Existence of solution for non-linear functional integral equations of two variables in banach algebra

Abstract

The aim of this article is to establish the existence of the solution of non-linear functional integral equations x(l, h) = (U(l, h, x(l, h)) + F (l, h, ∫0l ∫0h P(l, h, r, u, x(r, u))drdu, x(l, h)) × G (l, h, ∫0a ∫0a Q(l, h, r, u, x(r, u)) drdu, x(l, h)) of two variables, which is of the form of two operators in the setting of Banach algebra C ([0, a] × [0, a]), a > 0. Our methodology relies upon the measure of noncompactness related to the fixed point hypothesis. We have used the measure of noncompactness on C ([0, a] × [0, a]) and a fixed point theorem, which is a generalization of Darbo's fixed point theorem for the product of operators. We additionally illustrate our outcome with the help of an interesting example.