In this paper, under some conditions in the Banach space $C([0,\beta ],\mathbb{R})$, we establish the existence and uniqueness of the solution for the nonlinear integral equations involving the Riemann-Liouville fractional operator (RLFO). To establish the requirements for the existence and uniqueness of solutions, we apply the Leray-Schauder alternative and Banach's fixed point theorem. We analyze Hyers-Ulam-Rassias (H-U-R) and Hyers-Ulam (H-U) stability for the considered integral equations involving the RLFO in the space $C([0,\beta ],\mathbb{R})$. Also, we propose an effective and efficient computational method based on Laguerre polynomials to get the approximate numerical solutions of integral equations involving the RLFO. Five examples are given to interpret the method.

This work is devoted to the analysis of Hyers, Ulam, and Rassias types of stabilities for nonlinear fractional integral equations with $n$-product operators. In some special cases, our considered integral equation is related to an integral equation which arises in the study of the spread of an infectious disease that does not induce permanent immunity. $n$-product operators are described here in the sense of Riemann-Liouville fractional integrals of order ${\sigma }_i\in (0,1]$ for $i\in \{1,2,\dots ,n\}$. Sufficient conditions are provided to ensure Hyers-Ulam, $\lambda$-semi-Hyers-Ulam, and Hyers-Ulam-Rassias stabilities in the space of continuous real-valued functions defined on the interval $[0,a]$, where $0<a<\mathrm{\infty }$. Those conditions are established by applying the concept of fixed-point arguments within the framework of the Bielecki metric and its generalizations. Two examples are discussed to illustrate the established results.