Sylvester-type fractional delay differential equations: Representation, existence and stability
Communications in Nonlinear Science and Numerical Simulation, vol.162, 2026 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 162
- Publication Date: 2026
- Doi Number: 10.1016/j.cnsns.2026.110264
- Journal Name: Communications in Nonlinear Science and Numerical Simulation
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, Compendex, INSPEC, MathSciNet, zbMATH, Academic Search Ultimate (EBSCO), Technology Collection (ProQuest)
- Keywords: Delay, Explicit formula, Fractional differential matrix equations, Representation of solutions, Ulam-Hyers stability
- Azerbaijan State University of Economics (UNEC) Affiliated: Yes
Abstract
This paper investigates Sylvester-type fractional-delay differential equations, first addressing the linear system and subsequently extending the analysis to the semilinear system. For the linear system, we establish a fundamental lemma concerning the additivity property of auxiliary matrix functions, which is central to characterizing the solution kernel and forms the basis for stability analysis. Using classical techniques, we demonstrate that the delayed Mittag–Leffler-type matrix function acts as a fundamental solution, supported by several auxiliary lemmas. A key contribution is the derivation of an explicit integral representation for the solution via a variation-of-constants formula. This representation eliminates the restrictive commutativity assumptions such as B1F(t)=F(t)B1 and B1Φ(t)=Φ(t)B1 imposed in prior literature, thus generalizing existing results to arbitrary coefficient matrices without requiring any commutativity conditions. Furthermore, we provide a stability lemma and establish sufficient conditions for the Ulam–Hyers stability of the linear system. Subsequently, we extend the results obtained for the linear system to the semilinear system by exploiting the established framework to construct the solution via the Banach fixed-point theorem. Building upon these foundations, we rigorously analyze the existence, uniqueness, and stability of the solution to the semilinear system. Finally, the theoretical findings are illustrated and validated through a numerical example.