Title: The appropriate expression and non-uniqueness of solution for impulsive Katugampola fractional order system

Authors: Xianmin Zhang; Zuohua Liu; Yali He; Zuming Peng; Shixian Yang

Addresses: School of Mathematics and Statistics, Yangtze Normal University, Chongqing, 408100, China ' School of Chemistry and Chemical Engineering, Chongqing University, Chongqing, 400044, China ' School of Mathematics and Statistics, Yangtze Normal University, Chongqing, 408100, China ' School of Mathematics and Statistics, Yangtze Normal University, Chongqing, 408100, China ' School of Mathematics and Statistics, Yangtze Normal University, Chongqing, 408100, China

Abstract: For two impulsive generalised fractional order systems (IGFrOSs), we consider their conditions of fractional derivative and fractional integral by two new fractional order properties of piecewise function to find that the equivalent integral equations (EIEs) of the IGFrOSs are a combination of two integral equations (ɸ(t) and Φ_j(t)) with an arbitrary constant to reveal the non-uniqueness of the IGFrOSs' solution, and moreover, we give the appropriate expressions of the EIEs to easily verify that the EIEs satisfy the conditions of fractional derivative and fractional integral in the two IGFrOSs. Finally, we apply two numerical models to illustrate the EIEs and the non-uniqueness of solution of two IGFrOSs.

Keywords: impulsive fractional differential equations; equivalent integral equations; EIEs; initial value problems; non-uniqueness of solution.

DOI: 10.1504/IJDSDE.2024.144883

International Journal of Dynamical Systems and Differential Equations, 2024 Vol.13 No.4, pp.369 - 392

Received: 03 May 2023
Accepted: 25 Apr 2024
Published online: 06 Mar 2025 *