Some applications of Caristi's fixed point theorem in metric space

TL;DR

This paper explores applications of Caristi's fixed point theorem in metric spaces, addressing conjectures, deriving known theorems, and studying bounded solutions of functional equations.

Contribution

It provides partial solutions to several conjectures, shows how many fixed point theorems follow from Caristi's theorem, and investigates bounded solutions of functional equations.

Findings

Partial answers to conjectures by Reich, Mizoguchi, Takahashi, and Amini-Harandi.

Many fixed point theorems can be derived from Caristi's theorem.

Existence of bounded solutions for certain functional equations.

Abstract

In this work, partial answers to Reich, Mizoguchi and Takahashi, and Amini-Harandi's conjectures are presented via a light version of Caristi's fixed point theorem. Moreover, we introduce that many of known fixed point theorem can easily derived from the Caristi's theorem. Finally, existence of bounded solutions of a functional equation is studied.