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Existence of equilibria for multivalued mappings and its application to vectorial equilibria. (English) Zbl 1023.49014

Summary: We apply a new fixed-point theorem [L.-J. Lin and Z.-T. Yu, Nonlinear Anal., Theory Methods Appl. 43A, 987-999 (2001; Zbl 0989.47051)] and use various monotonicity and some coercivity conditions to establish equilibrium theorems for multimaps in generalized convex spaces of S. Park and H. Kim [J. Math. Anal. Appl. 197, 173-187 (1996; Zbl 0851.54039)]. As a simple consequence, we give a unified approach to vectorial equilibria for multimaps. We show that, from our results, some well-known classical results, such as the Ky Fan minimax inequality theorem and the Browder and Hartman-Stampacchia theorems concerning the existence for variational inequalities, can be derived easily.

MSC:

49J53 Set-valued and variational analysis
90C47 Minimax problems in mathematical programming
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
47H04 Set-valued operators
90C29 Multi-objective and goal programming
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