On the De Blasi Measure of Noncompactness and Solvability of a Delay Quadratic Functional Integro-Differential Equation

Abstract

Quadratic integro-differential equations have been discussed in many works, for instance. Some analytic results on the existence and the uniqueness of problem solutions to quadratic integro-differential equations have been investigated in different classes. Various techniques have been applied such as measure of noncompactness, Schauder’s fixed point theorem and Banach contraction mapping. Here, we shall investigate quadratic functional integro-differential equations with delay. To prove the existence of solutions of the quadratic integro-differential equations, we use the technique of De Blasi measure of noncompactness. Moreover, we study some uniqueness results and continuous dependence of the solution on the initial condition and on the delay function. Some examples are presented to verify our results.

Keywords

• quadratic integro-differential equation
• measure of noncompactness
• existence of monotonic integrable solutions