Asymptotic stability and approximate solutions to quadratic functional integral equations containing $\psi$-Riemann–Liouville fractional integral operator

This work includes the study of a quadratic functional integral equation by utilizing the generalized Riemann–Liouville fractional integral of order $\lambda>0$ with respect to an increasing and positive function $\psi$. The first aim of this study is to obtain the asymptotic stability of solutions for this generalized framework. The concepts of measure of noncompactness and fixed point theorem are used to prove this result. Moreover, the second aim of this study is to introduce a novel polynomial-based computational method to obtain approximate solutions for the considered problem. Besides, the results of the error analysis with discussions of the accuracy of solutions are presented in this article. To the best knowledge of the authors, this paper presents the first reference regarding the numerical methods for this generalized problem to obtain approximate solutions. Finally, two examples are discussed with the computational tables and convergence graphs to interpret the efficiency and applicability of the presented method.