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Some equilibrium problems in generalized convex spaces. (English) Zbl 1020.46026
Summary: We show how the fundamental theorems on equilibrium problems can be extended to generalized convex spaces. More precisely, most of important results in the KKM theory hold without assuming the linearity in topological vector spaces. Such examples are the KKM theorem, the minimax theorem and the intersection lemma of von Neumann, the Nash equilibrium theorem, various fixed point theorems, Ky Fan’s minimax inequality, variational inequalities, best approximation theorems, existence theorems for solutions of generalized quasi-equilibrium problems, and others.
MSC:
46N10Applications of functional analysis in optimization and programming
46A55Convex sets in topological linear spaces; Choquet theory
49J35Minimax problems (existence)
49J40Variational methods including variational inequalities
90C30Nonlinear programming