International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 673591 | 12 pages | https://doi.org/10.5402/2011/673591

Fixed Points and Existence Theorems of Maximal Elements with Applications in FC-Spaces

Academic Editor: E. Yee
Received16 Mar 2011
Accepted16 Apr 2011
Published06 Jul 2011

Abstract

We present some fixed point theorems and existence theorems of maximal elements in FC-space from which we derive several coincidence theorems. Applications of these results to generalized equilibrium problems and minimax theory will be given. Our results improve and generalize some recent results.

1. Introduction and Preliminaries

It is well known that many fixed point theorems, coincidence theorems, and maximal elements have been established in topological vector spaces, š»-spaces, and šŗ-convex spaces by many authors (see [1ā€“11]). In most of the known existence results of maximal elements and fixed point theorems, the convexity assumptions play a crucial role which strictly restricts the applicable area of these results. In fact, what we meet may not have convexity structure. Hence it is quite reasonable and valuable to study fixed point theorems, coincidence theorems, and maximal elements in general topological spaces without convexity structure.

In this paper, we will continue to study some problems in nonlinear analysis in š¹š¶-spaces. Some fixed point theorems and existence theorems of maximal elements are proved under noncompact setting of š¹š¶-spaces. We will apply these results to give some coincidence theorems. More specifically, the application of these results to generalized equilibrium problems and minimax theory will be given in š¹š¶-spaces. Our results improve and generalize the corresponding results in [1, 2, 12ā€“14].

Let š‘Œ be a nonempty set. We denote by 2š‘Œ and āŸØš‘ŒāŸ© the family of all subsets of š‘Œ and the family of all nonempty finite subsets of š‘Œ, respectively. For each š“āˆˆāŸØš‘‹āŸ©, we denote by |š“| the cardinality of š“. Let Ī”š‘› be the standard n-dimensional simplex with vertices {š‘’0,ā€¦,š‘’š‘›}. If š½ is a nonempty subset of {0,1,ā€¦,š‘›}, we denote by Ī”š½ the convex hull of the vertices {š‘’š‘—āˆ¶š‘—āˆˆš½}.

Let š‘‹ and š‘Œ be two sets, and, š‘‡āˆ¶š‘‹ā†’2š‘Œ be a set-valued mapping. We will use the following notations in the following material: (i)š‘‡(š‘„)={š‘¦āˆˆš‘Œāˆ¶š‘¦āˆˆš‘‡(š‘„)},(ii)ā‹ƒš‘‡(š“)=š‘„āˆˆš“š‘‡(š‘„),(iii)š‘‡āˆ’1(š‘¦)={š‘„āˆˆš‘‹āˆ¶š‘¦āˆˆš‘‡(š‘„)},(iv)š‘‡āˆ’1ā‹‚(šµ)={š‘„āˆˆš‘‹āˆ¶š‘‡(š‘„)šµā‰ āˆ…}.

For topological spaces š‘‹ and š‘Œ, a subset š“ of š‘‹ is said to be compactly open (resp., compactly closed) if for each nonempty compact subset š¾ of š‘‹, š“ā‹‚š¾ is open (resp., closed) in š¾. The compact closure of š“ and the compact interior of š“ (see [5]) are defined, respectively, by ī™īšš‘clš“={šµāŠ‚š‘‹āˆ¶š“āŠ‚šµandšµiscompactlyclosedinš‘‹},š‘intš“={šµāŠ‚š‘‹āˆ¶šµāŠ‚š“andšµiscompactlyopeninš‘‹}.(1.1)

It is easy to see that š‘cl(š‘‹ā§µš“)=š‘‹ā§µš‘intš“,intš“āŠ‚š‘intš“āŠ‚š“,š“āŠ‚š‘clš“āŠ‚clš“, š“ is compactly open in š‘‹ if and only if š“=š‘intš“. For each nonempty compact subset š¾ of š‘‹, ā‹‚š‘clš“š¾=clš¾ā‹‚(š“š¾) and ā‹‚š‘intš“š¾=intš¾ā‹‚(š“š¾), where clš¾ā‹‚(š“š¾) (resp., intš¾ā‹‚(š“š¾)) denotes the closure (resp., interior) of š“ā‹‚š¾ in š¾.

A set-valued mapping š‘‡āˆ¶š‘‹ā†’2š‘Œ is said to be transfer compactly open valued on š‘‹ (see [5]) if for each š‘„āˆˆš‘‹ and š‘¦āˆˆš‘‡(š‘„), there exists š‘„ā€²āˆˆš‘‹ such that š‘¦āˆˆš‘intš‘‡(š‘„ī…ž). It is clear that each transfer open valued correspondence is transfer compactly open valued. The inverse is not true in general.

Throughout this paper, all topological spaces are assumed to be Hausdorff.

Definition 1.1 (see [7]). (š‘‹,šœ‘š‘) is said to be an š¹š¶-space if š‘‹ is a topological space, and for each š‘={š‘„0,ā€¦,š‘„š‘›}āˆˆāŸØš‘‹āŸ©, where some elements in š‘ may be the same, there exists a continuous mapping šœ‘š‘āˆ¶Ī”š‘›ā†’š‘‹. A subset š· of (š‘‹,šœ‘š‘) is said to be an š¹š¶-subspace of š‘‹ if for each š‘={š‘„0,ā€¦,š‘„š‘›}āˆˆāŸØš‘‹āŸ© and for each {š‘„š‘–0,ā€¦,š‘„š‘–š‘˜ā‹‚š·}āŠ‚š‘, šœ‘š‘(Ī”š‘˜)āŠ‚š·, where Ī”š‘˜=co({š‘’š‘–š‘—āˆ¶š‘—=0,ā€¦,š‘˜}) and co denotes convex hull.
In many recent papers (see [9, 11, 15]), one studies one or more of the following generalized equilibrium problems
Let š‘‹ be a convex subset of a topological vector space, let š‘ and, š‘‰ be nonempty sets and let š¹āˆ¶š‘‹Ć—š‘ā†’2š‘‰,š¶āˆ¶š‘‹ā†’2š‘‰ be set-valued mappings. Find š‘„0āˆˆš‘‹ such that one of the following situations occurs:š¹(š‘„0,š‘§)āŠ†š¶(š‘„0) for all š‘§āˆˆš‘;š¹(š‘„0ā‹‚,š‘§)š¶(š‘„0)ā‰ āˆ… for all š‘§āˆˆš‘;š¹(š‘„0Ģø,š‘§)āŠ†š¶(š‘„0) for all š‘§āˆˆš‘;š¹(š‘„0ā‹‚,š‘§)š¶(š‘„0)=āˆ… for all š‘§āˆˆZ.ā€‰

In this paper, we will try a unified approach for all these problems considering a (binary) relation šœŒ on 2š‘‰ and looking for a point š‘„0āˆˆš‘‹ such that š¹(š‘„0,š‘§)šœŒš¶(š‘„0) for all š‘§āˆˆš‘.

Denote by šœŒš‘ the complementary relation of šœŒ, that is, for any š“,šµāŠ‚š‘‰, exactly one of the following assertions š“šœŒšµ,š“šœŒš‘šµ holds.

Since Fan [16] and Liu [17] extended the von Neumann-Sion principle to obtain two-function minimax inequalities, many such results involving two or more functions have appeared in the literature. We also obtain a very general minimax inequality of the following type: infš‘„āˆˆš‘‹ā„Ž(š‘„,š‘„)ā‰¤supš‘„āˆˆš‘‹infš‘§āˆˆš‘š‘“(š‘„,š‘§)+supš‘§āˆˆš‘infš‘„āˆˆš‘‹š‘”(š‘„,š‘§),(1.2) which is, to the best of our knowledge, different from all the minimax inequalities known in the literature.

2. Fixed Points and Maximal Elements

In order to prove our main results, we need the following definition and lemmas.

Definition 2.1. Let (š‘‹,šœ‘š‘) and (š‘,šœ‘āˆ—š‘€) be two š¹š¶-spaces. A set-valued mapping š‘‡āˆ¶š‘‹ā†’2š‘ is said to be weakly-š¹š¶ valued if for each š“={š‘„0,ā€¦,š‘„š‘›}āˆˆāŸØš‘‹āŸ©,š‘„āˆˆšœ‘š‘(Ī”š‘›) and šµ={š‘§0,ā€¦,š‘§š‘›}āˆˆāŸØš‘āŸ© with š‘§š‘–āˆˆš‘‡(š‘„š‘–)(š‘–=0,ā€¦,š‘›), we have ā‹‚šœ‘š‘‡(š‘„)āˆ—š‘€(Ī”ī…žš‘›)ā‰ āˆ…, where Ī”ī…žš‘› denotes the standard n-dimensional simplex corresponding, šµ={š‘§0,ā€¦,š‘§š‘›}āˆˆāŸØš‘āŸ©.

Lemma 2.2 (see [10]). Let š‘‹ and š‘Œ be two topological spaces, and let šŗāˆ¶š‘‹ā†’2š‘Œ be a set-valued mapping. Then the following statements are equivalent: (i)šŗāˆ’1āˆ¶š‘Œā†’2š‘‹ is transfer compactly open valued, and for all š‘„āˆˆš‘‹,šŗ(š‘„) is nonempty;(ii)ā‹ƒš‘‹=š‘¦āˆˆš‘Œš‘intšŗāˆ’1(š‘¦).

Theorem 2.3. Let (š‘‹,šœ‘š‘) be an š¹š¶-space, š¾ be a nonempty compact subset of š‘‹ and let š‘†,š‘„āˆ¶š‘‹ā†’2š‘‹ be set-valued mappings satisfying the following conditions: (i)for each š‘„āˆˆš‘‹ and š‘āˆˆāŸØš‘†(š‘„)āŸ©,šœ‘š‘(Ī”š‘˜)āŠ†š‘„(š‘„);(ii)ā‹ƒš‘‹=š‘¦āˆˆš‘‹š‘intš‘†āˆ’1(š‘¦);(iii)there exists a compact š¹š¶-subspace šæš‘ of š‘‹ containing š‘ such that šæš‘īšī€½ā§µš¾āŠ†š‘intš‘†āˆ’1(š‘¦)āˆ¶š‘¦āˆˆšæš‘ī€¾.(2.1) Then there exists š‘„āˆˆš‘‹ such that š‘„āˆˆš‘„(š‘„).

Proof. Since ā‹ƒš¾āŠ†š‘¦āˆˆš‘‹š‘intš‘†āˆ’1(š‘¦) and š¾ is a nonempty compact subset of š‘‹, there exists a finite set š‘={š‘¦1,ā€¦,š‘¦š‘›}āˆˆāŸØš‘‹āŸ© such that š¾=š‘›īšš‘–=1š‘intš‘†āˆ’1ī€·š‘¦š‘–ī€ø.(2.2) By condition (iii), there exists a compact š¹š¶-subspace šæš‘ of š‘‹ containing š‘ such that šæš‘īšī€½ā§µš¾āŠ†š‘intš‘†āˆ’1(š‘¦)āˆ¶š‘¦āˆˆšæš‘ī€¾.(2.3) It follows from (2.2) and š‘āŠ‚šæš‘ that šæš‘ī™īšī€½š¾āŠ†š‘intš‘†āˆ’1ī€¾āŠ†īšī€½(š‘¦)āˆ¶š‘¦āˆˆš‘š‘intš‘†āˆ’1(š‘¦)āˆ¶š‘¦āˆˆšæš‘ī€¾.(2.4) By (2.3) and (2.4), we have šæš‘āŠ†īšī€½š‘intš‘†āˆ’1(š‘¦)āˆ¶š‘¦āˆˆšæš‘ī€¾.(2.5) Since šæš‘ is compact, there exists a finite set š‘€={š‘§0,ā€¦,š‘§š‘š}āˆˆāŸØšæš‘āŸ© such that šæš‘=š‘šīšš‘–=0ī‚€š‘intš‘†āˆ’1ī€·š‘§š‘–ī€øī™šæš‘ī‚.(2.6) Since šæš‘ is also compact š¹š¶-subspace of š‘‹, there exists a continuous mapping šœ‘š‘€āˆ¶Ī”š‘šā†’š‘‹ such that šœ‘š‘€ī€·Ī”š½ī€øāŠ‚šæš‘,āˆ€š½āŠ‚{0,ā€¦,š‘š}.(2.7) We may assume that {šœ“š‘–}š‘šš‘–=0 is the continuous partition of unity subordinated to the open covering {š‘intš‘†āˆ’1(š‘§š‘–)ā‹‚šæš‘}š‘šš‘–=0 such that(1)for each š‘–=0,ā€¦,š‘š,šœ“š‘–āˆ¶šæš‘ā†’[0,1] is continuous;(2)for each š‘–=0,ā€¦,š‘š and š‘„āˆˆšæš‘,šœ“š‘–(š‘„)ā‰ 0āŸ¹š‘„āˆˆš‘intš‘†āˆ’1ī€·š‘§š‘–ī€øī™šæš‘āŠ‚š‘†āˆ’1ī€·š‘§š‘–ī€øāŸ¹ī€·š‘§š‘–ī€øāˆˆš‘†(š‘„),(2.8)(3) for each š‘„āˆˆšæš‘,āˆ‘š‘šš‘–=0šœ“š‘–(š‘„)=1.
Define a mapping šœ“āˆ¶šæš‘ā†’Ī”š‘š by šœ“(š‘„)=š‘šī“š‘–=0šœ“š‘–(š‘„)š‘’š‘–,āˆ€š‘„āˆˆšæš‘.(2.9) Hence šœ“ is continuous, and for each š‘„āˆˆšæš‘, ī“šœ“(š‘„)=š‘—āˆˆš½(š‘„)šœ“š‘—(š‘„)š‘’š‘—āˆˆĪ”š½(š‘„),āˆ€š‘„āˆˆšæš‘,(2.10) where š½(š‘„)={š‘—āˆˆ{0,ā€¦,š‘š}āˆ¶šœ“š‘—(š‘„)ā‰ 0}. It follows from (2.8) that {š‘§š‘–āˆ¶š‘–āˆˆš½(š‘„)}āŠ‚š‘†(š‘„). By condition (i) šœ‘š‘€āˆ˜šœ“(š‘„)āˆˆšœ‘š‘€ī€·Ī”š½(š‘„)ī€øāŠ†š‘„(š‘„),(2.11)š‘āŠ‚šæš‘āŠ‚š‘‹, and (2.7), we have, šœ“āˆ˜šœ‘š‘€āˆ¶Ī”š‘šā†’2Ī”š‘š has a fixed point š‘§āˆˆĪ”š‘š, that is, š‘§=šœ“āˆ˜šœ‘š‘€(š‘§). Let š‘„=šœ‘š‘€(š‘§). Then š‘„=šœ‘š‘€(š‘§)=šœ‘š‘€āˆ˜ī€·šœ“āˆ˜šœ‘š‘€ī€ø(š‘§)=šœ‘š‘€ī€·āˆ˜šœ“š‘„ī€øī€·āŠ‚š‘„š‘„ī€ø.(2.12) This completes the proof.

Lemma 2.4. Let (š‘‹,šœ‘š‘) and (š‘,šœ‘āˆ—š‘€) be two š¹š¶-spaces. let š‘ƒ,š‘‡āˆ¶š‘‹ā†’2š‘ be set-valued mappings. If for each š‘„āˆˆš‘‹,š‘ƒ(š‘„) is š¹š¶-subspace of š‘ and š‘‡ being weakly-š¹š¶ valued, then for each ā‹‚š‘„āˆˆš‘‹,{š‘¦āˆˆš‘‹āˆ¶š‘ƒ(š‘„)š‘‡(š‘¦)ā‰ āˆ…} is š¹š¶-subspace of š‘‹.

Proof. Let š‘„āˆˆš‘‹,{š‘¦1,ā€¦,š‘¦š‘›}āˆˆāŸØš‘‹āŸ© with ā‹‚š‘ƒ(š‘„)š‘‡(š‘¦š‘–)ā‰ āˆ… for each š‘–āˆˆ{1,ā€¦,š‘›}, and š‘¦āˆˆšœ‘š‘(Ī”š‘›). For each š‘–āˆˆ{1,ā€¦,š‘›}, let š‘§š‘–ā‹‚āˆˆš‘ƒ(š‘„)š‘‡(š‘¦š‘–). It follows from š‘‡ is weakly-š¹š¶ valued that ā‹‚šœ‘š‘‡(š‘¦)āˆ—š‘€(Ī”ī…žš‘›)ā‰ āˆ…. Since š‘„āˆˆš‘‹,š‘ƒ(š‘„) is š¹š¶-subspace of š‘, we have ā‹‚šœ‘āˆ…ā‰ š‘‡(š‘¦)āˆ—š‘€(Ī”ī…žš‘›ā‹‚)āŠ‚š‘‡(š‘¦)š‘ƒ(š‘„). Hence, for each ā‹‚š‘„āˆˆš‘‹,{š‘¦āˆˆš‘‹āˆ¶š‘ƒ(š‘„)š‘‡(š‘¦)ā‰ āˆ…} is š¹š¶-subspace of š‘‹. This completes the proof.

Theorem 2.5. Let (š‘‹,šœ‘š‘) be an š¹š¶-space, let š‘ be a nonempty set, and, let š‘ƒ,š‘‡āˆ¶š‘‹ā†’2š‘,š‘„āˆ¶š‘‹ā†’2š‘‹ be three set-valued mappings satisfying the following conditions: (i)for each š‘„āˆˆš‘‹,š‘„(š‘„) is š¹š¶-subspace of š‘‹;(ii)for each ā‹‚š‘„āˆˆš‘‹,{š‘¦āˆˆš‘‹āˆ¶š‘ƒ(š‘„)š‘‡(š‘¦)ā‰ āˆ…}āŠ†š‘„(š‘„);(iii)ā‹ƒš‘‹=š‘§āˆˆš‘‡(š‘‹)š‘intš‘ƒāˆ’1(š‘§);(iv)there exists a compact subset š¾ of š‘‹ such that for each š‘āˆˆāŸØš‘‹āŸ©, there exists a compact š¹š¶-subspace šæš‘ of š‘‹ containing š‘ such that šæš‘īšā§µš¾āŠ†š‘¦āˆˆšæš‘ī‚†ī™ī‚‡.š‘intš‘„āˆˆš‘‹āˆ¶š‘ƒ(š‘„)š‘‡(š‘¦)ā‰ āˆ…(2.13) Then there exists š‘„āˆˆš‘‹ such that š‘„āˆˆš‘„(š‘„).

Proof. Define a mapping š‘†āˆ¶š‘‹ā†’2š‘‹ by ī‚†ī™ī‚‡š‘†(š‘„)=š‘¦āˆˆš‘‹āˆ¶š‘ƒ(š‘„)š‘‡(š‘¦)ā‰ āˆ….(2.14) It is easy to see that for a family {š“š‘–}š‘–āˆˆš¼ of subsets of a topological space, ā‹ƒš‘–āˆˆš¼š‘intš“š‘–ā‹ƒāŠ†š‘int(š‘–āˆˆš¼š“š‘–). Thus, we have īšš‘¦āˆˆš‘‹š‘intš‘†āˆ’1īš(š‘¦)=š‘¦āˆˆš‘‹īƒ©īšš‘intš‘§āˆˆš‘‡(š‘¦)š‘ƒāˆ’1īƒŖāŠ‡īš(š‘§)š‘¦āˆˆš‘‹īšš‘§āˆˆš‘‡(š‘¦)š‘intš‘ƒāˆ’1īš(š‘§)=š‘§āˆˆš‘‡(š‘‹)š‘intš‘ƒāˆ’1(š‘§).(2.15) By condition (iii), it follows that ā‹ƒš‘‹=š‘¦āˆˆš‘‹š‘intš‘†āˆ’1(š‘¦). Clearly condition (iv) of Theorem 2.5 implies condition (iii) of Theorem 2.3. By condition (i) and (ii), for each š‘„āˆˆš‘‹ and š‘āˆˆāŸØš‘†(š‘„)āŸ©, we have šœ‘š‘(Ī”š‘˜)āŠ†š‘„(š‘„). Then all conditions of Theorem 2.3 are satisfied. By Theorem 2.3, there exists š‘„āˆˆš‘‹ such that š‘„āˆˆš‘„(š‘„). This completes the proof.

Particularly, when ā‹‚š‘„(š‘„)={š‘¦āˆˆš‘‹āˆ¶š‘ƒ(š‘„)š‘‡(š‘¦)ā‰ āˆ…}, Theorem 2.5 reduces to the following result.

Corollary 2.6. Let (š‘‹,šœ‘š‘) be an š¹š¶-space, let š‘ be a nonempty set, and let š‘ƒ,š‘‡āˆ¶š‘‹ā†’2š‘ be two set-valued mappings satisfying the following conditions: (i)ā‹ƒš‘‹=š‘§āˆˆš‘‡(š‘‹)š‘intš‘ƒāˆ’1(š‘§);(ii)for each š‘„āˆˆš‘‹,š‘„(š‘„) is š¹š¶-subspace of š‘‹;(iii)there exists a compact subset š¾ of š‘‹ such that for each š‘āˆˆāŸØš‘‹āŸ©, there exists a compact š¹š¶-subspace šæš‘ of š‘‹ containing š‘ such that šæš‘īšā§µš¾āŠ†š‘¦āˆˆšæš‘ī‚†ī™ī‚‡.š‘intš‘„āˆˆš‘‹āˆ¶š‘ƒ(š‘„)š‘‡(š‘¦)ā‰ āˆ…(2.16) Then there exists š‘„āˆˆš‘‹ such that š‘ƒ(ā‹‚š‘„)š‘‡(š‘„)ā‰ āˆ….

Remark 2.7. Theorem 2.3 generalizes Theorem 4 in [14]. Theorem 2.5 generalizes Theorem 1 in [2]. Corollary 2.6 generalizes Theorem 8 in [12], Theorem 3.6 in [13], and Theorem 2 in [2].

Corollary 2.8. Let (š‘‹,šœ‘š‘) be an š¹š¶-space, š‘ be a nonempty set and š‘ƒ,š‘‡āˆ¶š‘‹ā†’2š‘ be two set-valued mappings satisfying the following conditions: (i)š‘ƒāˆ’1 is transfer compactly open valued;(ii)for each ā‹‚š‘„āˆˆš‘‹,{š‘¦āˆˆš‘‹āˆ¶š‘ƒ(š‘„)š‘‡(š‘¦)ā‰ āˆ…} is š¹š¶-subspace of š‘‹;(iii)there exists a compact subset š¾ of š‘‹ such that for each š‘āˆˆāŸØš‘‹āŸ©, there exists a compact š¹š¶-subspace šæš‘ of š‘‹ containing š‘ such that šæš‘īšā§µš¾āŠ†š‘¦āˆˆšæš‘ī‚†ī™ī‚‡,š‘intš‘„āˆˆš‘‹āˆ¶š‘ƒ(š‘„)š‘‡(š‘¦)ā‰ āˆ…(2.17)(iv)for each ā‹‚š‘„āˆˆš‘‹,š‘ƒ(š‘„)š‘‡(š‘„)=āˆ….
Then at least one of the following assertions holds: (a)there exists š‘„āˆˆš‘‹ such that š‘ƒ(š‘„)=āˆ…,(b)there exists š‘§āˆˆš‘ such that š‘‡āˆ’1(š‘§)=āˆ….

Proof. By means of contradiction, suppose that for each š‘„āˆˆš‘‹,š‘ƒ(š‘„)ā‰ āˆ…, and for each š‘§āˆˆš‘,š‘‡āˆ’1(š‘§)ā‰ āˆ… (that is, š‘‡(š‘‹)=š‘). Then, by Lemma 2.2, we have īšš‘‹=š‘§āˆˆš‘š‘intš‘ƒāˆ’1īš(š‘§)=š‘§āˆˆš‘‡(š‘‹)š‘intš‘ƒāˆ’1(š‘§).(2.18) Then all conditions of Corollary 2.6 are satisfied, and it follows that there exists š‘„āˆˆš‘‹ such that š‘ƒ(ā‹‚š‘„)š‘‡(š‘„)ā‰ āˆ…, but this contradicts condition (iv). This completes the proof.

Corollary 2.9. In Corollary 2.8, if one assumes further that š‘‡(š‘‹)=š‘, then there exists š‘„āˆˆš‘‹ such that š‘ƒ(š‘„)=āˆ….

Corollary 2.10. Let (š‘‹,šœ‘š‘) be an š¹š¶-space, and let š‘ƒāˆ¶š‘‹ā†’2š‘‹ be a set-valued mapping satisfying the following conditions: (i)š‘ƒāˆ’1 is transfer compactly open valued;(ii)for each š‘„āˆˆš‘‹,š‘ƒ(š‘„) is š¹š¶-subspace of š‘‹;(iii)for each š‘„āˆˆš‘‹,š‘„āˆ‰š‘ƒ(š‘„);(iv)there exists a compact subset š¾ of š‘‹ such that for each š‘āˆˆāŸØš‘‹āŸ©, there exists a compact š¹š¶-subspace šæš‘ of š‘‹ containing š‘ such that šæš‘īšā§µš¾āŠ†š‘¦āˆˆšæš‘š‘intš‘ƒāˆ’1(š‘¦).(2.19) Then there exists š‘„āˆˆš‘‹ such that š‘ƒ(š‘„)=āˆ….

Proof. Define a mapping š‘‡āˆ¶š‘‹ā†’2š‘‹ by š‘‡(š‘„)={š‘„} for all š‘„āˆˆš‘‹. Then Corollary 2.10 follows from Corollary 2.8.

Remark 2.11. Corollary 2.8 generalizes Theorem 3 in [2]. Corollary 2.9 and Corollary 2.10 improve Corollary 1 and Corollary 2 in [2], respectively. If the š¹š¶-space š‘‹ is compact, condition (iii) in Theorem 2.3 is satisfied trivially. Consequently, condition (iv) of Theorem 2.5, condition (iv) of Corollary 2.10, condition (iii) of Corollary 2.6, condition (iii) of Corollary 2.8 and condition (iii) of Corollary 2.9 are all satisfied trivially.

Corollary 2.12. Let (š‘‹,šœ‘š‘) and (š‘,šœ‘āˆ—š‘€) be two š¹š¶-spaces and let š‘ƒ,š‘‡āˆ¶š‘‹ā†’2š‘ be two set-valued mappings satisfying conditions (i) and (iii) in Corollary 2.6. If for each š‘„āˆˆš‘‹,š‘ƒ(š‘„) is š¹š¶-subspace of š‘ and š‘‡ is weakly-š¹š¶ valued, then there exists š‘„āˆˆš‘‹ such that š‘ƒ(ā‹‚š‘„)š‘‡(š‘„)ā‰ āˆ….

Proof. By Lemma 2.4 and Corollary 2.6, we have that the conclusion of Corollary 2.12 holds.

Corollary 2.13. Let (š‘‹,šœ‘š‘) and (š‘,šœ‘āˆ—š‘€) be two š¹š¶-spaces and š‘ƒ,š‘‡āˆ¶š‘‹ā†’2š‘ be set-valued mappings satisfying conditions (i), (iii) and (iv) in Corollary 2.8. If for each š‘„āˆˆš‘‹,š‘ƒ(š‘„) is š¹š¶-subspace of š‘ and š‘‡ is weakly-š¹š¶ valued, then at least one of the following assertions holds: (a)there exists š‘„āˆˆš‘‹ such that š‘ƒ(š‘„)=āˆ…;(b)there exists š‘§āˆˆš‘ such that š‘‡āˆ’1(š‘§)=āˆ….

Proof. By Lemma 2.4 and Corollary 2.8, we have that the conclusion of Corollary 2.13 holds.

Remark 2.14. Corollary 2.12 generalizes Theorem 4 in [2]. Corollary 2.13 generalizes Theorem 5 in [2].

3. Generalized Equilibrium Theorems and Minimax Inequality

Inspired by Balaj and Lin [2], we have the following concept.

Definition 3.1. Let (š‘‹,šœ‘š‘) be an š¹š¶-space, let š‘ and š‘‰ be nonempty sets, let šœŒ be a relation on 2š‘‰, and let š¹āˆ¶š‘‹Ć—š‘ā†’2š‘‰,š¶āˆ¶š‘‹ā†’2š‘‰ be set-valued mappings. We say that š¹ is (š¶,šœŒ)-transfer compactly continuous in the first variable if for any š‘„āˆˆš‘‹,š‘§āˆˆ{š‘¢āˆˆš‘āˆ¶š¹(š‘„,š‘¢)šœŒš¶(š‘„)}, there exists š‘„āˆˆš‘‹ such that š‘§āˆˆš‘int{š‘¢āˆˆš‘āˆ¶š¹(š‘„,š‘¢)šœŒš¶(š‘„)}.

The following concept generalizes the corresponding notion in Ding [6, 8].

Definition 3.2. Let (š‘‹,šœ‘š‘) be an š¹š¶-space and š‘Œ be topological space. For each š‘¦āˆˆš‘Œ,š‘“āˆ¶š‘ŒĆ—š‘‹ā†’š‘… is said to be š¹š¶-quasiconvex (resp., š¹š¶-quasiconcave) in the first variable, if for each šœ†āˆˆš‘…, the set {š‘„āˆˆš‘‹āˆ¶š‘“(š‘¦,š‘„)<šœ†} (resp., {š‘„āˆˆš‘‹āˆ¶š‘“(š‘„,š‘¦)>šœ†}) is an š¹š¶-subspace of š‘‹.

The following concept generalizes the corresponding of notion in Chen and Chang [5].

Definition 3.3. Let š‘‹ and š‘Œ be two topological spaces, and let š‘“āˆ¶š‘ŒĆ—š‘‹ā†’š‘…āˆŖ{āˆ’āˆž,+āˆž} be a function. Then š‘“ is said to be š›¼-transfer compactly lower semicontinuous (in short, š›¼-transfer compactly l.s.c) in the first variable if for each š‘¦āˆˆš‘Œ with š‘„āˆˆ{š‘¢āˆˆš‘‹āˆ¶š‘“(š‘¦,š‘¢)>š›¼}, there exists š‘¦āˆˆš‘Œ such that š‘„āˆˆš‘int{š‘¢āˆˆš‘‹āˆ¶š‘“(š‘¦,š‘¢)>š›¼}. š‘“ is said to be š›¼-transfer compactly u.s.c in the first variable if and only if ā€“š‘“ is š›¼-transfer compactly l.s.c in the first variable. If š‘“ is š›¼-transfer compactly l.s.c in the first variable for each š›¼āˆˆš‘…, we say that š‘“ is transfer compactly l.s.c in the first variable.

Theorem 3.4. Let (š‘‹,šœ‘š‘) be an š¹š¶-space, let š‘ and š‘‰ be nonempty sets, and let šœŒ be a relation on 2š‘‰. Let š¹āˆ¶š‘‹Ć—š‘ā†’2š‘‰,š¶āˆ¶š‘‹ā†’2š‘‰, and š‘‡āˆ¶š‘‹ā†’2š‘ be set-valued mappings satisfying the following conditions: (i)š‘‡(š‘‹)=š‘;(ii)š¹ is (š¶,šœŒš‘)-transfer compactly continuous in the first variable;(iii) there exists a compact subset š¾ of š‘‹ such that for each š‘āˆˆāŸØš‘‹āŸ©, there exists a compact š¹š¶-subspace šæš‘ of š‘‹ containing š‘ such that šæš‘ī™ī™š‘¦āˆˆšæš‘š‘cl{š‘„āˆˆš‘‹āˆ¶š¹(š‘„,š‘§)šœŒš¶(š‘„),āˆ€š‘§āˆˆš‘‡(š‘¦)}āŠ†š¾,(3.1)(iv) for each š‘„āˆˆš‘‹, the set {š‘¦āˆˆš‘‹āˆ¶š¹(š‘„,š‘§)šœŒš‘š¶(š‘„),forsomeš‘§āˆˆš‘‡(š‘¦)} is š¹š¶-subspace of š‘‹;(v) for all š‘„āˆˆš‘‹ and any š‘§āˆˆš‘‡(š‘„),š¹(š‘„,š‘§)šœŒš¶(š‘„).Then there exists š‘„āˆˆš‘‹ such that š¹(š‘„,š‘§)šœŒš¶(š‘„) for all š‘§āˆˆš‘.

Proof. Let š‘ƒāˆ¶š‘‹Ć—š‘ā†’2š‘ be the set-valued mapping defined by š‘ƒ(š‘„)={š‘§āˆˆš‘āˆ¶š¹(š‘„,š‘§)šœŒš‘š¶(š‘„)}.(3.2) We prove that the mappings satisfy the conditions of Theorem 2.5. It is clear that condition (ii) is equivalent to the fact that the mapping š‘ƒāˆ’1 is transfer compactly open valued. By condition (iv), for each š‘„āˆˆš‘‹ the set ā‹‚{š‘¦āˆˆš‘‹āˆ¶š‘ƒ(š‘„)š‘‡(š‘¦)ā‰ āˆ…} is š¹š¶-subspace of š‘‹. Let š‘āˆˆāŸØš‘‹āŸ© and for all š‘„āˆˆšæš‘ā§µš¾. Then, by condition (iii) there exists š‘¦āˆˆšæš‘ such that ī€½š‘„š‘„āˆˆš‘‹ā§µš‘clī…žī€·š‘„āˆˆš‘‹āˆ¶š¹ī…žī€øī€·š‘„,š‘§šœŒš¶ī…žī€øī€¾ī€½š‘„,āˆ€š‘§āˆˆš‘‡(š‘¦)=š‘intī…žī€·š‘„āˆˆš‘‹āˆ¶š¹ī…žī€øšœŒ,š‘§š‘š¶ī€·š‘„ī…žī€øī€¾ī‚†š‘„,forsomeš‘§āˆˆš‘‡(š‘¦)=š‘intī…žī€·š‘„āˆˆš‘‹āˆ¶š‘ƒī…žī€øī™ī‚‡.š‘‡(š‘¦)ā‰ āˆ…(3.3)
Thus all the conditions of Theorem 2.5 are fulfilled, and by condition (i), š‘‡āˆ’1(š‘§)ā‰ āˆ… for all zāˆˆš‘. Hence, there exists š‘„āˆˆš‘‹ such that š‘ƒ(š‘„)=āˆ…, that is, š¹(š‘„,š‘§)šœŒš¶(š‘„) for all š‘§āˆˆš‘. This completes the proof.

Remark 3.5. Theorem 3.4 generalizes Theorem 6 in [2] in the following several aspects: (a) from šŗ-convex space to š¹š¶-space without linear structure; (b) from (š¶,šœŒš‘)-transfer continuous to (š¶,šœŒš‘)-transfer compactly continuous; (c) conditions (iii) and (iv) of Theorem 3.4 are weaker than conditions (iii) and (iv) of Theorem 6 in [2].

Corollary 3.6. Let (š‘‹,šœ‘š‘) and (š‘,šœ‘āˆ—š‘€) be two š¹š¶-spaces, let š‘‰ be nonempty set, and let šœŒ be a relation on 2š‘‰. Let š¹āˆ¶š‘‹Ć—š‘ā†’2š‘‰,š¶āˆ¶š‘‹ā†’2š‘‰, and š‘‡āˆ¶š‘‹ā†’2š‘ be set-valued mappings satisfying the following conditions: š‘ƒ,š‘‡āˆ¶š‘‹ā†’2š‘ are set-valued mappings satisfying conditions (i), (ii), (iii), and (v) in Theorem 3.4 and if for each š‘„āˆˆš‘‹,{š‘§āˆˆš‘āˆ¶š¹(š‘„,š‘§)šœŒš‘š¶(š‘„)} is š¹š¶-subspace of š‘ and š‘‡ is weakly-š¹š¶ valued, then there exists š‘„āˆˆš‘‹ such that š¹(š‘„,š‘§)šœŒš¶(š‘„) for all š‘§āˆˆš‘.

Proof. By Lemma 2.4 and Corollary 2.12, we have that the conclusion of Corollary 3.6 holds.

Remark 3.7. Corollary 3.6 generalizes Theorem 7 in [2] in the following several aspects: (a) from šŗ-convex space to š¹š¶-space without linear structure; (b) from (š¶,šœŒš‘)-transfer continuous to (š¶,šœŒš‘)-transfer compactly continuous; (c) from šŗ-weakly convex to weakly-š¹š¶ valued; (c) conditions (iii) and ā€œfor each š‘„āˆˆš‘‹,{š‘§āˆˆš‘āˆ¶š¹(š‘„,š‘§)šœŒš‘š¶(š‘„)} is š¹š¶-subspace of š‘ā€ of Corollary 3.6 are weaker than conditions (iii) and (vii) of Theorem 7 in [2].

Theorem 3.8. Let (š‘‹,šœ‘š‘) be a compact š¹š¶-space, let š‘ be nonempty set, let š‘“,š‘”āˆ¶š‘Ć—š‘‹ā†’š‘…,ā„Žāˆ¶š‘‹Ć—š‘‹ā†’š‘… be three functions, and let š›¼,š›½,š›¾ be three real numbers satisfying the following conditions: (i)for each š‘„āˆˆš‘‹, the set {š‘¦āˆˆš‘‹āˆ¶ā„Ž(š‘„,š‘¦)<š›¾} is š¹š¶-subspace of š‘‹;(ii)š‘“ is š›¼-transfer compactly u.s.c in the first variable;(iii)for each š‘„,š‘¦āˆˆš‘‹,š‘§āˆˆš‘, š‘“(š‘§,š‘„)<š›¼ and š‘”(š‘§,š‘¦)<š›½ā‡’ā„Ž(š‘„,š‘¦)<š›¾;(iv)ā„Ž(š‘„,š‘„)ā‰„š›¾ for all š‘„āˆˆš‘‹.Then, at least one of the following assertions holds: (a)there exists š‘„āˆˆš‘‹ such that š‘“(š‘§,š‘„)ā‰„š›¼ for all š‘§āˆˆš‘;(b)there exists š‘§āˆˆš‘ such that š‘”(š‘§,š‘„)ā‰„š›½ for all š‘„āˆˆš‘‹.

Proof. Let š‘ƒ,š‘‡āˆ¶š‘‹ā†’2š‘,š‘„āˆ¶š‘‹ā†’2š‘‹ be the set-valued mappings defined by š‘‡š‘ƒ(š‘„)={š‘§āˆˆš‘āˆ¶š‘“(š‘§,š‘„)<š›¼},(š‘„)={š‘§āˆˆš‘āˆ¶š‘”(š‘§,š‘„)<š›½},š‘„(š‘„)={š‘¦āˆˆš‘‹āˆ¶ā„Ž(š‘„,š‘¦)<š›¾}(3.4) for all š‘„āˆˆš‘‹. By means of contradiction, suppose that both assertions (a) and (b) would be false; that is(1)for each š‘„āˆˆš‘‹, there exists š‘§āˆˆš‘ such that š‘“(š‘§,š‘„)<š›¼,(2)for each š‘§āˆˆš‘, there exists š‘„āˆˆš‘‹ such that š‘”(š‘§,š‘„)<š›½.
By (2), š‘‡(š‘‹)=š‘. Arbitrarily choose š‘„āˆˆš‘‹. By (1) and condition (ii), it follows that š‘„āˆˆ{š‘¢āˆˆš‘‹āˆ¶š‘“(š‘§,š‘„)<š›¼}=š‘ƒāˆ’1(š‘§), and there exists š‘§āˆˆš‘ such that š‘„āˆˆš‘intš‘ƒāˆ’1(š‘§). Hence, we infer that ā‹ƒš‘‹=š‘§āˆˆš‘š‘intš‘ƒāˆ’1ā‹ƒ(š‘§)=š‘§āˆˆš‘‡(š‘‹)š‘intš‘ƒāˆ’1(š‘§). By condition (i), š‘„(š‘„) is š¹C-subspace of š‘‹, and by condition (iii), ā‹‚{š‘¦āˆˆš‘‹āˆ¶š‘ƒ(š‘„)š‘‡(š‘¦)ā‰ āˆ…}āŠ†š‘„(š‘„) for each š‘„āˆˆš‘‹. Taking into account Remark 2.11, all conditions of Theorem 2.3 are satisfied, and it follows that there exists š‘„āˆˆš‘‹ such that š‘„āˆˆš‘„(š‘„). Hence, we have ā„Ž(š‘„,š‘„)<š›¾, which contradicts condition (iv). This completes the proof.

Remark 3.9. Theorem 3.8 generalizes Theorem 12 in [2] in the following several aspects: (a) from šŗ-convex space to š¹š¶-space without linear structure; (b) from š›¼-transfer u.s.c to š›¼-transfer compactly u.s.c; (c) conditions (i) of Theorem 3.8 are weaker than conditions (i) of Theorem 12 in [2].

From Theorem 3.8, we may obtain the following minimax inequality.

Corollary 3.10. Let (š‘‹,šœ‘š‘) be a compact š¹š¶-space, let š‘ be nonempty set, and let š‘“,š‘”āˆ¶š‘Ć—š‘‹ā†’š‘…,ā„Žāˆ¶š‘‹Ć—š‘‹ā†’š‘… be three functions satisfying the following conditions: (i)ā„Ž(š‘„,ā‹…) is š¹š¶-quasiconvex, for each š‘„āˆˆš‘‹;(ii)š‘“ is transfer compactly u.s.c in the first variable;(iii)for each š‘„,š‘¦āˆˆš‘‹,š‘§āˆˆš‘, ā„Ž(š‘„,š‘¦)ā‰¤š‘“(š‘§,š‘„)+š‘”(š‘§,š‘¦).Then infš‘„āˆˆš‘‹ā„Ž(š‘„,š‘„)ā‰¤supš‘„āˆˆš‘‹infš‘§āˆˆš‘š‘“(š‘§,š‘„)+supš‘§āˆˆš‘infš‘„āˆˆš‘‹š‘”(š‘§,š‘„),(3.5) with the convention āˆž+(āˆ’āˆž)=āˆž.

Proof. We may suppose that infš‘„āˆˆš‘‹ā„Ž(š‘„,š‘„)>āˆ’āˆž,supš‘„āˆˆš‘‹infš‘§āˆˆš‘š‘“(š‘§,š‘„)<āˆž,supš‘§āˆˆš‘infš‘„āˆˆš‘‹š‘”(š‘§,š‘„)<āˆž.(3.6) By means of contradiction, suppose that infš‘„āˆˆš‘‹ā„Ž(š‘„,š‘„)>supš‘„āˆˆš‘‹infš‘§āˆˆš‘š‘“(š‘§,š‘„)+supš‘§āˆˆš‘infš‘„āˆˆš‘‹š‘”(š‘§,š‘„)(3.7) and choose š›¼,š›½,š›¾āˆˆš‘… such that supš‘„āˆˆš‘‹infš‘§āˆˆš‘š‘“(š‘§,š‘„)<š›¼,supš‘§āˆˆš‘infš‘„āˆˆš‘‹š‘”(š‘§,š‘„)<š›½,š›¾<infš‘„āˆˆš‘‹ā„Ž(š‘„,š‘„),š›¼+š›½<š›¾.(3.8)
It is easy to see that functions š‘“,š‘”, and ā„Ž satisfy all the conditions of Theorem 3.8. We prove that neither assertion (a) nor (b) of the conclusion of Theorem 3.8 can take place.If (a) happens, thenš›¼ā‰¤infš‘§āˆˆš‘š‘“ī€·š‘§,š‘„ī€øā‰¤supš‘„āˆˆš‘‹infš‘§āˆˆš‘š‘“(š‘§,š‘„),(3.9) a contradiction.If (b) happens, thensupš‘§āˆˆš‘infš‘„āˆˆš‘‹š‘”(š‘§,š‘„)ā‰„infš‘„āˆˆš‘‹š‘”ī€·ī€øš‘§,š‘„ā‰„š›½,(3.10) a contradiction again.
This completes the proof.

Remark 3.11. Corollary 3.10 generalizes Theorem 13 in [2] in the following several aspects: (a) from šŗ-convex space to š¹š¶-space without linear structure; (b) from šŗ-quasiconvex to š¹š¶-quasiconvex; (c) from transfer u.s.c to transfer compactly u.s.c.

Acknowledgment

This work is supported by the Scientific Research Foundation of CUIT under Grant KYTZ201114.

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Copyright Ā© 2011 Rong-Hua He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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