LAUSR

AbstractIn 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,$$u(s)=\phi_{i}(s)+ \int_{a}^{b}K_{i}\bigl(s, r,u(r)\bigr) \,dr, $$ where s∈(a,b)⊆R$s\in(a,b)\subseteq\mathbb{R}$; u,ϕi∈C((a,b),Rn)$u, \phi_{i}\in C((a,b),\mathbb{R}^{n})$ and Ki:(a,b)×(a,b)×Rn→Rn$K_{i}:(a,b)\times(a,b)\times \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, i=1,2,…,6$i=1,2,\ldots,6 $ and u(s)=pi(s)+λ∫0tm(s,r)gi(r,u(r))dr+μ∫0∞n(s,r)hi(r,u(r))dr,$$u(s)=p_{i}(s)+\lambda \int_{0}^{t}m(s, r)g_{i}\bigl(r,u(r) \bigr)\,dr+\mu \int_{0}^{\infty}n(s, r)h_{i}\bigl(r,u(r) \bigr)\,dr, $$ where s∈(0,∞)$s\in(0,\infty)$, λ,μ∈R$\lambda,\mu\in\mathbb{R}$, u, pi$p_{i}$, m(s,r)$m(s, r)$, n(s,r)$n(s, r)$, gi(r,u(r))$g_{i}(r,u(r))$ and hi(r,u(r))$h_{i}(r,u(r))$, i=1,2,…,6$i=1,2,\ldots,6$, are real-valued measurable functions both in s and r on (0,∞)$(0,\infty)$.