Existence of fixed points for pairs of mappings and application to Urysohn integral equations

TL;DR

This paper proves fixed point theorems for pairs of mappings in complex valued metric spaces and applies these results to establish the existence and uniqueness of solutions for a system of Urysohn integral equations.

Contribution

It introduces new fixed point results for weakly compatible mappings in complex metric spaces and applies them to solve integral equations.

Findings

Established common fixed point theorems in complex metric spaces.

Proved existence and uniqueness of solutions for the Urysohn integral system.

Extended fixed point theory to applications in integral equations.

Abstract

In this paper, we establish some common fixed point results for two pairs of weakly compatible mappings in the setting of $C$-complex valued metric space. Also, as application of the proved result, we obtain the existence and uniqueness of a common solution of the system of the Urysohn integral equations: \begin{eqnarray*} x(t)=\psi_i(t)+\int_{a}^{b}K_i(t,s,x(s))ds \end{eqnarray*} where $i=1, 2, 3, 4, a,b\in \mathbb{R}$ with $a\leq b, t\in [a,b], x, \psi_i\in C([a,b],\mathbb{R}^n)$ and $K_i:[a,b]\times [a,b]\times \mathbb{R}^n\rightarrow \mathbb{R}^n$ is a mapping for each $i=1, 2, 3, 4$.