Existence, uniqueness, and neural network approximation for Caputo q-fractional integro-differential equations with non-local conditions

This research explores the existence and uniqueness of solutions to fractional q -integro differential equations with non-local boundary conditions, combining both Caputo fractional q -derivatives and Riemann-Liouville fractional q -integral to model memory-dependent and discretized phenomena. Using Banach’s contraction principle and Krasnoselskii’s fixed point theorem, the study establishes sufficient conditions for well-posedness, ensuring uniqueness and the existence of solutions. Two examples are provided: one is solved analytically using the theorems, and the other is modelled on real-world memory-dependent systems. Additionally, artificial neural networks (ANNs) are employed to approximate solutions numerically, bridging theoretical analysis with computational efficiency. This dual approach both validates the theoretical framework and demonstrates its applicability across interdisciplinary fields, such as engineering and biology. It advances the study of differential equations by integrating advanced mathematics with machine learning.