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.


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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