A New Solution to the Rhoades' Open Problem with an Application

TL;DR

This paper introduces a novel approach using S-metrics to solve Rhoades' open problem on discontinuity at fixed points and applies this method to the fixed-circle problem, advancing fixed point theory.

Contribution

It provides a new solution to Rhoades' open problem and the fixed-circle problem using S-metrics and Zamfirescu-like mappings, offering fresh insights into fixed point analysis.

Findings

Solved Rhoades' open problem on discontinuity at fixed points.

Proposed a new solution to the fixed-circle problem.

Introduced the use of S-metrics in fixed point theory.

Abstract

We give a new solution to the Rhoades' open problem on the discontinuity at fixed point via the notion of an $S$-metric. To do this, we inspire with the notion of a Zamfirescu mapping. Also, we consider a recent problem called the "fixed-circle problem" and propose a new solution to this problem as an application of our technique.