Open Access

Some Maximal Elements' Theorems in -Spaces

Journal of Inequalities and Applications20092009:905605

https://doi.org/10.1155/2009/905605

Received: 30 March 2009

Accepted: 1 September 2009

Published: 28 September 2009

Abstract

Let be a finite or infinite index set, let be a topological space, and let be a family of -spaces. For each , let be a set-valued mapping. Some new existence theorems of maximal elements for a set-valued mapping and a family of set-valued mappings involving a better admissible set-valued mapping are established under noncompact setting of -spaces. Our results improve and generalize some recent results.

1. Introduction

It is well known that many existence theorems of maximal elements for various classes of set-valued mappings have been established in different spaces. For their applications to mathematical economies, generalized games, and other branches of mathematics, the reader may consult [112] and the references therein.

In most of the known existence results of maximal elements, the convexity assumptions play a crucial role which strictly restrict the applicable area of these results. In this paper, we will continue to study existence theorems of maximal elements in general topological spaces without convexity structure. We introduce a new class of generalized -majorized mappings for each which involve a set-valued mapping . The notion of generalized -majorized mappings unifies and generalizes the corresponding notions of -majorized mappings in [4]; -majorized mappings in [2, 13]; -majorized mappings in [14]. Some new existence theorems of maximal elements for generalized -majorized mappings are proved under noncompact setting of -spaces. Our results improve and generalize the corresponding results in [2, 4, 1416].

2. Preliminaries

Let and be two nonempty sets. 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 denote the standard -dimensional simplex with the vertices . If is a nonempty subset of we will denote by the convex hull of the vertices .

Let and be two sets, and let be a set-valued mapping. We will use the following notations in the sequel:

(i) ,

(ii) ,

(iii) .

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 [17]) are defined, respectively, by

(2.1)

It is easy to see that , is compactly open (resp., compactly closed) in if and only if (resp., ). For each nonempty compact subset of , and , where (resp., )) denotes the closure (resp., interior) of in . A set-valued mapping is transfer compactly open valued on (see [17]) if for each and , there exists such that . Let be transfer compactly open valued, then . It is clear that each transfer open valued correspondence is transfer compactly open valued. The inverse is not true in general.

The definition of -space and the class of better admissible mapping were introduced by Ding in [8]. Note that the class of better admissible mapping includes many important classes of mappings, for example, the class in [18], in [19] and so on as proper subclasses. Now we introduce the following definition.

Definition 2.1.

An -space is said to be an -space if for each , there exists a compact -subspace of containing .

be a -convex space, let the notion of -convex space was introduced by Ding in [4].

Lemma 2.2 ([8]).

Let be any index set. For each , let be an -space, and . Then is also an -space.

Let be a topological space, and let be any index set. For each , let be an -space, and let such that is an -space defined as in Lemma 2.2. Let and for each , let be a set-valued mapping. For each ,

(1) is said to be a generalized -mapping if

(a)for each and , , where is the projection of onto and ;

(b) is transfer compactly open in for each ;

(2) is said to be a generalized -majorant of at if is a generalized -mapping and there exists an open neighborhood of in such that for all ;

(3) is said to be a generalized -majorized if for each with , there exists a generalized -majorant of at , and for any , the mapping is transfer compactly open in ;

(4) is said to be a generalized -majorized if for each , there exists a generalized -majorant of at .

Then is said to be a family of generalized -mappings (resp., -majorant mappings) if for each is a generalized -mapping (resp., -majorant mapping).

If for each , let be a -convex space, a family of -mappings (resp., -majorant mappings) were introduced by Ding in [4]. Clearly, each family of generalized -mappings must be a family of generalized -majorant mappings. If is a single-valued mapping and is an -subspace of for each , then condition for each implies that condition (a) in (1) holds. Indeed, if (a) is false, then there exist , , and such that and hence for each . It follows from that where . It follows from being an -subspace of that which contradicts condition for each . Hence each -mapping (resp., -majorant mapping) introduced by Deguire et al. (see [2, page 934]) must be a generalized -mapping (resp., -majorant mapping). The inverse is not true in general.

3. Maximal Elements

In order to obtain our main results, we need the following lemmas.

Lemma 3.1 ([3]).

Let and be topological spaces, let be a nonempty compact subset of and let be a set-valued mapping such that for each . Then the following conditions are equivalent:

(1) have the compactly local intersection property;

(2)for each , there exists an open subset of (which may be empty) such that and ;

(3)there exists a set-valued mapping such that for each is open or empty in , and ;

(4)for each , there exists such that and ;

(5) is transfer compactly open valued on .

Lemma 3.2 ([8]).

Let be a topological space, and let be an -space, and such that

(i)for each and for each ,
(3.1)

(ii) is transfer compactly open valued;

(iii)there exists a nonempty set and for each , there exists a compact -subspace of   containing such that is empty or compact in , where denotes the complement of .

Then there exists a point such that .

Theorem 3.3.

Let be a topological space, let be a nonempty compact subset of , and let be an -space, and be a generalized -mapping such that

(i)for each , there exists a compact -subspace of containing such that for each .

Then there exists a point such that .

Proof.

Suppose that for each . Since is a generalized -mapping, is transfer compactly open valued. By Lemma 3.1, we have
(3.2)
Since is compact, there exists a finite set such that
(3.3)
By condition (i) and , there exists a compact -subspace of containing and is compact in , and hence we have
(3.4)

By using similar argument as in the proof of Lemma 3.2, we can show that there exists such that . Condition (i) implies that must be in . This completes the proof.

Remark 3.4.

Theorem 3.3 generalizes in [4, Theorem  2.2] in the following several aspects: (a) from -convex space to -space without linear structure; (b) from -mappings to generalized -mappings.

Theorem 3.5.

Let be a topological space, and let be an -space. Let and be a generalized -majorized mapping such that

(i)there exists a paracompact subset of such that ;

(ii)there exists a nonempty set and for each , there exists a compact -subspace of containing such that the set is empty or compact.

Then there exists a point such that .

Proof.

Suppose that for each . Since is a generalized -majorized, for each , there exists an open neighborhood of in and a generalized -mapping such that

(a) for each ,

(b)for each and ,

(c) is transfer compactly open in ,

(d)for any , the mapping is transfer compactly open in .

Since for each , it follows from condition (i) that is paracompact. By Dugundji in [20, Theorem  VIII.1.4], the open covering has an open precise locally finite refinement , and for each since is normal. For each , define a mapping by
(3.5)
Then for each , we have
(3.6)

Hence is transfer compactly open in by (c).

Now define a mapping by

(3.7)
We claim that is a generalized -mapping and for each . Indeed, for any nonempty compact subset of and each with , we may take any fixed . Since is locally finite, there exists an open neighborhood of in such that is a finite set. If , then , and hence for all which implies that for all . It follows that for each ,
(3.8)
For any nonempty compact subset of and each , if . Since is open in , it follows from (d) that there exists such that
(3.9)

This proves that is transfer compactly open valued in .

On the other hand, for each and , if , then . Since there exists such that and , we have , and hence by (b). Hence we have

(3.10)

for each and . This shows that is a generalized -mapping.

For each , if , then there exists an such that and . It follows from (a) that . Hence we have for each . By condition (ii), there exists a nonempty set and for each , there exists a compact -subspace of containing such that the set is empty or compact. Note that for each implies for each . Hence and is empty or compact. By Lemma 3.2, there exists a point such that , and hence which contradicts the assumption that for each . Therefore, there exists such that .

Theorem 3.6.

Let be a topological space, let be a nonempty compact subset of and be an -space. Let and be a generalized -majorized mapping such that

(i)there exists a paracompact subset of such that ;

(ii)for each , there exists a compact -subspace of containing such that for each .

Then there exists such that .

Proof.

Suppose that for each . By using similar argument as in the proof of Theorem 3.5, we can show that there exists a generalized -mapping such that for each . It follows from condition (ii) that for each . By Theorem 3.3, there exists such that , and hence which contradicts the assumption that for each . Therefore, there exists such that . Condition (ii) implies . This completes the proof.

Remark 3.7.

Theorem 3.5 generalizes [4, Theorem  2.3] in several aspects: Section 1(1) from -convex space to -space without linear structure; Section 1(2) from a -majorized mapping to a generalized -majorized mapping; Section 1(3) condition (ii) of Theorem 3.5 is weaker than condition (ii) of [4, Theorem  2.3]. If is compact, condition (i) is satisfied trivially. If is a compact -space, then by letting for all , conditions (i) and (ii) are satisfied automatically. Theorem 3.6 unifies and generalizes Shen's [14, Theorem  2.1, Corollary  2.2 and Theorem  2.3] in the following ways: Section 2(1) from -convex space to -space without linear structure; Section 2(2) from -majorized correspondences to generalized -majorized mapping; Section 2(3) condition (ii) of Theorem 3.6 is weaker than that in the corresponding results of Shen in [14]. Theorem 3.6 also generalizes in [4, Theorem  2.4], Ding in [15, Theorem  5.3], and Ding and Yuan in [16, Theorem  2.3] in several aspects.

Corollary 3.8.

Let be a compact topological space, and let be an -space. Let and be a generalized -majorized mapping. Then there exists a point such that .

Proof.

The conclusion of Corollary 3.8 follows from Theorem 3.6 with .

Corollary 3.9.

Let be a topological space, and let be an -space. Let be a compact mapping and be a generalized -majorized mapping. Then there exists a point such that .

Proof.

Since is a compact mapping, there exists a compact subset of such that . The mapping be the restriction of to . It is easy to see that is also generalized -majorized. By Corollary 3.8, there exists such that .

Remark 3.10.

Corollary 3.8 generalizes Deguire et al. [2, Theorem  1] in the following ways: (1.1) from a convex subset of Hausdorff topological vector space to an -space without linear structure; (1.2) from a -majorized mapping to a generalized -majorized mapping. Corollary 3.8 also generalizes [4, Corollary  2.3] from -convex space to -space and from a -majorized mapping to a generalized -majorized mapping. Corollary 3.9 generalizes [2, Theorem  2] and [4, Corollary  2.4] in several aspects.

Theorem 3.11.

Let be a topological space, and let be any index set. For each , let be an -space, and let such that is an -space defined as in Lemma 2.2. Let such that for each ,

(i)let be a generalized -majorized mapping;

(ii) ;

(iii)there exists a paracompact subset of such that ;

(iv)there exists a nonempty set and for each , there exists a compact -subspace of containing such that the set is empty or compact, where .

Then there exists such that for each .

Proof.

For each , . Define by
(3.11)

Then for each , if and only if . Let with , then there exists such that . By condition (ii), there exists such that . Since is generalized -majorized, there exist an open neighborhood of in and a generalized -majorant of at such that

(a) for all ,

(b)for each and ,
(3.12)

(c) is transfer compactly open in ,

(d)for each , the mapping is transfer compactly open in .

Without loss of generality, we can assume that . Hence, for each . Define by
(3.13)

We claim that is a generalized -majorant of at . Indeed, we have

()for each , ,

()for each and , if , then . It is easy to see that , so that , i.e., and hence by (b). It follows that
(3.14)
()for each , we have that
(3.15)

is transfer compactly open in by (c).

Hence is a generalized -majorant of at .

For each and , by (3.15), we have

(3.16)

It follows from (d) that is transfer compactly open in .

Hence is generalized -majorized. By condition (iii), we have

(3.17)
By condition (iv), there exists a nonempty set and for each , there exists a compact -subspace of containing . By the definition of , for each , we have
(3.18)

It follows from condition (iv) that is empty or compact and hence all conditions of Theorem 3.5 are satisfied. By Theorem 3.5, there exists such that which implies , that is, for each .

Theorem 3.12.

Let be a topological space, and let be any index set. For each , let be an -space, and let . Let be a compact mapping such that for each ,

(i)let be a generalized -majorized mapping;

(ii) .

Then there exists such that for each .

Proof.

Since for each , let be an -space, then for each , there exists a compact -subspace of containing . Let and , then is a compact -subspace of for each , is a compact -subspace of containing . Hence is also an -space.

For each , . Define

(3.19)

Then for each , if and only if . By using similar argument as in the proof of Theorem 3.11, we can show that is a generalized -majorized mapping. By Corollary 3.9, there exists such that , and so . Hence, we have for each .

Theorem 3.13.

Let be a topological space, let be a nonempty compact subset of and let be any index set. For each , let be an -space, and let such that is an -space defined as in Lemma 2.2. Let such that for each , be a generalized -mapping such that

(i)for each and , there exists a compact -subspace of containing and for each , there exists satisfying .

Then there exists such that for each .

Proof.

Suppose that the conclusion is not true, then for each , there exists such that . Since is a generalized -mapping, is transfer compactly open valued. By Lemma 3.1, we have
(3.20)
Since is compact, there exists a finite set such that for each , there exists with . It follows that for each , there exists a such that . We may take any fixed . For each , let . By condition (i), for each , there exists a compact -subspace of containing and for each , there exists satisfying . Hence for each , there exists such that . Let , then is a compact -subspace of and hence it is also a compact -space. Let , then is compact in . Define by . For each , we have
(3.21)
Since is transfer compactly open valued in for each and , so that we claim that is transfer open valued in . Noting that each is a generalized -mapping, for each and , we have
(3.22)

where .

Hence for each , is a generalized -mapping and hence it is also a generalized -majorized mapping. All conditions of Corollary 3.8 are satisfied. By Corollary 3.8, there exists such that for each , so we have which contradicts the fact that for each there exists such that . Therefore, there exists such that for each .

Remark 3.14.

Theorem 3.11 generalizes [4, Theorem  2.5] in several aspects. Theorem 3.12 improves [2, Theorem  3] from convex subsets of topological vector spaces to -spaces without linear structure and from a family of -majorized mappings to the family of generalized -majorized mappings. Theorem 3.13 generalizes [4, Theorem  2.6] in several aspects: (1.1) from -convex spaces to -spaces without linear structure; (1.2) from a -mapping to a generalized -mapping; (1.3) condition (i) of Theorem 3.13 is weaker than condition (i) of [4, Theorem  2.6]. Theorem 3.13 improves and generalizes [2, Theorem  7] in the following ways: (2.1) from nonempty convex subsets of Hausdorff topological vector spaces to -space without linear structure; (2.2) from the family of -majorized mappings to the family of generalized -majorized mappings.

Declarations

Acknowledgment

This work is supported by a Grant of the Natural Science Development Foundation of CUIT of China (no. CSRF200709).

Authors’ Affiliations

(1)
Department of Mathematics, Chengdu University of Information Technology
(2)
Department of Mathematics, Sichuan University

References

  1. Balaj M: Coincidence and maximal element theorems and their applications to generalized equilibrium problems and minimax inequalities. Nonlinear Analysis: Theory, Methods & Applications 2008,68(12):3962–3971. 10.1016/j.na.2007.04.033MathSciNetView ArticleMATHGoogle Scholar
  2. Deguire P, Tan KK, Yuan GX-Z: The study of maximal elements, fixed points for -majorized mappings and their applications to minimax and variational inequalities in product topological spaces. Nonlinear Analysis: Theory, Methods & Applications 1999,37(7):933–951. 10.1016/S0362-546X(98)00084-4MathSciNetView ArticleMATHGoogle Scholar
  3. Ding X-P: Maximal element principles on generalized convex spaces and their applications. In Set Valued Mappings with Applications in Nonlinear Analysis. Volume 4. Edited by: Argawal RP. Taylor & Francis, London, UK; 2002:149–174.Google Scholar
  4. Ding X-P: Maximal elements for -majorized mappings in product -convex spaces and applications—I. Applied Mathematics and Mechanics 2003,24(6):583–594.MathSciNetGoogle Scholar
  5. Ding X-P: Maximal elements for -majorized mappings in product -convex spaces and applications—II. Applied Mathematics and Mechanics 2003,24(9):899–905.MathSciNetGoogle Scholar
  6. Ding X-P, Xia FQ: Equilibria of nonparacompact generalized games with -majorized correspondence in -convex spaces. Nonlinear Analysis: Theory, Methods & Applications 2004,56(6):831–849. 10.1016/j.na.2003.10.015MathSciNetView ArticleMATHGoogle Scholar
  7. Ding X-P, Yao J-C, Lin L-J: Solutions of system of generalized vector quasi-equilibrium problems in locally -convex uniform spaces. Journal of Mathematical Analysis and Applications 2004,298(2):398–410. 10.1016/j.jmaa.2004.05.039MathSciNetView ArticleMATHGoogle Scholar
  8. Ding X-P: Maximal element theorems in product -spaces and generalized games. Journal of Mathematical Analysis and Applications 2005,305(1):29–42. 10.1016/j.jmaa.2004.10.060MathSciNetView ArticleMATHGoogle Scholar
  9. Ding X-P: Maximal elements of -majorized mappings in product -spaces and applications. I. Nonlinear Analysis: Theory, Methods & Applications 2007,67(3):963–973. 10.1016/j.na.2006.06.037MathSciNetView ArticleMATHGoogle Scholar
  10. Kim WK, Tan K-K: New existence theorems of equilibria and applications. Nonlinear Analysis: Theory, Methods & Applications 2001,47(1):531–542. 10.1016/S0362-546X(01)00198-5MathSciNetView ArticleMATHGoogle Scholar
  11. Lin L-J, Yu Z-T, Ansari QH, Lai L-P: Fixed point and maximal element theorems with applications to abstract economies and minimax inequalities. Journal of Mathematical Analysis and Applications 2003,284(2):656–671. 10.1016/S0022-247X(03)00385-8MathSciNetView ArticleMATHGoogle Scholar
  12. Singh SP, Tarafdar E, Watson B: A generalized fixed point theorem and equilibrium point of an abstract economy. Journal of Computational and Applied Mathematics 2000,113(1–2):65–71. 10.1016/S0377-0427(99)00244-7MathSciNetView ArticleMATHGoogle Scholar
  13. Yuan GX-Z: KKM Theory and Applications in Nonlinear Analysis, Monographs and Textbooks in Pure and Applied Mathematics. Volume 218. Marcel Dekker, New York, NY, USA; 1999:xiv+621.Google Scholar
  14. Shen Z-F: Maximal element theorems of -majorized correspondence and existence of equilibrium for abstract economies. Journal of Mathematical Analysis and Applications 2001,256(1):67–79. 10.1006/jmaa.2000.7285MathSciNetView ArticleMATHGoogle Scholar
  15. Ding X-P: Fixed points, minimax inequalities and equilibria of noncompact abstract economies. Taiwanese Journal of Mathematics 1998,2(1):25–55.MathSciNetMATHGoogle Scholar
  16. Ding X-P, Yuan GX-Z: The study of existence of equilibria for generalized games without lower semicontinuity in locally topological vector spaces. Journal of Mathematical Analysis and Applications 1998,227(2):420–438. 10.1006/jmaa.1998.6105MathSciNetView ArticleMATHGoogle Scholar
  17. Chen C-M, Chang T-H: Some results for the family and the -mapping. Journal of Mathematical Analysis and Applications 2007,329(1):92–101. 10.1016/j.jmaa.2006.06.040MathSciNetView ArticleMATHGoogle Scholar
  18. Park S: Coincidence theorems for the admissible multimaps on generalized convex spaces. Journal of the Korean Mathematical Society 2000,37(4):885–899.MathSciNetGoogle Scholar
  19. Park S, Kim H: Foundations of the KKM theory on generalized convex spaces. Journal of Mathematical Analysis and Applications 1997,209(2):551–571. 10.1006/jmaa.1997.5388MathSciNetView ArticleMATHGoogle Scholar
  20. Dugundji J: Topology. Allyn and Bacon, Boston, Mass, USA; 1966:xvi+447.MATHGoogle Scholar

Copyright

© R.-H. He and Y. Zhang. 2009

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