New operators through measure of non-compactness

In this article, we use two concepts, measure of non-compactness and Meir-Keeler condensing operators. The measure of non-compactness has been applied for existence of solution nonlinear integral equations, ordinary differential equations and system of differential equations in the case of finite and infinite dimensions by some authors. Also Meir-Keeler condensing operators are shown in some papers such as [7, 11]. By the above concepts we can generalize some theorems which was proved by other authors, especially Darbo’s fixed point theorem. The space of our solution contains of all convergence sequences with a finite limit, such that with suitable norm is a Banach space. To reach our subject, we prove some theorems by using measure of non-compactness and Meir-Keeler condensing operators to generalize some theorems which are created by some authors. For validity and application of our proposed theorems, we prove existence of solution for infinite system of second order differential equations with boundary conditions.