Petrov type extension for multivalued contraction mappings

TL;DR

This paper extends contraction mappings to multivalued triangle-based perimeters, analyzing their properties and fixed point results with new concepts and counterexamples.

Contribution

It introduces multivalued $ ext{lambda}$-contracting perimeters of triangles, generalizing Nadler's contraction and exploring their fixed point properties.

Findings

Multivalued $ ext{lambda}$-contracting perimeters are not necessarily multivalued $ ext{lambda}$-contractions.

The property of forming a triangle relates to the absence of period-2 points.

New fixed point results are established using the introduced concepts.

Abstract

In this paper, we introduce the concept of multivalued $\lambda $% -contracting perimeters of triangles. This concept generalizes the Nadler's contraction by considering triplets of points instead of pairs. Fundamental properties of such mappings are analyzed, including their continuity and their relationship to classical multivalued contractions. By means of a counterexample, we show that a mapping which satisfies the condition of multivalued $\lambda $-contracting perimeters of triangles is not necessarily a multivalued $\lambda $-contraction. We also introduce the notion of property of forming a triangle. Then, we investigate the relation between this property and the fact that there is no periodic point of prime period $2$ for any multivalued mapping. Furthermore, using the new concept and property mentioned above, we present some fixed point results for multivalued mappings. Finally, we provide an illustrative and comparative example.