Abstract and Applied AnalysisVolume 2011 (2011), Article ID 247236, 17 pagesdoi:10.1155/2011/247236

Research Article

Applications of an HIDS Theorem to the Existence of Fixed Point, Abstract Equilibria and Optimization Problems

Wei-Shih Du

Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan

Received 16 June 2011; Accepted 18 July 2011

Academic Editor: Josip E. Pečarić

Copyright © 2011 Wei-Shih Du. 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.

By applying hybrid inclusion and disclusion systems (HIDS), we establish several vectorial variants of system of Ekeland's variational principle on topological vector spaces, some existence theorems of system of parametric vectorial quasi-equilibrium problem, and an existence theorem of system of the Stampacchia-type vectorial equilibrium problem. As an application, a vectorial minimization theorem is also given. Moreover, we discuss some equivalence relations between our vectorial variant of Ekeland's variational principle, common fixed point theorem, and maximal element theorem.

1. Introduction

Let be a nonempty subset of a topological space (t.s., for short), and let be a function with for all . Then the scalar equilibrium problem in the sense of Blum and Oettli [1] is to find such that for all . The equilibrium problem was extensively investigated and generalized to the vectorial equilibrium problems for single-valued or multivalued maps and contains optimization problems, variational inequalities problems, saddle point problems, the Nash equilibrium problems, fixed point problems, complementary problems, bilevel problems, and semi-infinite problems as special cases and have some applications in mathematical program with equilibrium constraint; for detail one can refer to [1–4] and references therein.

The famous Ekeland's variational principle (EVP, for short) [5–7] is a forceful tool in various fields of applied mathematical analysis and nonlinear analysis. A number of generalizations in various different directions of these results for functions defined in metric (or quasimetric) spaces and more general in topological vector spaces have been investigated by several authors in the past; see [8–23] and references therein. It is wellknown that the original EVP is equivalent to Caristi's fixed point theorem, to Takahashi's nonconvex minimization theorem; and to the flower petal theorem for detail, see [14, 16–18, 20] and references therein. EVPs were extended to the vector case by using scalarization method and were applied to the study of efficiency (or approximative efficiency) and others; see, for example, [3, 8, 9, 12, 13, 15, 23].

Recently, the author first studied the following mathematical model about hybrid inclusion and disclusion systems (HIDS, for short) [11]. Let be any index set. For each , let be a nonempty closed convex subset of a Hausdorff topological vector space (t.v.s., for short) , , , , and let be multivalued maps. Hybrid inclusion and disclusion systems (HIDS) are defined as follows:

In fact, HIDS contains several important problems as special cases. Let be a nonempty subset of a topological space , and let be given. For each , let and be real t.v.s. with zero vector and , respectively.

Example 1.1. For each , let and be functions. If and are defined as follows: then HIDS will reduce to the following system of hybrid scalar equilibrium problem (P ₁): (P ₁) Find such that and for all and for all .

Example 1.2. For each , let and be multivalued maps with nonempty values, and let and be nonempty subsets of and , respectively. If and are defined as follows: then HIDS will reduce to the following problem (P ₂), which is an abstract equilibrium problem: (P ₂) Find such that and for all and for all .

Example 1.3. For each , let and be multivalued maps. If and are defined as follows: then HIDS will reduce to the following fixed point problem (P ₃): (P ₃) Find such that and for all for all .

Example 1.4. For each , let and be multivalued maps with nonempty values. If and are defined as follows: then HIDS will reduce to the following system of mixed type of parametric variational inclusion and disclusion problem (P ₄): (P ₄) Find such that and for all and for all .

In this paper, we study the existence theorems of system of parametric vectorial quasi-equilibrium problems, vectorial variants of system of Ekeland's variational principle (VSEVP, for short), and the existence theorems of system of the Stampacchia-type vectorial equilibrium problem by using an HIDS theorem established by the author [11]. Our results improve and generalize some theorems in [19] to the vector case. Till now, to my knowledge, there are extremely few results about Stampacchia-type vectorial equilibrium problem in the literature. The existence of VSEVP and the Stampacchia-type vectorial equilibrium problem are established by applying the HIDS theorem without the use of any scalarization method. So our results are completely different from [3, 8, 9, 12, 13, 15, 23]. As an application, a vectorial minimization theorem is also proved. Moreover, we prove some equivalence relations between our VSEVP, common fixed point theorem, and maximal element theorem.

2. Preliminaries

Let and be nonempty sets. A multivalued map is a function from to the power set of . We denote and let be defined by the condition that if and only if . We recall that a point is a maximal element of if . Let be a linear space with zero vector . A nonempty subset of is called a convex cone if and for all . A cone in is pointed if . Let be a real t.v.s., and let be a proper convex cone in , and . A point is called a vectorial minimal point of with respect to if for any , . The set of vectorial minimal point of is denoted by . The convex hull of and the closure of are denoted by and , respectively.

Definition 2.1. Let and be linear spaces, and let be a proper convex cone in . A map is called -convex if for any and , one has Clearly, if and are -convex and , then and are -convex.

Definition 2.2. Let be a nonempty convex subset of a vector space , let be a nonempty convex subset of a vector space , and let be a real t.v.s. Let and be multivalued maps such that for each , is a nonempty closed convex cone. For each fixed , is called -quasiconvex if for any , and , one has either or Now, we define the concept of vectorial upper and lower semicontinuous on t.v.s.

Definition 2.3. Let be a t.s., let be a t.v.s. with zero vector , and let be a pointed convex cone in . A map is said to be (i)vectorial lower semicontinuous with respect to (-v.l.s.c., for short) at if for any , there exists an open neighborhood of such that for all ,(ii)vectorial upper semicontinuous with respect to (-v.u.s.c., for short) at if for any , there exists an open neighborhood of such that for all . The function is called   -v.l.s.c. (resp., -v.u.s.c.) on if is -v.l.s.c. (resp., -v.u.s.c.) at every point of .

Proposition 2.4. Let be a t.s., let be a t.v.s with zero vector , and let be a pointed convex cone in . Let , be maps and . Then the following hold: (a) is -v.u.s.c. (resp., -v.l.s.c.) on is -v.l.s.c. (resp., -v.u.s.c.) on ; (b)if and are -v.u.s.c. (resp., -v.l.s.c.) on , then and are -v.u.s.c. (resp., -v.l.s.c.) on ; (c)if is -v.u.s.c. on , then(i) is open in for all ,(ii) is closed in for all ; (d)if is -v.l.s.c. on , then(iii) is open in for all ,(iv) is closed in for all .

Proof. Clearly, (a) and (b) hold from definition. To prove (c), it suffices to show (i). Suppose that is -v.u.s.c. on . Let and . Then . Since is -v.u.s.c. at , there exists an open neighborhood of such that for all . Hence is an open set in and (i) is proved. Obviously, (ii) is immediate from (i). It is easy to see that conclusion (d) follows from (a) and (c).

Remark 2.5. Let be a t.v.s. and with , where is the origin of . Let be a map. Hence in conclusion (c) (resp., (d)) of Proposition 2.4, we have In particular, if (the set of real numbers) and , then the -v.l.s.c. (resp. -v.u.s.c.) function is l.s.c. (resp. u.s.c.) in usual.The concept of -vectorial -function and -vectorial quasi-distance on topological spaces are introduced as follows.

Definition 2.6. Let and be t.v.s., let be the zero vector of , and let , a pointed convex cone in . A map is called (a)a -vectorial -function (-function, for short) if the following are satisfied: (VL1) for all ;(VL2) for any , is -convex;(VL3) for any , is -v.u.s.c. (b)a -vectorial quasi-distance if the following are satisfied:(VQD1) for all ;(VQD2) for any ;(VQD3) for any , is -convex and -v.l.s.c.;(VQD4) for any , is -v.u.s.c.If and letting be in (a) and (b), then the function is called a -function and quasi-distance, respectively, introduced by Lin and Du [19]. For examples and results of -function and quasi-distance, one can see [19].

Remark 2.7. (a) Obviously, a -vectorial quasi-distance is a -function, but the reverse is not true; (b) if and are -vectorial quasi-distances (resp., -functions) and , then and are -vectorial quasi-distances (resp., -functions);(c) if is a -v.l.s.c. and -convex function, then the function defined by is a -vectorial quasi-distance.

Lemma 2.8 . (see [24, 25]). Let and be the Hausdorff topological spaces, and let be a multivalued map. Then is l.s.c. at if and only if for any and for any net in converging to , there exists a subnet of and a net with such that for all .

3. The Existence of System of VSEVP and Abstract Equilibrium Problems

The following existence theorem for the solution of HIDS was established in [11].

Theorem 3.1 (HIDS theorem [11]). Let be any index set. For each , let be a nonempty closed convex subset of a Hausdorff t.v.s. . Let be a nonempty subset of a topological space , and . For each , let be a nonempty closed subset of , let be a multivalued map, and let be a multivalued map with nonempty values. For each , suppose that the following conditions are satisfied: (i)for each ;(ii)for each , and is convex;(iii)for each , and are open in ;(iv)there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and . Then there exists such that for each , and for all .

Example 3.2. Let and be Hausdorff t.v.s., let be a real t.v.s. with its zero vector , and .(a) Let be a multivalued map with nonempty values such that there exists such that and the map is closed. Then it is easy to see that is a nonempty closed subset of .(b) Let be a multivalued map with nonempty values and be a nonempty open set in . Suppose that(i) for each ,(ii) for each is -quasiconvex and for each , is l.s.c.Let be defined by Then for each . We claim that is open in for each . Indeed, let be given, and let . Then there exists a net in such that . Thus we have or . Clearly, . For any , since is l.s.c. at and , by Lemma 2.8, there exists a net with such that . Since is closed, . So . It implies , and hence is open in . Next, we show that for each , is convex. Let , . Then and . For any , let . Suppose to the contrary that there exists such that . By the -quasiconvexity of , either or which leads to a contradiction. Hence for each , is convex.Applying Theorem 3.1, we establish the following existence theorem of system of parametric vectorial quasi-equilibrium problem.

Theorem 3.3. Let be any index set. For each , let be a nonempty Hausdorff t.v.s., and let be a pointed convex cone in a t.v.s. with zero vector . Let . For each , let , be -vectorial quasi-distances and let be a multivalued map with nonempty values. Let with for all . For each , suppose that the following conditions are satisfied: (i)for each , and is open for all ;(ii)there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that . Then there exists such that for each , one has (a),(b) for all .

Proof. For each , define and by respectively. Thus, for each , we let and let be defined by Clearly, for each , for all , and hence for all . By the -vectorial lower semicontinuity of , is a nonempty closed subset of . By (i), for each , . For each , by the vectorial upper semicontinuity of , the set is open in , and hence is open in . For each , is convex from the -convexity of for all . By (ii), there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and . Therefore all the conditions of Theorem 3.1 are satisfied. Applying Theorem 3.1, there exists such that: (1),(2) for all .

Theorem 3.4. Let , , , , , , , and be the same as in Theorem 3.3. Let with for all . For each , suppose that the following conditions are satisfied: (i)for each , and is open for all ;(ii)there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that . Then there exists such that for each , one has (a),(b) for all .

Proof. For each , let be defined by . By Theorem 3.3, there exists such that for each , we have (1),(2) for all . We claim that for each , for all . For each , we have . If , then, by (VQD2) (in Definition 2.6(b)), we obtain which contradict with (1). Hence for all , and the proof is completed.

Remark 3.5. (i) Let be a nonempty Hausdorff t.v.s., and let be a nonempty pointed convex cone in a t.v.s. with zero vector . If a map satisfies that for each , is -quasiconvex, then for each with , is a nonempty convex subset in .(ii) In Theorem 3.3, if for each , is convex and for all , then conclusion (b) can be replaced with conclusion , where . Since the sum of two -vectorial quasi-distances is also a -vectorial quasi-distance, the following results related with system of VSEVP for vectorial quasi-distances in a Hausdorff t.v.s. are immediate from Theorem 3.3.

Theorem 3.6. Let , , , , , , , and be the same as in Theorem 3.3. Let with for all . For each , let be a -v.l.s.c. and -convex function and suppose that the following conditions are satisfied: (i)for each , and is open for all ;(ii)there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that . Then there exists such that for each , one has (a), (b) for all .

In Theorem 3.6, if and for all , then we have the following result.

Corollary 3.7. Let , , , and be the same as in Theorem 3.3. For each , let be a l.s.c. and convex function and let be a quasi-distance. Let with for all . Suppose that the following conditions are satisfied: (i)for each , and is open for all ;(ii)there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that . Then there exists such that for each , one has (a), (b) for all .

Theorem 3.8. Let , , , , , , , and be the same as in Theorem 3.3. Let with for all . For each , let be a -v.l.s.c. and -convex function and suppose that the following conditions are satisfied: (i)for each , and is open for all ;(ii)there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that . Then there exists such that for each , one has (a), (b) for all .

Corollary 3.9. Let , , , and be the same as in Theorem 3.3. For each , let be a l.s.c. and convex function and be a quasi-distance. Let with for all . Suppose that the following conditions are satisfied: (i)for each , and is open for all ;(ii)there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that . Then there exists such that for each , one has (a), (b) for all .

By using Theorem 3.1 again, we have the following result.

Theorem 3.10. Let , , , , , , , , and be the same as in Theorem 3.3. Suppose that there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that and . Then there exists such that for each , one has (a), (b) for all .

Proof. For each , let and multivalued maps , , and be defined as in the proof of Theorem 3.3. For each , define by Clearly, for each , is open in for all . By the -convexity of , is a nonempty convex subset of for all . Since is convex in and , we have for all . By our hypothesis, there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and . Thus all the conditions of Theorem 3.1 are satisfied, and the conclusion follows from Theorem 3.1.

Remark 3.11. In Theorem 3.10, the multivalued map and the condition “for each , ” are not assumed. So Theorems 3.10 and 3.3 are different.

Theorem 3.12. Let , , , , , , and be the same as in Theorem 3.3. Let with for all . Suppose that there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that and . Then there exists such that for each , one has (a), (b) for all .

Proof. For each , let . Applying Theorem 3.10 and following the same argument as in the proof of Theorem 3.4, one can prove the theorem.

Theorem 3.13. Let , , , , , , and be the same as in Theorem 3.3. Let with for all . For each , let be a -v.l.s.c. and -convex function. Suppose that there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that and . Then there exists such that for each , one has (a),(b) for all .

Corollary 3.14. Let , , and be the same as in Theorem 3.3. For each , let be a l.s.c. and convex function and be a quasi-distance. Let with for all . Suppose that there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that and . Then there exists such that for each , one has (a), (b) for all .

Remark 3.15. [19, Theorem  4.3] is a special case of Corollary 3.14.

Theorem 3.16. Let , , , , , , and be the same as in Theorem 3.3. Let with for all . For each , let be a -v.l.s.c. and -convex function, and suppose that there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that and . Then for each with for all , there exists such that for each , one has (a), (b) for all . In Theorem 3.16, if and for all , then we have the following system of Lin and Du's variant of system of Ekeland's variational principle in t.v.s.

Corollary 3.17. Let , , and be the same as in Theorem 3.3. For each , let be a l.s.c. and convex function and let be a quasi-distance. Let with for all . Suppose that there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that and . Then for each with for all , there exists such that for each , one has (a), (b) for all .

Remark 3.18. Corollary 3.17 generalizes [19, Theorem  4.1].

4. A Vectorial Minimization Theorem and Equivalent Formulations of VSEVP

Using Theorem 3.1 again, we also obtain an existence theorem of system of generalized vectorial equilibrium problem of the Stampacchia-type which can be regarded as a weak form of VSEVP for -vectorial -function in a Hausdorff t.v.s.

Theorem 4.1. Let be any index set. For each , let be a nonempty Hausdorff t.v.s. and be a nonempty pointed convex cone in a t.v.s. with zero vector . Let . For each , let , be -function. Suppose that (i), (ii)there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that . Then there exists such that for each , one has for all .

Proof. Let be given. For each , we define , and by respectively. Clearly, Using the same argument in the proof of Theorem 3.3, one can verify that all the conditions of Theorem 3.1 are satisfied. By Theorem 3.1, there exists such that for each , we have for all .

An existence theorem of system of generalized vector equilibrium problem is immediate from Theorem 4.1 if (the zero map) for all .

Theorem 4.2. Let be any index set. For each , let be a nonempty Hausdorff t.v.s., and let be a nonempty pointed convex cone in a t.v.s. with zero vector . Let . For each , let be a -function. Suppose that (i);(ii)there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that . Then there exists such that for each , one has for all .

Remark 4.3. (a) Notice that Theorems 4.1 and 4.2 are indeed equivalent since the sum of two -functions is also a -function.(b) In Theorem 4.1, the maps and are only assumed to be -functions which need not be -vectorial quasi-distances. So Theorem 4.1 is different from any theorem in Section 3 and is not a special case of any theorem in Section 3.

Theorem 4.4. Let , , , , , and be the same as in Theorem 4.1. For each , let be a -function and let be a -v.l.s.c. and -convex function. Suppose that (i), (ii)there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that . Then there exists such that for each , one has for all .

Remark 4.5. [19, Theorem  4.2] is a special case of Theorem 4.4.Applying Theorem 4.4, we obtain the following vectorial minimization theorem.

Theorem 4.6 (vectorial minimization theorem). Let , , , , , and be the same as in Theorem 4.1. For each , let be a -function and let be a -v.l.s.c. and -convex function. Suppose that (i), (ii)there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that ,(iii)for and with , there exists with such that . Then there exists such that for all .

Proof. Applying Theorem 4.4, there exists such that for each , we have for all . We claim that for all . Suppose to the contrary that there exists such that . Then, by our assumption, there exists with such that , which leads to a contradiction. Therefore for all .

The following scalar minimization theorem follows from Theorem 4.6 immediately.

Corollary 4.7. Let , , and be the same as in Theorem 4.1. For each , let be a l.s.c. and convex function and let be a -function. Suppose that (i), (ii)there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exist and such that ,(iii)for any and with there exists with such that . Then there exists such that for all .

Remark 4.8. (a) [19, Theorem  5.5] is a special case of Corollary 4.7. (b) Theorems 4.4 and 4.6 are equivalent if they further add the condition “for each , for all , .” Indeed, it suffices to show that Theorem 4.6 implies Theorem 4.4. Suppose that for each , there exists such that for some with . Then, by Theorem 4.6, there exists such that or But from our assumption, there exists such that for some with . It follows that which leads to a contradiction.

Now, we give some equivalent formulations of Theorem 4.1 (when is a singleton) as follows.

Theorem 4.9. Let be a Hausdorff t.v.s., let be a nonempty pointed convex cone in a t.v.s. with zero vector , , be -functions. Suppose that (1),(2)there exist a nonempty compact subset of and a nonempty compact convex subset of such that for each there exists such that . Then the following statements are equivalent. (i)(Vectorial Variant of Ekeland’s Variational Principle). There exists such that for all .(ii)(Common Fixed Point Theorem for a Family of Multivalued Maps). Let be an index set. For each , let be a multivalued map with nonempty values such that for each with , there exists with such that Then there exists such that . That is, has a common fixed point in .(iii)(Common Fixed Point Theorem for a Family of Single-Valued Maps). Let be an index set. For each , suppose that is a self-map satisfying for all . Then there exists such that for all ;(iv)(Maximal Element Theorem for a Family of Multivalued Maps). Let be an index set. For each , let be a multivalued map. Suppose that for each with , there exists with such that Then there exists such that for all .

Proof. By Theorem 4.1, conclusion (i) holds.(a) “(i) (ii)” By (i), there exists such that for all . We claim that for all . If for some , then, by hypothesis, there exists with such that which leads to a contradiction. Hence and is a common fixed point of . Suppose that for each , there exists with such that . Then for each , we can define a multivalued map by Clearly, for all . But, by (ii), has a fixed point in , a contradiction. So (i) is true.(b) “(ii) (iii)” Suppose (ii) holds. Under the assumption of (iii), for each , let be defined by . Then for each with , we have . By hypothesis of (iii), . Therefore, by (ii), there exists such that or for all , and hence (iii) is proved. Assume (iii) holds. Under the assumption of (ii), for each with , there exists with such that Define by Hence is a self-map of into satisfying for all . By (iii), there exists such that for all . This shows that (iii) implies (ii).(c) "(i) (iv)" Using (i), there exists such that for all . We want to show that for all . Suppose to the contrary that there exists such that . By hypothesis of (iv), there exists with such that which is a contradiction. Therefore for all . Suppose that for each , there exists with such that . For each , define a multivalued map by Then for all . But applying (iv), there exists such that , a contradiction. Hence (i) holds.

Remark 4.10. Theorem 4.9 improves and generalizes Theorems  4.2, 5.1, 5.2, 5.3, and 5.4 in [19].

Acknowledgment

This research was supported partially by grant no. NSC 99-2115-M-017-001 of the National Science Council of the Republic of China.

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