Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces |
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Authors: | Lai-Jiu Lin Wei-Shih Du |
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Institution: | Department of Mathematics, National Changhua University of Education, Changhua, 50058, Taiwan |
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Abstract: | In this paper, 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. |
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Keywords: | τ-Function Generalized Ekeland's variational principle Lower semicontinuous from above function Generalized Caristi's (common) fixed point theorem Nonconvex minimax theorem Nonconvex equilibrium theorem Generalized flower petal theorem |
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