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

Abstract

In the present manuscript, we establish some results on common fixed point for two weakly compatible pairs of 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 conditions $a\leq b, x, \psi_i\in C([a,b],\mathbb{R}^n), t\in [a,b]$ and $K_i:[a,b]\times [a,b]\times \mathbb{R}^n\rightarrow \mathbb{R}^n$ is a mapping for each $i=1, 2, 3, 4$.