Analyzing Single and Multi-valued Nonlinear Caputo Two-Term Fractional Differential Equation With Integral Boundary Conditions

This article primarily focuses on the single-valued and multi-valued cases of the class of nonlinear Caputo two-term fractional differential equation with three-point integral boundary conditions. In the single-valued case, we employ Schaefer’s fixed point theorem and the Banach fixed point theorem to establish results regarding the existence and uniqueness of solutions, using linear growth and Lipschitz conditions. Furthermore, we delve into the stability analysis of the single-valued problem using Ulam–Hyers and Ulam–Hyers–Rassias stabilities. In addition to the above, we address the multi-valued scenario and provide results on the existence of solutions. This is achieved by employing the Covitz–Nadler FPT and the nonlinear alternative for contractive maps. As an application of our fundamental findings, we present illustrative examples that validate our results. These examples have been implemented using MATLAB.