We defined admissible classes of maps which are general enough to include composites of maps appearing in nonlinear analysis or algebraic topology, and generalized convex spaces which are generalizations of many general convexity structures. In this paper we obtain a coincidence theorem for admissible maps defined on generalized convex spaces. Our new result is applied to obtain an abstract variational inequality, a KKM type theorem, and fixed point theorems. ᮊ

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