Lyapunov Functions and Lipschitz Stability for Riemann–Liouville Non-Instantaneous Impulsive Fractional Differential Equations

Ravi Agarwal,Snezhana Hristova,Donal O’Regan

doi:10.3390/sym13040730

Abstract

In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate definition of both the initial condition and the non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability is defined and studied for the given system. Two types of derivatives of the applied Lyapunov functions among the Riemann–Liouville fractional differential equations with non-instantaneous impulses are applied. Several sufficient conditions for the defined stability are obtained. Some comparison results are obtained. Several examples illustrate the theoretical results.

Highlights

One of them is Lipschitz stability, defined and studied for ordinary differential equations in [5]. This type of stability was studied for various types of differential equations and problems, such as nonlinear differential systems [6,7,8], impulsive differential equations with delays [9], fractional differential systems [10], Caputo fractional differential equations with non-instantaneous impulses [11], a piecewise linear
Fractional differential equations with non-instantaneous impulses is defined; two types of derivatives of Lyapunov functions among the RL fractional differential equations with non-instantaneous impulses are applied; comparison results with Lyapunov functions, scalar RL fractional equations with non-instantaneous impulses and both types of derivatives of Lyapunov functions are proved; sufficient conditions for generalized Lipschitz stability in time are obtained by the application of both types of derivatives of Lyapunov functions
We will obtain some sufficient conditions for generalized Lipschitz stability in time by Lyapunov functions and their two fractional derivatives

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. One of them is Lipschitz stability, defined and studied for ordinary differential equations in [5]. In this paper we define and study Lipschitz stability for Riemann–Liouville (RL) fractional differential equations with non-instantaneous impulses. Keeping in mind the above description, in this paper we will study the initial value problem (IVP) for the following system of nonlinear RL fractional differential equations with non-instantaneous impulses (NIRLFDE) of fractional order q ∈ (0, 1): RL q t i Dt x ( t ). It is called generalized Lipschitz stability in time This type of stability is connected with the singularity of the solution at both the initial. We use Lyapunov functions and two types of derivatives of these Lyapunov functions among the studied RL fractional equation with non-instantaneous impulses. The main contributions of the paper can be summarized as follows: for a nonlinear system with RL fractional derivatives of order q ∈ (0, 1) and noninstantaneous impulses we define in an appropriate way both the initial condition and the non-instantaneous impulsive conditions; generalized Lipschitz stability in time of the zero solution of a system of nonlinear RL fractional differential equations with non-instantaneous impulses is defined; two types of derivatives of Lyapunov functions among the RL fractional differential equations with non-instantaneous impulses are applied; comparison results with Lyapunov functions, scalar RL fractional equations with non-instantaneous impulses and both types of derivatives of Lyapunov functions are proved; sufficient conditions for generalized Lipschitz stability in time are obtained by the application of both types of derivatives of Lyapunov functions

Preliminaries

Lyapunov Functions and Comparison Results

Main Results

Conclusions