Analysis of Coupled \(\mathbb{G}\)-Caputo Variable-Order Fractional Differential Equations Subject to Nonlocal Integral Boundary Conditions

Abstract

This paper investigates a system of two coupled \(\mathbb{G}\)-Caputo fractional differential equations. The derivatives are of piecewise-constant variable order, and the system is complemented by nonlocal integral boundary conditions. The \(\mathbb{G}\)-Caputo framework, constructed from a strictly increasing differentiable kernel function \(\mathbb{G}\), encompasses the classical Caputo, Hadamard–Caputo and Katugampola–Caputo operators as particular cases. To the best of our knowledge, coupled \(\mathbb{G}\)-Caputo systems with piecewise-constant variable orders under nonlocal integral boundary conditions have not been treated previously. Furthermore, the proposed framework accommodates nonlinearities that may exhibit singular behavior at the initial point via a \(\mathbb{G}^{\delta_i}\)-weighted Lipschitz condition, thereby significantly extending the scope of existing bounded-growth theories. By reducing the problem to a system of coupled constant-order integral equations on a suitable partition, we establish existence via Darbo's fixed-point theorem and the Kuratowski measure of noncompactness, uniqueness through Banach's contraction principle, and Ulam–Hyers stability with an explicit stability constant. Three illustrative examples are provided: a Hadamard fractional system with constant orders, a variable-order thermo-diffusion model in a stratified medium, and a numerical experiment that confirms the predicted geometric convergence rate.