A cone-theoretic comparison principle for Caputo fractional differential equations

Abstract

This paper presents a comprehensive cone-theoretic comparison principle for Caputo fractional differential equations, thereby addressing a key gap in the qualitative theory of fractional systems. Although classical comparison techniques based on scalar and vector Lyapunov functions have been extended to fractional settings, the more general and powerful framework of cone-valued Lyapunov functions has received little attention. In this work, we develop a complete theoretical foundation for this extension. We establish core results on cone-preserving fractional differential inequalities, prove the existence and characterization of maximal solutions with respect to arbitrary closed convex cones, and derive a general comparison principle for Caputo fractional systems. The central result is a unified comparison theorem that enables stability analysis of fractional systems using cone-valued Lyapunov functions. This approach incorporates both scalar and vector Lyapunov methods as special cases, and can be naturally extended to systems whose dynamics follow non-standard partial orderings. In all, our results provide a robust theoretical basis for studying complex fractional-order systems that lie beyond the reach of traditional comparison techniques.