Existence of fixed points for pairs of mappings and application to Urysohn 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)=ψ_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 $a\le b,t\in [a,b],x,{\psi }_i\in C([a,b],{\mathbb{R}}^n)$ and $K_i:[a,b]\times [a,b]\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ is a mapping for each $i=1,2,3,4$.