Boundary value problem for a class of fractional integro-differential coupled systems with Hadamard fractional calculus and impulses

Kaihong Zhao,Yongzhi Liao,Leping Suo

doi:10.1186/s13661-019-1219-8

Kaihong Zhao, Yongzhi Liao + Show 1 more

Open Access

https://doi.org/10.1186/s13661-019-1219-8

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Abstract

This paper considers the boundary value problem for a class of fractional integro-differential coupled systems with Hadamard fractional calculus and impulses. Some sufficient conditions of the existence and uniqueness are obtained by means of the Banach contraction principle and Leray–Schauder alternative. We also give some interesting examples to illustrate the effectiveness of our main results.

Highlights

2 Preliminaries we introduce some notations and definitions of Hadamard fractional calculus and present preliminary results needed in our proofs later
The study of fractional differential equations has attracted the eyes of many scholars
Good papers involving the dynamics of the fractional differential equation are emerging in large numbers

Summary

Introduction

In [48], the authors considered the existence and finite-time stability results of Hadamard type impulsive fractional differential equations as follows:. 5. Definition 2.1 ([22]) For a ≥ 0, the left-sided Hadamard fractional integral of order α > 0 for a function u : (a, ∞) → R is defined as. Definition 2.2 ([22]) For a ≥ 0, the left-sided Riemann–Liouville type Hadamard fractional derivative of order α > 0 for a function u : (a, ∞) → R is defined by RLHDαa u(t). Lemma 2.1 ([22]) For a > 0, assume that u ∈ C(a, T) ∩ L1(a, T) with a left-sided Riemann– Liouville type Hadamard fractional derivative of order α > 0. The following properties hold: RLHDαa t ln a β–1

H Jaα t ln a β–1

H Jtαm g

Illustrative examples

Conclusions