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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:

54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology
47H10 Fixed-point theorems
91B50 General equilibrium theory
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
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