Some Generalizations of Fixed Point Theorems of Caristi Type and Mizoguchi–Takahashi Type Under Relaxed Conditions

Abstract

In this paper, we first study the approximate fixed point property for hybrid Caristi type and Mizoguchi–Takahashi type mappings on metric spaces. We give some new generalizations of Mizoguchi–Takahashi’s fixed point theorem and Caristi’s fixed point theorem under new relaxed conditions which are quite original in the existing literature. We present new generalized Ekeland’s variational principle, generalized Takahashi’s nonconvex minimization theorem and nonconvex maximal element theorem for uniformly below sequentially lower semicontinuous from above functions and essential distances. Their equivalence relationships are also established.