Common fixed point theorems for Hybrid generalized $(F,\varphi)$-contractions under common limit range property with applications

TL;DR

This paper establishes new common fixed point theorems for hybrid generalized contractions involving pairs of mappings under the common limit range property, with applications to functional equations and integral inclusions.

Contribution

It introduces generalized $(F,)$-contraction conditions for hybrid pairs of mappings and proves fixed point theorems that extend existing results in metric spaces.

Findings

Proved fixed point theorems for hybrid $(F,)$-contractions.

Applied results to solve functional equations in dynamic programming.

Provided an example illustrating the theoretical findings.

Abstract

We consider a relatively new hybrid generalized F-contraction involving a pair of mappings and utilize the same to prove a common fixed point theorem for a hybrid pair of occasionally coincidentally idempotent mappings satisfying generalized $(F,\varphi)$-contraction condition under common limit range property in complete metric spaces. A similar result involving a hybrid pair of mappings satisfying a Rational type Hardy-Rogers $(F,\varphi)$-contractive condition is also proved. Our results generalize and improve several results of the existing literature. As applications of our results, we prove two theorems for the existence of solutions of certain system of functional equations arising in dynamic programming, and Volterra integral inclusion besides providing an illustrative example.