In this article, we apply common fixed point results in incomplete metric spaces to examine the existence of a unique common solution for the following systems of Urysohn integral equations and Volterra-Hammerstein integral equations, respectively:
u(s)=ϕi(s)+∫abKi(s,r,u(r))dr,\documentclass[12pt]{minimal}
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\begin{document}$$u(s)=\phi_{i}(s)+ \int_{a}^{b}K_{i}\bigl(s, r,u(r)\bigr) \,dr, $$\end{document} where s∈(a,b)⊆R\documentclass[12pt]{minimal}
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\begin{document}$s\in(a,b)\subseteq\mathbb{R}$\end{document}; u,ϕi∈C((a,b),Rn)\documentclass[12pt]{minimal}
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\begin{document}$u, \phi_{i}\in C((a,b),\mathbb{R}^{n})$\end{document} and Ki:(a,b)×(a,b)×Rn→Rn\documentclass[12pt]{minimal}
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\begin{document}$K_{i}:(a,b)\times(a,b)\times \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$\end{document}, i=1,2,…,6\documentclass[12pt]{minimal}
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\begin{document}$i=1,2,\ldots,6 $\end{document} and
u(s)=pi(s)+λ∫0tm(s,r)gi(r,u(r))dr+μ∫0∞n(s,r)hi(r,u(r))dr,\documentclass[12pt]{minimal}
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\begin{document}$$u(s)=p_{i}(s)+\lambda \int_{0}^{t}m(s, r)g_{i}\bigl(r,u(r) \bigr)\,dr+\mu \int_{0}^{\infty}n(s, r)h_{i}\bigl(r,u(r) \bigr)\,dr, $$\end{document} where s∈(0,∞)\documentclass[12pt]{minimal}
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\begin{document}$s\in(0,\infty)$\end{document}, λ,μ∈R\documentclass[12pt]{minimal}
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\begin{document}$\lambda,\mu\in\mathbb{R}$\end{document}, u, pi\documentclass[12pt]{minimal}
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\begin{document}$p_{i}$\end{document}, m(s,r)\documentclass[12pt]{minimal}
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\begin{document}$m(s, r)$\end{document}, n(s,r)\documentclass[12pt]{minimal}
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\begin{document}$n(s, r)$\end{document}, gi(r,u(r))\documentclass[12pt]{minimal}
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\begin{document}$g_{i}(r,u(r))$\end{document} and hi(r,u(r))\documentclass[12pt]{minimal}
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\begin{document}$h_{i}(r,u(r))$\end{document}, i=1,2,…,6\documentclass[12pt]{minimal}
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\begin{document}$i=1,2,\ldots,6$\end{document}, are real-valued measurable functions both in s and r on (0,∞)\documentclass[12pt]{minimal}
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\begin{document}$(0,\infty)$\end{document}.
