Some generalizations of Darbo's theorem for solving a systems of functional-integral equations via measure of noncompactness

In this paper, using the concept of measure of noncompactness, which is a very useful and powerful tools in nonlinear functional analysis, metric fixed point theory and integral equations, we introduce a new contraction on a Banach space. For this purpose by using of a measure of noncompactness on a finite product space, we obtain some generalizations of Darbo’s fixed-point theorem. Then, with the obtained results, we present some theorems on the existence of coupled fixed point for a class of operators in a Banach space. Our results generalize and extend a lot of comparable results in the literature. Also as an application, we study the existence of solution for a class of the system of nonlinear functional integral equations, which the functions and operators in the related integral operators, satisfies in a particular contraction. Finllay a concrete example is also included, which demonstrates the applicability of the obtaind results.