Common solution of Urysohn integral equations with the help of common fixed point results in complex valued metric spaces

Abstract

The aim of this manuscript is to discuss the existence and uniqueness of common solution for the following system of Urysohn integral equations:

$$\begin{aligned} z(t)=\phi _{i}(t)+\int _{a}^{b}K_{i}(t,s,z(s)) ds, \end{aligned}$$

(0.1)

where $i=1,2,3,4$, $a,b\in \mathbb {R}$ with $a\le b$, $t\in [a,b]$, $z, \phi _{i} \in C([a,b],\mathbb {R}^n)$ and $K_{i}:[a,b]\times [a,b]\times \mathbb {R}^n\rightarrow \mathbb {R}^n$ is a given mapping for each $i=1,2,3,4$. For this intention we establish common fixed point results for two pairs of weakly compatible mappings satisfying the contractive condition of rational type in the frame work of complex valued metric spaces.