Existence of solutions of multi-term fractional differential equations with impulse effects on a half line

Yuji Liu

doi:10.15388/na.2017.5.7

Abstract

A class of boundary value problem for impulsive fractional differential equation on a half line is proposed. Some results on existence of solutions of this kind of boundary value problem for impulsive multi-term fractional differential equation on a half line are established by constructing a weighted Banach space, a completely continuous operator and using a fixed point theorem in the Banach space. Some unsuitable lemmas in recent published papers are pointed out. An example is given to illustrate the efficiency of the main theorems.

Highlights

Fractional differential equation is a generalization of ordinary differential equation to arbitrary noninteger orders
Recent investigations have shown that many physical systems can be represented more accurately through fractional derivative formulation [10, 13]
Impulsive fractional differential equations is an important area of study [1]

Summary

Introduction

Fractional differential equation is a generalization of ordinary differential equation to arbitrary noninteger orders. Authors in papers [2, 3, 6] studied the existence of solutions of the different initial value problems for the impulsive fractional differential equations. In recent paper [8], Liu studied existence of positive solutions for the following boundary value problems (BVP) for fractional impulsive differential equations: D0α+ u(t) = −f t, u(t) , t ∈ (0, 1), t = tk, k = 1, 2, . This motivates us to establish results on solutions of impulsive fractional differential equations with order α ∈ (1, 2).

Preliminary results

An example