A Suzuki-type fixed point theorem for nonlinear contractions

TL;DR

This paper extends fixed point theorems for nonlinear contractions by introducing admissible functions, providing conditions under which a unique fixed point exists, and characterizing metric completeness.

Contribution

It introduces a new fixed point theorem for nonlinear contractions using admissible functions, generalizing previous results by Lim and Geraghty.

Findings

Established a fixed point theorem for admissible functions.

Characterized metric completeness via fixed point conditions.

Unified previous contraction conditions under a new framework.

Abstract

We introduce the notion of admissible functions and show that the family of L-functions introduced by Lim in [Nonlinear Anal. 46(2001), 113--120] and the family of test functions introduced by Geraghty in [Proc. Amer. Math. Soc., 40(1973), 604--608] are admissible. Then we prove that if $\phi$ is an admissible function, $(X,d)$ is a complete metric space, and $T$ is a mapping on $X$ such that, for $\alpha(s)=\phi(s)/s$, the condition $1/(1+\alpha(d(x,Tx))) d(x,Tx) < d(x,y)$ implies $d(Tx,Ty) < \phi(d(x,y))$, for all $x,y\in X$, then $T$ has a unique fixed point. We also show that our fixed point theorem characterizes the metric completeness of $X$.