ISRN Applied MathematicsVolume 2011 (2011), Article ID 673591, 12 pagesdoi:10.5402/2011/673591

Research Article

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

Rong-Hua He

Department of Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610103, China

Received 16 March 2011; Accepted 16 April 2011

Academic Editors: S. Utyuzhnikov and E. Yee

Copyright © 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.

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, -spaces, and -convex spaces by many authors (see [1–11]). 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 -spaces. Some fixed point theorems and existence theorems of maximal elements are proved under noncompact setting of -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 -spaces. Our results improve and generalize the corresponding results in [1, 2, 12–14].

Let be a nonempty set. We denote by and the family of all subsets of and the family of all nonempty finite subsets of , respectively. For each , we denote by the cardinality of . Let be the standard n-dimensional simplex with vertices . If is a nonempty subset of , we denote by the convex hull of the vertices .

Let and be two sets, and, be a set-valued mapping. We will use the following notations in the following material: (i),(ii),(iii),(iv).

For topological spaces and , a subset of is said to be compactly open (resp., compactly closed) if for each nonempty compact subset of , is open (resp., closed) in . The compact closure of and the compact interior of (see [5]) are defined, respectively, by

It is easy to see that , is compactly open in if and only if . For each nonempty compact subset of , and , where (resp., ) denotes the closure (resp., interior) of in .

A set-valued mapping is said to be transfer compactly open valued on (see [5]) if for each and , there exists such that . 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]). is said to be an -space if is a topological space, and for each , where some elements in may be the same, there exists a continuous mapping . A subset of is said to be an -subspace of if for each and for each , , where and denotes convex hull.In many recent papers (see [9, 11, 15]), one studies one or more of the following generalized equilibrium problemsLet be a convex subset of a topological vector space, let and, be nonempty sets and let be set-valued mappings. Find such that one of the following situations occurs: for all ; for all ; for all ; for all . 

In this paper, we will try a unified approach for all these problems considering a (binary) relation on and looking for a point such that for all .

Denote by the complementary relation of , that is, for any , exactly one of the following assertions 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: 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 and be two -spaces. A set-valued mapping is said to be weakly- valued if for each and with , we have , where denotes the standard n-dimensional simplex corresponding, .

Lemma 2.2 (see [10]). Let and be two topological spaces, and let be a set-valued mapping. Then the following statements are equivalent: (i) is transfer compactly open valued, and for all is nonempty;(ii).

Theorem 2.3. Let be an -space, be a nonempty compact subset of and let be set-valued mappings satisfying the following conditions: (i)for each and ;(ii);(iii)there exists a compact -subspace of containing such that Then there exists such that .

Proof. Since and is a nonempty compact subset of , there exists a finite set such that By condition (iii), there exists a compact -subspace of containing such that It follows from (2.2) and that By (2.3) and (2.4), we have Since is compact, there exists a finite set such that Since is also compact -subspace of , there exists a continuous mapping such that We may assume that is the continuous partition of unity subordinated to the open covering such that(1)for each is continuous;(2)for each and ,(3) for each .Define a mapping by Hence is continuous, and for each , where . It follows from (2.8) that . By condition (i) , and (2.7), we have, has a fixed point , that is, . Let . Then This completes the proof.

Lemma 2.4. Let and be two -spaces. let be set-valued mappings. If for each is -subspace of and being weakly- valued, then for each is -subspace of .

Proof. Let with for each , and . For each , let . It follows from is weakly- valued that . Since is -subspace of , we have . Hence, for each is -subspace of . This completes the proof.

Theorem 2.5. Let be an -space, let be a nonempty set, and, let be three set-valued mappings satisfying the following conditions: (i)for each is -subspace of ;(ii)for each ;(iii);(iv)there exists a compact subset of such that for each , there exists a compact -subspace of containing such that Then there exists such that .

Proof. Define a mapping by It is easy to see that for a family of subsets of a topological space, . Thus, we have By condition (iii), it follows that . Clearly condition (iv) of Theorem 2.5 implies condition (iii) of Theorem 2.3. By condition (i) and (ii), for each and , we have . Then all conditions of Theorem 2.3 are satisfied. By Theorem 2.3, there exists such that . This completes the proof.

Particularly, when , Theorem 2.5 reduces to the following result.

Corollary 2.6. Let be an -space, let be a nonempty set, and let be two set-valued mappings satisfying the following conditions: (i);(ii)for each is -subspace of ;(iii)there exists a compact subset of such that for each , there exists a compact -subspace of containing such that Then there exists such that .

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 be an -space, be a nonempty set and be two set-valued mappings satisfying the following conditions: (i) is transfer compactly open valued;(ii)for each is -subspace of ;(iii)there exists a compact subset of such that for each , there exists a compact -subspace of containing such that (iv)for each . Then at least one of the following assertions holds: (a)there exists such that ,(b)there exists such that .

Proof. By means of contradiction, suppose that for each , and for each (that is, ). Then, by Lemma 2.2, we have Then all conditions of Corollary 2.6 are satisfied, and it follows that there exists such that , but this contradicts condition (iv). This completes the proof.

Corollary 2.9. In Corollary 2.8, if one assumes further that , then there exists such that .

Corollary 2.10. Let be an -space, and let be a set-valued mapping satisfying the following conditions: (i) is transfer compactly open valued;(ii)for each is -subspace of ;(iii)for each ;(iv)there exists a compact subset of such that for each , there exists a compact -subspace of containing such that Then there exists such that .

Proof. Define a mapping by for all . 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 -space 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 and be two -spaces and let be two set-valued mappings satisfying conditions (i) and (iii) in Corollary 2.6. If for each is -subspace of and is weakly- valued, then there exists such that .

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

Corollary 2.13. Let and be two -spaces and be set-valued mappings satisfying conditions (i), (iii) and (iv) in Corollary 2.8. If for each is -subspace of and is weakly- valued, then at least one of the following assertions holds: (a)there exists such that ;(b)there exists such that .

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 be an -space, let and be nonempty sets, let be a relation on , and let be set-valued mappings. We say that is -transfer compactly continuous in the first variable if for any , there exists such that .

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

Definition 3.2. Let be an -space and be topological space. For each is said to be -quasiconvex (resp., -quasiconcave) in the first variable, if for each , the set (resp., ) is an -subspace of .

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

Definition 3.3. Let and be two topological spaces, and let be a function. Then is said to be -transfer compactly lower semicontinuous (in short, -transfer compactly l.s.c) in the first variable if for each with , there exists such that . is said to be -transfer compactly u.s.c in the first variable if and only if – is -transfer compactly l.s.c in the first variable. If is -transfer compactly l.s.c in the first variable for each , we say that is transfer compactly l.s.c in the first variable.

Theorem 3.4. Let be an -space, let and be nonempty sets, and let be a relation on . Let , and be set-valued mappings satisfying the following conditions: (i);(ii) is -transfer compactly continuous in the first variable;(iii) there exists a compact subset of such that for each , there exists a compact -subspace of containing such that (iv) for each , the set is -subspace of ;(v) for all and any .Then there exists such that for all .

Proof. Let be the set-valued mapping defined by 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 is transfer compactly open valued. By condition (iv), for each the set is -subspace of . Let and for all . Then, by condition (iii) there exists such that Thus all the conditions of Theorem 2.5 are fulfilled, and by condition (i), for all . Hence, there exists such that , that is, for all . This completes the proof.

Remark 3.5. Theorem 3.4 generalizes Theorem 6 in [2] in the following several aspects: (a) from -convex space to -space without linear structure; (b) from -transfer continuous to -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 and be two -spaces, let be nonempty set, and let be a relation on . Let , and be set-valued mappings satisfying the following conditions: are set-valued mappings satisfying conditions (i), (ii), (iii), and (v) in Theorem 3.4 and if for each is -subspace of and is weakly- valued, then there exists such that for all .

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 -convex space to -space without linear structure; (b) from -transfer continuous to -transfer compactly continuous; (c) from -weakly convex to weakly- valued; (c) conditions (iii) and “for each is -subspace of ” of Corollary 3.6 are weaker than conditions (iii) and (vii) of Theorem 7 in [2].

Theorem 3.8. Let be a compact -space, let be nonempty set, let be three functions, and let be three real numbers satisfying the following conditions: (i)for each , the set is -subspace of ;(ii) is -transfer compactly u.s.c in the first variable;(iii)for each , and ;(iv) for all .Then, at least one of the following assertions holds: (a)there exists such that for all ;(b)there exists such that for all .

Proof. Let be the set-valued mappings defined by for all . By means of contradiction, suppose that both assertions (a) and (b) would be false; that is(1)for each , there exists such that ,(2)for each , there exists such that .By (2), . Arbitrarily choose . By (1) and condition (ii), it follows that , and there exists such that . Hence, we infer that . By condition (i), is -subspace of , and by condition (iii), for each . Taking into account Remark 2.11, all conditions of Theorem 2.3 are satisfied, and it follows that there exists such that . Hence, we have , 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 -convex space to -space without linear structure; (b) from -transfer u.s.c to -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 be a compact -space, let be nonempty set, and let be three functions satisfying the following conditions: (i) is -quasiconvex, for each ;(ii) is transfer compactly u.s.c in the first variable;(iii)for each , .Then with the convention .

Proof. We may suppose that By means of contradiction, suppose that and choose such that It is easy to see that functions , and 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, then a contradiction.If (b) happens, then 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 -convex space to -space without linear structure; (b) from -quasiconvex to -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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