Fixed point results for single and multi-valued three-points contractions

TL;DR

This paper introduces new fixed point theorems for single and multi-valued three-points contractions in metric spaces with three metrics, extending previous results and providing illustrative examples.

Contribution

It develops novel fixed point theorems for three-points contractions, including multi-valued mappings, in metric spaces with three metrics, generalizing recent work.

Findings

Established fixed point theorem for single-valued three-points contractions.

Extended fixed point results to multi-valued three-points contractions.

Provided examples validating the theoretical results.

Abstract

In this paper, we are concerned with the study of the existence of fixed points for single and multi-valued three-points contractions. Namely, we first introduce a new class of single-valued mappings defined on a metric space equipped with three metrics. A fixed point theorem is established for such mappings. The obtained result recovers that established recently by the second author [J. Fixed Point Theory Appl. 25 (2023) 74] for the class of single-valued mappings contracting perimeters of triangles. We next extend our study by introducing the class of multivalued three points contractions. A fixed point theorem, which is a multi-valued version of that obtained in the above reference, is established. Some examples showing the validity of our obtained results are provided.