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

Journal of Inequalities and Applications, Jan 2017

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 , where s ∈ ( a , b ) ⊆ R ; u , ϕ i ∈ C ( ( a , b ) , R n ) and K i : ( a , b ) × ( a , b ) × R n → R n , i = 1 , 2 , … , 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 , where s ∈ ( 0 , ∞ ) , λ , μ ∈ R , u, p i , m ( s , r ) , n ( s , r ) , g i ( r , u ( r ) ) and h i ( r , u ( r ) ) , i = 1 , 2 , … , 6 , are real-valued measurable functions both in s and r on ( 0 , ∞ ) . MSC: 47H10, 54H25.

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Mian Bahadur Zada, Muhammad Sarwar, Stojan Radenović. Existence of unique common solution to the system of non-linear integral equations via fixed point results in incomplete metric spaces, Journal of Inequalities and Applications, 2017, 22, DOI: 10.1186/s13660-016-1286-7