Boundedness Analysis for a Coupled $\phi$-Caputo Fractional System Using Measures of Noncompactness

Boundedness Analysis of $\phi$-Caputo Fractional Systems

Abstract

This paper investigates a system of two coupled nonlinear fractional differential equations with $\phi$-Caputo derivatives of orders
$q_1, q_2 \in (1,2)$. The system includes proportional delays and nonlocal initial conditions to capture memory and hereditary effects. Solutions are studied in a Banach space using Kuratowski-type measures of noncompactness combined with Sadovskii’s fixed point theorem. Existence and boundedness of solutions are established under suitable continuity, growth, and Lipschitz conditions. An integral reformulation and noncompactness techniques address challenges from nonlinearity and coupling. A numerical example illustrates the results, confirming the framework’s applicability. Future work will consider stability, bifurcation, and extensions to multivariate or spatially distributed models.