Existence Results for Impulsive Fractional Differential Inclusions with Two Different Caputo Fractional Derivatives

Dongdong Gao,Jianli Li

doi:10.1155/2019/1323176

Abstract

In this paper, we study the impulsive fractional differential inclusions with two different Caputo fractional derivatives and nonlinear integral boundary value conditions. Under certain assumptions, new criteria to guarantee the impulsive fractional impulsive fractional differential inclusion has at least one solution are established by using Bohnenblust-Karlin’s fixed point theorem. Also, some previous results will be significantly improved.

Highlights

In this paper, we consider the following fractional differential inclusions with impulsive effects: cDα0,t ( cDβ0,tu (t)) + λu (t) ∈ F (t, u (t)), a.e. t ∈ J = [0, 1], t ≠ tk, Δu = u (t+k ) − u (t−k ) = Ik (u), t = tk, k = 1, 2, . . . n, (1)au (0) + bu (1) = ∫ g (s, u) ds, [ c Dβ0,t u (t)] t=tk =ck, k = 0, 1, . . . , n, where 0 < α, β < 1, cDα0,t, and cDβ0,t represent the different Caputo fractional derivatives of orders α and β, respectively
We study the impulsive fractional differential inclusions with two different Caputo fractional derivatives and nonlinear integral boundary value conditions
As an extension of integer-order differential equations, fractional-order differential equations have been of great interest since the equations involving fractional derivatives always have better effects in applications than the traditional differential equations of integer order

Summary

Introduction

We consider the following fractional differential inclusions with impulsive effects: cDα0,t ( cDβ0,tu (t)) + λu (t) ∈ F (t, u (t)) , a.e. t ∈ J = [0, 1] , t ≠ tk, Δu (tk) = u (t+k ) − u (t−k ) = Ik (u (tk)) , t = tk, k = 1, 2, . In [12], the author investigates the following impulsive fractional differential equations with two different Caputo fractional derivatives with coefficients: cDα0,t ( cDβ0,tu (t)) + λu (t) = f (t, u (t)) , t ∈ J󸀠 = J \ {t1, . It is worth pointing out that there was no paper considering the impulsive fractional differential inclusions with two different Caputo fractional derivatives and nonlinear integral conditions by using Bohnenblust-Karlin’s fixed point theorem up to now, so our results are new.

Preliminaries

Main Results

An Example