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Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces. (English) Zbl 1101.49022
Summary: We introduce the concept of τ-function which generalizes the concept of w-distance studied in the literature. We establish a generalized Ekeland’s variational principle in the setting of lower semicontinuous from above and τ-functions. As applications of our Ekeland’s variational principle, we derive generalized Caristi’s (common) fixed point theorems, a generalized Takahashi’s nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces. We also prove that these theorems also imply our Ekeland’s variational principle.

49K27Optimal control problems in abstract spaces (optimality conditions)
49K35Minimax problems (optimality conditions)