Analysis of Coupled G-Caputo Variable-Order Fractional Differential Equations Subject to Nonlocal Integral Boundary Conditions

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Moumen A., Boulares H., Jarad F., Abdeljawad T., Saber H., Alraqad T., ...More

International Journal of Analysis and Applications, vol.24, 2026 (ESCI, Scopus)

  • Publication Type: Article / Article
  • Volume: 24
  • Publication Date: 2026
  • Doi Number: 10.28924/2291-8639-24-2026-286
  • Journal Name: International Journal of Analysis and Applications
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus
  • Keywords: coupled differential systems, Darbo theorem, G-Caputo derivative, Kuratowski measure, non-local integral boundary conditions, Ulam–Hyers stability, variable-order fractional operators, weighted singularity
  • Open Archive Collection: AVESIS Open Access Collection
  • Istanbul Gelisim University Affiliated: Yes

Abstract

This paper investigates a system of two coupled G-Caputo fractional differential equations. The derivatives are of piecewise-constant variable order, and the system is complemented by nonlocal integral boundary conditions. The G-Caputo framework, constructed from a strictly increasing differentiable kernel function G, encompasses the classical Caputo, Hadamard–Caputo and Katugampola–Caputo operators as particular cases. To the best of our knowledge, coupled 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 Gδi-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.