Orbital Lipschitzian mappings and semigroup actions on metric spaces

TL;DR

This paper investigates fixed points of families of mappings on metric spaces using weaker orbit Lipschitzian conditions, extending classical fixed point theories to semigroup actions.

Contribution

It introduces novel fixed point results under orbit Lipschitzian conditions for semigroup mappings, broadening the scope beyond traditional Lipschitz assumptions.

Findings

Established fixed point theorems under orbit Lipschitzian conditions

Extended classical fixed point results to semigroup actions

Results are new even for single mappings

Abstract

In this paper we study some results on common fixed points of families of mappings on metric spaces by imposing orbit Lipschitzian conditions on them. These orbit Lipschitzian conditions are weaker than asking the mappings to be Lipschitzian in the traditional way. We provide new results under the two classic approaches in the theory of fixed points for uniformly Lipschitzian mappings: the one under the normal structure property of the space (which can be regarded as the Cassini-Maluta's approach) and the one after the Lifschitz characteristic of the metric space (Lifschitz's approach). Although we focus on the case of semigroup of mappings, our results are new even when a mapping is considered by itself.