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

摘要

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 ) = ϕ i ( s ) + ∫ a b K i ( s , r , u ( r ) ) d r , $$u(s)=phi_{i}(s)+ int_{a}^{b}K_{i}igl(s, r,u(r)igr) ,dr, $$ where s ∈ ( a , b ) ⊆ R $sin(a,b)subseteqmathbb{R}$ ; u , ϕ i ∈ C ( ( a , b ) , R n ) $u, phi_{i}in C((a,b),mathbb{R}^{n})$ and K i : ( a , b ) × ( a , b ) × R n → R n $K_{i}:(a,b)imes(a,b)imes mathbb{R}^{n}ightarrowmathbb{R}^{n}$ , i = 1 , 2 , … , 6 $i=1,2,ldots,6 $ and