Qualitative analysis of solutions to mixed-order positive linear coupled systems with bounded or unbounded delays

TL;DR

This paper develops a comprehensive qualitative theory for mixed-order positive linear coupled systems with delays, establishing existence, positivity, stability, and attractivity criteria, supported by numerical simulations.

Contribution

It introduces new conditions for positivity, stability, and attractivity of mixed-order delay systems, including a novel comparison principle and asymptotic bounds.

Findings

Established existence and uniqueness of solutions.

Derived necessary and sufficient conditions for positivity.

Proved attractivity and stability criteria for the systems.

Abstract

This paper addresses the qualitative theory of mixed-order positive linear coupled systems with bounded or unbounded delays. First, we introduce a general result on the existence and uniqueness of solutions to mixed-order linear coupled systems with time-varying delays. Next, we obtain the necessary and sufficient criteria which characterize the positivity of a mixed-order delay linear coupled system. Our main contribution is in Section 5. More precisely, by using a smoothness property of solutions to fractional differential equations and developing a new appropriated comparison principle for solutions to mixed-order delayed positive systems, we prove the attractivity of mixed-order non-homogeneous linear positive coupled systems with bounded or unbounded delays. We also establish a necessary and sufficient condition to characterize the stability of homogeneous systems. As a consequence of these results, we show the smallest asymptotic bound of solutions to mixed-order delay non-homogeneous linear positive coupled systems where disturbances are continuous and bounded. Finally, we provide numerical simulations to illustrate the proposed theoretical results.