ResearchArticle A New Fixed Point Theorem and Applications · 2014-04-25 · 2 AbstractandAppliedAnalysis Inthesequel,wealsouse (𝑋,𝑌,{𝜑𝑁})todenote(𝑋,𝑌,Φ). Definition2.Let𝑆:𝑌→2𝑋beamultivaluedmapping

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 432402 5 pageshttpdxdoiorg1011552013432402

Research ArticleA New Fixed Point Theorem and Applications

Min Fang1 and Xie Ping Ding2

1 Department of Economic Mathematics South Western University of Finance and Economics Chengdu Sichuan 610074 China2 College of Mathematics and Software Science Sichuan Normal University Chengdu Sichuan 610068 China

Correspondence should be addressed to Xie Ping Ding xieping dinghotmaicom

Received 27 December 2012 Accepted 1 February 2013

Academic Editor Jen-Chih Yao

Copyright copy 2013 M Fang and X P Ding This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A new fixed point theorem is established under the setting of a generalized finitely continuous topological space (GFC-space)without the convexity structure As applications a weak KKM theorem and aminimax inequalities of Ky Fan type are also obtainedunder suitable conditions Our results are different from known results in the literature

1 Introduction

In the last decade the theory of fixed points has been investi-gated bymany authors see for example [1ndash11] and referencestherein which has been exploited in the existence study foralmost all areas of mathematics including optimization andapplications in economics Now there have been a lot ofgeneralizations of the fixed points theorem under differentassumptions and different underlying space and variousapplications have been given in different fields

On the other hand the weak KKM-type theorem intro-duced by Balaj [12] has attracted an increasing amountof attention and has been applied in many optimizationproblems so far see [12ndash14] and references therein

Inspired by the research works mentioned above weestablish a collectively fixed points theorem and a fixedpoint theorem As applications a weak KKM theorem and aminimax inequalities of Ky Fan type are also obtained undersuitable conditions Our results are new and different fromknown results in the literature

The rest of the paper is organized as follows In Section 2we first recall some definitions and theorems Section 3 isdevoted to a new collectively fixed points theorem undernoncompact situation on GFC-space and a new fixed pointtheorem In Section 4 we show a new weak KKM theoremin underlying GFC-space and by using the weak KKM

theorem a new minimax inequality of Ky Fan type isdeveloped

2 Preliminaries

Let 119883 be a topological space and 119862119863 sube 119883 Let int119862 andint119863119862 denote the interior of 119862 in 119883 and in 119863 respectively

Let ⟨119860⟩ denote the set of all nonempty finite subsets of a set119860and let Δ

119899denote the standard 119899-dimensional simplex with

vertices 1198900 1198901 119890

119899 Let119883 and119884 be two topological spaces

A mapping 119879 119883 rarr 2119884 is said to be upper semicontinuous

(usc) (resp lower semicontinuous (lsc)) if for every closedsubset 119861 of 119884 the set 119909 isin 119883 119879(119909) cap 119861 = 0 (resp 119909 isin 119883

119879(119909) sube 119861) is closedA subset 119860 of 119883 is said to be compactly open (resp

compactly closed) if for each nonempty compact subset119870 of119883 119860 cap 119870 is open (resp closed) in 119870

These following notionswere introduced byHai et al [15]

Definition 1 Let 119883 be a topological space 119884 a nonempty setandΦ a family of continuous mappings 120593 Δ

119899rarr 119883 119899 isin N

A triple (119883 119884Φ) is said to be a generalized finitely continuoustopological space (GFC-space) if and only if for each finitesubset119873 = 119910

0 1199101 119910

119899 of 119884 there is 120593

119873 Δ119899rarr 119883 of the

familyΦ

2 Abstract and Applied Analysis

In the sequel we also use (119883 119884 120593119873) to denote (119883 119884Φ)

Definition 2 Let 119878 119884 rarr 2119883 be a multivalued mapping

A subset 119863 of 119884 is called an 119878-subset of 119884 if and only if foreach 119873 = 119910

0 1199101 119910

119899 sube 119884 and each 119910

1198940 1199101198941 119910

119894119896 sube

119873 cap 119863 one has 120593119873(Δ119896) sub 119878(119863) where Δ

119896is the face of Δ

119899

corresponding to 1199101198940 1199101198941 119910

119894119896 that is the simplex with

vertices 1198901198940 1198901198941 119890

119894119896 Roughly speaking if119863 is an 119878-subset

of 119884 then (119878(119863) 119863Φ) is a GFC-spaceThe class of GFC-space contains a large number of spaces

with various kinds of generalized convexity structures suchas FC-space and G-convex space (see [15ndash17])

Definition 3 (see [8]) Let (119883 119884 120593119873) be a GFC-space and 119885

a nonempty set Let 119879 119883 rarr 2119885 and 119865 119884 rarr 2

119885 be twoset-valued mappings 119865 is called a weak KKM mapping withrespect to 119879 shortly weak T-KKMmapping if and only if foreach 119873 = 119910

0 1199101 119910

119899 sube 119884 119910

1198940 1199101198941 119910

119894119896 sube 119873 and

119909 isin 120593119873(Δ119896) 119879(119909) cap ⋃119896

119895=0119865(119910119894119895) = 0

Definition 4 (see [8]) Let119883 be aHausdorff space (119883 119884 120593119873)

aGFC-space119885 a topological space119879 119883 rarr 2119885119891 119884times119885 rarr

R cup minusinfin +infin and 119892 119883 times119885 rarr R cup minusinfin +infin Let 120582 isin R119891 is called (120582 119879 119892)-GFC quasiconvex if and only if for each119909 isin 119883 119911 isin 119879(119909) 119873 = 119910

0 1199101 119910

119899 isin ⟨119884⟩ and 119873

119896=

1199101198940 1199101198941 119910

119894119896 sube 119873 one has the implication 119891(119910

119894119895 119911) lt 120582

for all 119895 = 0 1 119896 implies that 119892(1199091015840 119911) lt 120582 for all 1199091015840 isin120593119873(Δ119896)

For 120582 isin R define 120573 isin R and 119867120582 119884 rarr 2

119885 by 120573 =

inf119909isin119883

sup119911isin119879(119909)

119892(119909 119911) and 119867120582(119910) = 119911 isin 119885 119891(119910 119911) ge 120582

respectively

Lemma 5 (see [8]) For 120582 lt 120573 if 119891 is (120582 119879 119892)-GFCquasiconvex then119867

120582is a weak T-KKM mapping

The following result is the obvious corollary of Theo-rem 31 of Khanh et al [8]

Lemma 6 Let (119883119894 119884119894 120593119873119894)119894isin119868

be a family of GFC-spacesand 119883 = prod

119894isin119868119883119894a compact Hausdorff space For each 119894 isin 119868

let119866119894 119883 rarr 2

119883119894 and119865119894 119883 rarr 2

119884119894 be such that the conditionshold as follows

(i) for each 119909 isin 119883 each 119873119894= 119910119894

0 119910119894

1 119910

119894

119899119894 sube 119884

119894and

each 1199101198941198940 119910119894

1198941 119910

119894

119894119896119894

sube 119873119894cap119865119894(119909) one has 120593

119873119894(Δ119896) sube

119866119894(119909) for all 119894 isin 119868

(ii) 119883 = ⋃119910119894isin119884119894

int119865minus1119894(119910119894) for all 119894 isin 119868

Then there exists 119909 = (119909119894)119894isin119868

isin 119883 such that 119909119894isin 119866119894(119909) for all

119894 isin 119868

3 Fixed Points Theorems

Let 119868 be an index set119883119894topological spaces119883 = prod

119894isin119868119883119894 and

119866119894 119883 rarr 2

119883119894 The collectively fixed points problem is to find119909 = (119909

119894)119894isin119868

isin 119883 such that 119909119894isin 119866119894(119909) for all 119894 isin 119868

Theorem 7 Let (119883119894 119884119894 120593119873119894)119894isin119868


119894isin119868119883119894a Hausdorff space For each 119894 isin 119868 let 119866

119894

119883 rarr 2119883119894 119865119894 119883 rarr 2

119884119894 and 119878119894 119884119894rarr 2119883119894 with the following

properties

(i) for each 119909 isin 119883 119873119894= 119910119894

0 119910119894

1 119910

119894

119899119894 sube 119884

119894 and

119910119894

1198950 119910119894

1198951 119910

119894

119895119896119894

sube 119873119894cap 119865119894(119909) one has 120593

119873119894(Δ119896) sube

119866119894(119909) for all 119894 isin 119868

(ii) for each compact subset 119870 of 119883 and each 119894 isin 119868 119870 sube

⋃119910119894isin119884119894

int119865minus1119894(119910119894)

(iii) there exists a nonempty compact subset 119870119894of 119883119894and

for each 119873119894isin ⟨119884119894⟩ there exists an 119878

119894-subset 119871

119873119894of 119884119894

containing119873119894with 119878

119894(119871119873119894) being compact such that

119878 (119871119873) 119870 sub ⋃

119910119894isin119871119873119894

int119865minus1119894(119910119894) (1)

where 119871119873

= prod119894isin119868119871119873119894 119870 = prod

119894isin119868119870119894 and 119878(119871

119873) =

prod119894isin119868119878119894(119871119873119894)


isin 119883 such that 119909119894isin 119866119894(119909) for

all 119894 isin 119868

Proof As 119870 is a compact subset of 119883 by the condition (ii)there exists a finite set119873

119894= 119910119894

0 119910119894

1 119910

119894

119899119894 sube 119884119894 such that

119870 sube

119899119894

⋃

119896=0

int119865minus1119894(119910119894

119894119896) (2)

By the condition (iii) there exists an 119878119894-subset 119871

119873119894of 119884119894

containing119873119894such that

119878 (119871119873) 119870 sub ⋃

119910119894isin119871119873119894

int119865minus1119894(119910119894) (3)

and it follows that

119878 (119871119873) sub ⋃

119910119894isin119871119873119894

int119865minus1119894(119910119894) (4)

We observe that the family 119878119894(119871119873119894) 119871119873119894 120593119873119894)119894isin119868

is a familyof GFC-space and 119878

119894(119871119873119894) is compact for each 119894 isin 119868 defining

set-valued mapping 119866lowast

119894 119878119894(119871119873119894) rarr 2

119878119894(119871119873119894) and 119865

lowast

119894

119878119894(119871119873119894) rarr 2

119871119873119894 as follows

119866lowast

119894(119909) = 119866119894 (119909) cap 119878119894 (119871119873119894

)

119865lowast

119894(119909) = 119865119894 (119909) cap 119871119873119894

(5)

We check assumptions (i) and (ii) of Lemma 6 for replaced119866119894

and 119865119894by 119866lowast119894and 119865lowast119894 respectively By (i) and the definition of

119878-subset for each 119909 isin 119878119894(119871119873119894) each 119873

119894= 119910119894

0 119910119894

1 119910

119894

119899119894 sube

119871119873119894and each 119910119894

1198950 119910119894

1198951 119910

119894

119895119896119894

sube 119873119894cap119865lowast

119894(119909) = 119873

119894cap119865119894(119909)cap119871

119873119894

we have

120593119873119894(Δ119896) sube 119866119894 (119909) cap 119878119894 (119871119873119894

) = 119866lowast

119894(119909) (6)

then assumption (i) of Lemma 6 is satisfied

Abstract and Applied Analysis 3

By (4) we have

119878 (119871119873) = ( ⋃

119910119894isin119871119873119894

int119865minus1119894(119910119894)) cap 119878 (119871

119873)

= ⋃

119910119894isin119871119873119894

int119878(119871119873)

(119865minus1

119894(119910119894) cap 119878 (119871

119873))

(7)

On the other hand for all 119910119894 isin 119871119873119894

(119865lowast

119894)minus1(119910119894) = 119909 isin 119883 119910

119894isin 119865119894 (119909) cap 119878 (119871119873)

= 119865minus1

119894(119910119894) cap 119878 (119871

119873)

(8)

Hence

119878 (119871119873) = ⋃

119910119894isin119871119873119894

int119878(119871119873)

(119865lowast)minus1(119910119894) (9)

Thus (ii) of Lemma 6 is also satisfied According to Lemma 6there exists a point 119909 = (119909

119894)119894isin119868


all 119894 isin 119868

Remark 8 Theorem 7 generalizes Theorem 34 of Ding [6]from FC-space to GFC-space and our condition (iii) isdifferent from its condition (iii) Theorem 7 also extendsTheorem 3 in [18] Note that Theorem 7 is the variation ofTheorem 32 in [8]

As a special case of Theorem 7 we have the followingfixed point theorem that will be used to prove a weak KKMtheorem in Section 4

Corollary 9 Let 119883 be the Hausdorff space (119883 119884 120593119873) a

GFC-space 119866 119883 rarr 2119883 119865 119883 rarr 2

119884 and 119878 119884 rarr 2119883

with the following properties

(i) for each 119909 isin 119883 119873 = 1199100 1199101 119910

119899 sube 119884 and

1199101198940 1199101198941 119910

119894119896 sube 119873 cap 119865(119909) one has 120593

119873(Δ119896) sube 119866(119909)

(ii) for each compact subset119870 of119883 119870 sube ⋃119910isin119884

int119865minus1(119910)

(iii) for each 119873 isin ⟨119884⟩ there exists an 119878-subset 119871119873of 119884

containing119873 with 119878(119871119873) being compact such that

119878 (119871119873) 119870 sub ⋃

119910isin119871119873

int119865minus1 (119910) (10)

Then there exists 119909 isin 119883 such that 119909 isin 119866(119909)

4 Applications

Theorem 10 Let119883 be a Hausdorff space (119883 119884 120593119873) a GFC-

space 119885 a nonempty set 119879 119883 rarr 2119885 119867 119884 rarr 2

119885 and119878 119884 rarr 2

119883 assume that

(i) H is a weak T-KKM mapping(ii) for each 119910 isin 119884 the set 119909 isin 119883 119879(119909) cap 119867(119910) = 0 is

compactly closed

(iii) there exists a compact 119870 of 119883 and for any 119873 isin ⟨119884⟩there exists an 119878-subset 119871

119873of 119884 containing 119873 with


119878 (119871119873) 119870 sub ⋃

119910isin119871119873

int 119909 isin 119883 119879 (119909) cap 119867 (119910) = 0 (11)

Then there exists a point 119909 isin 119883 such that 119879(119909) cap 119867(119910) = 0 foreach 119910 isin 119884

Proof Define 119865 119883 rarr 2119884 and 119866 119883 rarr 2

119883 by

119865 (119909) = 119910 isin 119884 119879 (119909) cap 119867 (119910) = 0

119866 (119909) = 1199091015840isin 119883 exist119910 isin 119865 (119909) 119879 (119909

1015840) cap 119867 (119910) = 0

(12)

Suppose the conclusion does not holdThen for each 119909 isin119883 there exists a 119910 isin 119884 such that

119879 (119909) cap 119867 (119910) = 0 (13)

It is easy to see that 119865 has nonempty values By (ii) for each119910 isin 119884

119865minus1(119910) = 119909 isin 119883 119879 (119909) cap 119867 (119910) = 0 (14)

is compactly open Then

119883 = ⋃

119910isin119884

int119865minus1 (119910) (15)

Since 119870 is a compact subset of 119883 then there exists 119873 isin ⟨119884⟩

such that

119870 sube ⋃

119910isin119873

int119865minus1 (119910) (16)

Then assumption (ii) of Corollary 9 is satisfiedIt follows from (iii) that there exists a compact119870 of119883 and

for any 119873 isin ⟨119884⟩ there exists a 119878-subset 119871119873of 119884 containing

119873 with 119878(119871119873) being compact such that

119878 (119871119873) 119870 sub ⋃

119910isin119871119873

int119865minus1 (119910) (17)

Therefore assumption (iii) of Corollary 9 is also satisfiedFurthermore 119866 has no fixed point Indeed if 119909 isin 119866(119909)

then there exists 119910 isin 119865(119909) such that

119879 (119909) cap 119867 (119910) = 0 (18)

which contracts the definition of 119865 Thus assumption (i) ofCorollary 9 must be violated that is there exist an 119909 isin 119883119873 = 119910

0 1199101 119910

119899 sube 119884 and

119873119896= 1199101198940 1199101198941 119910

119894119896 sube 119873 cap 119865 (119909) (19)

such that

120593119873(Δ119896) sube 119866 (119909) (20)


That is for each 119910 isin 119865(119909)

119879 (119909) cap 119867 (119910) = 0 (21)

Hence

119879 (119909) cap 119867(119873119896) = 0 (22)

On the other hand since 119867 is a weak T-KKM mappingand 119909 isin 120593

119873(Δ119896) we have

119879 (119909) cap 119867(119873119896) = 0 (23)

which is contradict This completes the proof

Remark 11 (1) Theorem 10 extends Theorem 1 in [13] fromthe G-convex space to GFC-space and our proof techniquesare different Theorem 10 also generalizes Theorem 41 of [8]from the compactness assumption to noncompact situation

(2) If119885 is a topological space condition (ii) inTheorem 10is fulfilled in any of the following cases (see [13])

(i) 119867 has closed values and 119879 is usc on each compactsubset of119883

(ii) 119867 has compactly closed values and 119879 is usc on eachcompact of subset of119883 and its values are compact


space 119885 a topological space 119879 119883 rarr 2119885 usc 119891 119884 times 119885 rarr

R cup minusinfin +infin and 119878 119884 rarr 2119883 assume that

(i) for each 119910 isin 119884 119891(119910 sdot) is usc on each compact subsetof 119885

(ii) 119891 is (120582 119879 119892)-GFC quasiconvex for all 120582 lt 120573 sufficientlyclose to 120573




119878 (119871119873) 119870 sub ⋃

119910isin119871119873

int 119909 isin 119883 119879 (119909) cap 119867120582 (119910) = 0 (24)

Then

inf119909isin119883

sup119911isin119879(119909)

119892 (119909 119911) le sup119909isin119883

inf119910isin119884

sup119911isin119879(119909)

119891 (119910 119911) (25)

Proof Let 120582 lt 120573 be arbitrary By Lemma 5 and condition (ii)119867120582is a weak T-KKM mapping It follows from condition (i)

that 119867120582has closed values Hence the set 119909 isin 119883 119879(119909) cap

119867120582(119910) = 0 is compactly closed for all 119910 isin 119884 (see Remark 11

(2)) Thus all the conditions of Theorem 10 are satisfied andso there exists an 119909 isin 119883 such that

119879 (119909) cap 119867120582 (119910) = 0 forall119910 isin 119884 (26)

This implies that 120582 le inf119910isin119884

sup119911isin119879(119909)

119891(119910 119911) and so

120582 le sup119909isin119883

inf119910isin119884

sup119911isin119879(119909)

119891 (119910 119911) (27)

Since 120582 lt 120573 is arbitrary we get the conclusionThis completesthe proof

Remark 13 Theorem 12 improves Theorem 42 of [8] fromthe compactness assumption to noncompact situationTheorem 12 also extendsTheorem 4 of [12] from compact G-convex space to noncompact GFC-space Our result includescorresponding earlier Fan-type minimax inequalities due toTan [19] Park [20] Liu [21] and Kim [22]

Acknowledgment

This work was supported by the University Research Founda-tion (JBK120926)

References

[1] E Tarafdar ldquoA fixed point theorem and equilibrium point of anabstract economyrdquo Journal of Mathematical Economics vol 20no 2 pp 211ndash218 1991

[2] K Q Lan and J Webb ldquoNew fixed point theorems for a familyof mappings and applications to problems on sets with convexsectionsrdquoProceedings of the AmericanMathematical Society vol126 no 4 pp 1127ndash1132 1998

[3] Q H Ansari and J-C Yao ldquoA fixed point theorem and itsapplications to a system of variational inequalitiesrdquo Bulletin ofthe Australian Mathematical Society vol 59 no 3 pp 433ndash4421999

[4] Q H Ansari A Idzik and J-C Yao ldquoCoincidence andfixed point theorems with applicationsrdquo Topological Methods inNonlinear Analysis vol 15 no 1 pp 191ndash202 2000

[5] X P Ding and K-K Tan ldquoFixed point theorems and equilibriaof noncompact generalized gamesrdquo in Fixed Point Theory andApplications pp 80ndash96 World Sicence Singapore 1992

[6] X P Ding ldquoContinuous selection collectively fixed points andsystem of coincidence theorems in product topological spacesrdquoActa Mathematica Sinica vol 22 no 6 pp 1629ndash1638 2006

[7] L-J Lin andQ H Ansari ldquoCollective fixed points andmaximalelements with applications to abstract economiesrdquo Journal ofMathematical Analysis andApplications vol 296 no 2 pp 455ndash472 2004

[8] P Q Khanh V S T Long and N H Quan ldquoContinu-ous selections collectively fixed points and weak Knaster-Kuratowski-Mazurkiewicz mappings in optimizationrdquo Journalof OptimizationTheory and Applications vol 151 no 3 pp 552ndash572 2011

[9] X PDing andLWang ldquoFixed pointsminimax inequalities andequilibria of noncompact abstract economies in FC-spacesrdquoNonlinear Analysis A vol 69 no 2 pp 730ndash746 2008

[10] X P Ding ldquoCollective fixed points generalized games andsystems of generalized quasi-variational inclusion problems intopological spacesrdquo Nonlinear Analysis A vol 73 no 6 pp1834ndash1841 2010

[11] G X-Z Yuan KKM Theory and Applications in NonlinearAnalysis vol 218 Marcel Dekker New York NY USA 1999

[12] M Balaj ldquoWeakly 119866-KKM mappings 119866-KKM property andminimax inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 294 no 1 pp 237ndash245 2004

[13] M Balaj and D OrsquoRegan ldquoWeak-equilibrium problems in 119866-convex spacesrdquo Rendiconti del Circolo Matematico di Palermovol 57 no 1 pp 103ndash117 2008

[14] G-S Tang Q-B Zhang and C-Z Cheng ldquo119882-119866-119865-KKMmapping intersection theorems and minimax inequalities in


FC-spacerdquo Journal of Mathematical Analysis and Applicationsvol 334 no 2 pp 1481ndash1491 2007

[15] N X Hai P Q Khanh and N H Quan ldquoSome existencetheorems in nonlinear analysis for mappings on GFC-spacesand applicationsrdquoNonlinear Analysis A vol 71 no 12 pp 6170ndash6181 2009

[16] P Q Khanh and N H Quan ldquoIntersection theorems coin-cidence theorems and maximal-element theorems in GFC-spacesrdquo Optimization vol 59 no 1 pp 115ndash124 2010

[17] P Q Khanh and N H Quan ldquoGeneral existence theoremsalternative theorems and applications to minimax problemsrdquoNonlinear Analysis vol 72 no 5 pp 2706ndash2715 2010

[18] S Park ldquoContinuous selection theorems in generalized convexspacesrdquo Numerical Functional Analysis and Optimization vol20 no 5-6 pp 567ndash583 1999

[19] K-K Tan ldquoComparison theorems on minimax inequalitiesvariational inequalities and fixed point theoremsrdquo Journal of theLondon Mathematical Society vol 28 no 3 pp 555ndash562 1983

[20] S Park ldquoGeneralized Fan-Browder fixed point theorems andtheir applicationsrdquo inCollection of PapersDedicated to J G Parkpp 51ndash77 1989

[21] F-C Liu ldquoOn a form of KKM principle and Sup Inf Supinequalities of von Neumann and of Ky Fan typerdquo Journal ofMathematical Analysis and Applications vol 155 no 2 pp 420ndash436 1991

[22] I Kim ldquoKKM theorem and minimax inequalities in 119866-convexspacesrdquo Nonlinear Analysis Forum vol 6 no 1 pp 135ndash1422001

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In the sequel we also use (119883 119884 120593119873) to denote (119883 119884Φ)

Definition 2 Let 119878 119884 rarr 2119883 be a multivalued mapping

A subset 119863 of 119884 is called an 119878-subset of 119884 if and only if foreach 119873 = 119910

0 1199101 119910

119899 sube 119884 and each 119910

1198940 1199101198941 119910

119894119896 sube

119873 cap 119863 one has 120593119873(Δ119896) sub 119878(119863) where Δ

119896is the face of Δ

119899

corresponding to 1199101198940 1199101198941 119910

119894119896 that is the simplex with

vertices 1198901198940 1198901198941 119890

119894119896 Roughly speaking if119863 is an 119878-subset

of 119884 then (119878(119863) 119863Φ) is a GFC-spaceThe class of GFC-space contains a large number of spaces

with various kinds of generalized convexity structures suchas FC-space and G-convex space (see [15ndash17])

Definition 3 (see [8]) Let (119883 119884 120593119873) be a GFC-space and 119885

a nonempty set Let 119879 119883 rarr 2119885 and 119865 119884 rarr 2

119885 be twoset-valued mappings 119865 is called a weak KKM mapping withrespect to 119879 shortly weak T-KKMmapping if and only if foreach 119873 = 119910

0 1199101 119910

119899 sube 119884 119910

1198940 1199101198941 119910

119894119896 sube 119873 and

119909 isin 120593119873(Δ119896) 119879(119909) cap ⋃119896

119895=0119865(119910119894119895) = 0

Definition 4 (see [8]) Let119883 be aHausdorff space (119883 119884 120593119873)

aGFC-space119885 a topological space119879 119883 rarr 2119885119891 119884times119885 rarr

R cup minusinfin +infin and 119892 119883 times119885 rarr R cup minusinfin +infin Let 120582 isin R119891 is called (120582 119879 119892)-GFC quasiconvex if and only if for each119909 isin 119883 119911 isin 119879(119909) 119873 = 119910

0 1199101 119910

119899 isin ⟨119884⟩ and 119873

119896=

1199101198940 1199101198941 119910

119894119896 sube 119873 one has the implication 119891(119910

119894119895 119911) lt 120582

for all 119895 = 0 1 119896 implies that 119892(1199091015840 119911) lt 120582 for all 1199091015840 isin120593119873(Δ119896)

For 120582 isin R define 120573 isin R and 119867120582 119884 rarr 2

119885 by 120573 =

inf119909isin119883

sup119911isin119879(119909)

119892(119909 119911) and 119867120582(119910) = 119911 isin 119885 119891(119910 119911) ge 120582

respectively

Lemma 5 (see [8]) For 120582 lt 120573 if 119891 is (120582 119879 119892)-GFCquasiconvex then119867

120582is a weak T-KKM mapping

The following result is the obvious corollary of Theo-rem 31 of Khanh et al [8]

Lemma 6 Let (119883119894 119884119894 120593119873119894)119894isin119868


119894isin119868119883119894a compact Hausdorff space For each 119894 isin 119868

let119866119894 119883 rarr 2

119883119894 and119865119894 119883 rarr 2

119884119894 be such that the conditionshold as follows

(i) for each 119909 isin 119883 each 119873119894= 119910119894

0 119910119894

1 119910

119894

119899119894 sube 119884

119894and

each 1199101198941198940 119910119894

1198941 119910

119894

119894119896119894

sube 119873119894cap119865119894(119909) one has 120593

119873119894(Δ119896) sube

119866119894(119909) for all 119894 isin 119868

(ii) 119883 = ⋃119910119894isin119884119894

int119865minus1119894(119910119894) for all 119894 isin 119868


isin 119883 such that 119909119894isin 119866119894(119909) for all

119894 isin 119868

3 Fixed Points Theorems

Let 119868 be an index set119883119894topological spaces119883 = prod

119894isin119868119883119894 and

119866119894 119883 rarr 2

119883119894 The collectively fixed points problem is to find119909 = (119909

119894)119894isin119868

isin 119883 such that 119909119894isin 119866119894(119909) for all 119894 isin 119868

Theorem 7 Let (119883119894 119884119894 120593119873119894)119894isin119868


119894isin119868119883119894a Hausdorff space For each 119894 isin 119868 let 119866

119894

119883 rarr 2119883119894 119865119894 119883 rarr 2

119884119894 and 119878119894 119884119894rarr 2119883119894 with the following

properties

(i) for each 119909 isin 119883 119873119894= 119910119894

0 119910119894

1 119910

119894

119899119894 sube 119884

119894 and

119910119894

1198950 119910119894

1198951 119910

119894

119895119896119894

sube 119873119894cap 119865119894(119909) one has 120593

119873119894(Δ119896) sube

119866119894(119909) for all 119894 isin 119868

(ii) for each compact subset 119870 of 119883 and each 119894 isin 119868 119870 sube

⋃119910119894isin119884119894

int119865minus1119894(119910119894)

(iii) there exists a nonempty compact subset 119870119894of 119883119894and

for each 119873119894isin ⟨119884119894⟩ there exists an 119878

119894-subset 119871

119873119894of 119884119894

containing119873119894with 119878


119878 (119871119873) 119870 sub ⋃

119910119894isin119871119873119894

int119865minus1119894(119910119894) (1)

where 119871119873

= prod119894isin119868119871119873119894 119870 = prod

119894isin119868119870119894 and 119878(119871

119873) =

prod119894isin119868119878119894(119871119873119894)



all 119894 isin 119868

Proof As 119870 is a compact subset of 119883 by the condition (ii)there exists a finite set119873

119894= 119910119894

0 119910119894

1 119910

119894

119899119894 sube 119884119894 such that

119870 sube

119899119894

⋃

119896=0

int119865minus1119894(119910119894

119894119896) (2)

By the condition (iii) there exists an 119878119894-subset 119871

119873119894of 119884119894

containing119873119894such that

119878 (119871119873) 119870 sub ⋃

119910119894isin119871119873119894

int119865minus1119894(119910119894) (3)

and it follows that

119878 (119871119873) sub ⋃

119910119894isin119871119873119894

int119865minus1119894(119910119894) (4)

We observe that the family 119878119894(119871119873119894) 119871119873119894 120593119873119894)119894isin119868

is a familyof GFC-space and 119878

119894(119871119873119894) is compact for each 119894 isin 119868 defining

set-valued mapping 119866lowast

119894 119878119894(119871119873119894) rarr 2

119878119894(119871119873119894) and 119865

lowast

119894

119878119894(119871119873119894) rarr 2

119871119873119894 as follows

119866lowast

119894(119909) = 119866119894 (119909) cap 119878119894 (119871119873119894

)

119865lowast

119894(119909) = 119865119894 (119909) cap 119871119873119894

(5)

We check assumptions (i) and (ii) of Lemma 6 for replaced119866119894

and 119865119894by 119866lowast119894and 119865lowast119894 respectively By (i) and the definition of

119878-subset for each 119909 isin 119878119894(119871119873119894) each 119873

119894= 119910119894

0 119910119894

1 119910

119894

119899119894 sube

119871119873119894and each 119910119894

1198950 119910119894

1198951 119910

119894

119895119896119894

sube 119873119894cap119865lowast

119894(119909) = 119873

119894cap119865119894(119909)cap119871

119873119894

we have

120593119873119894(Δ119896) sube 119866119894 (119909) cap 119878119894 (119871119873119894

) = 119866lowast

119894(119909) (6)

then assumption (i) of Lemma 6 is satisfied


By (4) we have

119878 (119871119873) = ( ⋃

119910119894isin119871119873119894

int119865minus1119894(119910119894)) cap 119878 (119871

119873)

= ⋃

119910119894isin119871119873119894

int119878(119871119873)

(119865minus1

119894(119910119894) cap 119878 (119871

119873))

(7)


(119865lowast

119894)minus1(119910119894) = 119909 isin 119883 119910

119894isin 119865119894 (119909) cap 119878 (119871119873)

= 119865minus1

119894(119910119894) cap 119878 (119871

119873)

(8)

Hence

119878 (119871119873) = ⋃

119910119894isin119871119873119894

int119878(119871119873)

(119865lowast)minus1(119910119894) (9)


119894)119894isin119868


all 119894 isin 119868





119884 and 119878 119884 rarr 2119883


(i) for each 119909 isin 119883 119873 = 1199100 1199101 119910

119899 sube 119884 and

1199101198940 1199101198941 119910

119894119896 sube 119873 cap 119865(119909) one has 120593

119873(Δ119896) sube 119866(119909)


int119865minus1(119910)



119878 (119871119873) 119870 sub ⋃

119910isin119871119873

int119865minus1 (119910) (10)


4 Applications



119885 and119878 119884 rarr 2

119883 assume that


compactly closed




119878 (119871119873) 119870 sub ⋃

119910isin119871119873

int 119909 isin 119883 119879 (119909) cap 119867 (119910) = 0 (11)



119883 by

119865 (119909) = 119910 isin 119884 119879 (119909) cap 119867 (119910) = 0

119866 (119909) = 1199091015840isin 119883 exist119910 isin 119865 (119909) 119879 (119909

1015840) cap 119867 (119910) = 0

(12)


119879 (119909) cap 119867 (119910) = 0 (13)


119865minus1(119910) = 119909 isin 119883 119879 (119909) cap 119867 (119910) = 0 (14)


119883 = ⋃

119910isin119884

int119865minus1 (119910) (15)


such that

119870 sube ⋃

119910isin119873

int119865minus1 (119910) (16)




119878 (119871119873) 119870 sub ⋃

119910isin119871119873

int119865minus1 (119910) (17)



119879 (119909) cap 119867 (119910) = 0 (18)


0 1199101 119910

119899 sube 119884 and

119873119896= 1199101198940 1199101198941 119910

119894119896 sube 119873 cap 119865 (119909) (19)

such that

120593119873(Δ119896) sube 119866 (119909) (20)



119879 (119909) cap 119867 (119910) = 0 (21)

Hence

119879 (119909) cap 119867(119873119896) = 0 (22)


119873(Δ119896) we have

119879 (119909) cap 119867(119873119896) = 0 (23)














119878 (119871119873) 119870 sub ⋃

119910isin119871119873

int 119909 isin 119883 119879 (119909) cap 119867120582 (119910) = 0 (24)

Then

inf119909isin119883

sup119911isin119879(119909)

119892 (119909 119911) le sup119909isin119883

inf119910isin119884

sup119911isin119879(119909)

119891 (119910 119911) (25)





119879 (119909) cap 119867120582 (119910) = 0 forall119910 isin 119884 (26)


sup119911isin119879(119909)

119891(119910 119911) and so

120582 le sup119909isin119883

inf119910isin119884

sup119911isin119879(119909)

119891 (119910 119911) (27)



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




By (4) we have

119878 (119871119873) = ( ⋃

119910119894isin119871119873119894

int119865minus1119894(119910119894)) cap 119878 (119871

119873)

= ⋃

119910119894isin119871119873119894

int119878(119871119873)

(119865minus1

119894(119910119894) cap 119878 (119871

119873))

(7)


(119865lowast

119894)minus1(119910119894) = 119909 isin 119883 119910

119894isin 119865119894 (119909) cap 119878 (119871119873)

= 119865minus1

119894(119910119894) cap 119878 (119871

119873)

(8)

Hence

119878 (119871119873) = ⋃

119910119894isin119871119873119894

int119878(119871119873)

(119865lowast)minus1(119910119894) (9)


119894)119894isin119868


all 119894 isin 119868





119884 and 119878 119884 rarr 2119883


(i) for each 119909 isin 119883 119873 = 1199100 1199101 119910

119899 sube 119884 and

1199101198940 1199101198941 119910

119894119896 sube 119873 cap 119865(119909) one has 120593

119873(Δ119896) sube 119866(119909)


int119865minus1(119910)



119878 (119871119873) 119870 sub ⋃

119910isin119871119873

int119865minus1 (119910) (10)


4 Applications



119885 and119878 119884 rarr 2

119883 assume that


compactly closed




119878 (119871119873) 119870 sub ⋃

119910isin119871119873

int 119909 isin 119883 119879 (119909) cap 119867 (119910) = 0 (11)



119883 by

119865 (119909) = 119910 isin 119884 119879 (119909) cap 119867 (119910) = 0

119866 (119909) = 1199091015840isin 119883 exist119910 isin 119865 (119909) 119879 (119909

1015840) cap 119867 (119910) = 0

(12)


119879 (119909) cap 119867 (119910) = 0 (13)


119865minus1(119910) = 119909 isin 119883 119879 (119909) cap 119867 (119910) = 0 (14)


119883 = ⋃

119910isin119884

int119865minus1 (119910) (15)


such that

119870 sube ⋃

119910isin119873

int119865minus1 (119910) (16)




119878 (119871119873) 119870 sub ⋃

119910isin119871119873

int119865minus1 (119910) (17)



119879 (119909) cap 119867 (119910) = 0 (18)


0 1199101 119910

119899 sube 119884 and

119873119896= 1199101198940 1199101198941 119910

119894119896 sube 119873 cap 119865 (119909) (19)

such that

120593119873(Δ119896) sube 119866 (119909) (20)



119879 (119909) cap 119867 (119910) = 0 (21)

Hence

119879 (119909) cap 119867(119873119896) = 0 (22)


119873(Δ119896) we have

119879 (119909) cap 119867(119873119896) = 0 (23)














119878 (119871119873) 119870 sub ⋃

119910isin119871119873

int 119909 isin 119883 119879 (119909) cap 119867120582 (119910) = 0 (24)

Then

inf119909isin119883

sup119911isin119879(119909)

119892 (119909 119911) le sup119909isin119883

inf119910isin119884

sup119911isin119879(119909)

119891 (119910 119911) (25)





119879 (119909) cap 119867120582 (119910) = 0 forall119910 isin 119884 (26)


sup119911isin119879(119909)

119891(119910 119911) and so

120582 le sup119909isin119883

inf119910isin119884

sup119911isin119879(119909)

119891 (119910 119911) (27)



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119879 (119909) cap 119867 (119910) = 0 (21)

Hence

119879 (119909) cap 119867(119873119896) = 0 (22)


119873(Δ119896) we have

119879 (119909) cap 119867(119873119896) = 0 (23)














119878 (119871119873) 119870 sub ⋃

119910isin119871119873

int 119909 isin 119883 119879 (119909) cap 119867120582 (119910) = 0 (24)

Then

inf119909isin119883

sup119911isin119879(119909)

119892 (119909 119911) le sup119909isin119883

inf119910isin119884

sup119911isin119879(119909)

119891 (119910 119911) (25)





119879 (119909) cap 119867120582 (119910) = 0 forall119910 isin 119884 (26)


sup119911isin119879(119909)

119891(119910 119911) and so

120582 le sup119909isin119883

inf119910isin119884

sup119911isin119879(119909)

119891 (119910 119911) (27)



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014










Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014



Documents

ResearchArticle A New Fixed Point Theorem and Applications · 2014-04-25 · 2 AbstractandAppliedAnalysis Inthesequel,wealsouse (𝑋,𝑌,{𝜑𝑁})todenote(𝑋,𝑌,Φ). Definition2.Let𝑆:𝑌→2𝑋beamultivaluedmapping