This article was downloaded by: [York University Libraries]On: 23 November 2014, At: 02:20Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Optimization: A Journal ofMathematical Programming andOperations ResearchPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gopt20

Intersection theorems, coincidencetheorems and maximal-elementtheorems in GFC-spacesPhan Quoc Khanh a & Nguyen Hong Quan ba Department of Mathematics , International University of Ho ChiMinh City , Linh Trung, Thu Duc, Ho Chi Minh City, Vietnamb Department Mathematics , Information Technology College of HoChi Minh City , Hoa Thanh, Tan Phu, Ho Chi Minh City, VietnamPublished online: 12 Feb 2010.

To cite this article: Phan Quoc Khanh & Nguyen Hong Quan (2010) Intersection theorems,coincidence theorems and maximal-element theorems in GFC-spaces, Optimization: AJournal of Mathematical Programming and Operations Research, 59:1, 115-124, DOI:10.1080/02331930903500324

To link to this article: http://dx.doi.org/10.1080/02331930903500324

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

2:20

23

Nov

embe

r 20

14

OptimizationVol. 59, No. 1, January 2010, 115–124

Intersection theorems, coincidence theorems and

maximal-element theorems in GFC-spaces

Phan Quoc Khanha* and Nguyen Hong Quanb

aDepartment of Mathematics, International University of Ho Chi Minh City, Linh Trung,Thu Duc, Ho Chi Minh City, Vietnam; bDepartment Mathematics,Information Technology College of Ho Chi Minh City, Hoa Thanh,

Tan Phu, Ho Chi Minh City, Vietnam

(Received 20 August 2008; final version received 19 March 2009)

We propose a definition of GFC-spaces to encompass G-convex spaces,FC-spaces and many recent existing spaces with generalized convexitystructures. Intersection, coincidence and maximal-element theorems arethen established under relaxed assumptions in GFC-spaces. These resultscontain, as true particular cases, a number of counterparts which wererecently developed in the literature.

Keywords: GFC-spaces; generalized T-KKM mappings; better admissiblemappings; compact openness; compact closedness

AMS Subject Classifications: 49J40; 47H04; 91B50; 47H10; 54H25

1. Introduction

Intersection theorems, coincidence theorems and maximal-element theorems areamong the fundamental theorems of nonlinear analysis and play crucial roles in theexistence study of wide-ranging problems of optimization and applied mathematics.Intersection theorems were applied in equilibrium problems and game theory (e.g. in[8,11]). For applications of coincidence theorems and fixed point theorems inexistence study, see [2,16,18,20,21,31] among others. Theorems on maximal elementswere employed in such investigations (e.g. in [1,9,13,15] and references therein).These theorems are closely related to other important theorems like fixed-pointtheorems, minimax theorems, invariant-point theorems and KKM-type theorems.Recently, these theorems have been generalized and extended along with general-izations of KKM mappings to general spaces with relaxed convexity structures. ForG-convex spaces [30], intersection theorems was studied in [8]; coincidence theoremswere established in [29,31]; maximal-element theorems were investigated in [6,7].FC-spaces were introduced in [7] with maximal-element theorems established. In[10–12] intersection and coincidence theorems in FC-spaces were considered. In thisarticle we propose a definition of a generalized FC-space (GFC-space in short) andestablish these three kinds of theorems. Our results contain several recent existing

*Corresponding author. Email: pqkhanh@hcmiu.edu.vn

ISSN 0233–1934 print/ISSN 1029–4945 online

� 2010 Taylor & Francis

DOI: 10.1080/02331930903500324

http://www.informaworld.com

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

2:20

23

Nov

embe

r 20

14

results in the literature as special cases. We leave applications of our results

to optimization-related problems, especially to studies of the solution existence, to

other works.The following several sentences were added in the final revision of this article,

since when finishing this revision our several developments of the results of the

present article have been completed. In [25] we applied generalized KKM properties

in GFC-spaces presented below to establish general KKM-type theorems. Work in

[22] was devoted to applications of these theorems to quasivariational inclusion

problems and minimax theorems. We developed stability properties related to KKM

theorems (and the so-called KKM points) in [23]. The three theorems dealt with this

article were improved and developed for product settings in [17,24].Our article is split into four sections. In the remaining part of this section we

propose a notion of GFC-spaces and generalize some definitions of known classes

of multivalued mappings from FC-spaces to GFC-spaces. Section 2 is devoted to

intersection theorems. In Section 3 some coincidence theorems and fixed-point

consequences are proved. Maximal-element theorems are developed in Section 4.For a set A, by hAi we denote the family of all finite subsets of A. If A�X,

X being a topological space, then A and Ac signify the closure and the complement

X \A, respectively, of A. A is called compactly open (compactly closed, respectively)

if for each nonempty compact subset K of X,A\K is open (closed, respectively) in K.

The compact interior and compact closure of A are defined by, respectively,

cintA ¼[fB � X : B � A and B is compactly open in X g,

cclA ¼\fB � X : B � A and B is compactly closed in X g:

For topological spaces X, Y and a multivalued mapping F :X! 2Y, F is said to be

upper semicontinuous (usc, in short) at x if, for each open subset U such that

U�F(x), there is a neighbourhood V of x such that U�F(V ). F is called

transfer-compactly-open-valued if, for each x2X and each compact subset K of

Y, y2F(x)\K implies the existence of x0 2X such that y2 intK(F(x0)\K ), where intK

stands for the interior in K, i.e. interior with respect to the topology of K induced

by the topology of Y. N stands for the set of all natural numbers. Dn, n2N, denotes

the n-simplex with the vertices being the unit vectors e1, e2, . . . , enþ1, which form

a basis of Rnþ1.

Definition 1.1 (i) Let X be a topological space, Y be a nonempty set and � be a

family of continuous mappings ’ :Dn!X, n2N. Then a triple (X,Y,�) is said to be

a generalized finitely continuous topological space (GFC-space in short) if for each

finite subset N¼ {y0, y1, . . . , yn}2 hY i there is ’N :Dn!X of the family �. Later we

also use (X,Y, {’N}) to denote (X,Y,�).(ii) Let S :Y! 2X be a multivalued mapping. A subset D of Y is called an

S-subset of Y if, for each N¼ {y0, y1, . . . , yn}2 hY i and each {yi0, yi1, . . . , yik}�N\D,

one has ’N :Dn!X of � such that ’N(Dk)�S(D), where Dk is the face of Dn

corresponding to {yi0, yi1, . . . , yik}.

Note that if Y¼X, then (X,Y,�) is rewritten as (X,�) and becomes an FC-space

[8]. If, in addition, S is the identity map then an S-subset of X coincides with a

FC-subspace of X [9]. Note also that FC-spaces and GFC-spaces have no convexity

116 P.Q. Khanh and N.H. Quan

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

2:20

23

Nov

embe

r 20

14

structure but they are generalizations of spaces with convexity structures (see e.g.[9]); in particular, of a nonempty convex subset of a vector space. If Y�X and(X,Y,�) is a G-convex space (� is a generalized convex hull operator, see [30]), and� is the family of continuous mappings ’N :Dn!�(N) as defined in [30], then(X,Y,�) is a GFC-space. Both G-convex space and FC-space are general andinclude many spaces mentioned in the literature, but they are incomparable. We haveseen that both of them are special cases of GFC-spaces. The notion of GFC-spacehelps us also to encompass many generalized KKM-mappings as shown afterDefinition 1.3.

Definition 1.2 Let (X,Y,�) be a GFC-space and Z be a topological space.A multivalued mapping T :X! 2Z is called better admissible if T is usc andcompact-valued such that for each N2 hY i and each continuous mapping :T(’N(Dn))!Dn, the composition � Tj’N(Dn) � ’N :Dn! 2Dn has a fixed point,where ’N2� is corresponding to N.

The class of all such better admissible mappings from X to Z is denoted byB(X,Y,Z). If Y¼X, we simply write B(X,Z). This class was proposed for theparticular case where X is a nonempty convex subset of a vector space in [27],extended later for G-convex spaces in [28] and for FC-spaces in [9].

Definition 1.3 Let (X,Y,�) be a GFC-space and Z be a topological space. LetF :Y! 2Z and T :X! 2Z be multivalued mappings. F is said to be a generalizedKKM mapping with respect to (wrt) T (T-KKM mapping in short) if for eachN¼ {y0, y1, . . . , yn}2 hY i and each {yi0, yi1, . . . , yik}�N one has Tð’NðDkÞÞ �Sk

j¼0 Fð yijÞ, where ’N2� is corresponding to N and Dk is the k-simplexcorresponding to {yi0, yi1, . . . , yik} in Definition 1.1.

T-KKM mappings were introduced for X being a convex subset of a topologicalvector space in [3] and extended for FC-spaces in [9]. Definition 1.3 includes thesedefinitions as particular cases. It encompasses also many other kinds of generalizedKKM mappings. We mention here some of them, while devoting [25] to generalizedKKM-type theorems in GFC-spaces. Let (X, {’N}) be an FC-space, Y be a nonemptyset and s :Y!X be a mapping. We define a GFC-space (X,Y, {’N}) by setting’N¼ ’s(N) for each N2 hY i. Then a generalized s-KKMmapping wrt T introduced in[10] becomes a T-KKM mapping by Definition 1.3. A multivalued mappingF :Y! 2X, being an R-KKM mapping as defined in [4], is a special case of T-KKMmappings on GFC-space when X¼Z and T is the identity map. The definition ofgeneralized KKM mappings wrt T in [26] is as well a particular case of Definition1.3.

Remark 1 Since a multivalued mapping S :Y! 2X is equivalent to a relation �(y, x)linking y2Y and x2X by setting: x2S(y) if and only if �(y, x) holds, theabove-mentioned notions related to multivalued mappings can be expressed in termsof relations. For instance, the definition of an S-subset of Y can be restated asfollows: for a relation � linking y2Y and x2X, a subset D of Y is called a �-subsetof Y if for each N¼ {y0, y1, . . . , yn}2 hY i and each {yi0, yi1, . . . , yik}�N\D, one has’NðDkÞ �

Sy2Dfx2X : �ð y, xÞ holds}. The class of better admissible relations

R(X,Y,Z) and, for a relation �(x, z) linking x2X and z2Z, an �-KKM relationF (y, z) linking y2Y and z2Z are defined similarly, corresponding to Definitions 1.2

Optimization 117

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

2:20

23

Nov

embe

r 20

14

and 1.3, respectively. Formulations of results involving multivalued mappings interms of relations may be very convenient when many variables and compositionsof mappings are involved (see, e.g. [19]), where variational inclusion problems arestated in terms of variational relations.

2. Intersection theorems

LEMMA 2.1 Let (X,Y, {’N}) be a GFC-space and Z be a topological space.Let F :Y! 2Z and T :X! 2Z be multivalued mappings. Assume that

(i) for each y2Y,F(y) is compactly closed;(ii) T2B(X,Y,Z) and F is T-KKM.

Then, for each N¼ {y0, y1, . . . , yn}2 hY i, Tð’NðDnÞÞ \T

yi 2NFð yiÞ 6¼ ;:

Proof Suppose to the contrary that N¼ {y0, y1, . . . , yn}2 hY i exists such that

Tð’NðDnÞÞ ¼[ni¼0

½ðZnFð yiÞÞ \ Tð’NðDnÞÞ�,

i.e. fðZnFð yiÞÞ \ Tð’NðDnÞÞgni¼0 is an open covering of the compact set T(’N(Dn)).

Let f igni¼0 be a continuous partition of unity associated with this covering and

:T(’N(Dn))!Dn be defined by ðzÞ ¼Pn

i¼0 iðzÞei: Then is continuous. Since Tis better admissible, there is a fixed point of �Tj’N(Dn)�’N, i.e. there is z02T(’N(Dn))such that z02T(’N( (z0))). We have

ðz0Þ ¼X

j2 Jðz0Þ

j ðz0Þej 2DJðz0Þ,

where J(z0)¼ { j2 {0, 1, . . . , n} : j (z0) 6¼ 0}. As F is T-KKM, we also have

z0 2Tð’Nð ðz0ÞÞÞ

� Tð’NðDJðz0ÞÞÞ

�[

j2 Jðz0Þ

Fð yj Þ:

Hence, there exists j2 J(z0), z02F(yj).On the other hand, by the definitions of J(z0) and of the partition f ig

ni¼0,

z0 2 fz2Tð’NðDnÞÞ : j ðzÞ 6¼ 0g

� ðZnFð yj ÞÞ \ Tð’NðDnÞÞ

� ðZnFð yj ÞÞ,

a contradiction. g

THEOREM 2.2 Let (X,Y, {’N}) be a GFC-space, Z be a topological space.Let T2B(X,Y,Z) and F :Y! 2Z satisfy the following conditions:

(i) for each y2Y,F(y) is compactly closed;(ii) F is T-KKM;(iii) either of the following three conditions holds:(iii1) there are N02 hY i and a nonempty compact subset K of Z such thatT

yi 2N0Fð yiÞ � K;

118 P.Q. Khanh and N.H. Quan

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

2:20

23

Nov

embe

r 20

14

(iii2) there is S :Y! 2X such that for each N2 hY i, there exists an S-subset LN of

Y, containing N, so that S(LN) is a compact subset and, for some nonempty

and compact subset K of Z, TðSðLNÞÞ \ ðT

y2LNFð yÞÞ � K;

(iii3) there are S :Y! 2X and a nonempty subset Y0�Y such that K :¼T

y2Y0Fð yÞ

is compact and that, for each N2 hY i, there exists an S-subset LN of Y

containing Y0[N so that S(LN) is compact.

Then K \ TðXÞ \ ðT

y2Y Fð yÞÞ 6¼ ;.

Proof Case of (iii1). For y2Y, set

Uð yÞ ¼ TðXÞ \

� \yi 2N0

Fð yiÞ

�\ Fð yÞ:

By (i) and (iii1), {U(y)}y2Y is a family of sets which are closed in K. For each N2 hY i,

setting M¼N[N0 and m¼ jNj þ jN0j, by Lemma 2.1 we have

; 6¼ Tð’MðDmÞÞ \

� \y2M

Fð yÞ

�

� TðXÞ \

� \y2M

Fð yÞ

�

¼\y2N

Uð yÞ:

Since K is compact, this implies that

; 6¼\y2Y

Uð yÞ � K \ TðXÞ \

� \y2Y

Fð yÞ

�:

Case of (iii2). By (i), fK \ TðXÞ \ Fð yÞgy2Y is a family of sets which are closed in K

(K is given in (iii2)). Suppose that

; ¼ K \ TðXÞ \

� \y2Y

Fð yÞ

�¼\y2Y

ðK \ TðXÞ \ Fð yÞÞ:

Then there exists N2 hY i such that

; ¼\y2N

ðK \ TðXÞ \ Fð yÞÞ

¼ K \ TðXÞ \

� \y2N

Fð yÞ

�,

i.e.

TðXÞ \

� \y2N

Fð yÞ

�� ZnK:

In view of the assumption (iii2),

TðSðLNÞÞ \

� \y2LN

Fð yÞ

�� K:

Optimization 119

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

2:20

23

Nov

embe

r 20

14

On the other hand,

TðSðLNÞÞ \

� \y2LN

Fð yÞ

�� TðXÞ \

� \y2N

Fð yÞ

�� ZnK:

Thus,

TðSðLNÞÞ \

� \y2LN

Fð yÞ

�¼ ;:

As LN is an S-subset of Y, by virtue of Lemma 2.1 we have, for each M2 hLNi,

; 6¼ Tð’MðDmÞÞ \

� \y2M

Fð yÞ

�

� TðSðLNÞÞ \

� \y2M

Fð yÞ

�:

By the compactness of T(S(LN)) (as T is usc, compact-valued and S(LN) is compact),this implies that

TðSðLNÞÞ \

� \y2LN

Fð yÞ

�6¼ ;,

a contradiction.

Case of (iii3). If K is compact, then fK \ TðXÞ \ Fð yÞgy2Y is a family of closed subsetsof K. Suppose that

K \ TðXÞ \

� \y2Y

Fð yÞ

�¼ ;:

Then there is N2 hY i such that

; ¼\y2N

ðK \ TðXÞ \ Fð yÞÞ

¼ TðXÞ \

� \y2Y0[N

Fð yÞ

�:

Therefore,

TðSðLNÞÞ \

� \y2LN

Fð yÞ

�� TðXÞ \

� \y2Y0[N

Fð yÞ

�¼ ;:

Now we can argue similarly as for the case (iii2) to get a contradiction. g

Note that Theorem 2.2 contains Theorem 3 of [31] and Theorems 1–3 of [5] asspecial cases for the case of G-convex spaces.

3. Coincidence theorems

THEOREM 3.1 Let (X,Y, {’N}) be a GFC-space and Z be a topological space. LetS :Y! 2X, T :X! 2Z and F :Z!2Y be multivalued mappings with T2B(X,Y,Z).

120 P.Q. Khanh and N.H. Quan

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

2:20

23

Nov

embe

r 20

14

Assume that

(i) for each x2X and each z2T(x), F(z) is an S-subset of Y;(ii) for each y2Y,F�1(y) contains a compactly open Oy (some Oy may be empty)

of Z such that K :¼S

y2Y Oy is nonempty and compact;(iii) either of the following three conditions holds:(iii1) there is N02 hY i such that

Ty2N0

Ocy � K;

(iii2) for each N2 hY i, there is an S-subset LN of Y, containing N such that S(LN) is

compact and TðSðLNÞÞ \ ðS

y2LNOc

yÞ � K;(iii3) K¼Z; there is a nonempty subset Y0 of Y such that

Ty2Y0

Ocy is compact or

empty; and for each N2 hY i there is an S-subset LN of Y containing Y0[N

so that S(LN) is compact.

Then a point ð �x, �y, �zÞ 2X� Y� Z exists such that �x2Sð �yÞ, �y2Fð �zÞ and �z2Tð �xÞ.

Proof Define a new multivalued mapping G :Y! 2Z by setting, 8y2Y, Gð yÞ ¼ Ocy,

which is compactly closed by (ii), i.e. assumption (i) of Theorem 2.2 for G in the place

of F is fulfilled. It is clear that (iii1), (iii2) and (iii3) imply the corresponding

assumptions of Theorem 2.2 for G. By the definition of K in (ii),

K \ TðXÞ \

� \y2Y

Gð yÞ

�� K \

� \y2Y

Ocy

�¼ ;,

which means that the conclusion of Theorem 2.2 for G in the place of F does not

hold. Therefore assumption (ii) of this theorem must be violated, i.e. G is not

T-KKM. This means that there are N¼ {y0, y1, . . . , yn}2 hY i and {yi0, yi1, . . . , yik}�N

such that

Tð’NðDkÞÞ 6�[kj¼0

Gð yij Þ ¼[kj¼0

Ocyij:

This in turn is equivalent to the existence of �x2 ’NðDkÞ and �z2Tð �xÞ such that �z2Oyij,

for all j¼ 0, 1, . . . , k. Since Oyij�F�1(yij) by (ii), yij 2Fð �zÞ, which is an S-subset of Y.

Hence

�x2 ’NðDkÞ � SðFð �zÞÞ,

which means that there is �y2Fð �zÞ such that �x2Sð �yÞ. g

Remark 1

(i) By setting H(y)¼F�1(y) for y2Y, Theorem 3.1 can be restated with the

conclusion that there is ð �x, �y, �zÞ 2X� Y� Z and �x2Sð �yÞ and �z2

Hð �yÞ \ Tð �xÞ. So Theorem 3.1 includes Theorem 1 of [29].(ii) For the special case, where X¼Y¼Z and T¼S is the identity mapping,

Theorem 3.1 becomes a fixed-point theorem for FC-spaces.

4. Maximal-element theorems

THEOREM 4.1 Let (X,Y, {’N}) be a GFC-space,Z be a topological space and K�Z

be nonempty and compact. Let F :Z! 2Y and T :X! 2Z be multivalued mapping such

Optimization 121

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

2:20

23

Nov

embe

r 20

14

that T2B(X,Y,Z) and the following assumptions are satisfied:

(i) for each y2Y,F�1(y) includes a compactly open subset Oy (some Oy may be

empty) of Z such thatS

y2YðOy \ KÞ ¼S

y2YðF�1ð yÞ \ KÞ;

(ii) for each N¼ {y0, y1, . . . , yn}2 hY i and each {yi0, yi1, . . . , yik}�N,

Tð’NðDkÞÞ \ ðTk

j¼0 OyijÞ ¼ ;;

(iii) either of the following conditions hold:(iii1) there is N02 hY i such that ZnK �

Syi 2N0

Oyi;

(iii2) there is a multivalued map S :Y! 2X such that, for each N2 hY i, there is

an S-subset LN of Y containing N so that S(LN) is compact and

TðSðLNÞÞnK �S

y2LNOy.

Then a point �z2K exists such that Fð �zÞ ¼ ;.

Proof We check the assumptions of Theorem 2.2 in order to apply it for, instead

of F, a new multivalued mapping G :Y! 2Z defined by Gð yÞ ¼ Ocy: Assumption (i)

is clearly fulfilled. For (ii), with arbitrary N¼ {y0, y1, . . . , yn}2 hY i and

{yi0, yi1, . . . , yik}�N, we have, by (ii) of Theorem 4.1,

Tð’NðDkÞÞ �[kj¼0

Ocyij¼[kj¼0

Gð yijÞ,

i.e. G is T-KKM as required. By (iii1) of this theorem, we obtain (iii1) since\yi 2N0

Gð yiÞ ¼\

yi 2N0

Ocyi� ZnðZnKÞ ¼ K:

From (iii2) of this theorem it follows that

TðSðLNÞÞ \

� \y2LN

Gð yÞ

�¼TðSðLNÞÞ \

�Zn

[y2LN

Oy

�

�TðSðLNÞÞ \ fZnðTðSðLNÞÞnKÞg

�K,

i.e. (iii2) is satisfied. Now that all the assumptions of Theorem 2.2 have been checked,

we obtain from this theorem

; 6¼ K \ TðXÞ \

� \y2Y

Gð yÞ

�¼K \ TðXÞ \

�Zn

[y2Y

Oy

�

�K \ TðXÞ \

�Zn

[y2Y

ðOy \ KÞ

�

¼K \ TðXÞ \

�Zn

[y2Y

ðF�1ð yÞ \ KÞ

�:

Therefore an element �z2K exists such that �z =2F�1ð yÞ \ K for every y2Y, i.e.

Fð �zÞ ¼ ;. g

Remark 1 Assumption (i) of Theorem 4.1 is satisfied with Oy¼ cintF�1(y) if F�1 is

transfer-compactly-open-valued (by Lemma 1.2 of [9]). This case of Theorem 4.1

extends Theorem 2.2 of [9] to the case of GFC-spaces.

122 P.Q. Khanh and N.H. Quan

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

2:20

23

Nov

embe

r 20

14

THEOREM 4.2 Let (X,Y, {’N}), Z,F and T be defined as in Theorem 4.1 such that(ii) is satisfied and (i) and (iii) are replaced, respectively, by

(i’) for each y2Y, F�1(y) contains a compactly open subset Oy, which may beempty, of Z such that, for each nonempty compact subset Z0 of Z,S

y2YðOy \ Z0Þ ¼S

y2YðF�1ð yÞ \ Z0Þ;

(iii’) there are a multivalued map S :Y! 2X and a nonempty subset Y0 of Y suchthat K :¼

Ty2Y0

Ocy is compact and that, for each N2 hY i, there exists an

S-subset LN of Y containing Y0[N so that S(LN) is compact.

Then an element �z2Z exists with Fð �zÞ ¼ ;:

Proof It is not hard to see that all the assumptions (i), (ii) and (iii3) of Theorem 2.2are fulfilled with G :Y! 2Z defined by Gð yÞ ¼ Oc

y in the place of F. By this theoremwe have

; 6¼ K \ TðXÞ \

� \y2Y

Gð yÞ

�� K \ TðXÞ \

�Zn

[y2Y

ðOy \ KÞ

�

¼ K \ TðXÞ \

�Zn

[y2Y

ðF�1ð yÞ \ KÞ

�:

Consequently, there is �z2K such that �z =2F�1ð yÞ for each y2Y, which means thatFð �zÞ ¼ ;: g

Similarly as mentioned in Remark 4.1, if F�1 is transfer-compactly-open-valued,by taking Oy¼ cintF�1(y), all the assumptions of Theorem 4.2 are satisfied. This caseof Theorem 4.2 includes Theorem 2.1 of [9] as a special case for the FC-space setting.

Acknowledgement

This work was supported by the National Foundation for Science and TechnologyDevelopment of Vietnam.

References

[1] L.Q. Anh and P.Q. Khanh, Existence conditions in symmetric multivalued vector

quasiequilibrium problems, Control Cyber. 36 (2007), pp. 519–530.[2] K.C. Border, Fixed-Point Theorems with Applications to Economics and Game Theory,

Cambridge University Press, Cambridge, 1985.

[3] T.H. Chang and C.L. Yen, KKM property and fixed-point theorems, J. Math. Anal. Appl.

203 (1996), pp. 21–28.[4] L. Deng and X. Xia, Generalized R-KKM theorems in topological spaces and their

applications, J. Math. Anal. Appl. 285 (2003), pp. 679–690.[5] X.P. Ding, Generalized G-KKM theorems in generalized convex spaces and their

applications, J. Math. Anal. Appl. 266 (2002), pp. 21–37.[6] X.P. Ding, Maximal elements of GB-majorized mappings in product G-convex spaces (I),

Appl. Math. Mech. 24 (2003), pp. 659–672.

[7] X.P. Ding, Maximal element of GB-majorized mappings in product G-convex spaces (II),

Appl. Math. Mech. 24 (2003), pp. 1017–1034.

Optimization 123

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

2:20

23

Nov

embe

r 20

14

[8] X.P. Ding, Nonempty intersection theorems and systems of generalized vector equilibriumproblems in G-convex spaces, Appl. Math. Mech. 25 (2004), pp. 618–626.

[9] X.P. Ding, Maximal-element theorems in product FC-spaces and generalized games,J. Math. Anal. Appl. 305 (2005), pp. 29–42.

[10] X.P. Ding, Generalized KKM type theorems in FC-spaces with applications (I), J. GlobalOptim. 36 (2006), pp. 581–596.

[11] X.P. Ding, Nonempty intersection theorems and generalized multi-objective games in

product FC-spaces, J. Global Optim. 37 (2007), pp. 63–73.[12] X.P. Ding, Generalized KKM type theorems in FC-spaces with applications (II), J. Global

Optim. 38 (2007), pp. 367–385.

[13] N.X. Hai and P.Q. Khanh, Systems of multivalued quasiequilibrium problems, Adv.Nonlin. Var. Inequalities 9 (2006), pp. 109–120.

[14] N.X. Hai and P.Q. Khanh, Existence of solutions to general quasi-equilibrium problems

and applications, J. Optim. Theory Appl. 133 (2007), pp. 317–327.[15] N.X. Hai and P.Q. Khanh, Systems of set-valued quasivariational inclusion problems,

J. Optim. Theory Appl. 135 (2007), pp. 55–67.[16] N.X. Hai, P.Q. Khanh, and N.H. Quan, On the existence of solutions to quasivariational

inclusion problems, J. Global Optim. 45 (2009), pp. 565–581.[17] N.X. Hai, P.Q. Khanh, and N.H. Quan, Some existence theorems in nonlinear

analysis for mappings on GFC-spaces and applications, Nonlinear Anal. TMA 71 (2009),

pp. 6170–6181.[18] A. Khaliq, Implicit vector quasi-equilibrium problems with applications to variational

inequalities, Nonlinear Anal. 63 (2005), pp. 1823–1831.

[19] P.Q. Khanh and D.T. Luc, Stability of solutions in parametric variational relationproblems, Set-Valued Anal. 16 (2008), pp. 1015–1035.

[20] P.Q. Khanh and L.M. Luu, Some existence results for vector quasivariational inequalitiesinvolving multifunctions and applications to traffic equilibrium problems, J. Global Optim.

32 (2005), pp. 551–568.[21] P.Q. Khanh and N.H. Quan, Existence conditions for quasivariational inclusion problems

in G-convex spaces, Acta Math. Vietnam 34 (2009), pp. 1–10.

[22] P.Q. Khanh and N.H. Quan, The solution existence of general inclusions using generalizedKKM theorems with applications to minimax problems, Submitted for publication.

[23] P.Q. Khanh and N.H. Quan, Generic stability and essential components of generalized

KKM points and applications, Submitted for publication.[24] P.Q. Khanh and N.H. Quan, General existence theorems, alternative theorems and

applications to minimax problems, Nonlinear Anal. TMA, OnlineFirst, 2009.

[25] P.Q. Khanh, N.H. Quan, and J.C. Yao, Generalized KKM-type theorems in GFC-spacesand applications, Nonlinear Anal. TMA 71 (2009), pp. 1227–1234.

[26] L.J. Lin, A KKM type theorem and its applications, Bull. Austral. Math. Soc. 59 (1999),pp. 481–493.

[27] S. Park, Coincidence theorems for admissible multimaps and their applications, NonlinearAnal. 30 (1977), pp. 4183–4191.

[28] S. Park, Fixed points of admissible maps on generalized convex spaces, J. Korean Math.

Soc. 37 (2000), pp. 885–899.[29] S. Park and H. Kim, Coincidence theorems for admissible multifunctions on generalized

convex spaces, J. Math. Anal. Appl. 197 (1996), pp. 173–187.

[30] S. Park and H. Kim, Foundations of the KKM theory on generalized convex spaces,J. Math. Anal. Appl. 209 (1997), pp. 551–571.

[31] S. Park and H. Kim, Coincidence theorems on product of generalized convex spaces andapplications to equilibria, J. Korean Math. Soc. 36 (1999), pp. 813–828.

124 P.Q. Khanh and N.H. Quan

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 0

2:20

23

Nov

embe

r 20

14