A fixed point theorem for contractive mappings that characterizes metric completeness

TL;DR

This paper presents a new fixed point theorem for contractive mappings that generalizes previous results and provides a characterization of metric completeness, advancing the theoretical understanding of metric spaces.

Contribution

It introduces a generalized fixed point theorem for contractive mappings that characterizes metric completeness, extending prior theorems by Suzuki and Geraghty.

Findings

The theorem characterizes metric completeness through fixed points.

It generalizes previous fixed point theorems.

Provides a new tool for analyzing metric spaces.

Abstract

Inspired by the work of Suzuki in [Proc. Amer. Math. Soc. 136 (2008), 1861--1869] we prove a fixed point theorem for contractive mappings that generalizes a theorem of Geraghty in [Proc. Amer. Math. Soc., 40 (1973), 604--608] and characterizes metric completeness.