New generalizations of Ekeland's variational principle and well-known fixed point theorems with applications to nonconvex optimization problems

Wei-Shih Du

Department of Mathematics

National Kaohsiung Normal University

wsdu@mail.nknu.edu.tw

In this talk, we establish new generalizations of Ekeland's variational principle, Caristi's fixed point theorem, Takahashi's nonconvex minimization theorem and nonconvex maximal element theorem for uniformly below sequentially lower semicontinuous from above functions and essential distances. New simultaneous generalizations of fixed point theorems of Mizoguchi-Takahashi type, Nadler type, Banach type, Kannan type, Chatterjea type and others are also presented. As applications, we concentrate on studying nonconvex optimization and minimax theorems in metric spaces.

Keyword: Nonconvex optimization, minimax theorem, Ekeland's variational principle, Caristi's (common) fixed point theorem, Takahashi's nonconvex minimization theorem, nonconvex maximal element theorem,  \mathcal{MT} -function (or  \mathcal{R} -function),  \mathcal{MT}(\lambda) -function, uniformly below sequentially lower semicontinuous from above, essential distance, Mizoguchi-Takahashi's fixed point theorem, Nadler's fixed point theorem, Banach contraction principle, Kannan's fixed point theorem, Chatterjea's fixed point theorem.