An Existence Results of a Product Type Fractional Functional Integral Equations Using Petryshyn's Fixed Point Theorem

Abstract

In this paper, we investigate the existence of solutions for a new class of nonlinear product-type fractional functional integral equations (FFIEs) involving the Riemann-Liouville fractional integral operator. To establish the existence of at least one solution, we employ Petryshyn's fixed-point theorem (PFPT) combined with the concept of the measure of noncompactness (MNC) in the Banach space $ C[0,a] $ C[0,a] of continuous functions. Unlike other approaches based on Darbo's or Schauder's fixed-point theorems in Banach algebras, our method does not require the operator to map a closed convex subset onto itself, nor does it rely on the commonly assumed "sublinear condition" for the functional involved in the equation. Therefore, our results generalize and unify several existing results in the literature under fewer conditions. Additionally, to support our theoretical findings, we provide an example of such nonlinear FFIEs, thereby illustrating the applicability of the proposed results.