Existence of unique common solution to the system of non-linear integral equations via fixed point results in incomplete metric spaces

Abstract

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, where s∈ (a, b) ⊆ R; u, ϕi∈ C((a, b) , Rn) and Ki: (a, b) × (a, b) × Rn→ Rn, i= 1 , 2 , … , 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, where s∈ (0 , ∞) , λ, μ∈ R, u, pi, m(s, r) , n(s, r) , gi(r, u(r)) and hi(r, u(r)) , i= 1 , 2 , … , 6 , are real-valued measurable functions both in s and r on (0 , ∞).