Fixed Point Theory and the Liouville–Caputo Integro-Differential FBVP with Multiple Nonlinear Terms

Abstract

This work is reserved for the study of a special category of boundary value problems ( $\mathbb{BVP}$ s) consisting of Liouville–Caputo integro-differential equations with multiple nonlinear terms. This fractional model and its boundary value conditions (BVCs) involve different simple $\mathbb{BVP}$ s, in which the second BVC as a linear combination of two Caputo derivatives of the unknown function equals a nonzero constant. The Banach principle gives a unique solution for this Liouville–Caputo $\mathbb{BVP}$ . Further, the Krasnoselskii and Leray–Schauder criteria give the existence property regarding solutions of the mentioned problem. For each theorem, we provide an example based on the required hypotheses and derive numerical data in the framework of tables and figures to show the consistency of results from different points of view.