Existence of solutions for some classes of integro-differential equations via measure of noncompactness

In this present paper, we introduce a new measure of noncompactness on the space consisting of all real functions which are $n$ times bounded and continuously differentiable on $\mathbb{R}_+$. As an application, we investigate the problem of the existence of solutions for some classes of the functional integral-differential equations which enables us to study the existence of solutions of nonlinear integro-differential equations. In our considerations we apply the technique of measures of noncompactness in conjunction with Darbo's fixed point theorem. Finally, we give some illustrative examples to verify the effectiveness and applicability of our results