An Effective Approach Towards the Existence of a Nonlinear Functional Integro-differential Equations

The purpose of this paper is to demonstrate the existence of a solution for a functional integro-differential equation. To achieve this, we employ noncompactness measures and utilize fundamental fixed point theorems, such as Petryshyn’s fixed theorem, in the framwork of Banach spaces. Then this paper introduces a robust approach, based on fixed point theory, for proving the existence of solutions to the integro-differential equation. Finally, we present several examples of equations to demonstrate the applicability of our results to a broad range of integral equations.