Abstract
In this paper, by applying the abstract maximal element principle of Lin and Du, we present some new existence theorems related with critical point theorem, maximal element theorem, generalized Ekeland’s variational principle and common (fuzzy) fixed point theorem for essential distances.
Highlights
Maximal element principle (MEP, for short) is a fascinating theory that has a wide range of applications in many fields of mathematics
Lin and Du [3,4,7] introduced the concepts of the sizing-up function and μ-bounded quasi-ordered set to define sufficient conditions for a nondecreasing sequence on a quasi-ordered set to have an upper bound and used them to establish an abstract MEP
We present some new existence theorems related with critical point theorem, generalized Ekeland’s variational principle, maximal element principle, and common fixed point theorem for essential distances by applying Theorem 1
Summary
Maximal element principle (MEP, for short) is a fascinating theory that has a wide range of applications in many fields of mathematics. (Takahashi’s nonconvex minimization theorem) Let ( M, d) be a complete metric space and f : M → It is well known that Caristi’s fixed point theorem, Takahashi’s nonconvex minimization theorem and Ekeland’s variational principle are logically equivalent; for detail, one can refer to [3,6,7,8,13,14,15,16,17,18,19,20,21,22,23,24]. By using Theorem 1, Du proved several versions of generalized Ekeland’s variational principle and maximal element principle and established their equivalent formulations in complete metric spaces, for detail, see [3,4]. We present some new existence theorems related with critical point theorem, generalized Ekeland’s variational principle, maximal element principle, and common (fuzzy) fixed point theorem for essential distances by applying Theorem 1
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