Hybrid Fractional Coupled Systems with Generalized Caputo Derivatives


== Abstract ==

<p>This paper establishes a comprehensive theoretical framework for a novel coupled system of nonlinear hybrid fractional differential equations involving generalized Caputo derivatives. The system&rsquo;s hybrid nature, coupled with the generality of the fractional operators, allows for modeling complex interdependent processes with memory effects that cannot be adequately captured by existing models.<br>
Using Krasnoselskii&rsquo;s fixed point theorem, we prove the existence of at least one solution under explicit coupling conditions. Under appropriate Lipschitz conditions, we establish uniqueness via Banach&rsquo;s contraction principle, deriving a quantitative condition involving the fractional orders, generalization parameters, and Lipschitz constants. We also conduct a rigorous analysis of Ulam-Hyers and Ulam-Hyers-Rassias stability, obtaining explicit stability constants that provide quantitative bounds on how perturbations in the input affect the solution.<br>
The theoretical results are validated through three carefully constructed numerical examples with explicit parameter values demonstrating existence, uniqueness, and Ulam-Hyers stability. A parameter sensitivity analysis confirms the robustness of the uniqueness condition across variations in fractional orders, generalization parameters, and interval lengths. The paper concludes with a discussion of limitations and directions for future research, including extensions to higher fractional orders, delay and impulsive effects, and numerical methods.</p>



== Document ==
<pdf>Media:Draft_content_562676460-3683-document.pdf</pdf>