Exploring existence and uniqueness in non-local Caputo q-fractional integro-differential equations

This study delves into a novel mathematical equation that integrates fractional q-derivatives with integral operations, specifically under the constraints of q-nonlocal conditions. Unlike previous research, the unique properties and potential applications of this equation in various real-world contexts have not been thoroughly explored, first, to establish the existence of solutions to this equation, and second, to demonstrate the uniqueness of such solutions. To achieve these aims, we employ the Schauder fixed point theorem, a fundamental tool in functional analysis. This theorem is vital, as it provides a rigorous framework for proving the existence of fixed points, which correspond to solutions of our equation. Furthermore, we will investigate the continuous dependence of solutions on initial conditions and parameters. This analysis is crucial, as it sheds light on how small changes in the input can affect the behavior of the solutions, thereby enhancing our understanding of the equation’s stability and robustness. To illustrate the practical implications of our theoretical findings, we present two concrete examples using neural networks, which demonstrated their ability to approximate nonlinear functions through systematic training. Rigorous application of existence and uniqueness theorem conditions ensures the theoretical soundness of the solutions obtained. By integrating these mathematical principles with practical computational techniques, the research establishes a solid link between theoretical foundations and applied numerical analysis, thereby reinforcing the reliability and effectiveness of the proposed method.