Common fixed points for set-valued contraction on a metric space with graph

TL;DR

This paper establishes common fixed point results for set-valued and single-valued mappings on metric spaces with a graph structure, extending existing theories and applying them to convergence and differential equations.

Contribution

It introduces new fixed point theorems for set-valued contractions with graphical structure, assuming closed-valued maps, and applies these to convergence of operators and fractional differential equations.

Findings

Proves common fixed point theorems under new conditions.

Demonstrates convergence of a nonlinear Bernstein operator.

Provides criteria for solutions to fractional differential equations.

Abstract

In this article, we derive a common fixed point result for a pair of single valued and set-valued mappings on a metric space having graphical structure. In this case, the set-valued map is assumed to be closed valued instead of closed and bounded valued. Several results regarding common fixed points and fixed points follow from the main theorem of this article. By applying our theorem, we deduce the convergence of the iterates for a nonlinear $q$-analogue Bernstein operator. Furthermore, we establish sufficient criteria for the occurrence of a solution to a fractional differential equation.