Symmetry analysis has been applied to solve many differential equations, although determining the symmetries can be computationally intensive compared to other solution methods. In this work, we study some operators which keep the set of solutions invariant. We discuss the existence of solutions for two initial value problems of a delay quadratic functional integro-differential equation of arbitrary (fractional) orders and its corresponding integer orders equation. The existence of the maximal and the minimal solutions is proved. The sufficient condition for the uniqueness of the solutions is given. The continuous dependence of the unique solution on some data is studied. The continuation of the arbitrary (fractional) orders problem to the integer order problem is investigated.

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.