119905 119880119863119909119880119863119909119909119880)

= inf119907isin119881

sup119906isin119880

119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907)

119867 (120574119905 119910 119901 119883 119906 119907)

=1

2tr (120590120590119879 (120574119905 119906 119907)119883) + 119901 sdot 119887 (120574119905 119906 119907)

+ 119891 (120574119905 119910 119901120590 (120574119905 119906 119907) 119906 119907)

(48)





C(Λ) is called


C12


0le119904le120575Λ 119905+119904 and

Γ(120574119905) = 119882(120574119905) one has

119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) ge 0 (49)


C12


0le119904le120575Λ 119905+119904 and

Γ(120574119905) = 119882(120574119905) one has

119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) le 0 (50)





C12

119897119887(Λ) denote

119865 (120574119904 119910 119911 119906 119907)

= 119863119904Γ (120574119904) +1

2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))

+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)

+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)

(51)



minus 1198891198841119906119907

(119904)

= 119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198851119906119907

(119904) 119889119861 (119904)

1198841119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]

(52)


1198841119906119907

(119904) = 119866120574119905119906119907

119904119905+120575[Γ (119883

120574119905119906119907

119905+120575)] minus Γ (119883

120574119905119906119907

119904) 119875-as (53)

Proof Note119866120574119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] is defined as119866120574

119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] =

119884119906119907


minus 119889119884119906119907

(119904) = 119891 (119883120574119905119906119907

119904 119884

119906119907(119904) 119885

119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 119885119906119907

(119904) 119889119861 (119904)

119884119906119907

(119905 + 120575) = Γ (119883120574119905119906119907

119905+120575) 119904 isin [119905 119905 + 120575]

(54)


119904) we have

119889 (119884119906119907

(119904) minus Γ (119883120574119905119906119907

119904)) = 119889119884

1119906119907(119904) (55)

Combined with 119884119906119907

(119905 + 120575) minus Γ(119883120574119905119906119907

119905+120575) = 0 = 119884

1119906119907(119905 + 120575) we



minus 1198891198842119906119907

(119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198852119906119907

(119904) 119889119861 (119904)

(56)

1198842119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)



(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816le 119862120575

32 119875-as (58)



119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905] le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172) (59)

combined with

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]

le 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

119887 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119903

10038161003816100381610038161003816100381610038161003816

2

| F119905]

+ 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

120590 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119861 (119903)

10038161003816100381610038161003816100381610038161003816

2

| F119905]

(60)

we have

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)


1205851 = 1205852 = 0

119892 (119904 119910 119911) = 119865 (119883120574119905119906119907

119904 119910 119911 119906119904 119907119904)

1205931 (119904) = 0

1205932 (119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906119904 119907119904)

minus 119865 (119883120574119905119906119907

119904 119884

2119906119907(119904) 119885

2119906119907(119904) 119906119904 119907119904)

(62)



119887 |1205932(119904)| le 1205880(|119883

120574119905119906119907

119904minus

120574119905(119905)|) we have

119864[int

119905+120575

119905

(100381610038161003816100381610038161198841119906119907

(119904) minus1198842119906119907

(119904)10038161003816100381610038161003816

2

+100381610038161003816100381610038161198851119906119907

(119904) minus1198852119906119907

(119904)10038161003816100381610038161003816

2

) 119889119904 | F119905]

le 119864[int

119905+120575

119905

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) 119889119904 | F119905]

le 119862120575119864[ sup119904isin[119905119905+120575]

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) | F119905]

le 1198621205752

(63)


So100381610038161003816100381610038161198841119906119907

(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816

=10038161003816100381610038161003816119864 [(119884

1119906119907(119904) minus 119884

2119906119907(119904)) | F119905]

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

119864 [int

119905+120575

119905

(119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904))

minus119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]

100381610038161003816100381610038161003816100381610038161003816

le 119862119864[int

119905+120575

119905

[1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

+100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816] 119889119904 | F119905]

le 119862119864[int

119905+120575

119905

1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) 119889119904 | F119905]

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

le 11986212057532

(64)


minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]

1198840 (119905 + 120575) = 0

(65)

where 1198650(120574119905 119910 119911) = sup119906isin119880

inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as

1198840 (119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) (66)


1198651 (120574119905 119910 119911 119906) = inf119907isin119881

119865 (120574119905 119910 119911 119906 119907)


(67)


minus1198891198843119906

(119904) = 1198651 (120574119905 1198843119906

(119904) 1198853119906

(119904) 119906 (119904)) 119889119904

minus1198853119906

(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]

1198843119906

(119905 + 120575) = 0

(68)


1198853119906

) solving (68) Also

1198843119906

(119905) = essinf119907isinV119905119905+120575

1198842119906119907


(69)


1198843119906

(119905) le 1198842119906119907

(119905) 119875-as for any 119907 isin V119905119905+120575

for every 119906 isin U119905119905+120575

(70)


4 R timesR119889


1198651 (120574119905 119910 119911 119906)

= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906

(71)

Set 1199074(119904) = 1199074(119884

3119906(119904) 119885

3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know

1199074isin V119905119905+120575 Therefore (1198843119906

1198853119906

) = (11988421199061199074

11988521199061199074


(119905) = 11988421199061199074


(119905) = essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-119886119904 for every 119906 isin U119905119905+120575


1198651(120574119905 119910 119911 119906) wealso derive

1198840 (119905) = esssup119906isinU119905119905+120575

1198843119906

(119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-as

(72)


119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575

32 119875-as

(73)



100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816

2

le 119862120575

119864 [int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904] le 119862120575

(74)



(119904)10038161003816100381610038161003816

le 119864 [int

119905+120575

119904

119865 (120574119905 1198842119906119907

(119903) 1198852119906119907

(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]

le 119862119864[int

119905+120575

119904

(1 +1003817100381710038171003817120574119905

10038171003817100381710038172+100381610038161003816100381610038161198842119906119907

(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816) 119889119903 | F119904]

le 119862120575 + 11986212057512

119864[int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904]

le 119862120575 119904 isin [119905 119905 + 120575]

(75)


119905|119885

2119906119907(119904)|

2

119889119904 |

F119905] le 1198621205752 119875-as Therefore

119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905]

le 1198621205752+ 119862120575

12(119864[int

119905+120575

119905

100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816

2

119889119903 | F119905])

12

le 11986212057532

119875-as

(76)



0le119904le120575Λ 119905+119904


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(77)

From 119882 ge Γ on ⋃0le119904le120575


119905119906120573(119906)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

120574119905119906120573(119906)

119905+120575)] minus Γ (120574119905) le 0 (78)

By Lemma 19 we have

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) le 0 119875-as (79)

Thus from Lemma 20

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as (80)


1198842119906119907

(119905) le 1198842119906120573(119906)

(119905) 120573 isin

B119905119905+120575 we have

esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905)

le essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as(81)

Thus by Lemma 21 1198840(119905) le 11986212057532


11986212057512

ge1

1205751198840 (119905) =

1

120575int

119905+120575

119905

1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)

sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)



Γ ge 119882 on⋃0le119904le120575


sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)


1198650 (120574119905 0 0) = sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)


119865 (120574119905 0 0 119906 120595 (119906)) le minus3

4120579 forall119906 isin 119880 (86)


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(87)



119905119906120573(119906)

119905119905+120575[sdot] (Lemma 7) we get

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

119906120573(119906)

119905+120575)] minus Γ (120574119905) ge 0 119875-as

(88)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) ge 0 119875-as (89)

in particular

esssup119906isinU119905119905+120575

1198841119906120595(119906)

(119905) ge 0 119875-as (90)


1119906120576120595(119906120576)(119905) ge minus120576120575


1198842119906120576120595(119906120576)(119905) ge minus119862120575

32minus 120576120575 119875-as (91)



1198842119906120576120595(119906120576)(119905)

= 119864 [int

119905+120575

119905

119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885

2119906120576120595(119906120576)(119904)

119906120576(119904) 120595 (119906

120576(119904))) 119889119904 | F119905]

le 119864[int

119905+120575

119905

(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816+ 119862

1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816

+119865 (120574119905 0 0 119906120576(119904) 120595 (119906

120576(119904)))) 119889119904 | F119905]

le 11986212057532

minus3

4120579120575 119875-as

(92)


minus 120576 le 11986212057512

minus


sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)



Appendix



119882120575 (120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (A1)




Lemma A2 119882120575(120574119905) le 119882(120574119905)





119882120575 (120574119905) le esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)] 119875-as

(A3)

We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907

119905119905+120575[119882(119883

120574119905119906119907



119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575

119868120575 (120574119905 1199061 1205731 (1199061))

= sup119894ge1

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) 119875-as

(A4)


119894 1205731(119906

1

119894)) +


(⋃119894minus1


119906120576

1= sum

119894ge11Γ119894

1199061


of 1205731 we have 1205731(119906120576

1) = sum

119894ge11Γ119894

1205731(1199061



120576

1 1205731(119906

120576

1)) = sum

119894ge11Γ119894

119868120575(120574119905 1199061

119894 1205731(119906

1

119894)) 119875-as So

119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum

119894ge1

1Γ119894

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) + 120576

= 119868120575 (120574119905 119906120576

1 1205731 (119906

120576

1)) + 120576

= 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119882(119883

120574119905119906120576

11205731(119906120576

1)

119905+120575)] + 120576

(A5)


120576

1oplus


119882(120574119905) le esssup1199062isinU119905+120575119879

119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817 for any 120574119905 120574119905 isin Λ

(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))

1003816100381610038161003816

le 1198621003817100381710038171003817120574119905 minus 120574

119905

1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879

(A7)


11205731(119906120576

1)

119905+120575that

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062)) 119875-as

(A8)



119895 119895 ge 1 sub U119905+120575119879 such that

esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

= sup119895ge1

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A9)


119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732(1199062)) le

119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732(119906

2


Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1


(ΩF119905+120575)-partition and 119906120576

2= sum

119895ge11Δ119895

1199062

119895isin U119905+120575119879 Due to


2) = sum

119895ge11Δ119895

1205732(1199062

119895)


1oplus 119906

120576

2) =

1205731(119906120576

1) oplus 1205732(119906

120576



119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

2 1205732 (119906

120576

2))

= 119884119883120574119905119906120576

11205731(119906120576

1)

119905+120575119906120576

21205732(119906120576

2)(119905 + 120575)

= sum

119895ge1

1Δ119895

119884119883120574119905119906120576

11205731(119906120576

1)

119905+1205751199062

1198951205732(1199062

119895)(119905 + 120575)

= sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A10)

So

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

le sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

2

119895 120573 (119906

120576

1oplus 119906

2

119895)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

120576

2 120573 (119906

120576

1oplus 119906

120576

2)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576 119875-as

(A11)

where 119906120576

= 119906120576

1oplus 119906

120576



119882120575 (120574119905)

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576] + 120576

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119866120574119905119906120576120573(119906120576)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119869 (120574119905 119906120576 120573 (119906

120576)) + (119862 + 1) 120576

= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576


119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576 119875-as

(A12)


119882120575 (120574119905) le essinf120573isinB119905119879

esssup119906isinU119905119879

119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576

= 119882(120574119905) + (119862 + 1) 120576

(A13)


Lemma A3 119882(120574119905) le 119882120575(120574119905)


119882120575 (120574119905) = essinf1205731isinB119905119905+120575

esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)]

= essinf1205731isinB119905119905+120575

119868120575 (120574119905 1205731)

(A14)


119894 119894 ge 1 sub B119905119905+120575 such

that

119882120575 (120574119905) = inf119894ge1

119868120575 (120574119905 1205731

119894) 119875-as (A15)

For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731

119894) minus 120576 le 119882120575(120574119905) isin


119897=1Π119897) isin F119905 119894 ge


1=

sum119894ge1

1Λ119894

1205731



1(1199061)) =

sum119894ge1

1Λ119894

119868120575(120574119905 1199061 1205731


119882120575 (120574119905) ge sum

119894ge1

1Π119894

119868120575 (120574119905 1205731

119894) minus 120576

ge sum

119894ge1

1Π119894

119868120575 (120574119905 1199061 1205731

119894(1199061)) minus 120576

= 119868120575 (120574119905 1199061 120573120576

1(1199061)) minus 120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

119875-as forall1199061 isin U119905119905+120575

(A16)




B119905+120575119879 such that

119882(119909119905+120575) ge esssup1199062isinU119905+120575119879

119869 (119909119905+120575 1199062 120573119909120576

(1199062)) minus 120576 119875-as

(A17)



1205741199051199061120573120576

1(1199061)

119905+120575] = sum

119894ge1120574119894

119905+12057511198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894

we have

100381610038161003816100381610038161003816119883

1205741199051199061120573120576

1(1199061)

119905+120575minus [119883

1205741199051199061120573120576

1(1199061)

119905+120575]100381610038161003816100381610038161003816le 120576


(A18)



(A17) and 120573120576

1199061

= sum119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894120573120574119894120576isin B119905+120575119879


120576

1(1199061) oplus 120573

120576

1199061

(1199062) 119906 isin





1015840 into 1199061 1199061015840

1isin

U119905119905+120575 1199062 1199061015840


1015840

1and 119906 = 1199062 oplus 119906

1015840

2

From 1199061 equiv 1199061015840

1on [[119905 119878 and (119905 + 120575)]] we get 120573120576

1(1199061) = 120573

120576

1(119906

1015840

1) on

[[119905 119878and(119905+120575)]] (note that 120573120576


side from 1199062 equiv 1199061015840

2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt

119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576

1(1199061)

119905+120575= 119883

1205741199051199061015840

1120573120576

1(1199061015840

1)

119905+120575


1199061

= 120573120576

11990610158401

on 119878 gt 119905 + 120575 and120573120576

1199061

(1199062) = 120573120576

11990610158401

(1199061015840


120573120576(119906) = 120573

120576

1(1199061) oplus 120573

120576

1199061

(1199062) = 120573120576

1(119906

1015840

1) oplus 120573

120576

11990610158401

(1199061015840

2) on [[119905 119878]] from




119882120575 (120574119905) ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575]) minus 119862120576] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575])] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119882(120574

119894

119905+120575)] minus 119862120576

119875-as

(A19)


119882120575 (120574119905)

ge1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))minus 120576]minus119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 ([119883

1205741199051199061120573120576

1(1199061)

119905+120575] 1199062 120573

120576

1199061

(1199062))] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062)) minus 119862120576] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062))] minus 119862120576

= 119866120574119905119906120573120576(119906)

119905119905+120575[119884

120574119905119906120573120576(119906)

(119905 + 120575)] minus 119862120576

= 119884120574119905119906120573120576(119906)


(A20)

Therefore

119882120575 (120574119905) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573120576(119906)) minus 119862120576


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) minus 119862120576

= 119882(120574119905) minus 119862120576 119875-as

(A21)



119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)

119905119905+120575[119882(119883

120574119905119906120576120573(119906120576)

119905+120575)] + 120576

119875-as(A22)


119880119905119905+120575

119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)

119905119905+120575[119882(119883

120574119905119906120573120576(119906)

119905+120575)] minus 120576

119875-as(A23)


inf120573isinB119905119879

sup119906isinU119905119879

119864[119869(120574119905 119906 120573(119906))]

Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905]le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172)

119864 [ sup119904isin[119905119879]

10038161003816100381610038161003816119883

120574119905119906119907

(119904) minus 119883120574119905119906119907

(119904)10038161003816100381610038161003816

2

| F119905]le 1198621003817100381710038171003817120574119905 minus 120574

119905

10038171003817100381710038172

119864 [ sup119904isin[119905119879]

1003816100381610038161003816119884120574119905119906119907

(119904)10038161003816100381610038162+int

119879

119905

1003816100381610038161003816119885120574119905119906119907

(119904)10038161003816100381610038162119889119904 | F119905]le 119862 (1+

100381710038171003817100381712057411990510038171003817100381710038172)

119864 [ sup119904isin[119905119879]

10038161003816100381610038161003816119884120574119905119906119907

(119904) minus 119884120574119905119906119907

(119904)10038161003816100381610038161003816

2

+int

119879

119905

10038161003816100381610038161003816119885120574119905119906119907

(119904) minus 119885120574119905119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]le 1198621003817100381710038171003817120574119905 minus 120574

119905

10038171003817100381710038172


following property


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817

(ii) 1003816100381610038161003816119882 (120574119905)1003816100381610038161003816 le 119862 (1 +

10038171003817100381710038171205741199051003817100381710038171003817)

(41)



119866120574119905119906119907

119904119905+120575[120578] =

120574119905119906119907

(119904) 119904 isin [119905 119905 + 120575] (42)

where 120578 isin 1198712(ΩF119905+120575

119905 119875R) (

120574119905119906119907

(119904) 119885120574119905119906119907

(119904))119905le119904le119905+120575


119889120574119905119906119907

(119904)

= minus119891 (119904 119883120574119905119906119907

119904

120574119905119906119907

(119904) 119885120574119905119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

+ 119885120574119905119906119907

(119904) 119889119861 (119904)

120574119905119906119907

(119905 + 120575) = 120578 119904 isin [119905 119905 + 120575]

(43)Also we have

119866120574119905119906119907

119905119879[Φ (119883

120574119905119906119907

119879)] = 119866

120574119905119906119907

119905119905+120575[119884

120574119905119906119907


0

119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (45)




119863119905119882(120574119905) + 119867minus(120574119905119882119863119909119882119863119909119909119882) = 0

119882 (120574119879) = Φ (120574119879) 120574119879 isin Λ

(46)

119863119905119880 (120574119905) + 119867+(120574119905 119880119863119909119880119863119909119909119880) = 0

119880 (120574119905) = Φ (120574119879) 120574119879 isin Λ

(47)

where

119867minus(120574119905119882119863119909119882119863119909119909119882)

= sup119906isin119880

inf119907isin119881

119867(120574119905119882119863119909119882119863119909119909119882119906 119907)

119867+(120574119905 119880119863119909119880119863119909119909119880)

= inf119907isin119881

sup119906isin119880

119867(120574119905 119880119863119909119880119863119909119909119880 119906 119907)

119867 (120574119905 119910 119901 119883 119906 119907)

=1

2tr (120590120590119879 (120574119905 119906 119907)119883) + 119901 sdot 119887 (120574119905 119906 119907)

+ 119891 (120574119905 119910 119901120590 (120574119905 119906 119907) 119906 119907)

(48)





C(Λ) is called


C12


0le119904le120575Λ 119905+119904 and

Γ(120574119905) = 119882(120574119905) one has

119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) ge 0 (49)


C12


0le119904le120575Λ 119905+119904 and

Γ(120574119905) = 119882(120574119905) one has

119863119905Γ (120574119905) + 119867minus(120574119905 Γ 119863119909Γ119863119909119909Γ) le 0 (50)





C12

119897119887(Λ) denote

119865 (120574119904 119910 119911 119906 119907)

= 119863119904Γ (120574119904) +1

2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))

+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)

+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)

(51)



minus 1198891198841119906119907

(119904)

= 119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198851119906119907

(119904) 119889119861 (119904)

1198841119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]

(52)


1198841119906119907

(119904) = 119866120574119905119906119907

119904119905+120575[Γ (119883

120574119905119906119907

119905+120575)] minus Γ (119883

120574119905119906119907

119904) 119875-as (53)

Proof Note119866120574119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] is defined as119866120574

119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] =

119884119906119907


minus 119889119884119906119907

(119904) = 119891 (119883120574119905119906119907

119904 119884

119906119907(119904) 119885

119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 119885119906119907

(119904) 119889119861 (119904)

119884119906119907

(119905 + 120575) = Γ (119883120574119905119906119907

119905+120575) 119904 isin [119905 119905 + 120575]

(54)


119904) we have

119889 (119884119906119907

(119904) minus Γ (119883120574119905119906119907

119904)) = 119889119884

1119906119907(119904) (55)

Combined with 119884119906119907

(119905 + 120575) minus Γ(119883120574119905119906119907

119905+120575) = 0 = 119884

1119906119907(119905 + 120575) we



minus 1198891198842119906119907

(119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198852119906119907

(119904) 119889119861 (119904)

(56)

1198842119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)



(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816le 119862120575

32 119875-as (58)



119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905] le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172) (59)

combined with

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]

le 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

119887 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119903

10038161003816100381610038161003816100381610038161003816

2

| F119905]

+ 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

120590 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119861 (119903)

10038161003816100381610038161003816100381610038161003816

2

| F119905]

(60)

we have

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)


1205851 = 1205852 = 0

119892 (119904 119910 119911) = 119865 (119883120574119905119906119907

119904 119910 119911 119906119904 119907119904)

1205931 (119904) = 0

1205932 (119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906119904 119907119904)

minus 119865 (119883120574119905119906119907

119904 119884

2119906119907(119904) 119885

2119906119907(119904) 119906119904 119907119904)

(62)



119887 |1205932(119904)| le 1205880(|119883

120574119905119906119907

119904minus

120574119905(119905)|) we have

119864[int

119905+120575

119905

(100381610038161003816100381610038161198841119906119907

(119904) minus1198842119906119907

(119904)10038161003816100381610038161003816

2

+100381610038161003816100381610038161198851119906119907

(119904) minus1198852119906119907

(119904)10038161003816100381610038161003816

2

) 119889119904 | F119905]

le 119864[int

119905+120575

119905

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) 119889119904 | F119905]

le 119862120575119864[ sup119904isin[119905119905+120575]

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) | F119905]

le 1198621205752

(63)


So100381610038161003816100381610038161198841119906119907

(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816

=10038161003816100381610038161003816119864 [(119884

1119906119907(119904) minus 119884

2119906119907(119904)) | F119905]

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

119864 [int

119905+120575

119905

(119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904))

minus119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]

100381610038161003816100381610038161003816100381610038161003816

le 119862119864[int

119905+120575

119905

[1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

+100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816] 119889119904 | F119905]

le 119862119864[int

119905+120575

119905

1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) 119889119904 | F119905]

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

le 11986212057532

(64)


minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]

1198840 (119905 + 120575) = 0

(65)

where 1198650(120574119905 119910 119911) = sup119906isin119880

inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as

1198840 (119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) (66)


1198651 (120574119905 119910 119911 119906) = inf119907isin119881

119865 (120574119905 119910 119911 119906 119907)


(67)


minus1198891198843119906

(119904) = 1198651 (120574119905 1198843119906

(119904) 1198853119906

(119904) 119906 (119904)) 119889119904

minus1198853119906

(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]

1198843119906

(119905 + 120575) = 0

(68)


1198853119906

) solving (68) Also

1198843119906

(119905) = essinf119907isinV119905119905+120575

1198842119906119907


(69)


1198843119906

(119905) le 1198842119906119907

(119905) 119875-as for any 119907 isin V119905119905+120575

for every 119906 isin U119905119905+120575

(70)


4 R timesR119889


1198651 (120574119905 119910 119911 119906)

= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906

(71)

Set 1199074(119904) = 1199074(119884

3119906(119904) 119885

3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know

1199074isin V119905119905+120575 Therefore (1198843119906

1198853119906

) = (11988421199061199074

11988521199061199074


(119905) = 11988421199061199074


(119905) = essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-119886119904 for every 119906 isin U119905119905+120575


1198651(120574119905 119910 119911 119906) wealso derive

1198840 (119905) = esssup119906isinU119905119905+120575

1198843119906

(119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-as

(72)


119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575

32 119875-as

(73)



100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816

2

le 119862120575

119864 [int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904] le 119862120575

(74)



(119904)10038161003816100381610038161003816

le 119864 [int

119905+120575

119904

119865 (120574119905 1198842119906119907

(119903) 1198852119906119907

(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]

le 119862119864[int

119905+120575

119904

(1 +1003817100381710038171003817120574119905

10038171003817100381710038172+100381610038161003816100381610038161198842119906119907

(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816) 119889119903 | F119904]

le 119862120575 + 11986212057512

119864[int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904]

le 119862120575 119904 isin [119905 119905 + 120575]

(75)


119905|119885

2119906119907(119904)|

2

119889119904 |

F119905] le 1198621205752 119875-as Therefore

119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905]

le 1198621205752+ 119862120575

12(119864[int

119905+120575

119905

100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816

2

119889119903 | F119905])

12

le 11986212057532

119875-as

(76)



0le119904le120575Λ 119905+119904


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(77)

From 119882 ge Γ on ⋃0le119904le120575


119905119906120573(119906)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

120574119905119906120573(119906)

119905+120575)] minus Γ (120574119905) le 0 (78)

By Lemma 19 we have

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) le 0 119875-as (79)

Thus from Lemma 20

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as (80)


1198842119906119907

(119905) le 1198842119906120573(119906)

(119905) 120573 isin

B119905119905+120575 we have

esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905)

le essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as(81)

Thus by Lemma 21 1198840(119905) le 11986212057532


11986212057512

ge1

1205751198840 (119905) =

1

120575int

119905+120575

119905

1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)

sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)



Γ ge 119882 on⋃0le119904le120575


sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)


1198650 (120574119905 0 0) = sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)


119865 (120574119905 0 0 119906 120595 (119906)) le minus3

4120579 forall119906 isin 119880 (86)


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(87)



119905119906120573(119906)

119905119905+120575[sdot] (Lemma 7) we get

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

119906120573(119906)

119905+120575)] minus Γ (120574119905) ge 0 119875-as

(88)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) ge 0 119875-as (89)

in particular

esssup119906isinU119905119905+120575

1198841119906120595(119906)

(119905) ge 0 119875-as (90)


1119906120576120595(119906120576)(119905) ge minus120576120575


1198842119906120576120595(119906120576)(119905) ge minus119862120575

32minus 120576120575 119875-as (91)



1198842119906120576120595(119906120576)(119905)

= 119864 [int

119905+120575

119905

119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885

2119906120576120595(119906120576)(119904)

119906120576(119904) 120595 (119906

120576(119904))) 119889119904 | F119905]

le 119864[int

119905+120575

119905

(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816+ 119862

1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816

+119865 (120574119905 0 0 119906120576(119904) 120595 (119906

120576(119904)))) 119889119904 | F119905]

le 11986212057532

minus3

4120579120575 119875-as

(92)


minus 120576 le 11986212057512

minus


sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)



Appendix



119882120575 (120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (A1)




Lemma A2 119882120575(120574119905) le 119882(120574119905)





119882120575 (120574119905) le esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)] 119875-as

(A3)

We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907

119905119905+120575[119882(119883

120574119905119906119907



119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575

119868120575 (120574119905 1199061 1205731 (1199061))

= sup119894ge1

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) 119875-as

(A4)


119894 1205731(119906

1

119894)) +


(⋃119894minus1


119906120576

1= sum

119894ge11Γ119894

1199061


of 1205731 we have 1205731(119906120576

1) = sum

119894ge11Γ119894

1205731(1199061



120576

1 1205731(119906

120576

1)) = sum

119894ge11Γ119894

119868120575(120574119905 1199061

119894 1205731(119906

1

119894)) 119875-as So

119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum

119894ge1

1Γ119894

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) + 120576

= 119868120575 (120574119905 119906120576

1 1205731 (119906

120576

1)) + 120576

= 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119882(119883

120574119905119906120576

11205731(119906120576

1)

119905+120575)] + 120576

(A5)


120576

1oplus


119882(120574119905) le esssup1199062isinU119905+120575119879

119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817 for any 120574119905 120574119905 isin Λ

(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))

1003816100381610038161003816

le 1198621003817100381710038171003817120574119905 minus 120574

119905

1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879

(A7)


11205731(119906120576

1)

119905+120575that

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062)) 119875-as

(A8)



119895 119895 ge 1 sub U119905+120575119879 such that

esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

= sup119895ge1

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A9)


119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732(1199062)) le

119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732(119906

2


Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1


(ΩF119905+120575)-partition and 119906120576

2= sum

119895ge11Δ119895

1199062

119895isin U119905+120575119879 Due to


2) = sum

119895ge11Δ119895

1205732(1199062

119895)


1oplus 119906

120576

2) =

1205731(119906120576

1) oplus 1205732(119906

120576



119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

2 1205732 (119906

120576

2))

= 119884119883120574119905119906120576

11205731(119906120576

1)

119905+120575119906120576

21205732(119906120576

2)(119905 + 120575)

= sum

119895ge1

1Δ119895

119884119883120574119905119906120576

11205731(119906120576

1)

119905+1205751199062

1198951205732(1199062

119895)(119905 + 120575)

= sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A10)

So

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

le sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

2

119895 120573 (119906

120576

1oplus 119906

2

119895)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

120576

2 120573 (119906

120576

1oplus 119906

120576

2)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576 119875-as

(A11)

where 119906120576

= 119906120576

1oplus 119906

120576



119882120575 (120574119905)

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576] + 120576

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119866120574119905119906120576120573(119906120576)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119869 (120574119905 119906120576 120573 (119906

120576)) + (119862 + 1) 120576

= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576


119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576 119875-as

(A12)


119882120575 (120574119905) le essinf120573isinB119905119879

esssup119906isinU119905119879

119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576

= 119882(120574119905) + (119862 + 1) 120576

(A13)


Lemma A3 119882(120574119905) le 119882120575(120574119905)


119882120575 (120574119905) = essinf1205731isinB119905119905+120575

esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)]

= essinf1205731isinB119905119905+120575

119868120575 (120574119905 1205731)

(A14)


119894 119894 ge 1 sub B119905119905+120575 such

that

119882120575 (120574119905) = inf119894ge1

119868120575 (120574119905 1205731

119894) 119875-as (A15)

For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731

119894) minus 120576 le 119882120575(120574119905) isin


119897=1Π119897) isin F119905 119894 ge


1=

sum119894ge1

1Λ119894

1205731



1(1199061)) =

sum119894ge1

1Λ119894

119868120575(120574119905 1199061 1205731


119882120575 (120574119905) ge sum

119894ge1

1Π119894

119868120575 (120574119905 1205731

119894) minus 120576

ge sum

119894ge1

1Π119894

119868120575 (120574119905 1199061 1205731

119894(1199061)) minus 120576

= 119868120575 (120574119905 1199061 120573120576

1(1199061)) minus 120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

119875-as forall1199061 isin U119905119905+120575

(A16)




B119905+120575119879 such that

119882(119909119905+120575) ge esssup1199062isinU119905+120575119879

119869 (119909119905+120575 1199062 120573119909120576

(1199062)) minus 120576 119875-as

(A17)



1205741199051199061120573120576

1(1199061)

119905+120575] = sum

119894ge1120574119894

119905+12057511198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894

we have

100381610038161003816100381610038161003816119883

1205741199051199061120573120576

1(1199061)

119905+120575minus [119883

1205741199051199061120573120576

1(1199061)

119905+120575]100381610038161003816100381610038161003816le 120576


(A18)



(A17) and 120573120576

1199061

= sum119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894120573120574119894120576isin B119905+120575119879


120576

1(1199061) oplus 120573

120576

1199061

(1199062) 119906 isin





1015840 into 1199061 1199061015840

1isin

U119905119905+120575 1199062 1199061015840


1015840

1and 119906 = 1199062 oplus 119906

1015840

2

From 1199061 equiv 1199061015840

1on [[119905 119878 and (119905 + 120575)]] we get 120573120576

1(1199061) = 120573

120576

1(119906

1015840

1) on

[[119905 119878and(119905+120575)]] (note that 120573120576


side from 1199062 equiv 1199061015840

2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt

119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576

1(1199061)

119905+120575= 119883

1205741199051199061015840

1120573120576

1(1199061015840

1)

119905+120575


1199061

= 120573120576

11990610158401

on 119878 gt 119905 + 120575 and120573120576

1199061

(1199062) = 120573120576

11990610158401

(1199061015840


120573120576(119906) = 120573

120576

1(1199061) oplus 120573

120576

1199061

(1199062) = 120573120576

1(119906

1015840

1) oplus 120573

120576

11990610158401

(1199061015840

2) on [[119905 119878]] from




119882120575 (120574119905) ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575]) minus 119862120576] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575])] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119882(120574

119894

119905+120575)] minus 119862120576

119875-as

(A19)


119882120575 (120574119905)

ge1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))minus 120576]minus119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 ([119883

1205741199051199061120573120576

1(1199061)

119905+120575] 1199062 120573

120576

1199061

(1199062))] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062)) minus 119862120576] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062))] minus 119862120576

= 119866120574119905119906120573120576(119906)

119905119905+120575[119884

120574119905119906120573120576(119906)

(119905 + 120575)] minus 119862120576

= 119884120574119905119906120573120576(119906)


(A20)

Therefore

119882120575 (120574119905) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573120576(119906)) minus 119862120576


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) minus 119862120576

= 119882(120574119905) minus 119862120576 119875-as

(A21)



119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)

119905119905+120575[119882(119883

120574119905119906120576120573(119906120576)

119905+120575)] + 120576

119875-as(A22)


119880119905119905+120575

119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)

119905119905+120575[119882(119883

120574119905119906120573120576(119906)

119905+120575)] minus 120576

119875-as(A23)


inf120573isinB119905119879

sup119906isinU119905119879

119864[119869(120574119905 119906 120573(119906))]

Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











C12

119897119887(Λ) denote

119865 (120574119904 119910 119911 119906 119907)

= 119863119904Γ (120574119904) +1

2tr (120590120590119879 (120574119904 119906 119907)119863119909119909Γ (120574119904))

+ 119863119909Γ (120574119904) sdot 119887 (120574119904 119906 119907)

+ 119891 (120574119904 119910 + Γ (120574119904) 119911 + 119863119909Γ (120574119904) sdot 120590 (120574119904 119906 119907) 119906 119907)

(51)



minus 1198891198841119906119907

(119904)

= 119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198851119906119907

(119904) 119889119861 (119904)

1198841119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575]

(52)


1198841119906119907

(119904) = 119866120574119905119906119907

119904119905+120575[Γ (119883

120574119905119906119907

119905+120575)] minus Γ (119883

120574119905119906119907

119904) 119875-as (53)

Proof Note119866120574119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] is defined as119866120574

119905119906119907

119904119905+120575[Γ(119883

120574119905119906119907

119905+120575)] =

119884119906119907


minus 119889119884119906119907

(119904) = 119891 (119883120574119905119906119907

119904 119884

119906119907(119904) 119885

119906119907(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 119885119906119907

(119904) 119889119861 (119904)

119884119906119907

(119905 + 120575) = Γ (119883120574119905119906119907

119905+120575) 119904 isin [119905 119905 + 120575]

(54)


119904) we have

119889 (119884119906119907

(119904) minus Γ (119883120574119905119906119907

119904)) = 119889119884

1119906119907(119904) (55)

Combined with 119884119906119907

(119905 + 120575) minus Γ(119883120574119905119906119907

119905+120575) = 0 = 119884

1119906119907(119905 + 120575) we



minus 1198891198842119906119907

(119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904)) 119889119904

minus 1198852119906119907

(119904) 119889119861 (119904)

(56)

1198842119906119907

(119905 + 120575) = 0 119904 isin [119905 119905 + 120575] (57)



(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816le 119862120575

32 119875-as (58)



119864[ sup119904isin[119905119879]

1003816100381610038161003816119883120574119905119906119907

(119904)10038161003816100381610038162| F119905] le 119862 (1 +

100381710038171003817100381712057411990510038171003817100381710038172) (59)

combined with

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905(119905)10038161003816100381610038162| F119905]

le 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

119887 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119903

10038161003816100381610038161003816100381610038161003816

2

| F119905]

+ 2119864[ sup119904isin[119905119905+120575]

10038161003816100381610038161003816100381610038161003816int

119904

119905

120590 (119883120574119905119906119907

119903 119906 (119903) 119907 (119903)) 119889119861 (119903)

10038161003816100381610038161003816100381610038161003816

2

| F119905]

(60)

we have

119864[ sup119904isin[119905119905+120575]

1003816100381610038161003816119883120574119905119906119907

(119904) minus 120574119905 (119905)10038161003816100381610038162| F119905] le 119862120575 (61)


1205851 = 1205852 = 0

119892 (119904 119910 119911) = 119865 (119883120574119905119906119907

119904 119910 119911 119906119904 119907119904)

1205931 (119904) = 0

1205932 (119904) = 119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906119904 119907119904)

minus 119865 (119883120574119905119906119907

119904 119884

2119906119907(119904) 119885

2119906119907(119904) 119906119904 119907119904)

(62)



119887 |1205932(119904)| le 1205880(|119883

120574119905119906119907

119904minus

120574119905(119905)|) we have

119864[int

119905+120575

119905

(100381610038161003816100381610038161198841119906119907

(119904) minus1198842119906119907

(119904)10038161003816100381610038161003816

2

+100381610038161003816100381610038161198851119906119907

(119904) minus1198852119906119907

(119904)10038161003816100381610038161003816

2

) 119889119904 | F119905]

le 119864[int

119905+120575

119905

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) 119889119904 | F119905]

le 119862120575119864[ sup119904isin[119905119905+120575]

1205882

0(1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905 (119905)

1003816100381610038161003816) | F119905]

le 1198621205752

(63)


So100381610038161003816100381610038161198841119906119907

(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816

=10038161003816100381610038161003816119864 [(119884

1119906119907(119904) minus 119884

2119906119907(119904)) | F119905]

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

119864 [int

119905+120575

119905

(119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904))

minus119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]

100381610038161003816100381610038161003816100381610038161003816

le 119862119864[int

119905+120575

119905

[1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

+100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816] 119889119904 | F119905]

le 119862119864[int

119905+120575

119905

1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) 119889119904 | F119905]

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

le 11986212057532

(64)


minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]

1198840 (119905 + 120575) = 0

(65)

where 1198650(120574119905 119910 119911) = sup119906isin119880

inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as

1198840 (119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) (66)


1198651 (120574119905 119910 119911 119906) = inf119907isin119881

119865 (120574119905 119910 119911 119906 119907)


(67)


minus1198891198843119906

(119904) = 1198651 (120574119905 1198843119906

(119904) 1198853119906

(119904) 119906 (119904)) 119889119904

minus1198853119906

(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]

1198843119906

(119905 + 120575) = 0

(68)


1198853119906

) solving (68) Also

1198843119906

(119905) = essinf119907isinV119905119905+120575

1198842119906119907


(69)


1198843119906

(119905) le 1198842119906119907

(119905) 119875-as for any 119907 isin V119905119905+120575

for every 119906 isin U119905119905+120575

(70)


4 R timesR119889


1198651 (120574119905 119910 119911 119906)

= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906

(71)

Set 1199074(119904) = 1199074(119884

3119906(119904) 119885

3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know

1199074isin V119905119905+120575 Therefore (1198843119906

1198853119906

) = (11988421199061199074

11988521199061199074


(119905) = 11988421199061199074


(119905) = essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-119886119904 for every 119906 isin U119905119905+120575


1198651(120574119905 119910 119911 119906) wealso derive

1198840 (119905) = esssup119906isinU119905119905+120575

1198843119906

(119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-as

(72)


119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575

32 119875-as

(73)



100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816

2

le 119862120575

119864 [int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904] le 119862120575

(74)



(119904)10038161003816100381610038161003816

le 119864 [int

119905+120575

119904

119865 (120574119905 1198842119906119907

(119903) 1198852119906119907

(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]

le 119862119864[int

119905+120575

119904

(1 +1003817100381710038171003817120574119905

10038171003817100381710038172+100381610038161003816100381610038161198842119906119907

(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816) 119889119903 | F119904]

le 119862120575 + 11986212057512

119864[int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904]

le 119862120575 119904 isin [119905 119905 + 120575]

(75)


119905|119885

2119906119907(119904)|

2

119889119904 |

F119905] le 1198621205752 119875-as Therefore

119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905]

le 1198621205752+ 119862120575

12(119864[int

119905+120575

119905

100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816

2

119889119903 | F119905])

12

le 11986212057532

119875-as

(76)



0le119904le120575Λ 119905+119904


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(77)

From 119882 ge Γ on ⋃0le119904le120575


119905119906120573(119906)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

120574119905119906120573(119906)

119905+120575)] minus Γ (120574119905) le 0 (78)

By Lemma 19 we have

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) le 0 119875-as (79)

Thus from Lemma 20

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as (80)


1198842119906119907

(119905) le 1198842119906120573(119906)

(119905) 120573 isin

B119905119905+120575 we have

esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905)

le essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as(81)

Thus by Lemma 21 1198840(119905) le 11986212057532


11986212057512

ge1

1205751198840 (119905) =

1

120575int

119905+120575

119905

1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)

sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)



Γ ge 119882 on⋃0le119904le120575


sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)


1198650 (120574119905 0 0) = sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)


119865 (120574119905 0 0 119906 120595 (119906)) le minus3

4120579 forall119906 isin 119880 (86)


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(87)



119905119906120573(119906)

119905119905+120575[sdot] (Lemma 7) we get

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

119906120573(119906)

119905+120575)] minus Γ (120574119905) ge 0 119875-as

(88)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) ge 0 119875-as (89)

in particular

esssup119906isinU119905119905+120575

1198841119906120595(119906)

(119905) ge 0 119875-as (90)


1119906120576120595(119906120576)(119905) ge minus120576120575


1198842119906120576120595(119906120576)(119905) ge minus119862120575

32minus 120576120575 119875-as (91)



1198842119906120576120595(119906120576)(119905)

= 119864 [int

119905+120575

119905

119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885

2119906120576120595(119906120576)(119904)

119906120576(119904) 120595 (119906

120576(119904))) 119889119904 | F119905]

le 119864[int

119905+120575

119905

(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816+ 119862

1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816

+119865 (120574119905 0 0 119906120576(119904) 120595 (119906

120576(119904)))) 119889119904 | F119905]

le 11986212057532

minus3

4120579120575 119875-as

(92)


minus 120576 le 11986212057512

minus


sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)



Appendix



119882120575 (120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (A1)




Lemma A2 119882120575(120574119905) le 119882(120574119905)





119882120575 (120574119905) le esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)] 119875-as

(A3)

We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907

119905119905+120575[119882(119883

120574119905119906119907



119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575

119868120575 (120574119905 1199061 1205731 (1199061))

= sup119894ge1

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) 119875-as

(A4)


119894 1205731(119906

1

119894)) +


(⋃119894minus1


119906120576

1= sum

119894ge11Γ119894

1199061


of 1205731 we have 1205731(119906120576

1) = sum

119894ge11Γ119894

1205731(1199061



120576

1 1205731(119906

120576

1)) = sum

119894ge11Γ119894

119868120575(120574119905 1199061

119894 1205731(119906

1

119894)) 119875-as So

119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum

119894ge1

1Γ119894

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) + 120576

= 119868120575 (120574119905 119906120576

1 1205731 (119906

120576

1)) + 120576

= 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119882(119883

120574119905119906120576

11205731(119906120576

1)

119905+120575)] + 120576

(A5)


120576

1oplus


119882(120574119905) le esssup1199062isinU119905+120575119879

119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817 for any 120574119905 120574119905 isin Λ

(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))

1003816100381610038161003816

le 1198621003817100381710038171003817120574119905 minus 120574

119905

1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879

(A7)


11205731(119906120576

1)

119905+120575that

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062)) 119875-as

(A8)



119895 119895 ge 1 sub U119905+120575119879 such that

esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

= sup119895ge1

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A9)


119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732(1199062)) le

119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732(119906

2


Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1


(ΩF119905+120575)-partition and 119906120576

2= sum

119895ge11Δ119895

1199062

119895isin U119905+120575119879 Due to


2) = sum

119895ge11Δ119895

1205732(1199062

119895)


1oplus 119906

120576

2) =

1205731(119906120576

1) oplus 1205732(119906

120576



119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

2 1205732 (119906

120576

2))

= 119884119883120574119905119906120576

11205731(119906120576

1)

119905+120575119906120576

21205732(119906120576

2)(119905 + 120575)

= sum

119895ge1

1Δ119895

119884119883120574119905119906120576

11205731(119906120576

1)

119905+1205751199062

1198951205732(1199062

119895)(119905 + 120575)

= sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A10)

So

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

le sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

2

119895 120573 (119906

120576

1oplus 119906

2

119895)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

120576

2 120573 (119906

120576

1oplus 119906

120576

2)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576 119875-as

(A11)

where 119906120576

= 119906120576

1oplus 119906

120576



119882120575 (120574119905)

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576] + 120576

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119866120574119905119906120576120573(119906120576)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119869 (120574119905 119906120576 120573 (119906

120576)) + (119862 + 1) 120576

= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576


119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576 119875-as

(A12)


119882120575 (120574119905) le essinf120573isinB119905119879

esssup119906isinU119905119879

119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576

= 119882(120574119905) + (119862 + 1) 120576

(A13)


Lemma A3 119882(120574119905) le 119882120575(120574119905)


119882120575 (120574119905) = essinf1205731isinB119905119905+120575

esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)]

= essinf1205731isinB119905119905+120575

119868120575 (120574119905 1205731)

(A14)


119894 119894 ge 1 sub B119905119905+120575 such

that

119882120575 (120574119905) = inf119894ge1

119868120575 (120574119905 1205731

119894) 119875-as (A15)

For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731

119894) minus 120576 le 119882120575(120574119905) isin


119897=1Π119897) isin F119905 119894 ge


1=

sum119894ge1

1Λ119894

1205731



1(1199061)) =

sum119894ge1

1Λ119894

119868120575(120574119905 1199061 1205731


119882120575 (120574119905) ge sum

119894ge1

1Π119894

119868120575 (120574119905 1205731

119894) minus 120576

ge sum

119894ge1

1Π119894

119868120575 (120574119905 1199061 1205731

119894(1199061)) minus 120576

= 119868120575 (120574119905 1199061 120573120576

1(1199061)) minus 120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

119875-as forall1199061 isin U119905119905+120575

(A16)




B119905+120575119879 such that

119882(119909119905+120575) ge esssup1199062isinU119905+120575119879

119869 (119909119905+120575 1199062 120573119909120576

(1199062)) minus 120576 119875-as

(A17)



1205741199051199061120573120576

1(1199061)

119905+120575] = sum

119894ge1120574119894

119905+12057511198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894

we have

100381610038161003816100381610038161003816119883

1205741199051199061120573120576

1(1199061)

119905+120575minus [119883

1205741199051199061120573120576

1(1199061)

119905+120575]100381610038161003816100381610038161003816le 120576


(A18)



(A17) and 120573120576

1199061

= sum119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894120573120574119894120576isin B119905+120575119879


120576

1(1199061) oplus 120573

120576

1199061

(1199062) 119906 isin





1015840 into 1199061 1199061015840

1isin

U119905119905+120575 1199062 1199061015840


1015840

1and 119906 = 1199062 oplus 119906

1015840

2

From 1199061 equiv 1199061015840

1on [[119905 119878 and (119905 + 120575)]] we get 120573120576

1(1199061) = 120573

120576

1(119906

1015840

1) on

[[119905 119878and(119905+120575)]] (note that 120573120576


side from 1199062 equiv 1199061015840

2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt

119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576

1(1199061)

119905+120575= 119883

1205741199051199061015840

1120573120576

1(1199061015840

1)

119905+120575


1199061

= 120573120576

11990610158401

on 119878 gt 119905 + 120575 and120573120576

1199061

(1199062) = 120573120576

11990610158401

(1199061015840


120573120576(119906) = 120573

120576

1(1199061) oplus 120573

120576

1199061

(1199062) = 120573120576

1(119906

1015840

1) oplus 120573

120576

11990610158401

(1199061015840

2) on [[119905 119878]] from




119882120575 (120574119905) ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575]) minus 119862120576] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575])] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119882(120574

119894

119905+120575)] minus 119862120576

119875-as

(A19)


119882120575 (120574119905)

ge1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))minus 120576]minus119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 ([119883

1205741199051199061120573120576

1(1199061)

119905+120575] 1199062 120573

120576

1199061

(1199062))] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062)) minus 119862120576] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062))] minus 119862120576

= 119866120574119905119906120573120576(119906)

119905119905+120575[119884

120574119905119906120573120576(119906)

(119905 + 120575)] minus 119862120576

= 119884120574119905119906120573120576(119906)


(A20)

Therefore

119882120575 (120574119905) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573120576(119906)) minus 119862120576


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) minus 119862120576

= 119882(120574119905) minus 119862120576 119875-as

(A21)



119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)

119905119905+120575[119882(119883

120574119905119906120576120573(119906120576)

119905+120575)] + 120576

119875-as(A22)


119880119905119905+120575

119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)

119905119905+120575[119882(119883

120574119905119906120573120576(119906)

119905+120575)] minus 120576

119875-as(A23)


inf120573isinB119905119879

sup119906isinU119905119879

119864[119869(120574119905 119906 120573(119906))]

Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










So100381610038161003816100381610038161198841119906119907

(119905) minus 1198842119906119907

(119905)10038161003816100381610038161003816

=10038161003816100381610038161003816119864 [(119884

1119906119907(119904) minus 119884

2119906119907(119904)) | F119905]

10038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

119864 [int

119905+120575

119905

(119865 (119883120574119905119906119907

119904 119884

1119906119907(119904) 119885

1119906119907(119904) 119906 (119904) 119907 (119904))

minus119865 (120574119905 1198842119906119907

(119904) 1198852119906119907

(119904) 119906 (119904) 119907 (119904))) 119889119904 | F119905]

100381610038161003816100381610038161003816100381610038161003816

le 119862119864[int

119905+120575

119905

[1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) +100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

+100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816] 119889119904 | F119905]

le 119862119864[int

119905+120575

119905

1205880 (1003816100381610038161003816119883

120574119905119906119907

119904minus 120574119905

1003816100381610038161003816) 119889119904 | F119905]

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198841119906119907

(119904) minus 1198842119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

+ 11986212057512

119864[int

119905+120575

119905

100381610038161003816100381610038161198851119906119907

(119904) minus 1198852119906119907

(119904)10038161003816100381610038161003816

2

119889119904 | F119905]

12

le 11986212057532

(64)


minus1198891198840 (119904) = 1198650 (120574119905 1198840 (119904) 0) 119889119904 119904 isin [119905 119905 + 120575]

1198840 (119905 + 120575) = 0

(65)

where 1198650(120574119905 119910 119911) = sup119906isin119880

inf119907isin119881 119865(120574119905 119910 119911 119906 119907) Then119875-as

1198840 (119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) (66)


1198651 (120574119905 119910 119911 119906) = inf119907isin119881

119865 (120574119905 119910 119911 119906 119907)


(67)


minus1198891198843119906

(119904) = 1198651 (120574119905 1198843119906

(119904) 1198853119906

(119904) 119906 (119904)) 119889119904

minus1198853119906

(119904) 119889119861 (119904) 119904 isin [119905 119905 + 120575]

1198843119906

(119905 + 120575) = 0

(68)


1198853119906

) solving (68) Also

1198843119906

(119905) = essinf119907isinV119905119905+120575

1198842119906119907


(69)


1198843119906

(119905) le 1198842119906119907

(119905) 119875-as for any 119907 isin V119905119905+120575

for every 119906 isin U119905119905+120575

(70)


4 R timesR119889


1198651 (120574119905 119910 119911 119906)

= 119865 (120574119905 119910 119911 119906 1199074(119910 119911 119906)) for any 119910 119911 119906

(71)

Set 1199074(119904) = 1199074(119884

3119906(119904) 119885

3119906(119904) 119906(119904)) 119904 isin [119905 119905 + 120575] we know

1199074isin V119905119905+120575 Therefore (1198843119906

1198853119906

) = (11988421199061199074

11988521199061199074


(119905) = 11988421199061199074


(119905) = essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-119886119904 for every 119906 isin U119905119905+120575


1198651(120574119905 119910 119911 119906) wealso derive

1198840 (119905) = esssup119906isinU119905119905+120575

1198843119906

(119905) = esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905) 119875-as

(72)


119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905] le 119862120575

32 119875-as

(73)



100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816

2

le 119862120575

119864 [int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904] le 119862120575

(74)



(119904)10038161003816100381610038161003816

le 119864 [int

119905+120575

119904

119865 (120574119905 1198842119906119907

(119903) 1198852119906119907

(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]

le 119862119864[int

119905+120575

119904

(1 +1003817100381710038171003817120574119905

10038171003817100381710038172+100381610038161003816100381610038161198842119906119907

(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816) 119889119903 | F119904]

le 119862120575 + 11986212057512

119864[int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904]

le 119862120575 119904 isin [119905 119905 + 120575]

(75)


119905|119885

2119906119907(119904)|

2

119889119904 |

F119905] le 1198621205752 119875-as Therefore

119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905]

le 1198621205752+ 119862120575

12(119864[int

119905+120575

119905

100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816

2

119889119903 | F119905])

12

le 11986212057532

119875-as

(76)



0le119904le120575Λ 119905+119904


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(77)

From 119882 ge Γ on ⋃0le119904le120575


119905119906120573(119906)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

120574119905119906120573(119906)

119905+120575)] minus Γ (120574119905) le 0 (78)

By Lemma 19 we have

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) le 0 119875-as (79)

Thus from Lemma 20

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as (80)


1198842119906119907

(119905) le 1198842119906120573(119906)

(119905) 120573 isin

B119905119905+120575 we have

esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905)

le essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as(81)

Thus by Lemma 21 1198840(119905) le 11986212057532


11986212057512

ge1

1205751198840 (119905) =

1

120575int

119905+120575

119905

1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)

sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)



Γ ge 119882 on⋃0le119904le120575


sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)


1198650 (120574119905 0 0) = sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)


119865 (120574119905 0 0 119906 120595 (119906)) le minus3

4120579 forall119906 isin 119880 (86)


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(87)



119905119906120573(119906)

119905119905+120575[sdot] (Lemma 7) we get

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

119906120573(119906)

119905+120575)] minus Γ (120574119905) ge 0 119875-as

(88)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) ge 0 119875-as (89)

in particular

esssup119906isinU119905119905+120575

1198841119906120595(119906)

(119905) ge 0 119875-as (90)


1119906120576120595(119906120576)(119905) ge minus120576120575


1198842119906120576120595(119906120576)(119905) ge minus119862120575

32minus 120576120575 119875-as (91)



1198842119906120576120595(119906120576)(119905)

= 119864 [int

119905+120575

119905

119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885

2119906120576120595(119906120576)(119904)

119906120576(119904) 120595 (119906

120576(119904))) 119889119904 | F119905]

le 119864[int

119905+120575

119905

(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816+ 119862

1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816

+119865 (120574119905 0 0 119906120576(119904) 120595 (119906

120576(119904)))) 119889119904 | F119905]

le 11986212057532

minus3

4120579120575 119875-as

(92)


minus 120576 le 11986212057512

minus


sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)



Appendix



119882120575 (120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (A1)




Lemma A2 119882120575(120574119905) le 119882(120574119905)





119882120575 (120574119905) le esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)] 119875-as

(A3)

We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907

119905119905+120575[119882(119883

120574119905119906119907



119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575

119868120575 (120574119905 1199061 1205731 (1199061))

= sup119894ge1

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) 119875-as

(A4)


119894 1205731(119906

1

119894)) +


(⋃119894minus1


119906120576

1= sum

119894ge11Γ119894

1199061


of 1205731 we have 1205731(119906120576

1) = sum

119894ge11Γ119894

1205731(1199061



120576

1 1205731(119906

120576

1)) = sum

119894ge11Γ119894

119868120575(120574119905 1199061

119894 1205731(119906

1

119894)) 119875-as So

119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum

119894ge1

1Γ119894

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) + 120576

= 119868120575 (120574119905 119906120576

1 1205731 (119906

120576

1)) + 120576

= 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119882(119883

120574119905119906120576

11205731(119906120576

1)

119905+120575)] + 120576

(A5)


120576

1oplus


119882(120574119905) le esssup1199062isinU119905+120575119879

119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817 for any 120574119905 120574119905 isin Λ

(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))

1003816100381610038161003816

le 1198621003817100381710038171003817120574119905 minus 120574

119905

1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879

(A7)


11205731(119906120576

1)

119905+120575that

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062)) 119875-as

(A8)



119895 119895 ge 1 sub U119905+120575119879 such that

esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

= sup119895ge1

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A9)


119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732(1199062)) le

119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732(119906

2


Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1


(ΩF119905+120575)-partition and 119906120576

2= sum

119895ge11Δ119895

1199062

119895isin U119905+120575119879 Due to


2) = sum

119895ge11Δ119895

1205732(1199062

119895)


1oplus 119906

120576

2) =

1205731(119906120576

1) oplus 1205732(119906

120576



119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

2 1205732 (119906

120576

2))

= 119884119883120574119905119906120576

11205731(119906120576

1)

119905+120575119906120576

21205732(119906120576

2)(119905 + 120575)

= sum

119895ge1

1Δ119895

119884119883120574119905119906120576

11205731(119906120576

1)

119905+1205751199062

1198951205732(1199062

119895)(119905 + 120575)

= sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A10)

So

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

le sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

2

119895 120573 (119906

120576

1oplus 119906

2

119895)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

120576

2 120573 (119906

120576

1oplus 119906

120576

2)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576 119875-as

(A11)

where 119906120576

= 119906120576

1oplus 119906

120576



119882120575 (120574119905)

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576] + 120576

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119866120574119905119906120576120573(119906120576)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119869 (120574119905 119906120576 120573 (119906

120576)) + (119862 + 1) 120576

= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576


119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576 119875-as

(A12)


119882120575 (120574119905) le essinf120573isinB119905119879

esssup119906isinU119905119879

119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576

= 119882(120574119905) + (119862 + 1) 120576

(A13)


Lemma A3 119882(120574119905) le 119882120575(120574119905)


119882120575 (120574119905) = essinf1205731isinB119905119905+120575

esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)]

= essinf1205731isinB119905119905+120575

119868120575 (120574119905 1205731)

(A14)


119894 119894 ge 1 sub B119905119905+120575 such

that

119882120575 (120574119905) = inf119894ge1

119868120575 (120574119905 1205731

119894) 119875-as (A15)

For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731

119894) minus 120576 le 119882120575(120574119905) isin


119897=1Π119897) isin F119905 119894 ge


1=

sum119894ge1

1Λ119894

1205731



1(1199061)) =

sum119894ge1

1Λ119894

119868120575(120574119905 1199061 1205731


119882120575 (120574119905) ge sum

119894ge1

1Π119894

119868120575 (120574119905 1205731

119894) minus 120576

ge sum

119894ge1

1Π119894

119868120575 (120574119905 1199061 1205731

119894(1199061)) minus 120576

= 119868120575 (120574119905 1199061 120573120576

1(1199061)) minus 120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

119875-as forall1199061 isin U119905119905+120575

(A16)




B119905+120575119879 such that

119882(119909119905+120575) ge esssup1199062isinU119905+120575119879

119869 (119909119905+120575 1199062 120573119909120576

(1199062)) minus 120576 119875-as

(A17)



1205741199051199061120573120576

1(1199061)

119905+120575] = sum

119894ge1120574119894

119905+12057511198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894

we have

100381610038161003816100381610038161003816119883

1205741199051199061120573120576

1(1199061)

119905+120575minus [119883

1205741199051199061120573120576

1(1199061)

119905+120575]100381610038161003816100381610038161003816le 120576


(A18)



(A17) and 120573120576

1199061

= sum119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894120573120574119894120576isin B119905+120575119879


120576

1(1199061) oplus 120573

120576

1199061

(1199062) 119906 isin





1015840 into 1199061 1199061015840

1isin

U119905119905+120575 1199062 1199061015840


1015840

1and 119906 = 1199062 oplus 119906

1015840

2

From 1199061 equiv 1199061015840

1on [[119905 119878 and (119905 + 120575)]] we get 120573120576

1(1199061) = 120573

120576

1(119906

1015840

1) on

[[119905 119878and(119905+120575)]] (note that 120573120576


side from 1199062 equiv 1199061015840

2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt

119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576

1(1199061)

119905+120575= 119883

1205741199051199061015840

1120573120576

1(1199061015840

1)

119905+120575


1199061

= 120573120576

11990610158401

on 119878 gt 119905 + 120575 and120573120576

1199061

(1199062) = 120573120576

11990610158401

(1199061015840


120573120576(119906) = 120573

120576

1(1199061) oplus 120573

120576

1199061

(1199062) = 120573120576

1(119906

1015840

1) oplus 120573

120576

11990610158401

(1199061015840

2) on [[119905 119878]] from




119882120575 (120574119905) ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575]) minus 119862120576] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575])] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119882(120574

119894

119905+120575)] minus 119862120576

119875-as

(A19)


119882120575 (120574119905)

ge1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))minus 120576]minus119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 ([119883

1205741199051199061120573120576

1(1199061)

119905+120575] 1199062 120573

120576

1199061

(1199062))] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062)) minus 119862120576] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062))] minus 119862120576

= 119866120574119905119906120573120576(119906)

119905119905+120575[119884

120574119905119906120573120576(119906)

(119905 + 120575)] minus 119862120576

= 119884120574119905119906120573120576(119906)


(A20)

Therefore

119882120575 (120574119905) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573120576(119906)) minus 119862120576


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) minus 119862120576

= 119882(120574119905) minus 119862120576 119875-as

(A21)



119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)

119905119905+120575[119882(119883

120574119905119906120576120573(119906120576)

119905+120575)] + 120576

119875-as(A22)


119880119905119905+120575

119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)

119905119905+120575[119882(119883

120574119905119906120573120576(119906)

119905+120575)] minus 120576

119875-as(A23)


inf120573isinB119905119879

sup119906isinU119905119879

119864[119869(120574119905 119906 120573(119906))]

Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119904)10038161003816100381610038161003816

le 119864 [int

119905+120575

119904

119865 (120574119905 1198842119906119907

(119903) 1198852119906119907

(119903) 119906 (119903) 119907 (119903)) 119889119903 | F119904]

le 119862119864[int

119905+120575

119904

(1 +1003817100381710038171003817120574119905

10038171003817100381710038172+100381610038161003816100381610038161198842119906119907

(119903)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816) 119889119903 | F119904]

le 119862120575 + 11986212057512

119864[int

119905+120575

119904

100381610038161003816100381610038161198852119906119907

(119903)10038161003816100381610038161003816

2

119889119903 | F119904]

le 119862120575 119904 isin [119905 119905 + 120575]

(75)


119905|119885

2119906119907(119904)|

2

119889119904 |

F119905] le 1198621205752 119875-as Therefore

119864[int

119905+120575

119905

(100381610038161003816100381610038161198842119906119907

(119904)10038161003816100381610038161003816+100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816) 119889119904 | F119905]

le 1198621205752+ 119862120575

12(119864[int

119905+120575

119905

100381610038161003816100381610038161198852119906119907

(119904)10038161003816100381610038161003816

2

119889119903 | F119905])

12

le 11986212057532

119875-as

(76)



0le119904le120575Λ 119905+119904


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(77)

From 119882 ge Γ on ⋃0le119904le120575


119905119906120573(119906)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

120574119905119906120573(119906)

119905+120575)] minus Γ (120574119905) le 0 (78)

By Lemma 19 we have

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) le 0 119875-as (79)

Thus from Lemma 20

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as (80)


1198842119906119907

(119905) le 1198842119906120573(119906)

(119905) 120573 isin

B119905119905+120575 we have

esssup119906isinU119905119905+120575

essinf119907isinV119905119905+120575

1198842119906119907

(119905)

le essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198842119906120573(119906)

(119905) le 11986212057532

119875-as(81)

Thus by Lemma 21 1198840(119905) le 11986212057532


11986212057512

ge1

1205751198840 (119905) =

1

120575int

119905+120575

119905

1198650 (120574119905 1198840 (119904) 0) 119889119904 120575 gt 0 (82)

sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (83)



Γ ge 119882 on⋃0le119904le120575


sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) ge 0 (84)


1198650 (120574119905 0 0) = sup119906isinU

inf119907isin119881

119865 (120574119905 0 0 119906 119907) le minus120579 lt 0 (85)


119865 (120574119905 0 0 119906 120595 (119906)) le minus3

4120579 forall119906 isin 119880 (86)


Γ (120574119905) = 119882(120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)]

(87)



119905119906120573(119906)

119905119905+120575[sdot] (Lemma 7) we get

essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[Γ (119883

119906120573(119906)

119905+120575)] minus Γ (120574119905) ge 0 119875-as

(88)


essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

1198841119906120573(119906)

(119905) ge 0 119875-as (89)

in particular

esssup119906isinU119905119905+120575

1198841119906120595(119906)

(119905) ge 0 119875-as (90)


1119906120576120595(119906120576)(119905) ge minus120576120575


1198842119906120576120595(119906120576)(119905) ge minus119862120575

32minus 120576120575 119875-as (91)



1198842119906120576120595(119906120576)(119905)

= 119864 [int

119905+120575

119905

119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885

2119906120576120595(119906120576)(119904)

119906120576(119904) 120595 (119906

120576(119904))) 119889119904 | F119905]

le 119864[int

119905+120575

119905

(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816+ 119862

1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816

+119865 (120574119905 0 0 119906120576(119904) 120595 (119906

120576(119904)))) 119889119904 | F119905]

le 11986212057532

minus3

4120579120575 119875-as

(92)


minus 120576 le 11986212057512

minus


sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)



Appendix



119882120575 (120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (A1)




Lemma A2 119882120575(120574119905) le 119882(120574119905)





119882120575 (120574119905) le esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)] 119875-as

(A3)

We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907

119905119905+120575[119882(119883

120574119905119906119907



119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575

119868120575 (120574119905 1199061 1205731 (1199061))

= sup119894ge1

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) 119875-as

(A4)


119894 1205731(119906

1

119894)) +


(⋃119894minus1


119906120576

1= sum

119894ge11Γ119894

1199061


of 1205731 we have 1205731(119906120576

1) = sum

119894ge11Γ119894

1205731(1199061



120576

1 1205731(119906

120576

1)) = sum

119894ge11Γ119894

119868120575(120574119905 1199061

119894 1205731(119906

1

119894)) 119875-as So

119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum

119894ge1

1Γ119894

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) + 120576

= 119868120575 (120574119905 119906120576

1 1205731 (119906

120576

1)) + 120576

= 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119882(119883

120574119905119906120576

11205731(119906120576

1)

119905+120575)] + 120576

(A5)


120576

1oplus


119882(120574119905) le esssup1199062isinU119905+120575119879

119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817 for any 120574119905 120574119905 isin Λ

(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))

1003816100381610038161003816

le 1198621003817100381710038171003817120574119905 minus 120574

119905

1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879

(A7)


11205731(119906120576

1)

119905+120575that

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062)) 119875-as

(A8)



119895 119895 ge 1 sub U119905+120575119879 such that

esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

= sup119895ge1

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A9)


119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732(1199062)) le

119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732(119906

2


Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1


(ΩF119905+120575)-partition and 119906120576

2= sum

119895ge11Δ119895

1199062

119895isin U119905+120575119879 Due to


2) = sum

119895ge11Δ119895

1205732(1199062

119895)


1oplus 119906

120576

2) =

1205731(119906120576

1) oplus 1205732(119906

120576



119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

2 1205732 (119906

120576

2))

= 119884119883120574119905119906120576

11205731(119906120576

1)

119905+120575119906120576

21205732(119906120576

2)(119905 + 120575)

= sum

119895ge1

1Δ119895

119884119883120574119905119906120576

11205731(119906120576

1)

119905+1205751199062

1198951205732(1199062

119895)(119905 + 120575)

= sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A10)

So

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

le sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

2

119895 120573 (119906

120576

1oplus 119906

2

119895)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

120576

2 120573 (119906

120576

1oplus 119906

120576

2)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576 119875-as

(A11)

where 119906120576

= 119906120576

1oplus 119906

120576



119882120575 (120574119905)

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576] + 120576

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119866120574119905119906120576120573(119906120576)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119869 (120574119905 119906120576 120573 (119906

120576)) + (119862 + 1) 120576

= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576


119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576 119875-as

(A12)


119882120575 (120574119905) le essinf120573isinB119905119879

esssup119906isinU119905119879

119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576

= 119882(120574119905) + (119862 + 1) 120576

(A13)


Lemma A3 119882(120574119905) le 119882120575(120574119905)


119882120575 (120574119905) = essinf1205731isinB119905119905+120575

esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)]

= essinf1205731isinB119905119905+120575

119868120575 (120574119905 1205731)

(A14)


119894 119894 ge 1 sub B119905119905+120575 such

that

119882120575 (120574119905) = inf119894ge1

119868120575 (120574119905 1205731

119894) 119875-as (A15)

For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731

119894) minus 120576 le 119882120575(120574119905) isin


119897=1Π119897) isin F119905 119894 ge


1=

sum119894ge1

1Λ119894

1205731



1(1199061)) =

sum119894ge1

1Λ119894

119868120575(120574119905 1199061 1205731


119882120575 (120574119905) ge sum

119894ge1

1Π119894

119868120575 (120574119905 1205731

119894) minus 120576

ge sum

119894ge1

1Π119894

119868120575 (120574119905 1199061 1205731

119894(1199061)) minus 120576

= 119868120575 (120574119905 1199061 120573120576

1(1199061)) minus 120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

119875-as forall1199061 isin U119905119905+120575

(A16)




B119905+120575119879 such that

119882(119909119905+120575) ge esssup1199062isinU119905+120575119879

119869 (119909119905+120575 1199062 120573119909120576

(1199062)) minus 120576 119875-as

(A17)



1205741199051199061120573120576

1(1199061)

119905+120575] = sum

119894ge1120574119894

119905+12057511198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894

we have

100381610038161003816100381610038161003816119883

1205741199051199061120573120576

1(1199061)

119905+120575minus [119883

1205741199051199061120573120576

1(1199061)

119905+120575]100381610038161003816100381610038161003816le 120576


(A18)



(A17) and 120573120576

1199061

= sum119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894120573120574119894120576isin B119905+120575119879


120576

1(1199061) oplus 120573

120576

1199061

(1199062) 119906 isin





1015840 into 1199061 1199061015840

1isin

U119905119905+120575 1199062 1199061015840


1015840

1and 119906 = 1199062 oplus 119906

1015840

2

From 1199061 equiv 1199061015840

1on [[119905 119878 and (119905 + 120575)]] we get 120573120576

1(1199061) = 120573

120576

1(119906

1015840

1) on

[[119905 119878and(119905+120575)]] (note that 120573120576


side from 1199062 equiv 1199061015840

2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt

119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576

1(1199061)

119905+120575= 119883

1205741199051199061015840

1120573120576

1(1199061015840

1)

119905+120575


1199061

= 120573120576

11990610158401

on 119878 gt 119905 + 120575 and120573120576

1199061

(1199062) = 120573120576

11990610158401

(1199061015840


120573120576(119906) = 120573

120576

1(1199061) oplus 120573

120576

1199061

(1199062) = 120573120576

1(119906

1015840

1) oplus 120573

120576

11990610158401

(1199061015840

2) on [[119905 119878]] from




119882120575 (120574119905) ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575]) minus 119862120576] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575])] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119882(120574

119894

119905+120575)] minus 119862120576

119875-as

(A19)


119882120575 (120574119905)

ge1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))minus 120576]minus119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 ([119883

1205741199051199061120573120576

1(1199061)

119905+120575] 1199062 120573

120576

1199061

(1199062))] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062)) minus 119862120576] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062))] minus 119862120576

= 119866120574119905119906120573120576(119906)

119905119905+120575[119884

120574119905119906120573120576(119906)

(119905 + 120575)] minus 119862120576

= 119884120574119905119906120573120576(119906)


(A20)

Therefore

119882120575 (120574119905) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573120576(119906)) minus 119862120576


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) minus 119862120576

= 119882(120574119905) minus 119862120576 119875-as

(A21)



119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)

119905119905+120575[119882(119883

120574119905119906120576120573(119906120576)

119905+120575)] + 120576

119875-as(A22)


119880119905119905+120575

119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)

119905119905+120575[119882(119883

120574119905119906120573120576(119906)

119905+120575)] minus 120576

119875-as(A23)


inf120573isinB119905119879

sup119906isinU119905119879

119864[119869(120574119905 119906 120573(119906))]

Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198842119906120576120595(119906120576)(119905)

= 119864 [int

119905+120575

119905

119865 (120574119905 1198842119906120576120595(119906120576)(119904) 119885

2119906120576120595(119906120576)(119904)

119906120576(119904) 120595 (119906

120576(119904))) 119889119904 | F119905]

le 119864[int

119905+120575

119905

(1198621003816100381610038161003816100381610038161198842119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816+ 119862

1003816100381610038161003816100381610038161198852119906120576120595(119906120576)(119904)

100381610038161003816100381610038161003816

+119865 (120574119905 0 0 119906120576(119904) 120595 (119906

120576(119904)))) 119889119904 | F119905]

le 11986212057532

minus3

4120579120575 119875-as

(92)


minus 120576 le 11986212057512

minus


sup119906isin119880

inf119907isin119881

119865 (120574119905 0 0 119906 119907) = 1198650 (120574119905 0 0) le 0 (93)



Appendix



119882120575 (120574119905) = essinf120573isinB119905119905+120575

esssup119906isinU119905119905+120575

119866120574119905119906120573(119906)

119905119905+120575[119882(119883

120574119905119906120573(119906)

119905+120575)] (A1)




Lemma A2 119882120575(120574119905) le 119882(120574119905)





119882120575 (120574119905) le esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)] 119875-as

(A3)

We put 119868120575(120574119905 119906 119907) = 119866120574119905119906119907

119905119905+120575[119882(119883

120574119905119906119907



119868120575 (120574119905 1205731) = esssup1199061isinU119905119905+120575

119868120575 (120574119905 1199061 1205731 (1199061))

= sup119894ge1

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) 119875-as

(A4)


119894 1205731(119906

1

119894)) +


(⋃119894minus1


119906120576

1= sum

119894ge11Γ119894

1199061


of 1205731 we have 1205731(119906120576

1) = sum

119894ge11Γ119894

1205731(1199061



120576

1 1205731(119906

120576

1)) = sum

119894ge11Γ119894

119868120575(120574119905 1199061

119894 1205731(119906

1

119894)) 119875-as So

119882120575 (120574119905) le 119868120575 (120574119905 1205731) le sum

119894ge1

1Γ119894

119868120575 (120574119905 1199061

119894 1205731 (119906

1

119894)) + 120576

= 119868120575 (120574119905 119906120576

1 1205731 (119906

120576

1)) + 120576

= 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119882(119883

120574119905119906120576

11205731(119906120576

1)

119905+120575)] + 120576

(A5)


120576

1oplus


119882(120574119905) le esssup1199062isinU119905+120575119879

119869 (120574119905 1199062 1205732 (1199062)) 119875-as (A6)


(i) 1003816100381610038161003816119882 (120574119905) minus 119882(120574119905)1003816100381610038161003816 le 119862

1003817100381710038171003817120574119905 minus 120574119905

1003817100381710038171003817 for any 120574119905 120574119905 isin Λ

(ii) 1003816100381610038161003816119869 (120574119905 1199062 1205732 (1199062)) minus 119869 (120574119905 1199062 1205732 (1199062))

1003816100381610038161003816

le 1198621003817100381710038171003817120574119905 minus 120574

119905

1003817100381710038171003817 119875-as for any 1199062 isin U119905+120575119879

(A7)


11205731(119906120576

1)

119905+120575that

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062)) 119875-as

(A8)



119895 119895 ge 1 sub U119905+120575119879 such that

esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

= sup119895ge1

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A9)


119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732(1199062)) le

119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732(119906

2


Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1


(ΩF119905+120575)-partition and 119906120576

2= sum

119895ge11Δ119895

1199062

119895isin U119905+120575119879 Due to


2) = sum

119895ge11Δ119895

1205732(1199062

119895)


1oplus 119906

120576

2) =

1205731(119906120576

1) oplus 1205732(119906

120576



119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

2 1205732 (119906

120576

2))

= 119884119883120574119905119906120576

11205731(119906120576

1)

119905+120575119906120576

21205732(119906120576

2)(119905 + 120575)

= sum

119895ge1

1Δ119895

119884119883120574119905119906120576

11205731(119906120576

1)

119905+1205751199062

1198951205732(1199062

119895)(119905 + 120575)

= sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A10)

So

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

le sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

2

119895 120573 (119906

120576

1oplus 119906

2

119895)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

120576

2 120573 (119906

120576

1oplus 119906

120576

2)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576 119875-as

(A11)

where 119906120576

= 119906120576

1oplus 119906

120576



119882120575 (120574119905)

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576] + 120576

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119866120574119905119906120576120573(119906120576)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119869 (120574119905 119906120576 120573 (119906

120576)) + (119862 + 1) 120576

= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576


119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576 119875-as

(A12)


119882120575 (120574119905) le essinf120573isinB119905119879

esssup119906isinU119905119879

119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576

= 119882(120574119905) + (119862 + 1) 120576

(A13)


Lemma A3 119882(120574119905) le 119882120575(120574119905)


119882120575 (120574119905) = essinf1205731isinB119905119905+120575

esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)]

= essinf1205731isinB119905119905+120575

119868120575 (120574119905 1205731)

(A14)


119894 119894 ge 1 sub B119905119905+120575 such

that

119882120575 (120574119905) = inf119894ge1

119868120575 (120574119905 1205731

119894) 119875-as (A15)

For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731

119894) minus 120576 le 119882120575(120574119905) isin


119897=1Π119897) isin F119905 119894 ge


1=

sum119894ge1

1Λ119894

1205731



1(1199061)) =

sum119894ge1

1Λ119894

119868120575(120574119905 1199061 1205731


119882120575 (120574119905) ge sum

119894ge1

1Π119894

119868120575 (120574119905 1205731

119894) minus 120576

ge sum

119894ge1

1Π119894

119868120575 (120574119905 1199061 1205731

119894(1199061)) minus 120576

= 119868120575 (120574119905 1199061 120573120576

1(1199061)) minus 120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

119875-as forall1199061 isin U119905119905+120575

(A16)




B119905+120575119879 such that

119882(119909119905+120575) ge esssup1199062isinU119905+120575119879

119869 (119909119905+120575 1199062 120573119909120576

(1199062)) minus 120576 119875-as

(A17)



1205741199051199061120573120576

1(1199061)

119905+120575] = sum

119894ge1120574119894

119905+12057511198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894

we have

100381610038161003816100381610038161003816119883

1205741199051199061120573120576

1(1199061)

119905+120575minus [119883

1205741199051199061120573120576

1(1199061)

119905+120575]100381610038161003816100381610038161003816le 120576


(A18)



(A17) and 120573120576

1199061

= sum119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894120573120574119894120576isin B119905+120575119879


120576

1(1199061) oplus 120573

120576

1199061

(1199062) 119906 isin





1015840 into 1199061 1199061015840

1isin

U119905119905+120575 1199062 1199061015840


1015840

1and 119906 = 1199062 oplus 119906

1015840

2

From 1199061 equiv 1199061015840

1on [[119905 119878 and (119905 + 120575)]] we get 120573120576

1(1199061) = 120573

120576

1(119906

1015840

1) on

[[119905 119878and(119905+120575)]] (note that 120573120576


side from 1199062 equiv 1199061015840

2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt

119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576

1(1199061)

119905+120575= 119883

1205741199051199061015840

1120573120576

1(1199061015840

1)

119905+120575


1199061

= 120573120576

11990610158401

on 119878 gt 119905 + 120575 and120573120576

1199061

(1199062) = 120573120576

11990610158401

(1199061015840


120573120576(119906) = 120573

120576

1(1199061) oplus 120573

120576

1199061

(1199062) = 120573120576

1(119906

1015840

1) oplus 120573

120576

11990610158401

(1199061015840

2) on [[119905 119878]] from




119882120575 (120574119905) ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575]) minus 119862120576] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575])] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119882(120574

119894

119905+120575)] minus 119862120576

119875-as

(A19)


119882120575 (120574119905)

ge1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))minus 120576]minus119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 ([119883

1205741199051199061120573120576

1(1199061)

119905+120575] 1199062 120573

120576

1199061

(1199062))] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062)) minus 119862120576] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062))] minus 119862120576

= 119866120574119905119906120573120576(119906)

119905119905+120575[119884

120574119905119906120573120576(119906)

(119905 + 120575)] minus 119862120576

= 119884120574119905119906120573120576(119906)


(A20)

Therefore

119882120575 (120574119905) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573120576(119906)) minus 119862120576


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) minus 119862120576

= 119882(120574119905) minus 119862120576 119875-as

(A21)



119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)

119905119905+120575[119882(119883

120574119905119906120576120573(119906120576)

119905+120575)] + 120576

119875-as(A22)


119880119905119905+120575

119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)

119905119905+120575[119882(119883

120574119905119906120573120576(119906)

119905+120575)] minus 120576

119875-as(A23)


inf120573isinB119905119879

sup119906isinU119905119879

119864[119869(120574119905 119906 120573(119906))]

Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119895 119895 ge 1 sub U119905+120575119879 such that

esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

= sup119895ge1

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A9)


119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732(1199062)) le

119869(119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732(119906

2


Δ 1 = Δ 1 Δ 119895 = Δ 119895 (⋃119895minus1


(ΩF119905+120575)-partition and 119906120576

2= sum

119895ge11Δ119895

1199062

119895isin U119905+120575119879 Due to


2) = sum

119895ge11Δ119895

1205732(1199062

119895)


1oplus 119906

120576

2) =

1205731(119906120576

1) oplus 1205732(119906

120576



119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

2 1205732 (119906

120576

2))

= 119884119883120574119905119906120576

11205731(119906120576

1)

119905+120575119906120576

21205732(119906120576

2)(119905 + 120575)

= sum

119895ge1

1Δ119895

119884119883120574119905119906120576

11205731(119906120576

1)

119905+1205751199062

1198951205732(1199062

119895)(119905 + 120575)

= sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

2

119895 1205732 (119906

2

119895)) 119875-as

(A10)

So

119882(119883120574119905119906120576

11205731(119906120576

1)

119905+120575)

le esssup1199062isinU119905+120575119879

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 1199062 1205732 (1199062))

le sum

119895ge1

1Δ119895

119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

2

119895 120573 (119906

120576

1oplus 119906

2

119895)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576

1oplus 119906

120576

2 120573 (119906

120576

1oplus 119906

120576

2)) + 120576

= 119869 (119883120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576 119875-as

(A11)

where 119906120576

= 119906120576

1oplus 119906

120576



119882120575 (120574119905)

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576)) + 120576] + 120576

le 119866120574119905119906120576

11205731(119906120576

1)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119866120574119905119906120576120573(119906120576)

119905119905+120575[119869 (119883

120574119905119906120576

11205731(119906120576

1)

119905+120575 119906

120576 120573 (119906

120576))] + (119862 + 1) 120576

= 119869 (120574119905 119906120576 120573 (119906

120576)) + (119862 + 1) 120576

= 119884120574119905119906120576120573(119906120576)(119905) + (119862 + 1) 120576


119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576 119875-as

(A12)


119882120575 (120574119905) le essinf120573isinB119905119879

esssup119906isinU119905119879

119884120574119905119906120573(119906)

(119905) + (119862 + 1) 120576

= 119882(120574119905) + (119862 + 1) 120576

(A13)


Lemma A3 119882(120574119905) le 119882120575(120574119905)


119882120575 (120574119905) = essinf1205731isinB119905119905+120575

esssup1199061isinU119905119905+120575

11986612057411990511990611205731(1199061)

119905119905+120575[119882(119883

12057411990511990611205731(1199061)

119905+120575)]

= essinf1205731isinB119905119905+120575

119868120575 (120574119905 1205731)

(A14)


119894 119894 ge 1 sub B119905119905+120575 such

that

119882120575 (120574119905) = inf119894ge1

119868120575 (120574119905 1205731

119894) 119875-as (A15)

For any 120576 gt 0 we set Π119894 = 119868120575(120574119905 1205731

119894) minus 120576 le 119882120575(120574119905) isin


119897=1Π119897) isin F119905 119894 ge


1=

sum119894ge1

1Λ119894

1205731



1(1199061)) =

sum119894ge1

1Λ119894

119868120575(120574119905 1199061 1205731


119882120575 (120574119905) ge sum

119894ge1

1Π119894

119868120575 (120574119905 1205731

119894) minus 120576

ge sum

119894ge1

1Π119894

119868120575 (120574119905 1199061 1205731

119894(1199061)) minus 120576

= 119868120575 (120574119905 1199061 120573120576

1(1199061)) minus 120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

119875-as forall1199061 isin U119905119905+120575

(A16)




B119905+120575119879 such that

119882(119909119905+120575) ge esssup1199062isinU119905+120575119879

119869 (119909119905+120575 1199062 120573119909120576

(1199062)) minus 120576 119875-as

(A17)



1205741199051199061120573120576

1(1199061)

119905+120575] = sum

119894ge1120574119894

119905+12057511198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894

we have

100381610038161003816100381610038161003816119883

1205741199051199061120573120576

1(1199061)

119905+120575minus [119883

1205741199051199061120573120576

1(1199061)

119905+120575]100381610038161003816100381610038161003816le 120576


(A18)



(A17) and 120573120576

1199061

= sum119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894120573120574119894120576isin B119905+120575119879


120576

1(1199061) oplus 120573

120576

1199061

(1199062) 119906 isin





1015840 into 1199061 1199061015840

1isin

U119905119905+120575 1199062 1199061015840


1015840

1and 119906 = 1199062 oplus 119906

1015840

2

From 1199061 equiv 1199061015840

1on [[119905 119878 and (119905 + 120575)]] we get 120573120576

1(1199061) = 120573

120576

1(119906

1015840

1) on

[[119905 119878and(119905+120575)]] (note that 120573120576


side from 1199062 equiv 1199061015840

2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt

119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576

1(1199061)

119905+120575= 119883

1205741199051199061015840

1120573120576

1(1199061015840

1)

119905+120575


1199061

= 120573120576

11990610158401

on 119878 gt 119905 + 120575 and120573120576

1199061

(1199062) = 120573120576

11990610158401

(1199061015840


120573120576(119906) = 120573

120576

1(1199061) oplus 120573

120576

1199061

(1199062) = 120573120576

1(119906

1015840

1) oplus 120573

120576

11990610158401

(1199061015840

2) on [[119905 119878]] from




119882120575 (120574119905) ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575]) minus 119862120576] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575])] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119882(120574

119894

119905+120575)] minus 119862120576

119875-as

(A19)


119882120575 (120574119905)

ge1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))minus 120576]minus119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 ([119883

1205741199051199061120573120576

1(1199061)

119905+120575] 1199062 120573

120576

1199061

(1199062))] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062)) minus 119862120576] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062))] minus 119862120576

= 119866120574119905119906120573120576(119906)

119905119905+120575[119884

120574119905119906120573120576(119906)

(119905 + 120575)] minus 119862120576

= 119884120574119905119906120573120576(119906)


(A20)

Therefore

119882120575 (120574119905) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573120576(119906)) minus 119862120576


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) minus 119862120576

= 119882(120574119905) minus 119862120576 119875-as

(A21)



119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)

119905119905+120575[119882(119883

120574119905119906120576120573(119906120576)

119905+120575)] + 120576

119875-as(A22)


119880119905119905+120575

119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)

119905119905+120575[119882(119883

120574119905119906120573120576(119906)

119905+120575)] minus 120576

119875-as(A23)


inf120573isinB119905119879

sup119906isinU119905119879

119864[119869(120574119905 119906 120573(119906))]

Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












B119905+120575119879 such that

119882(119909119905+120575) ge esssup1199062isinU119905+120575119879

119869 (119909119905+120575 1199062 120573119909120576

(1199062)) minus 120576 119875-as

(A17)



1205741199051199061120573120576

1(1199061)

119905+120575] = sum

119894ge1120574119894

119905+12057511198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894

we have

100381610038161003816100381610038161003816119883

1205741199051199061120573120576

1(1199061)

119905+120575minus [119883

1205741199051199061120573120576

1(1199061)

119905+120575]100381610038161003816100381610038161003816le 120576


(A18)



(A17) and 120573120576

1199061

= sum119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894120573120574119894120576isin B119905+120575119879


120576

1(1199061) oplus 120573

120576

1199061

(1199062) 119906 isin





1015840 into 1199061 1199061015840

1isin

U119905119905+120575 1199062 1199061015840


1015840

1and 119906 = 1199062 oplus 119906

1015840

2

From 1199061 equiv 1199061015840

1on [[119905 119878 and (119905 + 120575)]] we get 120573120576

1(1199061) = 120573

120576

1(119906

1015840

1) on

[[119905 119878and(119905+120575)]] (note that 120573120576


side from 1199062 equiv 1199061015840

2on [[119905 + 120575 119878 or (119905 + 120575)]](sub (119905 + 120575 119879] times 119878 gt

119905 + 120575) and on 119878 gt 119905 + 120575 we get 1198831205741199051199061120573120576

1(1199061)

119905+120575= 119883

1205741199051199061015840

1120573120576

1(1199061015840

1)

119905+120575


1199061

= 120573120576

11990610158401

on 119878 gt 119905 + 120575 and120573120576

1199061

(1199062) = 120573120576

11990610158401

(1199061015840


120573120576(119906) = 120573

120576

1(1199061) oplus 120573

120576

1199061

(1199062) = 120573120576

1(119906

1015840

1) oplus 120573

120576

11990610158401

(1199061015840

2) on [[119905 119878]] from




119882120575 (120574119905) ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882(119883

1205741199051199061120573120576

1(1199061)

119905+120575)] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575]) minus 119862120576] minus 120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119882([119883

1205741199051199061120573120576

1(1199061)

119905+120575])] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119882(120574

119894

119905+120575)] minus 119862120576

119875-as

(A19)


119882120575 (120574119905)

ge1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))minus 120576]minus119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[sum

119894ge1

11198831205741199051199061120573120576

1(1199061)

119905+120575isin119874119894119869 (120574

119894

119905+120575 1199062 120573

120574119894120576(1199062))] minus 119862120576

= 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 ([119883

1205741199051199061120573120576

1(1199061)

119905+120575] 1199062 120573

120576

1199061

(1199062))] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062)) minus 119862120576] minus 119862120576

ge 1198661205741199051199061120573120576

1(1199061)

119905119905+120575[119869 (119883

1205741199051199061120573120576

1(1199061)

119905+120575 1199062 120573

120576

1199061

(1199062))] minus 119862120576

= 119866120574119905119906120573120576(119906)

119905119905+120575[119884

120574119905119906120573120576(119906)

(119905 + 120575)] minus 119862120576

= 119884120574119905119906120573120576(119906)


(A20)

Therefore

119882120575 (120574119905) ge esssup119906isinU119905119879

119869 (120574119905 119906 120573120576(119906)) minus 119862120576


esssup119906isinU119905119879

119869 (120574119905 119906 120573 (119906)) minus 119862120576

= 119882(120574119905) minus 119862120576 119875-as

(A21)



119882(120574119905) (= 119882120575 (120574119905)) le 119866120574119905119906120576120573(119906120576)

119905119905+120575[119882(119883

120574119905119906120576120573(119906120576)

119905+120575)] + 120576

119875-as(A22)


119880119905119905+120575

119882(120574119905) (= 119882120575 (120574119905)) ge 119866120574119905119906120576120573120576(119906)

119905119905+120575[119882(119883

120574119905119906120573120576(119906)

119905+120575)] minus 120576

119875-as(A23)


inf120573isinB119905119879

sup119906isinU119905119879

119864[119869(120574119905 119906 120573(119906))]

Acknowledgments




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article The Dynamic Programming Method of