The Conjugate Gradient Viscosity Approximation Algorithm for Split …downloads.hindawi.com/journals/jfs/2018/8056276.pdf · 2019-07-30 · ,ndD1,D2>0.en JJ JJ J (1,1) 2 − 11 1

Research ArticleThe Conjugate Gradient Viscosity ApproximationAlgorithm for Split Generalized Equilibrium and VariationalInequality Problems

Meixia Li Haitao Che and Jingjing Tan

School of Mathematics and Information Science Weifang University Weifang Shandong 261061 China

Correspondence should be addressed to Meixia Li limeixia001163com

Received 16 July 2018 Revised 28 September 2018 Accepted 15 October 2018 Published 1 November 2018

Academic Editor Liguang Wang

Copyright copy 2018 Meixia Li et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we study a kind of conjugate gradient viscosity approximation algorithm for finding a common solution of splitgeneralized equilibrium problem and variational inequality problem Under mild conditions we prove that the sequence generatedby the proposed iterative algorithm converges strongly to the common solution The conclusion presented in this paper is thegeneralization extension and supplement of the previously known results in the corresponding references Some numerical resultsare illustrated to show the feasibility and efficiency of the proposed algorithm

1 Introduction

Let1198671 and1198672 be real Hilbert spaces with inner product ⟨sdot sdot⟩and norm sdot Let 119862 and 119876 be nonempty closed convexsubsets of 1198671 and 1198672 respectively Let 119909119899 be a sequencein 1198671 then 119909119899 997888rarr 119909 (respectively 119909119899 119909) will denotestrong (respectively weak) convergence of the sequence 119909119899Assume 119908120596(119909119896) = 119909 exist119909119896119895 119909 to stand for the weak 120596-limit set of 119909119896

The split feasibility problem (SFP) originally introducedby Censor and Elfving [1] is to find

119909 isin 119862 such that 119860119909 isin 119876 (1)

where 119860 1198671 997888rarr 1198672 is a bounded linear operator It servesas a model for many inverse problems where constraints areimposed on the solutions in the domain of a linear operatoras well as in these operatorrsquos ranges The applications of thesplit feasibility problem are very comprehensive such as CT inmedicine intelligence antennas and the electronic warningsystems in military the development of fast image processingtechnology and HDTV etc Many authors generalize SFPto a lot of important problems such as multiple-sets splitfeasibility problem split equality fixed point problem splitvariational inequality problem split variational inclusion

problem and split equilibrium problem and the theories andalgorithms are studied and details can be seen in [2ndash15] andreferences therein

The fixed point problem (FPP) for the mapping 119879 is tofind 119909 isin 119862 such that

119879119909 = 119909 (2)

We denote Fix(119879) fl 119909 isin 119862 119879119909 = 119909 the set of solution ofFPP

Let 119861 119862 997888rarr 1198671 be a nonlinear mappingThe variationalinequality problem (VIP) is to find 119909 isin 119862 such that

⟨119861119909 119910 minus 119909⟩ ge 0 forall119910 isin 119862 (3)

The solution set of VIP is denoted by VI(119862 119861) It is wellknown that if 119861 is strongly monotone and Lipschitz continu-ous mapping on 119862 then VIP has a unique solution

For finding a common problem of Fix(119879) cap VI(119862 119861)Takahashi and Toyoda [16] introduced the following iterativescheme

1199090 chosen arbitrary119909119899+1 = 120572119899119909119899 + (1 minus 120572119899) 119879119875119862 (119909119899 minus 120582119899119861119909119899) forall119899 ge 0 (4)

HindawiJournal of Function SpacesVolume 2018 Article ID 8056276 14 pageshttpsdoiorg10115520188056276

2 Journal of Function Spaces

where 119861 is 120588-inverse-strongly monotone 120572119899 is a sequencein (0 1) and 120582119899 is a sequence in (0 2120588) They showed thatif Fix(119879) capVI(119862 119861) = 0 then the sequence 119909119899 generated by(4) converges weakly to 1199110 isin Fix(119879) cap VI(119862 119861)

On the other hand there are several numerical methodsfor solving variational inequalities and related optimizationproblems see [5 17ndash24] and the references therein

In 2013 Kazmi and Rizvi [25] introduced the splitgeneralized equilibrium problem (SGEP) Let 1198651 ℎ1 119862 times119862 997888rarr 119877 and 1198652 ℎ2 119876 times 119876 997888rarr 119877 be nonlinear bifunctionsand 119860 1198671 997888rarr 1198672 be a bounded linear operator thenthe split generalized equilibrium problem (SGEP) is to find119909lowast isin 119862 such that

1198651 (119909lowast 119909) + ℎ1 (119909lowast 119909) ge 0 forall119909 isin 119862 (5)

and such that119910lowast = 119860119909lowast isin 119876 solves 1198652 (119910lowast 119910) + ℎ2 (119910lowast 119910) ge 0

forall119910 isin 119876 (6)

Denote the solution sets of generalized equilibrium prob-lem (GEP) (5) and GEP (6) by GEP(1198651 ℎ1) and GEP(1198652 ℎ2)respectively The solution set of SGEP is denoted by Γ =119909lowast isin GEP(1198651 ℎ1) 119860119909lowast isin GEP(1198652 ℎ2) They proposed thefollowing iterative method for finding a common solution ofsplit generalized equilibrium and fixed point problem

Let 119909119899 and 119906119899 be the sequences generated by 1199090 isin 119862and

119906119899 = 119879(1198651ℎ1)119903119899(119909119899 + 120575119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899) 119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899

+ ((1 minus 120573119899) 119868 minus 120572119899119863) 1119904119899 int119904119899

0

119879 (119904) 119906119899119889119904(7)

where 119878 = 119879(119904) 0 le 119904 lt infin is a nonexpansive semigroupon 119862 and 119865119894119909(119878) cap Γ = 0 119904119899 is a positive real sequence whichdiverges to +infin 120572119899 120573119899 sub (0 1) 119903119899 sub (0infin) and 120575 isin(0 1119871) 119871 is the spectral radius of the operator 119860lowast119860 119860lowast isthe adjoint of 119860 and

119879(1198651ℎ1)119903 (119909) = 119911 isin 119862 1198651 (119911 119910) + ℎ1 (119911 119910)+ 1119903 ⟨119910 minus 119911 119911 minus 119909⟩ ge 0 forall119910 isin 119862

(8)


(9)

Under suitable conditions they proved a strong conver-gence theorem for the sequence generated by the proposediterative scheme But the calculation of integral is generallynot easyTherefore it is necessary to reconsider the algorithmfor solving this kind of problem

Motivated by Kazmi and Rivi [25] as well as Che andLi [2] we introduce and study a kind of conjugate gradi-ent viscosity approximation algorithm for finding a com-mon solution of split generalized equilibrium problem and

variational inequality problem Under mild conditions weprove that the sequence generated by the proposed iterativealgorithm converges strongly to the common solution ofVI(119862 119861) and SGEPThe results presented in this paper are thegeneralization extension and supplement of the previouslyknown results in the corresponding references Numericalresults show the feasibility and efficiency of the proposedalgorithm

2 Preliminaries

In this section we introduce some concepts and results whichare needed in sequel

A mapping 119879 1198671 997888rarr 1198671 is called(1) contraction if there exists a constant 120572 isin (0 1) such

that1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 le 120572 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 1198671 (10)

If 120572 = 1 then 119879 is called nonexpansive(2) monotone if

⟨119879119909 minus 119879119910 119909 minus 119910⟩ ge 0 forall119909 119910 isin 1198671 (11)

(3) 120578-stronglymonotone if there exists a positive constant120578 such that

⟨119879119909 minus 119879119910 119909 minus 119910⟩ ge 120578 1003817100381710038171003817119909 minus 11991010038171003817100381710038172 forall119909 119910 isin 1198671 (12)

(4) 120572-inverse strongly monotone (120572-ism) if there exists aconstant 120572 gt 0 such that


(5) firmly nonexpansive if

⟨119879119909 minus 119879119910 119909 minus 119910⟩ ge 1003817100381710038171003817119879119909 minus 11987911991010038171003817100381710038172 forall119909 119910 isin 1198671 (14)

Amapping119875119862 is said to bemetric projection of1198671 onto119862if for every point 119909 isin 1198671 there exists a unique nearest pointin 119862 denoted by 119875119862119909 such that

1003817100381710038171003817119909 minus 1198751198621199091003817100381710038171003817 le 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119910 isin 119862 (15)

It is well known that 119875119862 is a nonexpansive mapping and ischaracterized by the following properties

1003817100381710038171003817119875119862119909 minus 11987511986211991010038171003817100381710038172 le ⟨119909 minus 119910 119875119862119909 minus 119875119862119910⟩ forall119909 119910 isin 1198671 (16)

⟨119909 minus 119875119862119909 119910 minus 119875119862119909⟩ le 0 forall119909 isin 1198671 119910 isin 119862 (17)

1003817100381710038171003817119909 minus 11991010038171003817100381710038172 ge 1003817100381710038171003817119909 minus 11987511986211990910038171003817100381710038172 + 1003817100381710038171003817119910 minus 11987511986211990910038171003817100381710038172 forall119909 isin 1198671 119910 isin 119862 (18)

and1003817100381710038171003817(119909 minus 119910) minus (119875119862119909 minus 119875119862119910)10038171003817100381710038172

ge 1003817100381710038171003817119909 minus 11991010038171003817100381710038172 minus 1003817100381710038171003817119875119862119909 minus 11987511986211991010038171003817100381710038172 forall119909 119910 isin 1198671(19)

Journal of Function Spaces 3

A linear bounded operator 119861 is strongly positive if thereexists a constant 120574 gt 0 with the property

⟨119861119909 119909⟩ ge 120574 1199092 forall119909 isin 1198671 (20)

A mapping 119879 1198671 997888rarr 1198671 is said to be averaged if andonly if it can be written as the average of the identity mappingand a nonexpansive mapping ie

119879 = (1 minus 120572) 119868 + 120572119878 (21)

where 120572 isin (0 1) 119878 1198671 997888rarr 1198671 is nonexpansiveand 119868 is the identity operator on 1198671 More precisely wesay that 119879 is 120572-averaged We note that averaged mapping isnonexpansive Furthermore firmly nonexpansive mapping(in particular projection on nonempty closed and convexsubset) is averaged

Let119861 be amonotonemapping of119862 into1198671 In the contextof the variational inequality problem the characterization ofprojection (17) implies the following relation

119906 isin VI (119862 119861) lArrrArr 119906 = 119875119862 (119906 minus 120582119861119906) 120582 gt 0 (22)

In the proof of our results we need the followingassumptions and lemmas

Lemma 1 (see [26]) If 119909 119910 119911 isin 1198671 then(i) 119909 + 1199102 le 1199092 + 2⟨119910 119909 + 119910⟩(ii) For any 120582 isin [0 1]

1003817100381710038171003817120582119909 + (1 minus 120582) 11991010038171003817100381710038172 = 120582 1199092 + (1 minus 120582) 100381710038171003817100381711991010038171003817100381710038172minus 120582 (1 minus 120582) 1003817100381710038171003817119909 minus 11991010038171003817100381710038172

(23)

(iii) For 119886 119887 119888 isin [0 1] with 119886 + 119887 + 119888 = 11003817100381710038171003817119886119909 + 119887119910 + 11988811991110038171003817100381710038172 = 119886 1199092 + 119887 100381710038171003817100381711991010038171003817100381710038172 + 119888 1199112

minus 119886119887 1003817100381710038171003817119909 minus 11991010038171003817100381710038172 minus 119886119888 119909 minus 1199112minus 119887119888 1003817100381710038171003817119910 minus 11991110038171003817100381710038172

(24)

Assumption 2 Let 119865 119862 times 119862 997888rarr 119877 be a bifunction satisfyingthe following assumption

(i) 119865(119909 119909) ge 0 forall119909 isin 119862(ii) 119865 is monotone ie 119865(119909 119910) + 119865(119910 119909) le 0 forall119909 isin 119862(iii) 119865 is upper hemicontinuous ie for each 119909 119910 119911 isin 119862

lim sup119905997888rarr0

119865 (119905119911 + (1 minus 119905) 119909 119910) le 119865 (119909 119910) (25)

(iv) For each 119909 isin 119862 fixed the function 119910 997891997888rarr 119865(119909 119910) isconvex and lower semicontinuous

Let ℎ 119862 times 119862 997888rarr 119877 such that(i) ℎ(119909 119909) ge 0 forall119909 isin 119862(ii) for each 119910 isin 119862 fixed the function 119909 997888rarr ℎ(119909 119910) is

upper semicontinuous(iii) for each 119909 isin 119862 fixed the function 119910 997888rarr ℎ(119909 119910) is

convex and lower semicontinuous

And assume that for fixed 119903 gt 0 and 119911 isin 119862 there existsa nonempty compact convex subset 119870 of 1198671 and 119909 isin 119862 cap 119870such that

119865 (119910 119909) + ℎ (119910 119909) + 1119903 ⟨119910 minus 119909 119909 minus 119911⟩ lt 0forall119910 isin 119862 119870

(26)

Lemma 3 (see [25]) Assume that 1198651 ℎ1 119862 times 119862 997888rarr 119877 satisfyAssumption 2 Let 119903 gt 0 and 119909 isin 1198671 en there exists 119911 isin 119862such that

1198651 (119911 119910) + ℎ1 (119911 119910) + 1119903 ⟨119910 minus 119911 119911 minus 119909⟩ ge 0forall119910 isin 119862

(27)

Lemma 4 (see [1]) Assume that the bifunctions 1198651 ℎ1 119862 times119862 997888rarr 119877 satisfy Assumption 2 and ℎ1 is monotone For 119903 gt 0and for all 119909 isin 1198671 define a mapping 119879(1198651ℎ1)119903 1198671 997888rarr 119862 as (8)en the following conclusions hold

(1) 119879(1198651ℎ1)119903 is single-valued(2) 119879(1198651ℎ1)119903 is firmly nonexpansive(3) 119865119894119909(119879(1198651ℎ1)119903 ) = 119866119864119875(1198651 ℎ1)(4) 119866119864119875(1198651 ℎ1) is compact and convex

Furthermore assume that 1198652 ℎ2 119876 times 119876 997888rarr 119877 satisfyAssumption 2 For 119904 gt 0 and for all 119908 isin 1198672 119879(1198652ℎ2)119904 1198672 997888rarr119876 is defined as (9) By Lemma 4 we easily observe that119879(1198652ℎ2)119904

is single-valued and firmly nonexpansive GEP(1198652 ℎ2 119876) iscompact and convex and Fix(119879(1198652ℎ2)119904 ) = GEP(1198652 ℎ2 119876)where GEP(1198652 ℎ2 119876) is the solution set of the followinggeneralized equilibrium problem which is to find 119910lowast isin 119876such that 1198652(119910lowast 119910) + ℎ2(119910lowast 119910) ge 0 forall119910 isin 119876

We observe that GEP(1198652 ℎ2) sub GEP(1198652 ℎ2 119876) Furtherit is easy to prove that Γ is a closed and convex set

Lemma 5 (see [27 28]) Assume that 119879 1198671 997888rarr 1198671 isnonexpansive operator For all (119909 119910) isin 1198671 times 1198671 the followinginequality is true

⟨(119909 minus 119879 (119909)) minus (119910 minus 119879 (119910)) 119879 (119910) minus 119879 (119909)⟩le 12 1003817100381710038171003817(119879 (119909) minus 119909) minus (119879 (119910) minus 119910)10038171003817100381710038172

(28)

And for all (119909 119910) isin 1198671 times 119865119894119909(119879) one has⟨119909 minus 119879 (119909) 119910 minus 119879 (119909)⟩ le 12 119879 (119909) minus 1199092 (29)

Lemma 6 (see [29]) Assume A is a strongly positive linearbounded operator on Hilbert space 1198671 with coefficient 120574 gt 0and 0 lt 120588 le 119860minus1 en 119868 minus 120588119860 le 1 minus 120588120574Lemma 7 (see [30 31]) Assume that 119886119899 is a sequence ofnonnegative real numbers such that

119886119899+1 le (1 minus 120574119899) 119886119899 + 120574119899120575119899 + 120573119899 119899 ge 0 (30)

where 120574119899 and 120573119899 are sequences in (0 1) and 120575119899 is asequence in R such that


(i)suminfin119899=0 120574119899 = infin(ii) lim sup119899997888rarrinfin120575119899 le 0 orsuminfin119899=0 |120574119899120575119899| lt infin(iii)suminfin119899=0 120573119899 lt infinen lim119899997888rarrinfin119886119899 = 0According to [28] it is easy to prove the following Lemma

Lemma 8 Let 1198651 ℎ1 119862 times 119862 997888rarr 119877 be bifunctions satisfyingAssumption 2 and for 119903 gt 0 the mapping 119879(1198651ℎ1)119903 is defined as(8) Let 119909 119910 isin 1198671 and 1199031 1199032 gt 0 en

10038171003817100381710038171003817119879(1198651 ℎ1)1199032119910 minus 119879(1198651ℎ1)1199031

11990910038171003817100381710038171003817 le 1003817100381710038171003817119910 minus 1199091003817100381710038171003817+ 10038161003816100381610038161003816100381610038161003816

1199032 minus 119903111990321003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817119879(1198651ℎ1)1199032

119910 minus 11991010038171003817100381710038171003817 (31)

Lemma 9 (see [32 33]) Let 119879 119867 997888rarr 119867 be given We havethe following

(i) 119879 is nonexpansive iff the complement 119868 minus 119879 is 12-ism(ii) if 119879 is v-ism then for 120574 gt 0 120574119879 is V120574-ism(iii) 119879 is averaged iff the complement 119868 minus 119879 is V-ism for

some V gt 12 indeed for 120572 isin (0 1) 119879 is 120572-averaged iff 119868 minus 119879is 12120572-ismLemma 10 (see [4 32]) Let the operators 119878 119879119881 119867 997888rarr 119867be given

(i) If 119879 = (1 minus 120572)119878 + 120572119881 where 119878 is averaged 119881 isnonexpansive and 120572 isin (0 1) then 119879 is averaged

(ii) 119879 is firmly nonexpansive iff the complement 119868 minus 119879 isfirmly nonexpansive

(iii) If 119879 = (1 minus 120572)119878 + 120572119881 where 119878 is firmly nonexpansive119881 is nonexpansive and 120572 isin (0 1) then 119879 is averaged(iv) e composite of finitely many averaged mappings is

averaged at is if each of the mappings 119879119894119873119894=1 is averagedthen so is the composite 1198791 119879119873 In particular if 1198791 is 1205721-averaged and 1198792 is 1205722-averaged where 1205721 1205722 isin (0 1) then thecomposite 11987911198792 is 120572-averaged where 120572 = 1205721 + 1205722 minus 12057211205722

(v) If themappings 119879119894119873119894=1 are averaged and have a commonfixed point then

119873⋂119894=1

Fix (119879119894) = Fix (1198791 119879119873) (32)

In the following we give the relation between the projec-tion operator and average mapping

Lemma 11 Assume that the variational inequality problem (3)is solvable If 119861 is 120573-ism from 119862 into 1198671 then 119875119862(119868 minus 120582119861) is(2120573 + 120582)4120573-averagedProof Note that 119861 is120573-ism which implies that 120582119861 is 120573120582-ismie 119868minus(119868minus120582119861) is120573120582-ism By Lemma 9(iii) we can see that 119868minus120582119861 is 1205822120573-averaged Since the projection 119875119862 is 12-averagedit is easy to see from Lemma 10 that the composite 119875119862(119868minus120582119861)is (2120573 + 120582)4120573-averaged for 0 lt 120582 lt 2120573 according to

12 + 1205822120573 minus 12 sdot 1205822120573 = 2120573 + 1205824120573 (33)

which completes the proof

As a result we have that for each 119899 119875119862(119868 minus 120582119899119861) is (2120573 +120582119899)4120573-averaged Therefore we can write

119875119862 (119868 minus 120582119899119861) = 2120573 minus 1205821198994120573 119868 + 2120573 + 1205821198994120573 119879119899= (1 minus 119887119899) 119868 + 119887119899119879119899

(34)

where 119879119899 is nonexpansive and 119887119899 = (2120573 + 120582119899)4120573 isin [12 1]Lemma 12 (see [28]) Let 119909119899 and 119911119899 be bounded sequencein a Banach space X and let 120573119899 be a sequence in [0 1] with0 lt lim inf119899997888rarrinfin120573119899 le lim sup119899997888rarrinfin120573119899 lt 1 Suppose 119909119899+1 =(1minus120573119899)119911119899+120573119899119909119899 for all integers 119899 ge 0 and lim sup119899997888rarrinfin(119911119899+1minus119911119899 minus 119909119899+1 minus 119909119899) le 0 en lim119899997888rarrinfin119911119899 minus 119909119899 = 03 Main Results

In this section we give the main results of this paper First wedescribe the algorithm for finding a common solution of splitgeneralized equilibrium and variational inequality problems

Throughout the rest of this paper let119891 be a contraction of1198671 into itself with coefficient 120578 isin (0 1)119860 be a bounded linearoperator 119861 be a 120573-inverse strongly monotone mapping from119862 into 1198671 and 119863 be a strongly positive linear bounded self-adjoint operator on1198671 with coefficient 120574 gt 0 and 0 lt 120574 lt 120574120578

Now we give the description of the algorithm

Algorithm 13 Let 1199090 isin 1198671 be arbitrary and 120572 gt 0 Assumethat 120572119899 120573119899 120574119899 sub (0 1) 120582119899 sub (0 2120573) 119903119899 sub (0infin) and120585 isin (0 1119871) where 119871 is the spectral radius of the operator119860119860lowast and 119860lowast is the adjoint of 119860 Calculate sequences 119906119899119910119899 and 119909119899 by the following iteration formula

119906119899 = 119879(1198651ℎ1)119903119899(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(35)

where 119889119899+1 = (1120572)(119879119899119906119899 minus 119906119899) + 120574119899119889119899 1198890 = (1120572)(11987901199060 minus 1199060)and 119879119899 is defined by (34)

As follows we propose the convergence analysis ofAlgorithm 13

Theorem14 Let1198671 and1198672 be two realHilbert spaces and119862 sub1198671 119876 sub 1198672 be nonempty closed convex subsets Let 1198651 ℎ1 119862 times 119862 997888rarr 119877 and 1198652 ℎ2 119876 times 119876 997888rarr 119877 satisfy Assumption 2ℎ1 ℎ2 are monotone and1198652 is upper semicontinuous in the firstargument Assume that Ω fl 119881119868(119862 119861) cap Γ = 0 and 119909119899 119906119899and 119910119899 are generated by (35) Suppose that 120572119899 120573119899 sub (0 1)120574119899 sub (0 12) 120582119899 sub (0 2120573) 119903119899 sub (0infin) satisfy thefollowing conditions

(C1) lim119899997888rarrinfin120572119899 = 0 suminfin119899=0 120572119899 = infin(C2) 0 lt lim inf119899997888rarrinfin120573119899 le lim sup119899997888rarrinfin120573119899 lt 1(C3) 120574119899 = 119900(120572119899)(C4) lim inf119899997888rarrinfin120582119899 gt 0 lim119899997888rarrinfin|120582119899+1 minus 120582119899| = 0(C5) lim119899997888rarrinfin(|119903119899+1 minus 119903119899|119903119899+1) = 0(C6) 119879119899119906119899 minus 119906119899 is bounded


en the sequence 119909119899 converges strongly to 119902 isin Ω where119902 = 119875Ω(119868 minus 119863 + 120574119891)119902 which is the unique solution of thevariational inequality problem

⟨(119863 minus 120574119891) 119902 119909 minus 119902⟩ ge 0 forall119909 isin Ω (36)

or equivalently 119902 is the unique solution to the minimizationproblem

min119909isinΩ

12 ⟨119863119909 119909⟩ minus ℎ (119909) (37)

where ℎ is a potential function for 120574119891 such that ℎ1015840(119909) = 120574119891(119909)for 119909 isin 1198671Proof Some equalities and inequalities in the following canbe obtained according to the proof of Theorem 1 in [25]However we give the detailed proof process in order to readhandily

From (C1) and (C2) without loss of generality we assumethat 120572119899 le (1 minus 120573119899)119863minus1 for all 119899 isin 119873 By Lemma 6 we have119868 minus 120588119863 le 1 minus 120588120574 if 0 lt 120588 le 119863minus1 Now suppose that119868 minus119863 le 1 minus 120574 Since119863 is a strongly positive linear boundedself-adjoint operator on1198671 we obtain

119863 = sup |⟨119863119909 119909⟩| 119909 isin 1198671 119909 = 1 (38)

Notice that

⟨((1 minus 120573119899) 119868 minus 120572119899119863)119909 119909⟩ = 1 minus 120573119899 minus 120572119899 ⟨119863119909 119909⟩ge 1 minus 120573119899 minus 120572119899 119863 ge 0 (39)

which shows that (1 minus 120573119899)119868 minus 120572119899119863 is positive definiteFurthermore we have1003817100381710038171003817(1 minus 120573119899) 119868 minus 1205721198991198631003817100381710038171003817

= sup 1003816100381610038161003816⟨((1 minus 120573119899) 119868 minus 120572119899119863)119909 119909⟩1003816100381610038161003816 119909 isin 1198671 119909= 1 = sup 1 minus 120573119899 minus 120572119899 ⟨119863119909 119909⟩ 119909 isin 1198671 119909 = 1le 1 minus 120573119899 minus 120572119899120574

(40)

Since 119891 is a contraction mapping with constant 120578 isin (0 1) forall 119909 119910 isin 1198671 we have1003817100381710038171003817119875Ω (119868 minus 119863 + 120574119891) (119909) minus 119875Ω (119868 minus 119863 + 120574119891) (119910)1003817100381710038171003817

le 1003817100381710038171003817(119868 minus 119863 + 120574119891) (119909) minus (119868 minus 119863 + 120574119891) (119910)1003817100381710038171003817le 119868 minus 119863 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 + 120574 1003817100381710038171003817119891 (119909) minus 119891 (119910)1003817100381710038171003817le (1 minus 120574) 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 + 120574120578 1003817100381710038171003817119909 minus 1199101003817100381710038171003817le (1 minus (120574 minus 120574120578)) 1003817100381710038171003817119909 minus 1199101003817100381710038171003817

(41)

which implies that 119875Ω(119868 minus 119863 + 120574119891) is a contraction mappingfrom 1198671 into itself It follows from the Banach contractionprinciple that there exists an element 119902 isin Ω such that 119902 =119875Ω(119868 minus 119863 + 120574119891)119902Step 1 (we show that 119909119899 is bounded) Let 119909lowast isin Ω ie 119909lowast isin Γwe have 119909lowast = 119879(1198651ℎ1)119903119899

119909lowast and 119860119909lowast = 119879(1198652 ℎ2)119903119899119860119909lowast

In the following we compute1003817100381710038171003817119906119899 minus 119909lowast10038171003817100381710038172 = 10038171003817100381710038171003817119879(1198651ℎ1)119903119899

(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899minus 119868)119860119909119899)

minus 119909lowast100381710038171003817100381710038172 = 10038171003817100381710038171003817119879(1198651ℎ1)119903119899(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899)minus 119879(1198651ℎ1)119903119899

119909lowast100381710038171003817100381710038172 le 10038171003817100381710038171003817119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899minus 119868)119860119909119899

minus 119909lowast100381710038171003817100381710038172 = 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 + 1205852 10038171003817100381710038171003817119860lowast (119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

+ 2120585 ⟨119909119899 minus 119909lowast 119860lowast (119879(1198652ℎ2)119903119899minus 119868)119860119909119899⟩

(42)

Thus we have1003817100381710038171003817119906119899 minus 119909lowast10038171003817100381710038172le 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172

+ 1205852 ⟨(119879(1198652ℎ2)119903119899minus 119868)119860119909119899 119860119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899⟩+ 2120585 ⟨119909119899 minus 119909lowast 119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899⟩ (43)

On the other hand we have

1205852 ⟨(119879(1198652ℎ2)119903119899minus 119868)119860119909119899 119860119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899⟩le 1198711205852 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899

minus 119868)119860119909119899100381710038171003817100381710038172 (44)

and

2120585 ⟨119909119899 minus 119909lowast 119860lowast (119879(1198652ℎ2)119903119899minus 119868)119860119909119899⟩

= 2120585 ⟨119860 (119909119899 minus 119909lowast) (119879(1198652 ℎ2)119903119899minus 119868)119860119909119899⟩

= 2120585 ⟨119860 (119909119899 minus 119909lowast) + (119879(1198652 ℎ2)119903119899minus 119868)119860119909119899

minus (119879(1198652ℎ2)119903119899minus 119868)119860119909119899 (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899⟩= 2120585 ⟨119879(1198652ℎ2)119903119899

119860119909119899 minus 119860119909lowast (119879(1198652ℎ2)119903119899minus 119868)119860119909119899⟩

minus 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

le 2120585 12 10038171003817100381710038171003817(119879(1198652 ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

minus 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

le minus120585 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

(45)

where the first inequality is derived from (29)From (43)-(45) we have1003817100381710038171003817119906119899 minus 119909lowast10038171003817100381710038172 le 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172

+ 120585 (119871120585 minus 1) 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

(46)

Noticing that 120585 isin (0 1119871) we obtain1003817100381710038171003817119906119899 minus 119909lowast10038171003817100381710038172 le 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 (47)


As follows we prove that 119889119899 is bounded The proofis by induction It is true trivially for 119899 = 0 Let 1198721 =max1198890 (2120572)sup119899isin119873119879119899119906119899 minus 119906119899 From (C6) it is shownthat 1198721 lt infin Assume that 119889119899 le 1198721 for some 119899 we provethat it holds for 119899+1 According to the triangle inequality weobtain

1003817100381710038171003817119889119899+11003817100381710038171003817 = 1003817100381710038171003817100381710038171003817 1120572 (119879119899119906119899 minus 119906119899) + 1205741198991198891198991003817100381710038171003817100381710038171003817le 1120572 1003817100381710038171003817119879119899119906119899 minus 1199061198991003817100381710038171003817 + 120574119899 10038171003817100381710038171198891198991003817100381710038171003817 le 1120572 sdot 12057221198721 + 11987212= 1198721

(48)

which implies that 119889119899 le 1198721 for all 119899 isin 119873 ie 119889119899 isbounded

It is easy to see that 119909lowast isin 119881119868(119862 119861) according to 119909lowast isin ΩBy (22) we have 119875119862(119868 minus 120582119861)119909lowast = 119909lowast which together with(34) implies that

119887119899119879119899119909lowast = 119875119862 (119868 minus 120582119861) 119909lowast minus (1 minus 119887119899) 119909lowast = 119887119899119909lowast (49)

that is

119879119899119909lowast = 119909lowast (50)

By the definition of 119910119899 (47) and 119879119899 being nonexpan-sive we have

1003817100381710038171003817119910119899 minus 119909lowast1003817100381710038171003817 = 1003817100381710038171003817119906119899 + 120572119889119899+1 minus 119909lowast1003817100381710038171003817= 1003817100381710038171003817119879119899 (119906119899) + 120572120574119899119889119899 minus 119879119899119909lowast1003817100381710038171003817le 1003817100381710038171003817119906119899 minus 119909lowast1003817100381710038171003817 + 1205721205741198991198721le 1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817 + 1205721205741198991198721

(51)

As a result it follows from (51) Lemma 6and the fact that120572119899 997888rarr 0 and 120574119899 = 119900(120572119899) that when 119899 is large enough1003817100381710038171003817119909119899+1 minus 119909lowast1003817100381710038171003817 = 1003817100381710038171003817120572119899 (120574119891 (119909119899) minus 119863119909lowast) + 120573119899 (119909119899 minus 119909lowast)+ ((1 minus 120573119899) 119868 minus 120572119899119863) (119910119899 minus 119909lowast)1003817100381710038171003817 le 120572119899 1003817100381710038171003817120574119891 (119909119899)minus 119863119909lowast1003817100381710038171003817 + 120573119899 1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817 + (1 minus 120573119899 minus 120572119899120574)sdot (1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817 + 1205721205741198991198721) le 120572119899120574 1003817100381710038171003817119891 (119909119899) minus 119891 (119909lowast)1003817100381710038171003817+ 120572119899 1003817100381710038171003817120574119891 (119909lowast) minus 119863119909lowast1003817100381710038171003817 + 120573119899 1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817 + (1 minus 120573119899minus 120572119899120574) (1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817 + 1205721205741198991198721) le 120572119899120574120578 1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817+ 120572119899 1003817100381710038171003817120574119891 (119909lowast) minus 119863119909lowast1003817100381710038171003817 + (1 minus 120572119899120574) 1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817 + (1minus 120573119899 minus 120572119899120574) 1205721205741198991198721 le (1 minus 120572119899 (120574 minus 120574120578)) 1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817+ 120572119899 (120574 minus 120574120578) 1205721198721 +

1003817100381710038171003817120574119891 (119909lowast) minus 119863119909lowast1003817100381710038171003817120574 minus 120574120578le max1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817 1205721198721 +

1003817100381710038171003817120574119891 (119909lowast) minus 119863119909lowast1003817100381710038171003817120574 minus 120574120578

(52)

where the third inequality is true because 120573119899 isin (0 1) 120572119899 997888rarr0 and 120574119899 = 119900(120572119899) As a result(1 minus 120573119899 minus 120572119899120574) 1205721205741198991198721 le 1205721205721198991198721 (53)

when 119899 is large enoughHence 119909119899 is bounded and so are 119906119899 and 119910119899

Step 2 (we show that lim119899997888rarrinfin119909119899+1 minus 119909119899 = 0) Since 119879(1198651ℎ1)119903119899+1

and 119879(1198652ℎ2)119903119899+1both are firmly nonexpansive for 120585 isin (0 1119871) the

mapping 119879(1198651 ℎ1)119903119899+1(119868 + 120585119860lowast(119879(1198652ℎ2)119903119899+1

minus 119868)119860) is nonexpansive see[34 35] Noticing that 119906119899 = 119879(1198651ℎ1)119903119899

(119909119899 +120585119860lowast(119879(1198652ℎ2)119903119899minus119868)119860119909119899)

and 119906119899+1 = 119879(1198651ℎ1)119903119899+1(119909119899+1 + 120585119860lowast(119879(1198652ℎ2)119903119899+1

minus 119868)119860119909119899+1) we havefrom Lemma 8 that1003817100381710038171003817119906119899+1 minus 1199061198991003817100381710038171003817

le 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1(119909119899+1 + 120585119860lowast (119879(1198652ℎ2)119903119899+1

minus 119868)119860119909119899+1)minus 119879(1198651ℎ1)119903119899+1

(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899+1minus 119868)119860119909119899)10038171003817100381710038171003817

+ 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899+1

minus 119868)119860119909119899)minus 119879(1198651ℎ1)119903119899

(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899minus 119868)119860119909119899)10038171003817100381710038171003817 le 1003817100381710038171003817119909119899+1

minus 1199091198991003817100381710038171003817 + 10038171003817100381710038171003817119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899+1minus 119868)119860119909119899

minus (119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899minus 119868)119860119909119899)10038171003817100381710038171003817 +

100381610038161003816100381610038161003816100381610038161 minus119903119899119903119899+1

10038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1

(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899+1minus 119868)119860119909119899)

minus (119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899+1minus 119868)119860119909119899)10038171003817100381710038171003817 le 1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817

+ 120585 119860 10038171003817100381710038171003817119879(1198652 ℎ2)119903119899+1119860119909119899 minus 119879(1198652ℎ2)119903119899

11986011990911989910038171003817100381710038171003817 + 120577119899 le 1003817100381710038171003817119909119899+1minus 1199091198991003817100381710038171003817 + 120585 119860 100381610038161003816100381610038161003816100381610038161 minus

119903119899119903119899+11003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817119879(1198652ℎ2)119903119899+1

119860119909119899 minus 11986011990911989910038171003817100381710038171003817 + 120577119899= 1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 + 120585 119860 120590119899 + 120577119899

(54)

where 120590119899 fl |1 minus 119903119899119903119899+1|119879(1198652ℎ2)119903119899+1119860119909119899 minus 119860119909119899 and 120577119899 fl |1 minus

119903119899119903119899+1|119879(1198651ℎ1)119903119899+1(119909119899+120585119860lowast(119879(1198652ℎ2)119903119899

minus119868)119860119909119899)minus(119909119899+120585119860lowast(119879(1198652ℎ2)119903119899+1minus119868)119860119909119899)

Furthermore one has1003817100381710038171003817119910119899+1 minus 1199101198991003817100381710038171003817 = 1003817100381710038171003817119906119899+1 + 120572119889119899+2 minus (119906119899 + 120572119889119899+1)1003817100381710038171003817le 1003817100381710038171003817119879119899+1119906119899+1 minus 1198791198991199061198991003817100381710038171003817

+ 120572 1003817100381710038171003817120574119899+1119889119899+1 minus 1205741198991198891198991003817100381710038171003817le 1003817100381710038171003817119879119899+1119906119899+1 minus 119879119899+11199061198991003817100381710038171003817

+ 1003817100381710038171003817119879119899+1119906119899 minus 1198791198991199061198991003817100381710038171003817+ 120572 1003817100381710038171003817120574119899+1119889119899+1 minus 1205741198991198891198991003817100381710038171003817

le 1003817100381710038171003817119906119899+1 minus 1199061198991003817100381710038171003817 + 1003817100381710038171003817119879119899+1119906119899 minus 1198791198991199061198991003817100381710038171003817+ 1205721198721 (120574119899+1 + 120574119899)

(55)


It follows from (34) that

1003817100381710038171003817119879119899+1119906119899 minus 1198791198991199061198991003817100381710038171003817 =100381710038171003817100381710038171003817100381710038171003817119875119862 (119868 minus 120582119899+1119861) minus (1 minus 119887119899+1) 119868119887119899+1 119906119899

minus 119875119862 (119868 minus 120582119899119861) minus (1 minus 119887119899) 119868119887119899 119906119899100381710038171003817100381710038171003817100381710038171003817

= 1003817100381710038171003817100381710038171003817100381710038174120573119875119862 (119868 minus 120582119899+1119861) minus (2120573 minus 120582119899+1) 1198682120573 + 120582119899+1 119906119899

minus 4120573119875119862 (119868 minus 120582119899119861) minus (2120573 minus 120582119899) 1198682120573 + 120582119899 119906119899100381710038171003817100381710038171003817100381710038171003817 =

1003817100381710038171003817100381710038171003817100381710038174120573119875119862 (119868 minus 120582119899+1119861)2120573 + 120582119899+1 119906119899

minus 4120573119875119862 (119868 minus 120582119899119861)2120573 + 120582119899 119906119899 + (2120573 minus 120582119899)2120573 + 120582119899 119906119899 minus(2120573 minus 120582119899+1)2120573 + 120582119899+1 119906119899

100381710038171003817100381710038171003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817

4120573 [(2120573 + 120582119899) 119875119862 (119868 minus 120582119899+1119861) minus (2120573 + 120582119899+1) 119875119862 (119868 minus 120582119899119861)](2120573 + 120582119899+1) (2120573 + 120582119899)sdot 119906119899

100381710038171003817100381710038171003817100381710038171003817 +100381710038171003817100381710038171003817100381710038171003817

4120573 (120582119899+1 minus 120582119899)(2120573 + 120582119899+1) (2120573 + 120582119899)119906119899100381710038171003817100381710038171003817100381710038171003817

le 1003817100381710038171003817100381710038171003817100381710038174120573 (120582119899 minus 120582119899+1) 119875119862 (119868 minus 120582119899+1119861)(2120573 + 120582119899+1) (2120573 + 120582119899) 119906119899

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817

4120573 (2120573 + 120582119899+1) [119875119862 (119868 minus 120582119899+1119861) minus 119875119862 (119868 minus 120582119899119861)](2120573 + 120582119899+1) (2120573 + 120582119899) 119906119899100381710038171003817100381710038171003817100381710038171003817

+ 1003817100381710038171003817100381710038171003817100381710038174120573 (120582119899+1 minus 120582119899)(2120573 + 120582119899+1) (2120573 + 120582119899)119906119899

100381710038171003817100381710038171003817100381710038171003817 le1120573 1003816100381610038161003816120582119899 minus 120582119899+11003816100381610038161003816

sdot (1003817100381710038171003817119875119862 (119868 minus 120582119899+1119861) 1199061198991003817100381710038171003817 + 4120573 10038171003817100381710038171198611199061198991003817100381710038171003817+ 10038171003817100381710038171199061198991003817100381710038171003817) le 1198722 1003816100381610038161003816120582119899+1 minus 1205821198991003816100381610038161003816

(56)

where 1198722 = sup119899isin119873(1120573)(119875119862(119868 minus 120582119899+1119861)119906119899 + 4120573119861119906119899 +119906119899)Hence from (54)-(56) we get

1003817100381710038171003817119910119899+1 minus 1199101198991003817100381710038171003817 le 1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 + 120585 119860 120590119899 + 120577119899+1198722 1003816100381610038161003816120582119899+1 minus 1205821198991003816100381610038161003816 + 1205721198721 (120574119899+1 + 120574119899) (57)

Set 119909119899+1 = (1 minus 120573119899)119911119899 + 120573119899119909119899 it follows that119911119899 = 119909119899+1 minus 1205731198991199091198991 minus 120573119899

= 120572119899120574119891 (119909119899) + ((1 minus 120573119899) 119868 minus 120572119899119863)1199101198991 minus 120573119899 (58)

As a result

1003817100381710038171003817119911119899+1 minus 1199111198991003817100381710038171003817= 100381710038171003817100381710038171003817100381710038171003817

120572119899+1120574119891 (119909119899+1) + ((1 minus 120573119899+1) 119868 minus 120572119899+1119863)119910119899+11 minus 120573119899+1minus 120572119899120574119891 (119909119899) + ((1 minus 120573119899) 119868 minus 120572119899119863)1199101198991 minus 120573119899

100381710038171003817100381710038171003817100381710038171003817

= 100381710038171003817100381710038171003817100381710038171003817120572119899+1 (120574119891 (119909119899+1) minus 119863119910119899+1)1 minus 120573119899+1

minus 120572119899 (120574119891 (119909119899) minus 119863119910119899)1 minus 120573119899 + 119910119899+1 minus 119910119899100381710038171003817100381710038171003817100381710038171003817

le 120572119899+11 minus 120573119899+11003817100381710038171003817120574119891 (119909119899+1) minus 119863119910119899+11003817100381710038171003817

+ 1205721198991 minus 1205731198991003817100381710038171003817120574119891 (119909119899) minus 1198631199101198991003817100381710038171003817 + 1003817100381710038171003817119910119899+1 minus 1199101198991003817100381710038171003817

le 120572119899+11 minus 120573119899+11003817100381710038171003817120574119891 (119909119899+1) minus 119863119910119899+11003817100381710038171003817

+ 1205721198991 minus 1205731198991003817100381710038171003817120574119891 (119909119899) minus 1198631199101198991003817100381710038171003817 + 1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 + 120585 119860

sdot 120590119899 + 120577119899 +1198722 1003816100381610038161003816120582119899+1 minus 1205821198991003816100381610038161003816 + 1205721198721 (120574119899+1 + 120574119899) (59)

Letting 119899 997888rarr infin from (C1)-(C5) we have

lim sup119899997888rarrinfin

(1003817100381710038171003817119911119899+1 minus 1199111198991003817100381710038171003817 minus 1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817) le 0 (60)

By Lemma 12 we obtain

lim119899997888rarrinfin

1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (61)

Furtherlim119899997888rarrinfin

1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 = lim119899997888rarrinfin

(1 minus 120573119899) 1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (62)

Step 3 (we show that lim119899997888rarrinfin119906119899 minus 119909119899 = 0) Since 119909lowast isin Ω119909lowast = 119879(1198651ℎ1)119903119899119909lowast and 119879(1198651ℎ1)119903119899

is firmly nonexpansive we obtain

1003817100381710038171003817119906119899 minus 119909lowast10038171003817100381710038172 = 10038171003817100381710038171003817119879(1198651ℎ1)119903119899(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899

minus 119868)119860119909119899)minus 119909lowast100381710038171003817100381710038172 = 10038171003817100381710038171003817119879(1198651ℎ1)119903119899

(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899minus 119868)119860119909119899)

minus 119879(1198651ℎ1)119903119899119909lowast100381710038171003817100381710038172 le ⟨119906119899 minus 119909lowast 119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899

minus 119868)sdot 119860119909119899 minus 119909lowast⟩ = 12 1003817100381710038171003817119906119899 minus 119909lowast10038171003817100381710038172 + 10038171003817100381710038171003817119909119899+ 120585119860lowast (119879(1198652 ℎ2)119903119899

minus 119868)119860119909119899 minus 119909lowast100381710038171003817100381710038172 minus 10038171003817100381710038171003817(119906119899 minus 119909lowast)minus [119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899 minus 119909lowast]100381710038171003817100381710038172 = 12 1003817100381710038171003817119906119899minus 119909lowast10038171003817100381710038172 + 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119906119899 minus 11990911989910038171003817100381710038172 + 2120585⟨119906119899minus 119909lowast 119860lowast (119879(1198652 ℎ2)119903119899

minus 119868)119860119909119899⟩

(63)

Hence we obtain1003817100381710038171003817119906119899 minus 119909lowast10038171003817100381710038172

le 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119906119899 minus 11990911989910038171003817100381710038172+ 2120585 1003817100381710038171003817119860 (119906119899 minus 119909lowast)1003817100381710038171003817 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899

minus 119868)11986011990911989910038171003817100381710038171003817 (64)


Furthermore

1003817100381710038171003817119910119899 minus 119909lowast10038171003817100381710038172 = 1003817100381710038171003817119906119899 + 120572119889119899+1 minus 119909lowast10038171003817100381710038172le 1003817100381710038171003817119906119899 minus 119909lowast10038171003817100381710038172 + 2120572120574119899 ⟨119910119899 minus 119909lowast 119889119899⟩le 1003817100381710038171003817119906119899 minus 119909lowast10038171003817100381710038172 +1198723120574119899

(65)

where1198723 = sup119899isin1198732120572⟨119910119899 minus 119909lowast 119889119899⟩By Lemma 1 (119894119894119894) (46) and (65) we have

1003817100381710038171003817119909119899+1 minus 119909lowast10038171003817100381710038172 = 1003817100381710038171003817120572119899120574119891 (119909119899) + 120573119899119909119899+ ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899 minus 119909lowast10038171003817100381710038172= 1003817100381710038171003817120572119899 (120574119891 (119909119899) minus 119863119909lowast) + 120573119899 (119909119899 minus 119909lowast)+ ((1 minus 120573119899) 119868 minus 120572119899119863) (119910119899 minus 119909lowast)10038171003817100381710038172le (120572119899 1003817100381710038171003817120574119891 (119909119899) minus 119863119909lowast1003817100381710038171003817 + 120573119899 1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817+ (1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119910119899 minus 119909lowast1003817100381710038171003817)2= (120572119899120574 10038171003817100381710038171003817100381710038171003817

1120574 (120574119891 (119909119899) minus 119863119909lowast)10038171003817100381710038171003817100381710038171003817 + 120573119899 1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817+ (1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119910119899 minus 119909lowast1003817100381710038171003817)

2 le 120572119899120574 11205742 1003817100381710038171003817120574119891 (119909119899)

minus 119863119909lowast10038171003817100381710038172 + 120573119899 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 + (1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119910119899minus 119909lowast10038171003817100381710038172 le 120572119899120574 1003817100381710038171003817120574119891 (119909119899) minus 119863119909lowast10038171003817100381710038172 + 120573119899 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172

+ (1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119910119899 minus 119909lowast10038171003817100381710038172 le 120572119899120574 1003817100381710038171003817120574119891 (119909119899)minus 119863119909lowast10038171003817100381710038172 + 120573119899 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 + (1 minus 120573119899 minus 120572119899120574)sdot (1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 + 120585 (119871120585 minus 1) 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899

minus 119868)119860119909119899100381710038171003817100381710038172+1198723120574119899) = 120572119899120574 1003817100381710038171003817120574119891 (119909119899) minus 119863119909lowast10038171003817100381710038172 + 120573119899 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172+ (1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 minus (1 minus 120573119899 minus 120572119899120574) 120585 (1minus 119871120585) 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899

minus 119868)119860119909119899100381710038171003817100381710038172 + (1 minus 120573119899 minus 120572119899120574)1198723120574119899

(66)

As a result

(1 minus 120573119899 minus 120572119899120574) 120585 (1 minus 119871120585) 10038171003817100381710038171003817(119879(1198652 ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

le 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119899+1 minus 119909lowast10038171003817100381710038172+ 120572119899120574 1003817100381710038171003817120574119891 (119909119899) minus 119863119909lowast10038171003817100381710038172 + (1 minus 120573119899 minus 120572119899120574)1198723120574119899

le (1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817 + 1003817100381710038171003817119909119899+1 minus 119909lowast1003817100381710038171003817) 1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817+ 120572119899120574 1003817100381710038171003817120574119891 (119909119899) minus 119863119909lowast10038171003817100381710038172 + (1 minus 120573119899 minus 120572119899120574)1198723120574119899

(67)

According to120572119899 997888rarr 0 120574119899 = 119900(120572119899) (1minus120573119899minus120572119899120574)120585(1minus119871120585) gt0 and lim119899997888rarrinfin119909119899+1 minus 119909119899 = 0 we obtainlim119899997888rarrinfin

10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)11986011990911989910038171003817100381710038171003817 = 0 (68)

From (65) (64) and Lemma 1 (119894119894119894) we obtain1003817100381710038171003817119909119899+1 minus 119909lowast10038171003817100381710038172 = 1003817100381710038171003817120572119899120574119891 (119909119899) + 120573119899119909119899+ ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899 minus 119909lowast10038171003817100381710038172= 1003817100381710038171003817120572119899 (120574119891 (119909119899) minus 119863119909lowast) + 120573119899 (119909119899 minus 119909lowast)+ ((1 minus 120573119899) 119868 minus 120572119899119863) (119910119899 minus 119909lowast)10038171003817100381710038172le (120572119899 1003817100381710038171003817120574119891 (119909119899) minus 119863119909lowast1003817100381710038171003817 + 120573119899 1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817+ (1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119910119899 minus 119909lowast1003817100381710038171003817)2= (120572119899120574 10038171003817100381710038171003817100381710038171003817

1120574 (120574119891 (119909119899) minus 119863119909lowast)10038171003817100381710038171003817100381710038171003817 + 120573119899 1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817

+ (1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119910119899 minus 119909lowast1003817100381710038171003817)2 le 120572119899120574 1

1205742 1003817100381710038171003817120574119891 (119909119899)minus 119863119909lowast10038171003817100381710038172 + 120573119899 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 + (1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119910119899minus 119909lowast10038171003817100381710038172 le 120572119899120574 1003817100381710038171003817120574119891 (119909119899) minus 119863119909lowast10038171003817100381710038172 + 120573119899 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172

+ (1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119910119899 minus 119909lowast10038171003817100381710038172 le 120572119899120574 1003817100381710038171003817120574119891 (119909119899)minus 119863119909lowast10038171003817100381710038172 + 120573119899 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 + (1 minus 120573119899 minus 120572119899120574)sdot (1003817100381710038171003817119906119899 minus 119909lowast10038171003817100381710038172 +1198723120574119899) le 120572119899120574 1003817100381710038171003817120574119891 (119909119899) minus 119863119909lowast10038171003817100381710038172+ 120573119899 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 + (1 minus 120573119899 minus 120572119899120574) (1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172minus 1003817100381710038171003817119906119899 minus 11990911989910038171003817100381710038172+ 2120585 1003817100381710038171003817119860 (119906119899 minus 119909lowast)1003817100381710038171003817 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899

minus 119868)11986011990911989910038171003817100381710038171003817 +1198723120574119899)= 120572119899120574 1003817100381710038171003817120574119891 (119909119899) minus 119863119909lowast10038171003817100381710038172 + 120573119899 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 + (1 minus 120573119899minus 120572119899120574) 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 minus (1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119906119899 minus 11990911989910038171003817100381710038172+ 2120585 (1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119860 (119906119899 minus 119909lowast)1003817100381710038171003817sdot 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899


(69)

Therefore one has

(1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119906119899 minus 11990911989910038171003817100381710038172 le 120572119899120574 1003817100381710038171003817120574119891 (119909119899) minus 119863119909lowast10038171003817100381710038172+ (1 minus 120572119899120574) 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119899+1 minus 119909lowast10038171003817100381710038172+ 2120585 (1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119860 (119906119899 minus 119909lowast)1003817100381710038171003817


sdot 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)11986011990911989910038171003817100381710038171003817 + (1 minus 120573119899 minus 120572119899120574)1198723120574119899

le 120572119899120574 1003817100381710038171003817120574119891 (119909119899) minus 119863119909lowast10038171003817100381710038172 + 1003817100381710038171003817119909119899 minus 119909lowast10038171003817100381710038172minus 1003817100381710038171003817119909119899+1 minus 119909lowast10038171003817100381710038172 + 2120585 (1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119860 (119906119899 minus 119909lowast)1003817100381710038171003817sdot 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899

minus 119868)11986011990911989910038171003817100381710038171003817 + (1 minus 120573119899 minus 120572119899120574)1198723120574119899le 120572119899120574 1003817100381710038171003817120574119891 (119909119899) minus 119863119909lowast10038171003817100381710038172 + 1003817100381710038171003817119909119899 minus 119909119899+11003817100381710038171003817sdot (1003817100381710038171003817119909119899 minus 119909lowast1003817100381710038171003817 + 1003817100381710038171003817119909119899+1 minus 119909lowast1003817100381710038171003817) + 2120585 (1 minus 120573119899 minus 120572119899120574)sdot 1003817100381710038171003817119860 (119906119899 minus 119909lowast)1003817100381710038171003817 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899

minus 119868)11986011990911989910038171003817100381710038171003817+ (1 minus 120573119899 minus 120572119899120574)1198723120574119899

(70)

According to 120572119899 997888rarr 0 120574119899 = 119900(120572119899) (62) and (68) we obtain


1003817100381710038171003817119906119899 minus 1199091198991003817100381710038171003817 = 0 (71)

Step 4 (we show that lim119899997888rarrinfin119879119899119906119899 minus119906119899 = 0) From (58) wehave

119911119899 minus 119910119899 = 1205721198991 minus 120573119899 120574119891 (119909119899) minus1205721198991 minus 120573119899119863119910119899 (72)

Hence

1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817 le 1205721198991205741 minus 1205731198991003816100381610038161003816119891 (119909119899)1003816100381610038161003816 + 1205721198991 minus 120573119899

10038171003817100381710038171198631199101198991003817100381710038171003817 (73)

From (C1) and (C2) one has


1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817 = 0 (74)

By (74) and (61) we have


1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (75)

Combining (71) and (75) one has


1003817100381710038171003817119906119899 minus 1199101198991003817100381710038171003817 = 0 (76)

Noting that

119910119899 minus 119906119899 = 119879119899119906119899 minus 119906119899 + 120572120574119899119889119899 (77)

one has

119879119899119906119899 minus 119906119899 = 119910119899 minus 119906119899 minus 120572120574119899119889119899 (78)

It follows from (76) and 120572119899 997888rarr 0 120574119899 = 119900(120572119899) thatlim119899997888rarrinfin

1003817100381710038171003817119879119899119906119899 minus 1199061198991003817100381710038171003817 = 0 (79)

We claim that


⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (80)

where 119902 is the unique solution of the variational inequality⟨(119863 minus 120574119891)119902 119909 minus 119902⟩ ge 0 forall119909 isin ΩTo show this inequality we choose a subsequence 119909119899119894 of119909119899 such that


⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩= lim119894997888rarrinfin

⟨(119863 minus 120574119891) 119902 119902 minus 119909119899119894⟩ (81)

Since 119909119899119894 is bounded there exists a subsequence of 119909119899119894which converges weakly to 119911 isin 119862 Without loss of generalitywe can assume that 119909119899119894 119911Step 5 (we show that 119911 isin Ω) First we show that 119911 isin VI(119862 119861)

Let119872 1198671 997888rarr 21198671 be a set-valued mapping defined by

119872V = 119861V + 119873119862V V isin 1198620 V notin 119862 (82)

where119873119862V fl 119911 isin 1198671 ⟨Vminus119906 119911⟩ ge 0 forall119906 isin 119862 is the normalcone to 119862 at V isin 119862 Then 119872 is maximal monotone and 0 isin119872V if and only if V isin VI(119862 119861) (see [36]) Let (V 119906) isin 119866(119872)Therefore we have

119906 isin 119872V = 119861V + 119873119862V (83)

and so

119906 minus 119861V isin 119873119862V (84)

According to 119906119899 isin 119862 we obtain⟨V minus 119906119899 119906 minus 119861V⟩ ge 0 (85)

On the other hand according to

119875119862 ((119868 minus 120582119899119861) 119906119899) = 119906119899 + 119887119899 (119879119899119906119899 minus 119906119899) (86)

where 119887119899 = (2120573 + 120582119899)4120573 for forall119899 isin 119873 and V isin 1198671 we have⟨V minus 119906119899 minus 119887119899 (119879119899119906119899 minus 119906119899) 119906119899 + 119887119899 (119879119899119906119899 minus 119906119899)minus (119906119899 minus 120582119899119861119906119899)⟩ ge 0 (87)

Therefore

⟨V minus 119906119899 119887119899 (119879119899119906119899 minus 119906119899) + 120582119899119861119906119899⟩ minus 1003817100381710038171003817119887119899 (119879119899119906119899 minus 119906119899)10038171003817100381710038172minus ⟨119887119899 (119879119899119906119899 minus 119906119899) 120582119899119861119906119899⟩ ge 0 (88)

As a result

⟨V minus 119906119899 119887119899120582119899 (119879119899119906119899 minus 119906119899) + 119861119906119899⟩minus ⟨119887119899 (119879119899119906119899 minus 119906119899) 119861119906119899⟩ ge 0

(89)


Furthermore according to (85) and (89) for forall119899 isin 119873 onehas

⟨V minus 119906119899 119906⟩ ge ⟨V minus 119906119899 119861V⟩minus ⟨V minus 119906119899 119887119899120582119899 (119879119899119906119899 minus 119906119899) + 119861119906119899⟩+ ⟨119887119899 (119879119899119906119899 minus 119906119899) 119861119906119899⟩

= ⟨V minus 119906119899 119861V minus 119861119906119899⟩minus ⟨V minus 119906119899 119887119899120582119899 (119879119899119906119899 minus 119906119899)⟩+ ⟨119887119899 (119879119899119906119899 minus 119906119899) 119861119906119899⟩

(90)

Replacing 119899 by 119899119894 one has⟨V minus 119906119899119894 119906⟩ ge ⟨V minus 119906119899119894 119861V minus 119861119906119899119894⟩

minus ⟨V minus 119906119899119894 119887119899119894120582119899119894 (119879119899119894119906119899119894 minus 119906119899119894)⟩+ ⟨119887119899119894 (119879119899119894119906119899119894 minus 119906119899119894) 119861119906119899119894⟩

(91)

Since 119906119899 minus 119909119899 997888rarr 0 and 119909119899119894 119911 we have 119906119899119894 119911 Notingthat 119861 is 120573-ism from (91) and (79) we have

⟨V minus 119911 119906⟩ ge 0 (92)

Since119872 is maximal monotone one has 119911 isin 119872minus10 Hence119911 isin 119881119868(119862 119861)Next we prove 119911 isin ΓAccording to Algorithm 13 we have

119906119899119894 = 119879(1198651 ℎ1)119903119899119894(119909119899119894 + 120585119860lowast (119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894) (93)

By (8) for any 119908 isin 119862 one has0 le 1198651 (119906119899119894 119908) + ℎ1 (119906119899119894 119908) + 1119903119899119894 ⟨119908 minus 119906119899119894 119906119899119894

minus (119909119899119894 + 120585119860lowast (119879(1198652ℎ2)119903119899119894minus 119868)119860119909119899119894)⟩ = 1198651 (119906119899119894 119908)

+ ℎ1 (119906119899119894 119908) + 1119903119899119894 ⟨119908 minus 119906119899119894 119906119899119894 minus 119909119899119894⟩ minus 120585119903119899119894 ⟨119860119908minus 119860119906119899119894 (119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894⟩ le 1198651 (119906119899119894 119908)+ ℎ1 (119906119899119894 119908) + 1119903119899119894

10038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817 + 12058511990311989911989410038171003817100381710038171003817119860119908

minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894minus 119868)119860119909119899119894100381710038171003817100381710038171003817

(94)

According to the monotonicity of 1198651 we have1198651 (119908 119906119899119894)

le ℎ1 (119906119899119894 119908) + 111990311989911989410038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817

+ 12058511990311989911989410038171003817100381710038171003817119860119908 minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894100381710038171003817100381710038171003817(95)

FromAssumption 2 (119894V) on 119865 and (119894119894) on ℎ (68) and (71)one has

1198651 (119908 119911) le ℎ1 (119911 119908) (96)

It follows from the monotonicity of ℎ1 that1198651 (119908 119911) + ℎ1 (119908 119911) le 0 forall119908 isin 119862 (97)

For any 119905 isin (0 1] and 119908 isin 119862 let 119908119905 = 119905119908 + (1 minus 119905)119911 Since119911 isin 119862 and 119862 is convex we obtain that 119908119905 isin 119862 Hence1198651 (119908119905 119911) + ℎ1 (119908119905 119911) le 0 (98)

From Assumption 2 (119894) (119894V) on 119865 and (119894) (119894119894119894) on ℎ we have0 le 1198651 (119908119905 119908119905) + ℎ1 (119908119905 119908119905)le 119905 [1198651 (119908119905 119908) + ℎ1 (119908119905 119908)]

+ (1 minus 119905) [1198651 (119908119905 119911) + ℎ1 (119908119905 119911)]le 119905 [1198651 (119908119905 119908) + ℎ1 (119908119905 119908)]

(99)

which implies that

1198651 (119908119905 119908) + ℎ1 (119908119905 119908) ge 0 forall119908 isin 119862 (100)

Letting 119905 997888rarr 0 and by Assumption 2 (119894119894119894) on 119865 and (119894119894) on ℎwe obtain

1198651 (119911 119908) + ℎ1 (119911 119908) ge 0 forall119908 isin 119862 (101)

that is 119911 isin GEP(1198651 ℎ1)As follows we prove 119860119911 isin GEP(1198652 ℎ2)Since119860 is a bounded linear operator one has119860119909119899119894 119860119911Now set 120577119899119894 = 119860119909119899119894minus119879(1198652ℎ2)119903119899119894

119860119909119899119894 It follows from (68) thatlim119894997888rarrinfin120577119899119894 = 0 Since119860119909119899119894 minus120577119899119894 = 119879(1198652ℎ2)119903119899119894

119860119909119899119894 by (9) we have1198652 (119860119909119899119894 minus 120577119899119894 ) + ℎ2 (119860119909119899119894 minus 120577119899119894 )

+ 1119903119899119894 ⟨ minus (119860119909119899119894 minus 120577119899119894) (119860119909119899119894 minus 120577119899119894) minus 119860119909119899119894⟩ge 0 forall isin 119876

(102)

Furthermore one has

1198652 (119860119909119899119894 minus 120577119899119894 ) + ℎ2 (119860119909119899119894 minus 120577119899119894 )+ 1119903119899119894 ⟨ minus 119860119909119899119894 + 120577119899119894 minus120577119899119894⟩ ge 0 forall isin 119876 (103)


From the upper semicontinuity of 1198652(119909 119910) and ℎ2(119909 119910)on 119909 we have

1198652 (119860119911 ) + ℎ2 (119860119911 ) ge 0 forall isin 119876 (104)

which means that 119860119911 isin GEP(1198652 ℎ2) As a result 119911 isin ΓTherefore 119911 isin ΩSince 119902 = 119875Ω(119868 minus 119863 + 120574119891)119902 is the unique solution of the

variational inequality problem ⟨(119863minus120574119891)119902 119909minus119902⟩ ge 0 forall119909 isin Ωby (81) and 119911 isin Ω we have


⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (105)

Step 6 (finally we show that 119909119899 converges strongly to 119902) It isobvious that1003817100381710038171003817119910119899 minus 1199021003817100381710038171003817 = 1003817100381710038171003817119879119899119906119899 minus 119902 + 1205721205741198991198891198991003817100381710038171003817

le 1003817100381710038171003817119879119899119906119899 minus 1198791198991199021003817100381710038171003817 + 120572120574119899 10038171003817100381710038171198891198991003817100381710038171003817le 1003817100381710038171003817119906119899 minus 1199021003817100381710038171003817 + 1205721198721120574119899 = 1003817100381710038171003817119906119899 minus 1199021003817100381710038171003817 +1198724120574119899le 1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817 +1198724120574119899

(106)

where1198724 = 1205721198721 And the first inequality is true because 119902 isinΩ and 119879119899119902 = 119902 according to the same reasoning to equality(50) The last inequality is obtained by 119902 isin Ω and the samereasoning to inequality (47)Thus fromLemma 1 (119894) we have1003817100381710038171003817119909119899+1 minus 11990210038171003817100381710038172 = 1003817100381710038171003817120572119899120574119891 (119909119899) + 120573119899119909119899

+ ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899 minus 11990210038171003817100381710038172= 1003817100381710038171003817120572119899 (120574119891 (119909119899) minus 119863119902) + 120573119899 (119909119899 minus 119902)+ ((1 minus 120573119899) 119868 minus 120572119899119863) (119910119899 minus 119902)10038171003817100381710038172 le 1003817100381710038171003817120573119899 (119909119899 minus 119902)+ ((1 minus 120573119899) 119868 minus 120572119899119863) (119910119899 minus 119902)10038171003817100381710038172 + 2120572119899 ⟨120574119891 (119909119899)minus 119863119902 119909119899+1 minus 119902⟩ le (120573119899 1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817+ (1 minus 120573119899 minus 120572119899120574) 1003817100381710038171003817119910119899 minus 119902)1003817100381710038171003817)2 + 2120572119899120574 ⟨119891 (119909119899)minus 119891 (119902) 119909119899+1 minus 119902⟩ + 2120572119899 ⟨120574119891 (119902) minus 119863119902 119909119899+1 minus 119902⟩le ((1 minus 120572119899120574) 1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817 + (1 minus 120573119899 minus 120572119899120574)1198724120574119899)2+ 2120572119899120574120578 1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817 1003817100381710038171003817119909119899+1 minus 1199021003817100381710038171003817 + 2120572119899 ⟨120574119891 (119902)minus 119863119902 119909119899+1 minus 119902⟩ le (1 minus 120572119899120574)2 1003817100381710038171003817119909119899 minus 11990210038171003817100381710038172 +1198725120574119899+ 120572119899120574120578 1003817100381710038171003817119909119899 minus 11990210038171003817100381710038172 + 120572119899120574120578 1003817100381710038171003817119909119899+1 minus 11990210038171003817100381710038172+ 2120572119899 ⟨120574119891 (119902) minus 119863119902 119909119899+1 minus 119902⟩

(107)

where1198725 = sup119899isin1198732(1 minus 120572119899120574)(1 minus 120573119899 minus 120572119899120574)1198724119909119899 minus 119902 +(1 minus 120573119899 minus 120572119899120574)211987224120574119899As a result

(1 minus 120572119899120574120578) 1003817100381710038171003817119909119899+1 minus 11990210038171003817100381710038172le (1 minus 2120572119899120574 + 12057221198991205742 + 120572119899120574120578) 1003817100381710038171003817119909119899 minus 11990210038171003817100381710038172 +1198725120574119899

+ 2120572119899 ⟨120574119891 (119902) minus 119863119902 119909119899+1 minus 119902⟩ (108)

which implies that

1003817100381710038171003817119909119899+1 minus 11990210038171003817100381710038172 le (1 minus 2 (120574 minus 120574120578) 1205721198991 minus 120572119899120574120578 ) 1003817100381710038171003817119909119899 minus 11990210038171003817100381710038172

+ 2 (120574 minus 120574120578) 1205721198991 minus 120572119899120574120578 (1205721198991205742 1003817100381710038171003817119909119899 minus 11990210038171003817100381710038172 +1198725 (120574119899120572119899)2 (120574 minus 120574120578)+ 1120574 minus 120574120578 ⟨120574119891 (119902) minus 119863119902 119909119899+1 minus 119902⟩) = (1 minus 119905119899)sdot 1003817100381710038171003817119909119899 minus 11990210038171003817100381710038172 + 119905119899120575119899

(109)

where 119905119899 = 2(120574 minus 120574120578)120572119899(1 minus 120572119899120574120578) 120575119899 = (1205721198991205742119909119899 minus 1199022 +1198725(120574119899120572119899))2(120574 minus 120574120578) + (1(120574 minus 120574120578))⟨120574119891(119902) minus 119863119902 119909119899+1 minus 119902⟩According to (80) (1198621) (1198623) and 120578120574 lt 120574 we havesuminfin119899=0 119905119899 = infin and lim sup119899997888rarrinfin120575119899 le 0By Lemma 7 119909119899 997888rarr 119902 which completes the proof

4 Consequently Results

In the above section we discuss the iterative algorithm andprove the strong convergence theorem for finding a commonsolution of split generalized equilibrium and variationalinequality problems In this section we give some corollarieswhich can find a common solution of the special issuesobtained from split generalized equilibrium and variationalinequality problems

If ℎ1 = ℎ2 = 0 then SGEP (5)-(6) reduces to the followingsplit equilibrium problem (SEP)

Let 1198651 119862 times 119862 997888rarr 119877 and 1198652 119876 times 119876 997888rarr 119877 be nonlinearbifunctions and119860 1198671 997888rarr 1198672 be a bounded linear operatorthen SEP is to find 119909lowast isin 119862 such that

1198651 (119909lowast 119909) ge 0 forall119909 isin 119862 (110)

and such that

119910lowast = 119860119909lowast isin 119876 solves 1198652 (119910lowast 119910) ge 0 forall119910 isin 119876 (111)

The solution set of SEP (110)-(111) is denoted by Γ1 And1198791198651119903 (119909) = 119911 isin 119862 1198651 (119911 119910) + 1119903 ⟨119910 minus 119911 119911 minus 119909⟩

ge 0 forall119910 isin 119862 (112)

1198791198652119904 (119908) = 119889 isin 119876 1198652 (119889 119890) + 1119904 ⟨119890 minus 119889 119889 minus 119908⟩ge 0 forall119890 isin 119876

(113)

According toTheorem 14 we can obtain the following corol-lary

Corollary 15 Let 1198671 and 1198672 be two real Hilbert spaces and119862 sub 1198671 119876 sub 1198672 be nonempty closed convex subsets Let 1198651 119862 times 119862 997888rarr 119877 and 1198652 119876 times 119876 997888rarr 119877 satisfy Assumption 2 and1198652 is upper semicontinuous in the first argument Assume that


Ω fl VI(119862 119861) cap Γ1 = 0 1199090 isin 1198671 and 119906119899 119910119899 and 119909119899 aregenerated by the following iterative scheme

119906119899 = 1198791198651119903119899 (119909119899 + 120585119860lowast (1198791198652119903119899 minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(114)

where 119889119899+1 = (1120572)(119879119899119906119899 minus 119906119899) + 120574119899119889119899 1198890 = (1120572)(11987901199060 minus 1199060)120572 gt 0 and 119879119899 is defined by (34) Suppose that 120572119899 120573119899 sub(0 1) 120574119899 sub (0 12) 120582119899 sub (0 2120573) 119903119899 sub (0infin)satisfy conditions (C1)-(C6) ineorem 14en sequence 119909119899converges strongly to 119902 isin Ω where 119902 = 119875Ω(119868 minus119863+120574119891)119902 whichis the unique solution of the variational inequality problem



min119909isin119876

12 ⟨119863119909 119909⟩ minus ℎ (119909) (116)

where ℎ is a potential function for 120574119891 such that ℎ1015840(119909) = 120574119891(119909)for 119909 isin 1198671

Furthermore if 1198651 = 1198652 = 119865 1198671 = 1198672 = 119867 and 119860 =0 then SEP (110)-(111) reduce to the following equilibriumproblem (EP)

Let 119865 119862 times 119862 997888rarr 119877 be nonlinear bifunction then EP isto find 119909lowast isin 119862 such that


The solution set of EP (117) is denoted by Γ2 And119879119865119903 (119909) = 119911 isin 119862 119865 (119911 119910) + 1119903 ⟨119910 minus 119911 119911 minus 119909⟩

ge 0 forall119910 isin 119862 (118)

According to Corollary 15 let 119863 = 119868 and we can obtain thefollowing corollary

Corollary 16 Let 119867 be real Hilbert space and 119862 sub 119867 benonempty closed convex subset Let 119865 119862 times 119862 997888rarr 119877 satisfyAssumption 2 Assume that Ω fl VI(119862 119861) cap Γ2 = 0 1199090 isin 1198671and 119906119899 119910119899 and 119909119899 are generated by the following iterativescheme

119906119899 = 119879119865119903119899 (119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + (1 minus 120573119899 minus 120572119899) 119910119899(119)

where 119889119899+1 = (1120572)(119879119899119906119899 minus 119906119899) + 120574119899119889119899 1198890 = (1120572)(11987901199060 minus 1199060)120572 gt 0 and 119879119899 is defined by (34) Suppose that 120572119899 120573119899 sub(0 1) 120574119899 sub (0 12) 120582119899 sub (0 2120573) 119903119899 sub (0infin)satisfy conditions (C1)-(C6) ineorem 14en sequence 119909119899converges strongly to 119902 isin Ω where 119902 = 119875Ω120574119891(119902)

Table 1 The final value and the cpu time for different initial points

Init Fina Sec119909 = (07060 00318)119879 119909 = (49924 39939)119879 114119909 = (54722 13862)119879 119909 = (49924 39939)119879 135119909 = (890903 959291)119879 119909 = (49924 39939)119879 1365 Numerical Examples

In this section we show some insight into the behavior ofAlgorithm 13 The whole codes are written in Matlab 70 Allthe numerical results are carried out on a personal LenovoThinkpad computer with Intel(R) Core(TM) i7-6500U CPU250GHz and RAM 800GB

Example 1 In the variational inequality problem (3) as well asthe split generalized equilibriumproblem (5) and (6) let1198671 =1198672 = 1198772 119862 = (1199091 1199092)119879 isin 1198772 | 1 le 1199091 le 5 05 le 1199092 le 4119876 = (1199101 1199102)119879 isin 1198772 | 2 le 1199101 le 10 3 le 1199102 le 24 1198651(119909 119910) =ℎ1(119909 119910) = 119890119879(119909minus119910) forall119909 119910 isin 119862 1198652(119909 119910) = ℎ2(119909 119910) = 5119890119879(119909minus119910) forall119909 119910 isin 119876 and119860 = ( 2 00 6 )119861 = ( 12 0

0 12) where 119890 = (1 1)119879

It is easy to see that the optimal solution of Example 1is 119909lowast = (5 4)119879 Now we use Algorithm 13 to compute thesolution of the problem Let120572 = 12 120585 = 17119863 = ( 910 0

0 910)120574 = 12 119903119899 = 110 + 12119899 120582119899 = 1119899 120572119899 = 1radic119899 120573119899 =110 + 11198992 120574119899 = 11198992 and 119891(119909) = 1199092 forall119909 isin 1198671 The

stopping criterion is 119909119899+1 minus 119909119899 le 120598 For 120598 = 10minus7 anddifferent initial points which are presented randomly such as

119909 = rand (2 1) 119909 = 10 lowast rand (2 1) 119909 = 1000 lowast rand (2 1)

(120)

separately Table 1 shows the initial value the final value andthe cpu time for the above three cases We denote Init Finaand Sec the initial value the final value and the cpu time inseconds respectively

From Table 1 we can see that the final value 119909 is notinfluenced by the initial value

To show the changing tendency of the final value 119909 fordifferent 120598 Table 2 gives the different 120598 the initial value thefinal value and the cpu time in seconds

Now we further express the status that the final value 119909tends to optimal solution 119909lowast through Figures 1 and 2 Wecarry out 100 experiments for different 120598 from 10minus7 to 10minus9From Figures 1 and 2 we know that the final value 119909 tends tothe optimal solution 119909lowast when 120598 tends to 0 which illustratesthe efficiency of Algorithm 13

6 Conclusion

In this paper we study the split generalized equilibriumproblem and variational inequality problem For findingtheir common solution we propose a kind of conjugategradient viscosity approximation algorithm Under mildconditions we prove that the sequence generated by the


Table 2 The final value and the cpu time for different 120598120598 Init Fina Sec10minus5 119909 = (80028 14189)119879 119909 = (49640 39712)119879 00810minus6 119909 = (74313 39223)119879 119909 = (49835 39868)119879 03010minus7 119909 = (54722 13862)119879 119909 = (49924 39939)119879 135

02 04 06 08 10the value of epsilon x 10minus7

4992

4993

4994

4995

4996

4997

4998

4999

the fi

rst c

oord

inat

e of fi

nal v

alue

Figure 1 The behaviors of first coordinate of 119909 for different 120598

02 04 06 08 10the value of epsilon

3993

3994

3995

3996

3997

3998

3999

the s

econ

d co

ordi

nate

of fi

nal v

alue

x 10minus7

Figure 2 The behaviors of the second coordinate of 119909 for different120598

iterative algorithm converges strongly to the common solu-tion In comparison to [25] the authors introduce and studyan effective iterative algorithm to approximate a commonsolution of a split generalized equilibrium problem and afixed point problem Under suitable conditions they proveda strong convergence theorem for the sequence generatedby the iterative scheme Furthermore we can study theiterative algorithm for finding a common solution of thesplit generalized equilibrium problem variational inequalityproblem and fixed point problem

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This project is supported by the Natural Science Foundationof China (Grants nos 11401438 and 11571120) and Shan-dong Provincial Natural Science Foundation (Grants nosZR2013FL032 and ZR2017LA002) and the Project of Shan-dong Province Higher Educational Science and TechnologyProgram (Grant no J14LI52)

References

[1] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquoNumerical Algorithmsvol 8 no 2ndash4 pp 221ndash239 1994

[2] H Che and M Li ldquoThe conjugate gradient method for splitvariational inclusion and constrained convex minimizationproblemsrdquo Applied Mathematics and Computation vol 290 pp426ndash438 2016

[3] S Y Cho W Li and S M Kang ldquoConvergence analysis of aniterative algorithm for monotone operatorsrdquo Journal of Inequal-ities and Applications 2013199 14 pages 2013

[4] P L Combettes ldquoSolving monotone inclusions via compo-sitions of nonexpansive averaged operatorsrdquo Optimization AJournal of Mathematical Programming and Operations Researchvol 53 no 5-6 pp 475ndash504 2004

[5] F Facchinei and J-S Pang Finite-dimensional variationalinequalities and complementarity problems vol 1 Springer NewYork NY USA 2003

[6] N Fang and Y Gong ldquoViscosity iterative methods for splitvariational inclusion problems and fixed point problems ofa nonexpansive mappingrdquo Communications in Optimizationeory vol 2016 15 pages 2016

[7] Meixia Li Xiping Kao and Haitao Che ldquoA simultaneousiteration algorithm for solving extended split equality fixedpoint problemrdquo Mathematical Problems in Engineering vol2017 Article ID 9737062 9 pages 2017

[8] B Liu B Qu and N Zheng ldquoA successive projection algorithmfor solving the multiple-sets split feasibility problemrdquo Numer-ical Functional Analysis and Optimization vol 35 no 11 pp1459ndash1466 2014

[9] HMahdioui andOChadli ldquoOn a SystemofGeneralizedMixedEquilibrium Problems Involving Variational-Like Inequali-ties in Banach Spaces Existence and Algorithmic Aspectsrdquo


Advances in Operations Research vol 2012 Article ID 84348618 pages 2012

[10] X Qin B A Bin Dehaish and S Y Cho ldquoViscosity splittingmethods for variational inclusion and fixed point problems inHilbert spacesrdquo Journal of Nonlinear Sciences and ApplicationsJNSA vol 9 no 5 pp 2789ndash2797 2016

[11] B Qu and H Chang ldquoRemark on the successive projec-tion algorithm for the multiple-sets split feasibility problemrdquoNumerical Functional Analysis and Optimization vol 38 no 12pp 1614ndash1623 2017

[12] B Qu B Liu and N Zheng ldquoOn the computation of the step-size for the CQ-like algorithms for the split feasibility problemrdquoApplied Mathematics and Computation vol 262 pp 218ndash2232015

[13] X Wang ldquoAlternating proximal penalization algorithm for themodified multiple-sets split feasibility problemsrdquo Journal ofInequalities and Applications vol 48 pp 1ndash8 2018

[14] M Zhang ldquoStrong convergence of a viscosity iterative algorithmin Hilbert spacesrdquo Journal of Nonlinear Functional Analysis vol2014 16 pages 2014

[15] H Zhang and Y Wang ldquoA new CQ method for solving splitfeasibility problemrdquo Frontiers of Mathematics in China vol 5no 1 pp 37ndash46 2010

[16] W Takahashi andM Toyoda ldquoWeak convergence theorems fornonexpansive mappings and monotone mappingsrdquo Journal ofOptimization eory and Applications vol 118 no 2 pp 417ndash428 2003

[17] A S Antipin ldquoMethods for solving variational inequalities withrelated constraintsrdquo Computational Mathematics and Mathe-matical Physics vol 40 no 9 pp 1291ndash1307 2000

[18] H Che Y Wang and M Li ldquoA smoothing inexact Newtonmethod for 1198750 nonlinear complementarity problemrdquo Frontiersof Mathematics in China vol 7 no 6 pp 1043ndash1058 2012

[19] H Chen and Y Wang ldquoA family of higher-order convergentiterative methods for computing the Moore-Penrose inverserdquoAppliedMathematics and Computation vol 218 no 8 pp 4012ndash4016 2011

[20] GMKorpelevich ldquoAn extragradientmethod for finding saddlepoints and for other problemsrdquo Ekonomika i MatematicheskieMetody vol 12 no 4 pp 747ndash756 1976

[21] S Lian and L Zhang ldquoA simple smooth exact penalty functionfor smooth optimization problemrdquo Journal of Systems Science ampComplexity vol 25 no 3 pp 521ndash528 2012

[22] M Sun Y Wang and J Liu ldquoGeneralized Peaceman-Rachfordsplitting method for multiple-block separable convex program-ming with applications to robust PCArdquo Calcolo A Quarterly onNumerical Analysis and eory of Computation vol 54 no 1pp 77ndash94 2017

[23] G Wang and H-t Che ldquoGeneralized strict feasibility andsolvability for generalized vector equilibrium problem with set-valued map in reflexive Banach spacesrdquo Journal of Inequalitiesand Applications vol 66 pp 1ndash11 2012

[24] B Zhong H Dong L Kong and J Tao ldquoGeneralized nonlinearcomplementarity problems with order 1198750 and 1198770 propertiesrdquoPositivity An International Mathematics Journal Devoted toeory and Applications of Positivity vol 18 no 2 pp 413ndash4232014

[25] K R Kazmi and S H Rizvi ldquoIterative approximation of acommon solution of a split generalized equilibrium problemand a fixed point problem for nonexpansive semigrouprdquoMath-ematical Sciences vol 7 pp 1ndash10 2013

[26] S Chang ldquoSome problems and results in the study of nonlinearanalysisrdquo Nonlinear Analysis vol 30 no 7 pp 4197ndash4208 1997

[27] K Shimoji and W Takahashi ldquoStrong convergence to commonfixed points of infinite nonexpansive mappings and applica-tionsrdquo Taiwanese Journal of Mathematics vol 5 no 2 pp 387ndash404 2001

[28] T Suzuki ldquoStrong convergence theorems for infinite families ofnonexpansive mappings in general Banach spacesrdquo Fixed Pointeory and Applications vol 2005 no 1 pp 103ndash123 2005

[29] G Marino and H K Xu ldquoA general iterative method for non-expansive mappings in Hilbert spacesrdquo Journal of MathematicalAnalysis and Applications vol 318 no 1 pp 43ndash52 2006

[30] L S Liu ldquoIshikawa and Mann iterative process with errors fornonlinear strongly accretive mappings in Banach spacesrdquo Jour-nal of Mathematical Analysis and Applications vol 194 no 1 pp114ndash125 1995

[31] H K Xu ldquoViscosity approximation methods for nonexpansivemappingsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 279ndash291 2004

[32] C Byrne ldquoA unified treatment of some iterative algorithms insignal processing and image reconstructionrdquo Inverse Problemsvol 20 no 1 pp 103ndash120 2004

[33] C Martinez-Yanes and H Xu ldquoStrong convergence of the CQmethod for fixed point iteration processesrdquoNonlinear Analysiseory Methods amp Applications vol 64 no 11 pp 2400ndash24112006

[34] C Byrne Y Censor A Gibali and S Reich ldquoThe split commonnull point problemrdquo Journal of Nonlinear and Convex Analysisvol 13 no 4 pp 759ndash775 2012

[35] A Moudafi ldquoThe split common fixed-point problem for demi-contractive mappingsrdquo Inverse Problems vol 26 6 pages 2010

[36] R T Rockafellar ldquoMonotone operators and the proximal pointalgorithmrdquo SIAM Journal on Control and Optimization vol 14no 5 pp 877ndash898 1976

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where 119861 is 120588-inverse-strongly monotone 120572119899 is a sequencein (0 1) and 120582119899 is a sequence in (0 2120588) They showed thatif Fix(119879) capVI(119862 119861) = 0 then the sequence 119909119899 generated by(4) converges weakly to 1199110 isin Fix(119879) cap VI(119862 119861)

On the other hand there are several numerical methodsfor solving variational inequalities and related optimizationproblems see [5 17ndash24] and the references therein

In 2013 Kazmi and Rizvi [25] introduced the splitgeneralized equilibrium problem (SGEP) Let 1198651 ℎ1 119862 times119862 997888rarr 119877 and 1198652 ℎ2 119876 times 119876 997888rarr 119877 be nonlinear bifunctionsand 119860 1198671 997888rarr 1198672 be a bounded linear operator thenthe split generalized equilibrium problem (SGEP) is to find119909lowast isin 119862 such that

1198651 (119909lowast 119909) + ℎ1 (119909lowast 119909) ge 0 forall119909 isin 119862 (5)

and such that119910lowast = 119860119909lowast isin 119876 solves 1198652 (119910lowast 119910) + ℎ2 (119910lowast 119910) ge 0

forall119910 isin 119876 (6)

Denote the solution sets of generalized equilibrium prob-lem (GEP) (5) and GEP (6) by GEP(1198651 ℎ1) and GEP(1198652 ℎ2)respectively The solution set of SGEP is denoted by Γ =119909lowast isin GEP(1198651 ℎ1) 119860119909lowast isin GEP(1198652 ℎ2) They proposed thefollowing iterative method for finding a common solution ofsplit generalized equilibrium and fixed point problem

Let 119909119899 and 119906119899 be the sequences generated by 1199090 isin 119862and

119906119899 = 119879(1198651ℎ1)119903119899(119909119899 + 120575119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899) 119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899

+ ((1 minus 120573119899) 119868 minus 120572119899119863) 1119904119899 int119904119899

0

119879 (119904) 119906119899119889119904(7)

where 119878 = 119879(119904) 0 le 119904 lt infin is a nonexpansive semigroupon 119862 and 119865119894119909(119878) cap Γ = 0 119904119899 is a positive real sequence whichdiverges to +infin 120572119899 120573119899 sub (0 1) 119903119899 sub (0infin) and 120575 isin(0 1119871) 119871 is the spectral radius of the operator 119860lowast119860 119860lowast isthe adjoint of 119860 and


(8)


(9)

Under suitable conditions they proved a strong conver-gence theorem for the sequence generated by the proposediterative scheme But the calculation of integral is generallynot easyTherefore it is necessary to reconsider the algorithmfor solving this kind of problem

Motivated by Kazmi and Rivi [25] as well as Che andLi [2] we introduce and study a kind of conjugate gradi-ent viscosity approximation algorithm for finding a com-mon solution of split generalized equilibrium problem and

variational inequality problem Under mild conditions weprove that the sequence generated by the proposed iterativealgorithm converges strongly to the common solution ofVI(119862 119861) and SGEPThe results presented in this paper are thegeneralization extension and supplement of the previouslyknown results in the corresponding references Numericalresults show the feasibility and efficiency of the proposedalgorithm

2 Preliminaries

In this section we introduce some concepts and results whichare needed in sequel

A mapping 119879 1198671 997888rarr 1198671 is called(1) contraction if there exists a constant 120572 isin (0 1) such

that1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 le 120572 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 1198671 (10)

If 120572 = 1 then 119879 is called nonexpansive(2) monotone if

⟨119879119909 minus 119879119910 119909 minus 119910⟩ ge 0 forall119909 119910 isin 1198671 (11)

(3) 120578-stronglymonotone if there exists a positive constant120578 such that


(4) 120572-inverse strongly monotone (120572-ism) if there exists aconstant 120572 gt 0 such that


(5) firmly nonexpansive if

⟨119879119909 minus 119879119910 119909 minus 119910⟩ ge 1003817100381710038171003817119879119909 minus 11987911991010038171003817100381710038172 forall119909 119910 isin 1198671 (14)

Amapping119875119862 is said to bemetric projection of1198671 onto119862if for every point 119909 isin 1198671 there exists a unique nearest pointin 119862 denoted by 119875119862119909 such that

1003817100381710038171003817119909 minus 1198751198621199091003817100381710038171003817 le 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119910 isin 119862 (15)

It is well known that 119875119862 is a nonexpansive mapping and ischaracterized by the following properties

1003817100381710038171003817119875119862119909 minus 11987511986211991010038171003817100381710038172 le ⟨119909 minus 119910 119875119862119909 minus 119875119862119910⟩ forall119909 119910 isin 1198671 (16)

⟨119909 minus 119875119862119909 119910 minus 119875119862119909⟩ le 0 forall119909 isin 1198671 119910 isin 119862 (17)

1003817100381710038171003817119909 minus 11991010038171003817100381710038172 ge 1003817100381710038171003817119909 minus 11987511986211990910038171003817100381710038172 + 1003817100381710038171003817119910 minus 11987511986211990910038171003817100381710038172 forall119909 isin 1198671 119910 isin 119862 (18)

and1003817100381710038171003817(119909 minus 119910) minus (119875119862119909 minus 119875119862119910)10038171003817100381710038172

ge 1003817100381710038171003817119909 minus 11991010038171003817100381710038172 minus 1003817100381710038171003817119875119862119909 minus 11987511986211991010038171003817100381710038172 forall119909 119910 isin 1198671(19)



⟨119861119909 119909⟩ ge 120574 1199092 forall119909 isin 1198671 (20)


119879 = (1 minus 120572) 119868 + 120572119878 (21)







(23)

(iii) For 119886 119887 119888 isin [0 1] with 119886 + 119887 + 119888 = 11003817100381710038171003817119886119909 + 119887119910 + 11988811991110038171003817100381710038172 = 119886 1199092 + 119887 100381710038171003817100381711991010038171003817100381710038172 + 119888 1199112


(24)




119865 (119905119911 + (1 minus 119905) 119909 119910) le 119865 (119909 119910) (25)







(26)



(27)








(28)



119886119899+1 le (1 minus 120574119899) 119886119899 + 120574119899120575119899 + 120573119899 119899 ge 0 (30)





10038171003817100381710038171003817119879(1198651 ℎ1)1199032119910 minus 119879(1198651ℎ1)1199031

11990910038171003817100381710038171003817 le 1003817100381710038171003817119910 minus 1199091003817100381710038171003817+ 10038161003816100381610038161003816100381610038161003816

1199032 minus 119903111990321003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817119879(1198651ℎ1)1199032

119910 minus 11991010038171003817100381710038171003817 (31)









119873⋂119894=1

Fix (119879119894) = Fix (1198791 119879119873) (32)



12 + 1205822120573 minus 12 sdot 1205822120573 = 2120573 + 1205824120573 (33)




(34)






119906119899 = 119879(1198651ℎ1)119903119899(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(35)









min119909isinΩ

12 ⟨119863119909 119909⟩ minus ℎ (119909) (37)



119863 = sup |⟨119863119909 119909⟩| 119909 isin 1198671 119909 = 1 (38)

Notice that




(40)



(41)




(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899minus 119868)119860119909119899)


minus 119868)119860119909119899)minus 119879(1198651ℎ1)119903119899




(42)


+ 1205852 ⟨(119879(1198652ℎ2)119903119899minus 119868)119860119909119899 119860119860lowast (119879(1198652ℎ2)119903119899


minus 119868)119860119909119899⟩ (43)


1205852 ⟨(119879(1198652ℎ2)119903119899minus 119868)119860119909119899 119860119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899⟩le 1198711205852 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899

minus 119868)119860119909119899100381710038171003817100381710038172 (44)

and




minus (119879(1198652ℎ2)119903119899minus 119868)119860119909119899 (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899⟩= 2120585 ⟨119879(1198652ℎ2)119903119899


minus 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

le 2120585 12 10038171003817100381710038171003817(119879(1198652 ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

minus 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

le minus120585 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

(45)


+ 120585 (119871120585 minus 1) 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

(46)





(48)




that is

119879119899119909lowast = 119909lowast (50)



(51)




(52)







(119909119899 +120585119860lowast(119879(1198652ℎ2)119903119899minus119868)119860119909119899)

and 119906119899+1 = 119879(1198651ℎ1)119903119899+1(119909119899+1 + 120585119860lowast(119879(1198652ℎ2)119903119899+1


le 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1(119909119899+1 + 120585119860lowast (119879(1198652ℎ2)119903119899+1

minus 119868)119860119909119899+1)minus 119879(1198651ℎ1)119903119899+1

(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899+1minus 119868)119860119909119899)10038171003817100381710038171003817

+ 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899+1

minus 119868)119860119909119899)minus 119879(1198651ℎ1)119903119899

(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899minus 119868)119860119909119899)10038171003817100381710038171003817 le 1003817100381710038171003817119909119899+1



100381610038161003816100381610038161003816100381610038161 minus119903119899119903119899+1

10038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1

(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899+1minus 119868)119860119909119899)


+ 120585 119860 10038171003817100381710038171003817119879(1198652 ℎ2)119903119899+1119860119909119899 minus 119879(1198652ℎ2)119903119899

11986011990911989910038171003817100381710038171003817 + 120577119899 le 1003817100381710038171003817119909119899+1minus 1199091198991003817100381710038171003817 + 120585 119860 100381610038161003816100381610038161003816100381610038161 minus

119903119899119903119899+11003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817119879(1198652ℎ2)119903119899+1

119860119909119899 minus 11986011990911989910038171003817100381710038171003817 + 120577119899= 1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 + 120585 119860 120590119899 + 120577119899

(54)


119903119899119903119899+1|119879(1198651ℎ1)119903119899+1(119909119899+120585119860lowast(119879(1198652ℎ2)119903119899



+ 120572 1003817100381710038171003817120574119899+1119889119899+1 minus 1205741198991198891198991003817100381710038171003817le 1003817100381710038171003817119879119899+1119906119899+1 minus 119879119899+11199061198991003817100381710038171003817

+ 1003817100381710038171003817119879119899+1119906119899 minus 1198791198991199061198991003817100381710038171003817+ 120572 1003817100381710038171003817120574119899+1119889119899+1 minus 1205741198991198891198991003817100381710038171003817

le 1003817100381710038171003817119906119899+1 minus 1199061198991003817100381710038171003817 + 1003817100381710038171003817119879119899+1119906119899 minus 1198791198991199061198991003817100381710038171003817+ 1205721198721 (120574119899+1 + 120574119899)

(55)







1003817100381710038171003817100381710038171003817100381710038174120573119875119862 (119868 minus 120582119899+1119861)2120573 + 120582119899+1 119906119899


100381710038171003817100381710038171003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817


100381710038171003817100381710038171003817100381710038171003817 +100381710038171003817100381710038171003817100381710038171003817

4120573 (120582119899+1 minus 120582119899)(2120573 + 120582119899+1) (2120573 + 120582119899)119906119899100381710038171003817100381710038171003817100381710038171003817

le 1003817100381710038171003817100381710038171003817100381710038174120573 (120582119899 minus 120582119899+1) 119875119862 (119868 minus 120582119899+1119861)(2120573 + 120582119899+1) (2120573 + 120582119899) 119906119899

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817


+ 1003817100381710038171003817100381710038171003817100381710038174120573 (120582119899+1 minus 120582119899)(2120573 + 120582119899+1) (2120573 + 120582119899)119906119899

100381710038171003817100381710038171003817100381710038171003817 le1120573 1003816100381610038161003816120582119899 minus 120582119899+11003816100381610038161003816


(56)





As a result

1003817100381710038171003817119911119899+1 minus 1199111198991003817100381710038171003817= 100381710038171003817100381710038171003817100381710038171003817


100381710038171003817100381710038171003817100381710038171003817

= 100381710038171003817100381710038171003817100381710038171003817120572119899+1 (120574119891 (119909119899+1) minus 119863119910119899+1)1 minus 120573119899+1


le 120572119899+11 minus 120573119899+11003817100381710038171003817120574119891 (119909119899+1) minus 119863119910119899+11003817100381710038171003817


le 120572119899+11 minus 120573119899+11003817100381710038171003817120574119891 (119909119899+1) minus 119863119910119899+11003817100381710038171003817


sdot 120590119899 + 120577119899 +1198722 1003816100381610038161003816120582119899+1 minus 1205821198991003816100381610038161003816 + 1205721198721 (120574119899+1 + 120574119899) (59)






1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (61)



(1 minus 120573119899) 1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (62)





(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899minus 119868)119860119909119899)





minus 119868)119860119909119899⟩

(63)



minus 119868)11986011990911989910038171003817100381710038171003817 (64)


Furthermore


(65)




2 le 120572119899120574 11205742 1003817100381710038171003817120574119891 (119909119899)





(66)

As a result




(67)


10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)11986011990911989910038171003817100381710038171003817 = 0 (68)








(69)

Therefore one has







(70)



1003817100381710038171003817119906119899 minus 1199091198991003817100381710038171003817 = 0 (71)



Hence


10038171003817100381710038171198631199101198991003817100381710038171003817 (73)



1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817 = 0 (74)



1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (75)



1003817100381710038171003817119906119899 minus 1199101198991003817100381710038171003817 = 0 (76)

Noting that

119910119899 minus 119906119899 = 119879119899119906119899 minus 119906119899 + 120572120574119899119889119899 (77)

one has



1003817100381710038171003817119879119899119906119899 minus 1199061198991003817100381710038171003817 = 0 (79)

We claim that


⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (80)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899119894⟩ (81)



119872V = 119861V + 119873119862V V isin 1198620 V notin 119862 (82)


119906 isin 119872V = 119861V + 119873119862V (83)

and so

119906 minus 119861V isin 119873119862V (84)



119875119862 ((119868 minus 120582119899119861) 119906119899) = 119906119899 + 119887119899 (119879119899119906119899 minus 119906119899) (86)


Therefore


As a result


(89)





(90)



(91)


⟨V minus 119911 119906⟩ ge 0 (92)


119906119899119894 = 119879(1198651 ℎ1)119903119899119894(119909119899119894 + 120585119860lowast (119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894) (93)




minus 119868)119860119909119899119894⟩ le 1198651 (119906119899119894 119908)+ ℎ1 (119906119899119894 119908) + 1119903119899119894

10038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817 + 12058511990311989911989410038171003817100381710038171003817119860119908

minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894minus 119868)119860119909119899119894100381710038171003817100381710038171003817

(94)


le ℎ1 (119906119899119894 119908) + 111990311989911989410038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817

+ 12058511990311989911989410038171003817100381710038171003817119860119908 minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894100381710038171003817100381710038171003817(95)


1198651 (119908 119911) le ℎ1 (119911 119908) (96)




+ (1 minus 119905) [1198651 (119908119905 119911) + ℎ1 (119908119905 119911)]le 119905 [1198651 (119908119905 119908) + ℎ1 (119908119905 119908)]

(99)

which implies that

1198651 (119908119905 119908) + ℎ1 (119908119905 119908) ge 0 forall119908 isin 119862 (100)


1198651 (119911 119908) + ℎ1 (119911 119908) ge 0 forall119908 isin 119862 (101)





(102)

Furthermore one has




1198652 (119860119911 ) + ℎ2 (119860119911 ) ge 0 forall isin 119876 (104)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (105)



(106)



(107)



+ 2120572119899 ⟨120574119891 (119902) minus 119863119902 119909119899+1 minus 119902⟩ (108)

which implies that



(109)







and such that



ge 0 forall119910 isin 119862 (112)


(113)





119906119899 = 1198791198651119903119899 (119909119899 + 120585119860lowast (1198791198652119903119899 minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(114)




min119909isin119876

12 ⟨119863119909 119909⟩ minus ℎ (119909) (116)






ge 0 forall119910 isin 119862 (118)



119906119899 = 119879119865119903119899 (119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + (1 minus 120573119899 minus 120572119899) 119910119899(119)






0 12) where 119890 = (1 1)119879





(120)





6 Conclusion





4992

4993

4994

4995

4996

4997

4998

4999

the fi

rst c

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3993

3994

3995

3996

3997

3998

3999

the s

econ

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of fi

nal v

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x 10minus7



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018










⟨119861119909 119909⟩ ge 120574 1199092 forall119909 isin 1198671 (20)


119879 = (1 minus 120572) 119868 + 120572119878 (21)







(23)

(iii) For 119886 119887 119888 isin [0 1] with 119886 + 119887 + 119888 = 11003817100381710038171003817119886119909 + 119887119910 + 11988811991110038171003817100381710038172 = 119886 1199092 + 119887 100381710038171003817100381711991010038171003817100381710038172 + 119888 1199112


(24)




119865 (119905119911 + (1 minus 119905) 119909 119910) le 119865 (119909 119910) (25)







(26)



(27)








(28)



119886119899+1 le (1 minus 120574119899) 119886119899 + 120574119899120575119899 + 120573119899 119899 ge 0 (30)





10038171003817100381710038171003817119879(1198651 ℎ1)1199032119910 minus 119879(1198651ℎ1)1199031

11990910038171003817100381710038171003817 le 1003817100381710038171003817119910 minus 1199091003817100381710038171003817+ 10038161003816100381610038161003816100381610038161003816

1199032 minus 119903111990321003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817119879(1198651ℎ1)1199032

119910 minus 11991010038171003817100381710038171003817 (31)









119873⋂119894=1

Fix (119879119894) = Fix (1198791 119879119873) (32)



12 + 1205822120573 minus 12 sdot 1205822120573 = 2120573 + 1205824120573 (33)




(34)






119906119899 = 119879(1198651ℎ1)119903119899(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(35)









min119909isinΩ

12 ⟨119863119909 119909⟩ minus ℎ (119909) (37)



119863 = sup |⟨119863119909 119909⟩| 119909 isin 1198671 119909 = 1 (38)

Notice that




(40)



(41)




(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899minus 119868)119860119909119899)


minus 119868)119860119909119899)minus 119879(1198651ℎ1)119903119899




(42)


+ 1205852 ⟨(119879(1198652ℎ2)119903119899minus 119868)119860119909119899 119860119860lowast (119879(1198652ℎ2)119903119899


minus 119868)119860119909119899⟩ (43)


1205852 ⟨(119879(1198652ℎ2)119903119899minus 119868)119860119909119899 119860119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899⟩le 1198711205852 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899

minus 119868)119860119909119899100381710038171003817100381710038172 (44)

and




minus (119879(1198652ℎ2)119903119899minus 119868)119860119909119899 (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899⟩= 2120585 ⟨119879(1198652ℎ2)119903119899


minus 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

le 2120585 12 10038171003817100381710038171003817(119879(1198652 ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

minus 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

le minus120585 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

(45)


+ 120585 (119871120585 minus 1) 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

(46)





(48)




that is

119879119899119909lowast = 119909lowast (50)



(51)




(52)







(119909119899 +120585119860lowast(119879(1198652ℎ2)119903119899minus119868)119860119909119899)

and 119906119899+1 = 119879(1198651ℎ1)119903119899+1(119909119899+1 + 120585119860lowast(119879(1198652ℎ2)119903119899+1


le 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1(119909119899+1 + 120585119860lowast (119879(1198652ℎ2)119903119899+1

minus 119868)119860119909119899+1)minus 119879(1198651ℎ1)119903119899+1

(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899+1minus 119868)119860119909119899)10038171003817100381710038171003817

+ 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899+1

minus 119868)119860119909119899)minus 119879(1198651ℎ1)119903119899

(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899minus 119868)119860119909119899)10038171003817100381710038171003817 le 1003817100381710038171003817119909119899+1



100381610038161003816100381610038161003816100381610038161 minus119903119899119903119899+1

10038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1

(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899+1minus 119868)119860119909119899)


+ 120585 119860 10038171003817100381710038171003817119879(1198652 ℎ2)119903119899+1119860119909119899 minus 119879(1198652ℎ2)119903119899

11986011990911989910038171003817100381710038171003817 + 120577119899 le 1003817100381710038171003817119909119899+1minus 1199091198991003817100381710038171003817 + 120585 119860 100381610038161003816100381610038161003816100381610038161 minus

119903119899119903119899+11003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817119879(1198652ℎ2)119903119899+1

119860119909119899 minus 11986011990911989910038171003817100381710038171003817 + 120577119899= 1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 + 120585 119860 120590119899 + 120577119899

(54)


119903119899119903119899+1|119879(1198651ℎ1)119903119899+1(119909119899+120585119860lowast(119879(1198652ℎ2)119903119899



+ 120572 1003817100381710038171003817120574119899+1119889119899+1 minus 1205741198991198891198991003817100381710038171003817le 1003817100381710038171003817119879119899+1119906119899+1 minus 119879119899+11199061198991003817100381710038171003817

+ 1003817100381710038171003817119879119899+1119906119899 minus 1198791198991199061198991003817100381710038171003817+ 120572 1003817100381710038171003817120574119899+1119889119899+1 minus 1205741198991198891198991003817100381710038171003817

le 1003817100381710038171003817119906119899+1 minus 1199061198991003817100381710038171003817 + 1003817100381710038171003817119879119899+1119906119899 minus 1198791198991199061198991003817100381710038171003817+ 1205721198721 (120574119899+1 + 120574119899)

(55)







1003817100381710038171003817100381710038171003817100381710038174120573119875119862 (119868 minus 120582119899+1119861)2120573 + 120582119899+1 119906119899


100381710038171003817100381710038171003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817


100381710038171003817100381710038171003817100381710038171003817 +100381710038171003817100381710038171003817100381710038171003817

4120573 (120582119899+1 minus 120582119899)(2120573 + 120582119899+1) (2120573 + 120582119899)119906119899100381710038171003817100381710038171003817100381710038171003817

le 1003817100381710038171003817100381710038171003817100381710038174120573 (120582119899 minus 120582119899+1) 119875119862 (119868 minus 120582119899+1119861)(2120573 + 120582119899+1) (2120573 + 120582119899) 119906119899

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817


+ 1003817100381710038171003817100381710038171003817100381710038174120573 (120582119899+1 minus 120582119899)(2120573 + 120582119899+1) (2120573 + 120582119899)119906119899

100381710038171003817100381710038171003817100381710038171003817 le1120573 1003816100381610038161003816120582119899 minus 120582119899+11003816100381610038161003816


(56)





As a result

1003817100381710038171003817119911119899+1 minus 1199111198991003817100381710038171003817= 100381710038171003817100381710038171003817100381710038171003817


100381710038171003817100381710038171003817100381710038171003817

= 100381710038171003817100381710038171003817100381710038171003817120572119899+1 (120574119891 (119909119899+1) minus 119863119910119899+1)1 minus 120573119899+1


le 120572119899+11 minus 120573119899+11003817100381710038171003817120574119891 (119909119899+1) minus 119863119910119899+11003817100381710038171003817


le 120572119899+11 minus 120573119899+11003817100381710038171003817120574119891 (119909119899+1) minus 119863119910119899+11003817100381710038171003817


sdot 120590119899 + 120577119899 +1198722 1003816100381610038161003816120582119899+1 minus 1205821198991003816100381610038161003816 + 1205721198721 (120574119899+1 + 120574119899) (59)






1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (61)



(1 minus 120573119899) 1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (62)





(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899minus 119868)119860119909119899)





minus 119868)119860119909119899⟩

(63)



minus 119868)11986011990911989910038171003817100381710038171003817 (64)


Furthermore


(65)




2 le 120572119899120574 11205742 1003817100381710038171003817120574119891 (119909119899)





(66)

As a result




(67)


10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)11986011990911989910038171003817100381710038171003817 = 0 (68)








(69)

Therefore one has







(70)



1003817100381710038171003817119906119899 minus 1199091198991003817100381710038171003817 = 0 (71)



Hence


10038171003817100381710038171198631199101198991003817100381710038171003817 (73)



1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817 = 0 (74)



1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (75)



1003817100381710038171003817119906119899 minus 1199101198991003817100381710038171003817 = 0 (76)

Noting that

119910119899 minus 119906119899 = 119879119899119906119899 minus 119906119899 + 120572120574119899119889119899 (77)

one has



1003817100381710038171003817119879119899119906119899 minus 1199061198991003817100381710038171003817 = 0 (79)

We claim that


⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (80)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899119894⟩ (81)



119872V = 119861V + 119873119862V V isin 1198620 V notin 119862 (82)


119906 isin 119872V = 119861V + 119873119862V (83)

and so

119906 minus 119861V isin 119873119862V (84)



119875119862 ((119868 minus 120582119899119861) 119906119899) = 119906119899 + 119887119899 (119879119899119906119899 minus 119906119899) (86)


Therefore


As a result


(89)





(90)



(91)


⟨V minus 119911 119906⟩ ge 0 (92)


119906119899119894 = 119879(1198651 ℎ1)119903119899119894(119909119899119894 + 120585119860lowast (119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894) (93)




minus 119868)119860119909119899119894⟩ le 1198651 (119906119899119894 119908)+ ℎ1 (119906119899119894 119908) + 1119903119899119894

10038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817 + 12058511990311989911989410038171003817100381710038171003817119860119908

minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894minus 119868)119860119909119899119894100381710038171003817100381710038171003817

(94)


le ℎ1 (119906119899119894 119908) + 111990311989911989410038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817

+ 12058511990311989911989410038171003817100381710038171003817119860119908 minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894100381710038171003817100381710038171003817(95)


1198651 (119908 119911) le ℎ1 (119911 119908) (96)




+ (1 minus 119905) [1198651 (119908119905 119911) + ℎ1 (119908119905 119911)]le 119905 [1198651 (119908119905 119908) + ℎ1 (119908119905 119908)]

(99)

which implies that

1198651 (119908119905 119908) + ℎ1 (119908119905 119908) ge 0 forall119908 isin 119862 (100)


1198651 (119911 119908) + ℎ1 (119911 119908) ge 0 forall119908 isin 119862 (101)





(102)

Furthermore one has




1198652 (119860119911 ) + ℎ2 (119860119911 ) ge 0 forall isin 119876 (104)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (105)



(106)



(107)



+ 2120572119899 ⟨120574119891 (119902) minus 119863119902 119909119899+1 minus 119902⟩ (108)

which implies that



(109)







and such that



ge 0 forall119910 isin 119862 (112)


(113)





119906119899 = 1198791198651119903119899 (119909119899 + 120585119860lowast (1198791198652119903119899 minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(114)




min119909isin119876

12 ⟨119863119909 119909⟩ minus ℎ (119909) (116)






ge 0 forall119910 isin 119862 (118)



119906119899 = 119879119865119903119899 (119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + (1 minus 120573119899 minus 120572119899) 119910119899(119)






0 12) where 119890 = (1 1)119879





(120)





6 Conclusion





4992

4993

4994

4995

4996

4997

4998

4999

the fi

rst c

oord

inat

e of fi

nal v

alue



3993

3994

3995

3996

3997

3998

3999

the s

econ

d co

ordi

nate

of fi

nal v

alue

x 10minus7



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018











10038171003817100381710038171003817119879(1198651 ℎ1)1199032119910 minus 119879(1198651ℎ1)1199031

11990910038171003817100381710038171003817 le 1003817100381710038171003817119910 minus 1199091003817100381710038171003817+ 10038161003816100381610038161003816100381610038161003816

1199032 minus 119903111990321003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817119879(1198651ℎ1)1199032

119910 minus 11991010038171003817100381710038171003817 (31)









119873⋂119894=1

Fix (119879119894) = Fix (1198791 119879119873) (32)



12 + 1205822120573 minus 12 sdot 1205822120573 = 2120573 + 1205824120573 (33)




(34)






119906119899 = 119879(1198651ℎ1)119903119899(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(35)









min119909isinΩ

12 ⟨119863119909 119909⟩ minus ℎ (119909) (37)



119863 = sup |⟨119863119909 119909⟩| 119909 isin 1198671 119909 = 1 (38)

Notice that




(40)



(41)




(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899minus 119868)119860119909119899)


minus 119868)119860119909119899)minus 119879(1198651ℎ1)119903119899




(42)


+ 1205852 ⟨(119879(1198652ℎ2)119903119899minus 119868)119860119909119899 119860119860lowast (119879(1198652ℎ2)119903119899


minus 119868)119860119909119899⟩ (43)


1205852 ⟨(119879(1198652ℎ2)119903119899minus 119868)119860119909119899 119860119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899⟩le 1198711205852 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899

minus 119868)119860119909119899100381710038171003817100381710038172 (44)

and




minus (119879(1198652ℎ2)119903119899minus 119868)119860119909119899 (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899⟩= 2120585 ⟨119879(1198652ℎ2)119903119899


minus 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

le 2120585 12 10038171003817100381710038171003817(119879(1198652 ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

minus 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

le minus120585 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

(45)


+ 120585 (119871120585 minus 1) 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

(46)





(48)




that is

119879119899119909lowast = 119909lowast (50)



(51)




(52)







(119909119899 +120585119860lowast(119879(1198652ℎ2)119903119899minus119868)119860119909119899)

and 119906119899+1 = 119879(1198651ℎ1)119903119899+1(119909119899+1 + 120585119860lowast(119879(1198652ℎ2)119903119899+1


le 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1(119909119899+1 + 120585119860lowast (119879(1198652ℎ2)119903119899+1

minus 119868)119860119909119899+1)minus 119879(1198651ℎ1)119903119899+1

(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899+1minus 119868)119860119909119899)10038171003817100381710038171003817

+ 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899+1

minus 119868)119860119909119899)minus 119879(1198651ℎ1)119903119899

(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899minus 119868)119860119909119899)10038171003817100381710038171003817 le 1003817100381710038171003817119909119899+1



100381610038161003816100381610038161003816100381610038161 minus119903119899119903119899+1

10038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1

(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899+1minus 119868)119860119909119899)


+ 120585 119860 10038171003817100381710038171003817119879(1198652 ℎ2)119903119899+1119860119909119899 minus 119879(1198652ℎ2)119903119899

11986011990911989910038171003817100381710038171003817 + 120577119899 le 1003817100381710038171003817119909119899+1minus 1199091198991003817100381710038171003817 + 120585 119860 100381610038161003816100381610038161003816100381610038161 minus

119903119899119903119899+11003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817119879(1198652ℎ2)119903119899+1

119860119909119899 minus 11986011990911989910038171003817100381710038171003817 + 120577119899= 1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 + 120585 119860 120590119899 + 120577119899

(54)


119903119899119903119899+1|119879(1198651ℎ1)119903119899+1(119909119899+120585119860lowast(119879(1198652ℎ2)119903119899



+ 120572 1003817100381710038171003817120574119899+1119889119899+1 minus 1205741198991198891198991003817100381710038171003817le 1003817100381710038171003817119879119899+1119906119899+1 minus 119879119899+11199061198991003817100381710038171003817

+ 1003817100381710038171003817119879119899+1119906119899 minus 1198791198991199061198991003817100381710038171003817+ 120572 1003817100381710038171003817120574119899+1119889119899+1 minus 1205741198991198891198991003817100381710038171003817

le 1003817100381710038171003817119906119899+1 minus 1199061198991003817100381710038171003817 + 1003817100381710038171003817119879119899+1119906119899 minus 1198791198991199061198991003817100381710038171003817+ 1205721198721 (120574119899+1 + 120574119899)

(55)







1003817100381710038171003817100381710038171003817100381710038174120573119875119862 (119868 minus 120582119899+1119861)2120573 + 120582119899+1 119906119899


100381710038171003817100381710038171003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817


100381710038171003817100381710038171003817100381710038171003817 +100381710038171003817100381710038171003817100381710038171003817

4120573 (120582119899+1 minus 120582119899)(2120573 + 120582119899+1) (2120573 + 120582119899)119906119899100381710038171003817100381710038171003817100381710038171003817

le 1003817100381710038171003817100381710038171003817100381710038174120573 (120582119899 minus 120582119899+1) 119875119862 (119868 minus 120582119899+1119861)(2120573 + 120582119899+1) (2120573 + 120582119899) 119906119899

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817


+ 1003817100381710038171003817100381710038171003817100381710038174120573 (120582119899+1 minus 120582119899)(2120573 + 120582119899+1) (2120573 + 120582119899)119906119899

100381710038171003817100381710038171003817100381710038171003817 le1120573 1003816100381610038161003816120582119899 minus 120582119899+11003816100381610038161003816


(56)





As a result

1003817100381710038171003817119911119899+1 minus 1199111198991003817100381710038171003817= 100381710038171003817100381710038171003817100381710038171003817


100381710038171003817100381710038171003817100381710038171003817

= 100381710038171003817100381710038171003817100381710038171003817120572119899+1 (120574119891 (119909119899+1) minus 119863119910119899+1)1 minus 120573119899+1


le 120572119899+11 minus 120573119899+11003817100381710038171003817120574119891 (119909119899+1) minus 119863119910119899+11003817100381710038171003817


le 120572119899+11 minus 120573119899+11003817100381710038171003817120574119891 (119909119899+1) minus 119863119910119899+11003817100381710038171003817


sdot 120590119899 + 120577119899 +1198722 1003816100381610038161003816120582119899+1 minus 1205821198991003816100381610038161003816 + 1205721198721 (120574119899+1 + 120574119899) (59)






1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (61)



(1 minus 120573119899) 1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (62)





(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899minus 119868)119860119909119899)





minus 119868)119860119909119899⟩

(63)



minus 119868)11986011990911989910038171003817100381710038171003817 (64)


Furthermore


(65)




2 le 120572119899120574 11205742 1003817100381710038171003817120574119891 (119909119899)





(66)

As a result




(67)


10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)11986011990911989910038171003817100381710038171003817 = 0 (68)








(69)

Therefore one has







(70)



1003817100381710038171003817119906119899 minus 1199091198991003817100381710038171003817 = 0 (71)



Hence


10038171003817100381710038171198631199101198991003817100381710038171003817 (73)



1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817 = 0 (74)



1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (75)



1003817100381710038171003817119906119899 minus 1199101198991003817100381710038171003817 = 0 (76)

Noting that

119910119899 minus 119906119899 = 119879119899119906119899 minus 119906119899 + 120572120574119899119889119899 (77)

one has



1003817100381710038171003817119879119899119906119899 minus 1199061198991003817100381710038171003817 = 0 (79)

We claim that


⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (80)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899119894⟩ (81)



119872V = 119861V + 119873119862V V isin 1198620 V notin 119862 (82)


119906 isin 119872V = 119861V + 119873119862V (83)

and so

119906 minus 119861V isin 119873119862V (84)



119875119862 ((119868 minus 120582119899119861) 119906119899) = 119906119899 + 119887119899 (119879119899119906119899 minus 119906119899) (86)


Therefore


As a result


(89)





(90)



(91)


⟨V minus 119911 119906⟩ ge 0 (92)


119906119899119894 = 119879(1198651 ℎ1)119903119899119894(119909119899119894 + 120585119860lowast (119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894) (93)




minus 119868)119860119909119899119894⟩ le 1198651 (119906119899119894 119908)+ ℎ1 (119906119899119894 119908) + 1119903119899119894

10038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817 + 12058511990311989911989410038171003817100381710038171003817119860119908

minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894minus 119868)119860119909119899119894100381710038171003817100381710038171003817

(94)


le ℎ1 (119906119899119894 119908) + 111990311989911989410038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817

+ 12058511990311989911989410038171003817100381710038171003817119860119908 minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894100381710038171003817100381710038171003817(95)


1198651 (119908 119911) le ℎ1 (119911 119908) (96)




+ (1 minus 119905) [1198651 (119908119905 119911) + ℎ1 (119908119905 119911)]le 119905 [1198651 (119908119905 119908) + ℎ1 (119908119905 119908)]

(99)

which implies that

1198651 (119908119905 119908) + ℎ1 (119908119905 119908) ge 0 forall119908 isin 119862 (100)


1198651 (119911 119908) + ℎ1 (119911 119908) ge 0 forall119908 isin 119862 (101)





(102)

Furthermore one has




1198652 (119860119911 ) + ℎ2 (119860119911 ) ge 0 forall isin 119876 (104)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (105)



(106)



(107)



+ 2120572119899 ⟨120574119891 (119902) minus 119863119902 119909119899+1 minus 119902⟩ (108)

which implies that



(109)







and such that



ge 0 forall119910 isin 119862 (112)


(113)





119906119899 = 1198791198651119903119899 (119909119899 + 120585119860lowast (1198791198652119903119899 minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(114)




min119909isin119876

12 ⟨119863119909 119909⟩ minus ℎ (119909) (116)






ge 0 forall119910 isin 119862 (118)



119906119899 = 119879119865119903119899 (119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + (1 minus 120573119899 minus 120572119899) 119910119899(119)






0 12) where 119890 = (1 1)119879





(120)





6 Conclusion





4992

4993

4994

4995

4996

4997

4998

4999

the fi

rst c

oord

inat

e of fi

nal v

alue



3993

3994

3995

3996

3997

3998

3999

the s

econ

d co

ordi

nate

of fi

nal v

alue

x 10minus7



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018












min119909isinΩ

12 ⟨119863119909 119909⟩ minus ℎ (119909) (37)



119863 = sup |⟨119863119909 119909⟩| 119909 isin 1198671 119909 = 1 (38)

Notice that




(40)



(41)




(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899minus 119868)119860119909119899)


minus 119868)119860119909119899)minus 119879(1198651ℎ1)119903119899




(42)


+ 1205852 ⟨(119879(1198652ℎ2)119903119899minus 119868)119860119909119899 119860119860lowast (119879(1198652ℎ2)119903119899


minus 119868)119860119909119899⟩ (43)


1205852 ⟨(119879(1198652ℎ2)119903119899minus 119868)119860119909119899 119860119860lowast (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899⟩le 1198711205852 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899

minus 119868)119860119909119899100381710038171003817100381710038172 (44)

and




minus (119879(1198652ℎ2)119903119899minus 119868)119860119909119899 (119879(1198652ℎ2)119903119899

minus 119868)119860119909119899⟩= 2120585 ⟨119879(1198652ℎ2)119903119899


minus 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

le 2120585 12 10038171003817100381710038171003817(119879(1198652 ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

minus 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

le minus120585 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

(45)


+ 120585 (119871120585 minus 1) 10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)119860119909119899100381710038171003817100381710038172

(46)





(48)




that is

119879119899119909lowast = 119909lowast (50)



(51)




(52)







(119909119899 +120585119860lowast(119879(1198652ℎ2)119903119899minus119868)119860119909119899)

and 119906119899+1 = 119879(1198651ℎ1)119903119899+1(119909119899+1 + 120585119860lowast(119879(1198652ℎ2)119903119899+1


le 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1(119909119899+1 + 120585119860lowast (119879(1198652ℎ2)119903119899+1

minus 119868)119860119909119899+1)minus 119879(1198651ℎ1)119903119899+1

(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899+1minus 119868)119860119909119899)10038171003817100381710038171003817

+ 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899+1

minus 119868)119860119909119899)minus 119879(1198651ℎ1)119903119899

(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899minus 119868)119860119909119899)10038171003817100381710038171003817 le 1003817100381710038171003817119909119899+1



100381610038161003816100381610038161003816100381610038161 minus119903119899119903119899+1

10038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1

(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899+1minus 119868)119860119909119899)


+ 120585 119860 10038171003817100381710038171003817119879(1198652 ℎ2)119903119899+1119860119909119899 minus 119879(1198652ℎ2)119903119899

11986011990911989910038171003817100381710038171003817 + 120577119899 le 1003817100381710038171003817119909119899+1minus 1199091198991003817100381710038171003817 + 120585 119860 100381610038161003816100381610038161003816100381610038161 minus

119903119899119903119899+11003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817119879(1198652ℎ2)119903119899+1

119860119909119899 minus 11986011990911989910038171003817100381710038171003817 + 120577119899= 1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 + 120585 119860 120590119899 + 120577119899

(54)


119903119899119903119899+1|119879(1198651ℎ1)119903119899+1(119909119899+120585119860lowast(119879(1198652ℎ2)119903119899



+ 120572 1003817100381710038171003817120574119899+1119889119899+1 minus 1205741198991198891198991003817100381710038171003817le 1003817100381710038171003817119879119899+1119906119899+1 minus 119879119899+11199061198991003817100381710038171003817

+ 1003817100381710038171003817119879119899+1119906119899 minus 1198791198991199061198991003817100381710038171003817+ 120572 1003817100381710038171003817120574119899+1119889119899+1 minus 1205741198991198891198991003817100381710038171003817

le 1003817100381710038171003817119906119899+1 minus 1199061198991003817100381710038171003817 + 1003817100381710038171003817119879119899+1119906119899 minus 1198791198991199061198991003817100381710038171003817+ 1205721198721 (120574119899+1 + 120574119899)

(55)







1003817100381710038171003817100381710038171003817100381710038174120573119875119862 (119868 minus 120582119899+1119861)2120573 + 120582119899+1 119906119899


100381710038171003817100381710038171003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817


100381710038171003817100381710038171003817100381710038171003817 +100381710038171003817100381710038171003817100381710038171003817

4120573 (120582119899+1 minus 120582119899)(2120573 + 120582119899+1) (2120573 + 120582119899)119906119899100381710038171003817100381710038171003817100381710038171003817

le 1003817100381710038171003817100381710038171003817100381710038174120573 (120582119899 minus 120582119899+1) 119875119862 (119868 minus 120582119899+1119861)(2120573 + 120582119899+1) (2120573 + 120582119899) 119906119899

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817


+ 1003817100381710038171003817100381710038171003817100381710038174120573 (120582119899+1 minus 120582119899)(2120573 + 120582119899+1) (2120573 + 120582119899)119906119899

100381710038171003817100381710038171003817100381710038171003817 le1120573 1003816100381610038161003816120582119899 minus 120582119899+11003816100381610038161003816


(56)





As a result

1003817100381710038171003817119911119899+1 minus 1199111198991003817100381710038171003817= 100381710038171003817100381710038171003817100381710038171003817


100381710038171003817100381710038171003817100381710038171003817

= 100381710038171003817100381710038171003817100381710038171003817120572119899+1 (120574119891 (119909119899+1) minus 119863119910119899+1)1 minus 120573119899+1


le 120572119899+11 minus 120573119899+11003817100381710038171003817120574119891 (119909119899+1) minus 119863119910119899+11003817100381710038171003817


le 120572119899+11 minus 120573119899+11003817100381710038171003817120574119891 (119909119899+1) minus 119863119910119899+11003817100381710038171003817


sdot 120590119899 + 120577119899 +1198722 1003816100381610038161003816120582119899+1 minus 1205821198991003816100381610038161003816 + 1205721198721 (120574119899+1 + 120574119899) (59)






1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (61)



(1 minus 120573119899) 1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (62)





(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899minus 119868)119860119909119899)





minus 119868)119860119909119899⟩

(63)



minus 119868)11986011990911989910038171003817100381710038171003817 (64)


Furthermore


(65)




2 le 120572119899120574 11205742 1003817100381710038171003817120574119891 (119909119899)





(66)

As a result




(67)


10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)11986011990911989910038171003817100381710038171003817 = 0 (68)








(69)

Therefore one has







(70)



1003817100381710038171003817119906119899 minus 1199091198991003817100381710038171003817 = 0 (71)



Hence


10038171003817100381710038171198631199101198991003817100381710038171003817 (73)



1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817 = 0 (74)



1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (75)



1003817100381710038171003817119906119899 minus 1199101198991003817100381710038171003817 = 0 (76)

Noting that

119910119899 minus 119906119899 = 119879119899119906119899 minus 119906119899 + 120572120574119899119889119899 (77)

one has



1003817100381710038171003817119879119899119906119899 minus 1199061198991003817100381710038171003817 = 0 (79)

We claim that


⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (80)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899119894⟩ (81)



119872V = 119861V + 119873119862V V isin 1198620 V notin 119862 (82)


119906 isin 119872V = 119861V + 119873119862V (83)

and so

119906 minus 119861V isin 119873119862V (84)



119875119862 ((119868 minus 120582119899119861) 119906119899) = 119906119899 + 119887119899 (119879119899119906119899 minus 119906119899) (86)


Therefore


As a result


(89)





(90)



(91)


⟨V minus 119911 119906⟩ ge 0 (92)


119906119899119894 = 119879(1198651 ℎ1)119903119899119894(119909119899119894 + 120585119860lowast (119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894) (93)




minus 119868)119860119909119899119894⟩ le 1198651 (119906119899119894 119908)+ ℎ1 (119906119899119894 119908) + 1119903119899119894

10038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817 + 12058511990311989911989410038171003817100381710038171003817119860119908

minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894minus 119868)119860119909119899119894100381710038171003817100381710038171003817

(94)


le ℎ1 (119906119899119894 119908) + 111990311989911989410038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817

+ 12058511990311989911989410038171003817100381710038171003817119860119908 minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894100381710038171003817100381710038171003817(95)


1198651 (119908 119911) le ℎ1 (119911 119908) (96)




+ (1 minus 119905) [1198651 (119908119905 119911) + ℎ1 (119908119905 119911)]le 119905 [1198651 (119908119905 119908) + ℎ1 (119908119905 119908)]

(99)

which implies that

1198651 (119908119905 119908) + ℎ1 (119908119905 119908) ge 0 forall119908 isin 119862 (100)


1198651 (119911 119908) + ℎ1 (119911 119908) ge 0 forall119908 isin 119862 (101)





(102)

Furthermore one has




1198652 (119860119911 ) + ℎ2 (119860119911 ) ge 0 forall isin 119876 (104)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (105)



(106)



(107)



+ 2120572119899 ⟨120574119891 (119902) minus 119863119902 119909119899+1 minus 119902⟩ (108)

which implies that



(109)







and such that



ge 0 forall119910 isin 119862 (112)


(113)





119906119899 = 1198791198651119903119899 (119909119899 + 120585119860lowast (1198791198652119903119899 minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(114)




min119909isin119876

12 ⟨119863119909 119909⟩ minus ℎ (119909) (116)






ge 0 forall119910 isin 119862 (118)



119906119899 = 119879119865119903119899 (119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + (1 minus 120573119899 minus 120572119899) 119910119899(119)






0 12) where 119890 = (1 1)119879





(120)





6 Conclusion





4992

4993

4994

4995

4996

4997

4998

4999

the fi

rst c

oord

inat

e of fi

nal v

alue



3993

3994

3995

3996

3997

3998

3999

the s

econ

d co

ordi

nate

of fi

nal v

alue

x 10minus7



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018











(48)




that is

119879119899119909lowast = 119909lowast (50)



(51)




(52)







(119909119899 +120585119860lowast(119879(1198652ℎ2)119903119899minus119868)119860119909119899)

and 119906119899+1 = 119879(1198651ℎ1)119903119899+1(119909119899+1 + 120585119860lowast(119879(1198652ℎ2)119903119899+1


le 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1(119909119899+1 + 120585119860lowast (119879(1198652ℎ2)119903119899+1

minus 119868)119860119909119899+1)minus 119879(1198651ℎ1)119903119899+1

(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899+1minus 119868)119860119909119899)10038171003817100381710038171003817

+ 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899+1

minus 119868)119860119909119899)minus 119879(1198651ℎ1)119903119899

(119909119899 + 120585119860lowast (119879(1198652 ℎ2)119903119899minus 119868)119860119909119899)10038171003817100381710038171003817 le 1003817100381710038171003817119909119899+1



100381610038161003816100381610038161003816100381610038161 minus119903119899119903119899+1

10038161003816100381610038161003816100381610038161003816sdot 10038171003817100381710038171003817119879(1198651ℎ1)119903119899+1

(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899+1minus 119868)119860119909119899)


+ 120585 119860 10038171003817100381710038171003817119879(1198652 ℎ2)119903119899+1119860119909119899 minus 119879(1198652ℎ2)119903119899

11986011990911989910038171003817100381710038171003817 + 120577119899 le 1003817100381710038171003817119909119899+1minus 1199091198991003817100381710038171003817 + 120585 119860 100381610038161003816100381610038161003816100381610038161 minus

119903119899119903119899+11003816100381610038161003816100381610038161003816100381610038171003817100381710038171003817119879(1198652ℎ2)119903119899+1

119860119909119899 minus 11986011990911989910038171003817100381710038171003817 + 120577119899= 1003817100381710038171003817119909119899+1 minus 1199091198991003817100381710038171003817 + 120585 119860 120590119899 + 120577119899

(54)


119903119899119903119899+1|119879(1198651ℎ1)119903119899+1(119909119899+120585119860lowast(119879(1198652ℎ2)119903119899



+ 120572 1003817100381710038171003817120574119899+1119889119899+1 minus 1205741198991198891198991003817100381710038171003817le 1003817100381710038171003817119879119899+1119906119899+1 minus 119879119899+11199061198991003817100381710038171003817

+ 1003817100381710038171003817119879119899+1119906119899 minus 1198791198991199061198991003817100381710038171003817+ 120572 1003817100381710038171003817120574119899+1119889119899+1 minus 1205741198991198891198991003817100381710038171003817

le 1003817100381710038171003817119906119899+1 minus 1199061198991003817100381710038171003817 + 1003817100381710038171003817119879119899+1119906119899 minus 1198791198991199061198991003817100381710038171003817+ 1205721198721 (120574119899+1 + 120574119899)

(55)







1003817100381710038171003817100381710038171003817100381710038174120573119875119862 (119868 minus 120582119899+1119861)2120573 + 120582119899+1 119906119899


100381710038171003817100381710038171003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817


100381710038171003817100381710038171003817100381710038171003817 +100381710038171003817100381710038171003817100381710038171003817

4120573 (120582119899+1 minus 120582119899)(2120573 + 120582119899+1) (2120573 + 120582119899)119906119899100381710038171003817100381710038171003817100381710038171003817

le 1003817100381710038171003817100381710038171003817100381710038174120573 (120582119899 minus 120582119899+1) 119875119862 (119868 minus 120582119899+1119861)(2120573 + 120582119899+1) (2120573 + 120582119899) 119906119899

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817


+ 1003817100381710038171003817100381710038171003817100381710038174120573 (120582119899+1 minus 120582119899)(2120573 + 120582119899+1) (2120573 + 120582119899)119906119899

100381710038171003817100381710038171003817100381710038171003817 le1120573 1003816100381610038161003816120582119899 minus 120582119899+11003816100381610038161003816


(56)





As a result

1003817100381710038171003817119911119899+1 minus 1199111198991003817100381710038171003817= 100381710038171003817100381710038171003817100381710038171003817


100381710038171003817100381710038171003817100381710038171003817

= 100381710038171003817100381710038171003817100381710038171003817120572119899+1 (120574119891 (119909119899+1) minus 119863119910119899+1)1 minus 120573119899+1


le 120572119899+11 minus 120573119899+11003817100381710038171003817120574119891 (119909119899+1) minus 119863119910119899+11003817100381710038171003817


le 120572119899+11 minus 120573119899+11003817100381710038171003817120574119891 (119909119899+1) minus 119863119910119899+11003817100381710038171003817


sdot 120590119899 + 120577119899 +1198722 1003816100381610038161003816120582119899+1 minus 1205821198991003816100381610038161003816 + 1205721198721 (120574119899+1 + 120574119899) (59)






1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (61)



(1 minus 120573119899) 1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (62)





(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899minus 119868)119860119909119899)





minus 119868)119860119909119899⟩

(63)



minus 119868)11986011990911989910038171003817100381710038171003817 (64)


Furthermore


(65)




2 le 120572119899120574 11205742 1003817100381710038171003817120574119891 (119909119899)





(66)

As a result




(67)


10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)11986011990911989910038171003817100381710038171003817 = 0 (68)








(69)

Therefore one has







(70)



1003817100381710038171003817119906119899 minus 1199091198991003817100381710038171003817 = 0 (71)



Hence


10038171003817100381710038171198631199101198991003817100381710038171003817 (73)



1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817 = 0 (74)



1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (75)



1003817100381710038171003817119906119899 minus 1199101198991003817100381710038171003817 = 0 (76)

Noting that

119910119899 minus 119906119899 = 119879119899119906119899 minus 119906119899 + 120572120574119899119889119899 (77)

one has



1003817100381710038171003817119879119899119906119899 minus 1199061198991003817100381710038171003817 = 0 (79)

We claim that


⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (80)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899119894⟩ (81)



119872V = 119861V + 119873119862V V isin 1198620 V notin 119862 (82)


119906 isin 119872V = 119861V + 119873119862V (83)

and so

119906 minus 119861V isin 119873119862V (84)



119875119862 ((119868 minus 120582119899119861) 119906119899) = 119906119899 + 119887119899 (119879119899119906119899 minus 119906119899) (86)


Therefore


As a result


(89)





(90)



(91)


⟨V minus 119911 119906⟩ ge 0 (92)


119906119899119894 = 119879(1198651 ℎ1)119903119899119894(119909119899119894 + 120585119860lowast (119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894) (93)




minus 119868)119860119909119899119894⟩ le 1198651 (119906119899119894 119908)+ ℎ1 (119906119899119894 119908) + 1119903119899119894

10038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817 + 12058511990311989911989410038171003817100381710038171003817119860119908

minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894minus 119868)119860119909119899119894100381710038171003817100381710038171003817

(94)


le ℎ1 (119906119899119894 119908) + 111990311989911989410038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817

+ 12058511990311989911989410038171003817100381710038171003817119860119908 minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894100381710038171003817100381710038171003817(95)


1198651 (119908 119911) le ℎ1 (119911 119908) (96)




+ (1 minus 119905) [1198651 (119908119905 119911) + ℎ1 (119908119905 119911)]le 119905 [1198651 (119908119905 119908) + ℎ1 (119908119905 119908)]

(99)

which implies that

1198651 (119908119905 119908) + ℎ1 (119908119905 119908) ge 0 forall119908 isin 119862 (100)


1198651 (119911 119908) + ℎ1 (119911 119908) ge 0 forall119908 isin 119862 (101)





(102)

Furthermore one has




1198652 (119860119911 ) + ℎ2 (119860119911 ) ge 0 forall isin 119876 (104)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (105)



(106)



(107)



+ 2120572119899 ⟨120574119891 (119902) minus 119863119902 119909119899+1 minus 119902⟩ (108)

which implies that



(109)







and such that



ge 0 forall119910 isin 119862 (112)


(113)





119906119899 = 1198791198651119903119899 (119909119899 + 120585119860lowast (1198791198652119903119899 minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(114)




min119909isin119876

12 ⟨119863119909 119909⟩ minus ℎ (119909) (116)






ge 0 forall119910 isin 119862 (118)



119906119899 = 119879119865119903119899 (119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + (1 minus 120573119899 minus 120572119899) 119910119899(119)






0 12) where 119890 = (1 1)119879





(120)





6 Conclusion





4992

4993

4994

4995

4996

4997

4998

4999

the fi

rst c

oord

inat

e of fi

nal v

alue



3993

3994

3995

3996

3997

3998

3999

the s

econ

d co

ordi

nate

of fi

nal v

alue

x 10minus7



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018














1003817100381710038171003817100381710038171003817100381710038174120573119875119862 (119868 minus 120582119899+1119861)2120573 + 120582119899+1 119906119899


100381710038171003817100381710038171003817100381710038171003817le 100381710038171003817100381710038171003817100381710038171003817


100381710038171003817100381710038171003817100381710038171003817 +100381710038171003817100381710038171003817100381710038171003817

4120573 (120582119899+1 minus 120582119899)(2120573 + 120582119899+1) (2120573 + 120582119899)119906119899100381710038171003817100381710038171003817100381710038171003817

le 1003817100381710038171003817100381710038171003817100381710038174120573 (120582119899 minus 120582119899+1) 119875119862 (119868 minus 120582119899+1119861)(2120573 + 120582119899+1) (2120573 + 120582119899) 119906119899

100381710038171003817100381710038171003817100381710038171003817+ 100381710038171003817100381710038171003817100381710038171003817


+ 1003817100381710038171003817100381710038171003817100381710038174120573 (120582119899+1 minus 120582119899)(2120573 + 120582119899+1) (2120573 + 120582119899)119906119899

100381710038171003817100381710038171003817100381710038171003817 le1120573 1003816100381610038161003816120582119899 minus 120582119899+11003816100381610038161003816


(56)





As a result

1003817100381710038171003817119911119899+1 minus 1199111198991003817100381710038171003817= 100381710038171003817100381710038171003817100381710038171003817


100381710038171003817100381710038171003817100381710038171003817

= 100381710038171003817100381710038171003817100381710038171003817120572119899+1 (120574119891 (119909119899+1) minus 119863119910119899+1)1 minus 120573119899+1


le 120572119899+11 minus 120573119899+11003817100381710038171003817120574119891 (119909119899+1) minus 119863119910119899+11003817100381710038171003817


le 120572119899+11 minus 120573119899+11003817100381710038171003817120574119891 (119909119899+1) minus 119863119910119899+11003817100381710038171003817


sdot 120590119899 + 120577119899 +1198722 1003816100381610038161003816120582119899+1 minus 1205821198991003816100381610038161003816 + 1205721198721 (120574119899+1 + 120574119899) (59)






1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (61)



(1 minus 120573119899) 1003817100381710038171003817119911119899 minus 1199091198991003817100381710038171003817 = 0 (62)





(119909119899 + 120585119860lowast (119879(1198652ℎ2)119903119899minus 119868)119860119909119899)





minus 119868)119860119909119899⟩

(63)



minus 119868)11986011990911989910038171003817100381710038171003817 (64)


Furthermore


(65)




2 le 120572119899120574 11205742 1003817100381710038171003817120574119891 (119909119899)





(66)

As a result




(67)


10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)11986011990911989910038171003817100381710038171003817 = 0 (68)








(69)

Therefore one has







(70)



1003817100381710038171003817119906119899 minus 1199091198991003817100381710038171003817 = 0 (71)



Hence


10038171003817100381710038171198631199101198991003817100381710038171003817 (73)



1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817 = 0 (74)



1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (75)



1003817100381710038171003817119906119899 minus 1199101198991003817100381710038171003817 = 0 (76)

Noting that

119910119899 minus 119906119899 = 119879119899119906119899 minus 119906119899 + 120572120574119899119889119899 (77)

one has



1003817100381710038171003817119879119899119906119899 minus 1199061198991003817100381710038171003817 = 0 (79)

We claim that


⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (80)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899119894⟩ (81)



119872V = 119861V + 119873119862V V isin 1198620 V notin 119862 (82)


119906 isin 119872V = 119861V + 119873119862V (83)

and so

119906 minus 119861V isin 119873119862V (84)



119875119862 ((119868 minus 120582119899119861) 119906119899) = 119906119899 + 119887119899 (119879119899119906119899 minus 119906119899) (86)


Therefore


As a result


(89)





(90)



(91)


⟨V minus 119911 119906⟩ ge 0 (92)


119906119899119894 = 119879(1198651 ℎ1)119903119899119894(119909119899119894 + 120585119860lowast (119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894) (93)




minus 119868)119860119909119899119894⟩ le 1198651 (119906119899119894 119908)+ ℎ1 (119906119899119894 119908) + 1119903119899119894

10038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817 + 12058511990311989911989410038171003817100381710038171003817119860119908

minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894minus 119868)119860119909119899119894100381710038171003817100381710038171003817

(94)


le ℎ1 (119906119899119894 119908) + 111990311989911989410038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817

+ 12058511990311989911989410038171003817100381710038171003817119860119908 minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894100381710038171003817100381710038171003817(95)


1198651 (119908 119911) le ℎ1 (119911 119908) (96)




+ (1 minus 119905) [1198651 (119908119905 119911) + ℎ1 (119908119905 119911)]le 119905 [1198651 (119908119905 119908) + ℎ1 (119908119905 119908)]

(99)

which implies that

1198651 (119908119905 119908) + ℎ1 (119908119905 119908) ge 0 forall119908 isin 119862 (100)


1198651 (119911 119908) + ℎ1 (119911 119908) ge 0 forall119908 isin 119862 (101)





(102)

Furthermore one has




1198652 (119860119911 ) + ℎ2 (119860119911 ) ge 0 forall isin 119876 (104)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (105)



(106)



(107)



+ 2120572119899 ⟨120574119891 (119902) minus 119863119902 119909119899+1 minus 119902⟩ (108)

which implies that



(109)







and such that



ge 0 forall119910 isin 119862 (112)


(113)





119906119899 = 1198791198651119903119899 (119909119899 + 120585119860lowast (1198791198652119903119899 minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(114)




min119909isin119876

12 ⟨119863119909 119909⟩ minus ℎ (119909) (116)






ge 0 forall119910 isin 119862 (118)



119906119899 = 119879119865119903119899 (119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + (1 minus 120573119899 minus 120572119899) 119910119899(119)






0 12) where 119890 = (1 1)119879





(120)





6 Conclusion





4992

4993

4994

4995

4996

4997

4998

4999

the fi

rst c

oord

inat

e of fi

nal v

alue



3993

3994

3995

3996

3997

3998

3999

the s

econ

d co

ordi

nate

of fi

nal v

alue

x 10minus7



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









Furthermore


(65)




2 le 120572119899120574 11205742 1003817100381710038171003817120574119891 (119909119899)





(66)

As a result




(67)


10038171003817100381710038171003817(119879(1198652ℎ2)119903119899minus 119868)11986011990911989910038171003817100381710038171003817 = 0 (68)








(69)

Therefore one has







(70)



1003817100381710038171003817119906119899 minus 1199091198991003817100381710038171003817 = 0 (71)



Hence


10038171003817100381710038171198631199101198991003817100381710038171003817 (73)



1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817 = 0 (74)



1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (75)



1003817100381710038171003817119906119899 minus 1199101198991003817100381710038171003817 = 0 (76)

Noting that

119910119899 minus 119906119899 = 119879119899119906119899 minus 119906119899 + 120572120574119899119889119899 (77)

one has



1003817100381710038171003817119879119899119906119899 minus 1199061198991003817100381710038171003817 = 0 (79)

We claim that


⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (80)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899119894⟩ (81)



119872V = 119861V + 119873119862V V isin 1198620 V notin 119862 (82)


119906 isin 119872V = 119861V + 119873119862V (83)

and so

119906 minus 119861V isin 119873119862V (84)



119875119862 ((119868 minus 120582119899119861) 119906119899) = 119906119899 + 119887119899 (119879119899119906119899 minus 119906119899) (86)


Therefore


As a result


(89)





(90)



(91)


⟨V minus 119911 119906⟩ ge 0 (92)


119906119899119894 = 119879(1198651 ℎ1)119903119899119894(119909119899119894 + 120585119860lowast (119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894) (93)




minus 119868)119860119909119899119894⟩ le 1198651 (119906119899119894 119908)+ ℎ1 (119906119899119894 119908) + 1119903119899119894

10038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817 + 12058511990311989911989410038171003817100381710038171003817119860119908

minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894minus 119868)119860119909119899119894100381710038171003817100381710038171003817

(94)


le ℎ1 (119906119899119894 119908) + 111990311989911989410038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817

+ 12058511990311989911989410038171003817100381710038171003817119860119908 minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894100381710038171003817100381710038171003817(95)


1198651 (119908 119911) le ℎ1 (119911 119908) (96)




+ (1 minus 119905) [1198651 (119908119905 119911) + ℎ1 (119908119905 119911)]le 119905 [1198651 (119908119905 119908) + ℎ1 (119908119905 119908)]

(99)

which implies that

1198651 (119908119905 119908) + ℎ1 (119908119905 119908) ge 0 forall119908 isin 119862 (100)


1198651 (119911 119908) + ℎ1 (119911 119908) ge 0 forall119908 isin 119862 (101)





(102)

Furthermore one has




1198652 (119860119911 ) + ℎ2 (119860119911 ) ge 0 forall isin 119876 (104)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (105)



(106)



(107)



+ 2120572119899 ⟨120574119891 (119902) minus 119863119902 119909119899+1 minus 119902⟩ (108)

which implies that



(109)







and such that



ge 0 forall119910 isin 119862 (112)


(113)





119906119899 = 1198791198651119903119899 (119909119899 + 120585119860lowast (1198791198652119903119899 minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(114)




min119909isin119876

12 ⟨119863119909 119909⟩ minus ℎ (119909) (116)






ge 0 forall119910 isin 119862 (118)



119906119899 = 119879119865119903119899 (119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + (1 minus 120573119899 minus 120572119899) 119910119899(119)






0 12) where 119890 = (1 1)119879





(120)





6 Conclusion





4992

4993

4994

4995

4996

4997

4998

4999

the fi

rst c

oord

inat

e of fi

nal v

alue



3993

3994

3995

3996

3997

3998

3999

the s

econ

d co

ordi

nate

of fi

nal v

alue

x 10minus7



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018













(70)



1003817100381710038171003817119906119899 minus 1199091198991003817100381710038171003817 = 0 (71)



Hence


10038171003817100381710038171198631199101198991003817100381710038171003817 (73)



1003817100381710038171003817119911119899 minus 1199101198991003817100381710038171003817 = 0 (74)



1003817100381710038171003817119909119899 minus 1199101198991003817100381710038171003817 = 0 (75)



1003817100381710038171003817119906119899 minus 1199101198991003817100381710038171003817 = 0 (76)

Noting that

119910119899 minus 119906119899 = 119879119899119906119899 minus 119906119899 + 120572120574119899119889119899 (77)

one has



1003817100381710038171003817119879119899119906119899 minus 1199061198991003817100381710038171003817 = 0 (79)

We claim that


⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (80)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899119894⟩ (81)



119872V = 119861V + 119873119862V V isin 1198620 V notin 119862 (82)


119906 isin 119872V = 119861V + 119873119862V (83)

and so

119906 minus 119861V isin 119873119862V (84)



119875119862 ((119868 minus 120582119899119861) 119906119899) = 119906119899 + 119887119899 (119879119899119906119899 minus 119906119899) (86)


Therefore


As a result


(89)





(90)



(91)


⟨V minus 119911 119906⟩ ge 0 (92)


119906119899119894 = 119879(1198651 ℎ1)119903119899119894(119909119899119894 + 120585119860lowast (119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894) (93)




minus 119868)119860119909119899119894⟩ le 1198651 (119906119899119894 119908)+ ℎ1 (119906119899119894 119908) + 1119903119899119894

10038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817 + 12058511990311989911989410038171003817100381710038171003817119860119908

minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894minus 119868)119860119909119899119894100381710038171003817100381710038171003817

(94)


le ℎ1 (119906119899119894 119908) + 111990311989911989410038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817

+ 12058511990311989911989410038171003817100381710038171003817119860119908 minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894100381710038171003817100381710038171003817(95)


1198651 (119908 119911) le ℎ1 (119911 119908) (96)




+ (1 minus 119905) [1198651 (119908119905 119911) + ℎ1 (119908119905 119911)]le 119905 [1198651 (119908119905 119908) + ℎ1 (119908119905 119908)]

(99)

which implies that

1198651 (119908119905 119908) + ℎ1 (119908119905 119908) ge 0 forall119908 isin 119862 (100)


1198651 (119911 119908) + ℎ1 (119911 119908) ge 0 forall119908 isin 119862 (101)





(102)

Furthermore one has




1198652 (119860119911 ) + ℎ2 (119860119911 ) ge 0 forall isin 119876 (104)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (105)



(106)



(107)



+ 2120572119899 ⟨120574119891 (119902) minus 119863119902 119909119899+1 minus 119902⟩ (108)

which implies that



(109)







and such that



ge 0 forall119910 isin 119862 (112)


(113)





119906119899 = 1198791198651119903119899 (119909119899 + 120585119860lowast (1198791198652119903119899 minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(114)




min119909isin119876

12 ⟨119863119909 119909⟩ minus ℎ (119909) (116)






ge 0 forall119910 isin 119862 (118)



119906119899 = 119879119865119903119899 (119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + (1 minus 120573119899 minus 120572119899) 119910119899(119)






0 12) where 119890 = (1 1)119879





(120)





6 Conclusion





4992

4993

4994

4995

4996

4997

4998

4999

the fi

rst c

oord

inat

e of fi

nal v

alue



3993

3994

3995

3996

3997

3998

3999

the s

econ

d co

ordi

nate

of fi

nal v

alue

x 10minus7



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018












(90)



(91)


⟨V minus 119911 119906⟩ ge 0 (92)


119906119899119894 = 119879(1198651 ℎ1)119903119899119894(119909119899119894 + 120585119860lowast (119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894) (93)




minus 119868)119860119909119899119894⟩ le 1198651 (119906119899119894 119908)+ ℎ1 (119906119899119894 119908) + 1119903119899119894

10038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817 + 12058511990311989911989410038171003817100381710038171003817119860119908

minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894minus 119868)119860119909119899119894100381710038171003817100381710038171003817

(94)


le ℎ1 (119906119899119894 119908) + 111990311989911989410038171003817100381710038171003817119908 minus 11990611989911989410038171003817100381710038171003817 10038171003817100381710038171003817119906119899119894 minus 11990911989911989410038171003817100381710038171003817

+ 12058511990311989911989410038171003817100381710038171003817119860119908 minus 11986011990611989911989410038171003817100381710038171003817 100381710038171003817100381710038171003817(119879(1198652ℎ2)119903119899119894

minus 119868)119860119909119899119894100381710038171003817100381710038171003817(95)


1198651 (119908 119911) le ℎ1 (119911 119908) (96)




+ (1 minus 119905) [1198651 (119908119905 119911) + ℎ1 (119908119905 119911)]le 119905 [1198651 (119908119905 119908) + ℎ1 (119908119905 119908)]

(99)

which implies that

1198651 (119908119905 119908) + ℎ1 (119908119905 119908) ge 0 forall119908 isin 119862 (100)


1198651 (119911 119908) + ℎ1 (119911 119908) ge 0 forall119908 isin 119862 (101)





(102)

Furthermore one has




1198652 (119860119911 ) + ℎ2 (119860119911 ) ge 0 forall isin 119876 (104)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (105)



(106)



(107)



+ 2120572119899 ⟨120574119891 (119902) minus 119863119902 119909119899+1 minus 119902⟩ (108)

which implies that



(109)







and such that



ge 0 forall119910 isin 119862 (112)


(113)





119906119899 = 1198791198651119903119899 (119909119899 + 120585119860lowast (1198791198652119903119899 minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(114)




min119909isin119876

12 ⟨119863119909 119909⟩ minus ℎ (119909) (116)






ge 0 forall119910 isin 119862 (118)



119906119899 = 119879119865119903119899 (119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + (1 minus 120573119899 minus 120572119899) 119910119899(119)






0 12) where 119890 = (1 1)119879





(120)





6 Conclusion





4992

4993

4994

4995

4996

4997

4998

4999

the fi

rst c

oord

inat

e of fi

nal v

alue



3993

3994

3995

3996

3997

3998

3999

the s

econ

d co

ordi

nate

of fi

nal v

alue

x 10minus7



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018










1198652 (119860119911 ) + ℎ2 (119860119911 ) ge 0 forall isin 119876 (104)




⟨(119863 minus 120574119891) 119902 119902 minus 119909119899⟩ le 0 (105)



(106)



(107)



+ 2120572119899 ⟨120574119891 (119902) minus 119863119902 119909119899+1 minus 119902⟩ (108)

which implies that



(109)







and such that



ge 0 forall119910 isin 119862 (112)


(113)





119906119899 = 1198791198651119903119899 (119909119899 + 120585119860lowast (1198791198652119903119899 minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(114)




min119909isin119876

12 ⟨119863119909 119909⟩ minus ℎ (119909) (116)






ge 0 forall119910 isin 119862 (118)



119906119899 = 119879119865119903119899 (119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + (1 minus 120573119899 minus 120572119899) 119910119899(119)






0 12) where 119890 = (1 1)119879





(120)





6 Conclusion





4992

4993

4994

4995

4996

4997

4998

4999

the fi

rst c

oord

inat

e of fi

nal v

alue



3993

3994

3995

3996

3997

3998

3999

the s

econ

d co

ordi

nate

of fi

nal v

alue

x 10minus7



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018










119906119899 = 1198791198651119903119899 (119909119899 + 120585119860lowast (1198791198652119903119899 minus 119868)119860119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + ((1 minus 120573119899) 119868 minus 120572119899119863)119910119899(114)




min119909isin119876

12 ⟨119863119909 119909⟩ minus ℎ (119909) (116)






ge 0 forall119910 isin 119862 (118)



119906119899 = 119879119865119903119899 (119909119899) 119910119899 = 119906119899 + 120572119889119899+1

119909119899+1 = 120572119899120574119891 (119909119899) + 120573119899119909119899 + (1 minus 120573119899 minus 120572119899) 119910119899(119)






0 12) where 119890 = (1 1)119879





(120)





6 Conclusion





4992

4993

4994

4995

4996

4997

4998

4999

the fi

rst c

oord

inat

e of fi

nal v

alue



3993

3994

3995

3996

3997

3998

3999

the s

econ

d co

ordi

nate

of fi

nal v

alue

x 10minus7



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018











4992

4993

4994

4995

4996

4997

4998

4999

the fi

rst c

oord

inat

e of fi

nal v

alue



3993

3994

3995

3996

3997

3998

3999

the s

econ

d co

ordi

nate

of fi

nal v

alue

x 10minus7



Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018












































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

The Conjugate Gradient Viscosity Approximation Algorithm for Split …downloads.hindawi.com/journals/jfs/2018/8056276.pdf · 2019-07-30 · ,ndD1,D2>0.en JJ JJ J (1,1) 2 − 11 1