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)=\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 $i=1, 2, 3, 4$.