Some Existence Results for High Order Fractional Impulsive Differential Equation on Infinite Interval

Zhaocai Hao,Tian Wang,Chuanjun Chen

doi:10.1155/2020/6018273

Abstract

In this paper, we consider the high order impulsive differential equation on infinite interval D 0 + α u t + f t , u t , J 0 + β u t , D 0 + α − 1 u t = 0 , t ∈ 0 , ∞ ∖ t k k = 1 m △ u t k = I k u t k , t = t k , k = 1 , … , m u 0 = u ′ 0 = ⋯ = u n − 2 0 = 0 , D 0 + α − 1 u ∞ = u 0 By applying Schauder fixed points and Altman fixed points, we obtain some new results on the existence of solutions. The nonlinear term of the equation contains fractional integral operator J β u t and lower order derivative operator D 0 + α − 1 u t . An example is presented to illustrate our results.

Highlights

We are concerned with the following impulsive differential equation on infinite interval:
In [13], Liu investigated the existence of solutions for higher order impulsive fractional differential equations given by
Motivated by the aforementioned work, we studied existence of solution of problem (1) by Schauder’s fixedpoint theorem and Altman’s fixed-point theorem. e main features of this paper are as follows

Summary

Introduction

We are concerned with the following impulsive differential equation on infinite interval:. Wang et al studied the existence and multiplicity of solutions for impulsive fractional boundary value problem with p-Laplacian in [4], and Liu considered fractional impulsive differential equations using bifurcation techniques in [5]. In [13], Liu investigated the existence of solutions for higher order impulsive fractional differential equations given by. To the best of our knowledge, there are few articles involving the impulsive fractional order differential equations on the infinite interval. If the nonlinear term contained fractional integral and t ∈ [0, ∞), it will bring new obstacles to solve the problem.

Preliminaries and Lemmas

Main Results

Example