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Collective fixed points and maximal elements with applications to abstract economies. (English) Zbl 1051.54028
Summary: We first establish collective fixed points theorems for a family of multivalued maps with or without assuming that the product of these multivalued maps is $\Phi$-condensing. As an application of our collective fixed points theorem, we derive a coincidence theorem for two families of multivalued maps defined on product spaces. Then we give some existence results for maximal elements for a family of $L_S$-majorized multivalued maps whose product is $\Phi$-condensing. We also prove some existence results for maximal elements for a family of multivalued maps which are not $L_S$-majorized but their product is $\Phi$-condensing. As applications of our results, some existence results for equilibria of abstract economies are also derived. The results of this paper are more general than those given in the literature.

MSC:
54H25Fixed-point and coincidence theorems in topological spaces
54C60Set-valued maps (general topology)
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
91B50General equilibrium theory in economics
47N10Applications of operator theory in optimization, convex analysis, programming, economics
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References:
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