Best Proximity Sets and Equilibrium Pairs for a Finite Family of Multimaps
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Research Article Best Proximity Sets and Equilibrium Pairs for a Finite Family of Multimaps M. A. Al-Thagafi and Naseer Shahzad Department of Mathematics, King AbdulAziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Correspondence should be addressed to M. A. Al-Thagafi, [email protected] Received 12 May 2008; Accepted 16 October 2008 Recommended by Jerzy Jezierski We establish the existence of a best proximity pair for which the best proximity set is nonempty for a finite family of multimaps whose product is either an Aκc -multimap or a multimap T : A → 2B such that both T and S ◦ T are closed and have the KKM property for each Kakutani multimap S : B → 2A . As applications, we obtain existence theorems of equilibrium pairs for free n-person games as well as for free 1-person games. Our results extend and improve several well-known and recent results. Copyright q 2008 M. A. Al-Thagafi and N. Shahzad. 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.
1. Introduction Let E : E, · be a normed space, A a nonempty subset of E, and T : A → E a singlevalued map. Whenever the equation T x x has no solution in A, it is natural to ask if there exists an approximate solution. Fan 1 provided sufficient conditions for the existence of an approximate solution a ∈ A called a best approximant such that a − T a dT a, A : inf{dT a, x : x ∈ A},
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where A is compact and convex and T is continuous. However, there is no guarantee that such an approximate solution is optimal. For suitable subsets A and B of E and multimap T : A → 2B , Sadiq Basha and Veeramani 2 provided sufficient conditions for the existence of an optimal solution a, T a called a best proximity pair such that da, T a dA, B : inf{x − y : x ∈ A, y ∈ B}.
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Srinivasan and Veeramani 3, 4 extended these results and obtained existence theorems of equilibrium pairs for constrained generalized games. Kim and Lee 5, 6 generalized
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Fixed Point Theory and Applications
Srinivasan and Veeramani results and obtained existence theorems of equilibrium pairs for free n-person games. Recently, Al-Thagafi and Shahzad 7 generalized and extended the above results to Kakutani multimaps. In this paper, we establish the existence of a best proximity pair for which the best proximity set is nonempty for a finite family of multimaps whose product is either an Aκc multimap or a multimap T : A → 2B such that both T and S ◦ T are closed and have the KKM property for each Kakutani multimap S : B → 2A. As applications, we obtain existence theorems of equilibrium pairs for free n-person games as well as free 1-person games. Our results extend and improve several well-known and recent results. 2. Preliminaries Throughout, E : E, · is a normed space, A and B are nonempty subsets of E, 2A is the family of all subsets of A, coA is the convex hull of A in E, i
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