Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses

Danfeng Luo,Zhiguo Luo

doi:10.1186/s13662-019-2101-9

Danfeng Luo, Zhiguo Luo

Open Access

https://doi.org/10.1186/s13662-019-2101-9

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Journal: Advances in Difference Equations	Publication Date: Apr 25, 2019
Citations: 13	License type: open-access

Affiliation: Hunan Normal University

Abstract

In this paper, we mainly consider the existence and finite-time stability of solutions for a kind of ψ-Hilfer fractional differential equations involving time-varying delays and non-instantaneous impulses. By Schauder’s fixed point theorem, the contraction mapping principle and the Lagrange mean-value theorem, we present new constructive results as regards existence and uniqueness of solutions. In addition, under some new criteria and by applying the generalized Gronwall inequality, we deduce that the solutions of the addressed equation have finite-time stability. Some results in the literature can be generalized and improved. As an application, three typical examples are delineated to demonstrate the effectiveness of our theoretical results.

Highlights

Fractional differential equations play an important role in many fields, especially in biological medicine, dynamics mechanic, population dynamics and communication engineering
We are concerned with the following ψ-Hilfer fractional differential equation with time-varying delays and non-instantaneous impulses:
Compared with some recent results in the literature, such as [6, 8,9,10,11,12,13] and some others, the chief contributions of our study contain at least the following four issues: (1) In [6, 8,9,10], authors discussed several types of stability except the finite-time stability, and we first introduce the definition of finite-time stability into the ψ-Hilfer fractional differential equation with non-instantaneous impulses

Summary

Introduction

Fractional differential equations play an important role in many fields, especially in biological medicine, dynamics mechanic, population dynamics and communication engineering. In [25], the authors presented finite-time stability results of nonlinear fractional delay differential equations under mild conditions on the nonlinear term. We are concerned with the following ψ-Hilfer fractional differential equation with time-varying delays and non-instantaneous impulses:

Results

Conclusion