The non-uniqueness of solution for initial value problem of impulsive differential equations involving higher order Katugampola fractional derivative

Xian-Min Zhang

doi:10.1186/s13662-020-2536-z

Abstract

In this paper we consider the initial value problem for some impulsive differential equations with higher order Katugampola fractional derivative (fractional order q in (1,2]). The systems of impulsive higher order fractional differential equations can involve one or two kinds of impulses, and by analyzing the error between the approximate solution and exact solution it is found that these impulsive systems are equivalent to some integral equations with one or two undetermined constants correspondingly, which uncover the non-uniqueness of solution to these impulsive systems. Some numerical examples are offered to explain the obtained results.

Highlights

1 Introduction Fractional calculus serves as an important tool to characterize hereditary properties in many fields of science and engineering
The potential application in quantum mechanics was considered for some properties of the Katugampola fractional derivative in [34], and the existence and uniqueness of solutions was studied for fractional Langevin equation with the nonlocal Katugampola fractional integral conditions in [35]
Impulsive differential equations are used in description of some processes with impulsive effects [36], and the subject of impulsive fractional differential equations (IFrDE) has been getting an enormous amount of attention recently [37,38,39,40,41,42,43,44,45]

Summary

Introduction

Fractional calculus serves as an important tool to characterize hereditary properties in many fields of science and engineering (such as chaotic behavior, epidemiology, thermal science, hydrology, and biology [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]). By (4.5) the exact solution of (1.2) as t ∈ (t1, t2] satisfies lim x(t) = y(t0, t) for t ∈ Is the Hadamard fractional derivative, we κ(J1(x(t1–))) = ξ J1(x(t1–)) (here ξ is an arbitrary constant) by applying Lemma 3.3 in [44] to (1.2) and (4.12) with ρ → 0+. Proof First, we prove the necessity that the solution of (1.1) satisfies (4.25) by the mathematical induction. By Lemmas 4.1 and 4.3, the exact solution x(t) of (1.1) as t ∈ If x(·) ∈ IC([t0, T], R), x(t) is a solution of (1.4) if, and only if, x(t) satisfies the following integral equation:.

Examples

Conclusion