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

• Main Authors: Mian Bahadur Zada, Muhammad Sarwar, Stojan Radenović
• Format: Article
• Language: English
• Published: SpringerOpen 2017-01-01
• Series: Journal of Inequalities and Applications
• Subjects: common fixed point; weakly compatible maps; common ( C L R ) $(CLR)$ -property; common ( E . A ) $(E.A)$ -property; Urysohn integral equations; Volterra-Hammerstein integral equations
• Online Access: http://link.springer.com/article/10.1186/s13660-016-1286-7

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 ) = ϕ i ( s ) + ∫ a b K i ( s , r , u ( r ) ) d r , $$u(s)=\phi_{i}(s)+ \int_{a}^{b}K_{i}\bigl(s, r,u(r)\bigr) \,dr, $$ where s ∈ ( a , b ) ⊆ R $s\in(a,b)\subseteq\mathbb{R}$ ; u , 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 $K_{i}:(a,b)\times(a,b)\times \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ , i = 1 , 2 , … , 6 $i=1,2,\ldots,6 $ and u ( s ) = p i ( s ) + λ ∫ 0 t m ( s , r ) g i ( r , u ( r ) ) d r + μ ∫ 0 ∞ n ( s , r ) h i ( r , u ( r ) ) d r , $$u(s)=p_{i}(s)+\lambda \int_{0}^{t}m(s, r)g_{i}\bigl(r,u(r) \bigr)\,dr+\mu \int_{0}^{\infty}n(s, r)h_{i}\bigl(r,u(r) \bigr)\,dr, $$ where s ∈ ( 0 , ∞ ) $s\in(0,\infty)$ , λ , μ ∈ R $\lambda,\mu\in\mathbb{R}$ , u, p i $p_{i}$ , m ( s , r ) $m(s, r)$ , n ( s , r ) $n(s, r)$ , g i ( r , u ( r ) ) $g_{i}(r,u(r))$ and h i ( 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 , ∞ ) $(0,\infty)$ .