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

Authors: Dongdong Gao; Jianli Li
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## 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.


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## Body

## 1. Introduction

In this paper, we consider the following fractional differential inclusions with impulsive effects:(1)cD0,tαcD0,tβut+λut∈Ft,ut,a.e.t∈J=0,1,t≠tk,Δutk=utk+-utk-=Ikutk,t=tk,k=1,2,…n,au0+bu1=∫01gs,uds,cD0,tβutt=tk=ck,k=0,1,…,n,where 0<α,β<1, cD0,tα, and cD0,tβ represent the different Caputo fractional derivatives of orders α and β, respectively. F:J×R→P(R) is a multivalued map, P(R) is the family of all nonempty subsets of R, and g:J×R→R is a given continuous function. 0=t0<t1<⋯<tn+1=1, a>0,b≥0,0≤ck≤c,k=0,1,…,n are real constants and λ is a given positive parameter. u(tk+)=limh→0+u(tk+h) and u(tk-)=limh→0-u(tk+h) represent the right and left limits of u(t) at t=tk,k=1,2,…n.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. Due to these significant applications in various sciences, such as physics, engineering, chemistry, and biology, fractional differential equations have received much attention by researchers during the past two decades. Up to now, fractional boundary value problems are still heated research topics. That is why, more and more considerations by many people have been paid to study the existence of solutions for fractional boundary value problems; we refer readers to [1–12].However, the articles of fractional boundary value problems with two different Caputo fractional derivatives are not many. More precisely, in [10], the authors have studied the following impulsive fractional Langevin equations with two different Caputo fractional derivatives:(2)cDtβcDtα+λxt=ft,xt,t∈J′=J∖t1,…,tm,J≔0,1,Δutk≔utk+-utk-=Ik,Ik∈R,x0=0,xηk=0,x1=0,ηk∈tk,tk+1,k=1,2,…m-1,where f:J×R→R is a given function, 0<α,β<1 and 0<α+β<1, 0=t0<t1<⋯<tm+1=1, λ>0, u(tk+)=limh→0+u(tk+h), and u(tk-)=limh→0-u(tk+h) represent the right and left limits of u(t) at t=tk,k=1,2,…m.Then, in [11], the authors considered the following nonlinear Langevin inclusions with two different Caputo fractional derivatives:(3)cDpcDq+λxt∈Ft,xt,0<t<1,x0=∑i=1nβiIμixζ,x1=∑i=1nαiIνixη,0<ζ<η<1,where 0<p,q<1, λ is a real number, Ik is the Riemann-Liouville fractional integral of order k>0(k=νi,μi;i=1,2,…,n), and α,β are constants.In [12], the author investigates the following impulsive fractional differential equations with two different Caputo fractional derivatives with coefficients:(4)cD0,tαcD0,tβut+λut=ft,ut,t∈J′=J∖t1,…,tm,Δutk=utk+-utk-=yk,k=1,2,…m,au0+bu1=c,cD0,tβutt=tk=dk,k=0,1,2,…,m,where J=[0,1], f∈C(J×R,R),0<α,β<1,yk∈R,λ>0, a>0,b≥0,c≥0,dk≥0 are real constants.To the best of our knowledge, integral boundary conditions appear in population dynamics and cellular systems; it has constituted a very interesting and important class of problems. However, fractional boundary value problems with integral boundary conditions have not received so much attention as periodic boundary conditions, so the main aim in this paper is intended as an attempt to establish some criteria of existence of solutions for (1). 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. Also, we improve some previous results.The arrangement of the rest paper is as follows. In Section2, some preliminaries and results which are applied in the later paper are presented. In Section 3, the main proof of theorems will be vividly shown. In Section 4, a corresponding example is given to illustrate the obtained results in Section 3.


## 2. Preliminaries

In this section, we recall some basic knowledge of definitions and lemmas that we shall use in the rest of the paper.LetC(J,R) denote a Banach space of continuous functions from J into R with the norm(5)u=supt∈J⁡utfor u∈C(J,R). Also, we denote the function space by(6)PCJ,R=u:u∈Ctk,tk+1,Rutk+=utkk=1,…,mwith the norm uPC=supt∈Jut. Clearly, PC(J,R) is Banach spaces.LetL1(J,R) be a Banach space of measurable functions y:J→R which are Lebesgue integrable and normed by(7)yL1=∫01ytdt.Let(X,·) be a Banach space. We give following notations for convenience: let(8)PclX=Y∈PX:Yisclosed,PbX=Y∈PX:Yisbounded,PcpX=Y∈PX:Yiscompact,Pcp,cX=Y∈PX:Yiscompactandconvex,and BCC(X) denote the set of all nonempty bounded, closed, and convex subset of X.A multivalued mapG:X→2X(i) is convex (closed) valued ifG(x) is convex (closed) for all x∈X;(ii) is bounded on bounded sets ifG(B)=⋃x∈BG(x) is bounded in X for any bounded set B of X(i.e.supx∈B⁡sup⁡y:y∈Gx<∞);(iii) is called upper semicontinuous (u.s.c.) onX if, for each x0∈X, the set G(x0) is nonempty closed subset of X, and if, for each open set N of X containing G(x0), there exists an open neighborhood N0 of x0 such that G(N0)⊆N;(iv) is said to be completely continuous ifG(B) is relatively compact for every bounded subset B of X;(v) is completely continuous with nonempty compact values; thenG is u.s.c. if and only if G has a closed graph; i.e., xn→x∗,yn→y∗,yn∈G(xn) imply y∗∈G(x∗).(vi) has a fixed point if there isx∈X such that x∈G(x).Definition 1.
A multivalued mapF:J×R→P(R) isCarathéodory if(i)t↦F(t,u) is measurable for each u∈R,(ii)u↦F(t,u) is upper semicontinuous for almost all t∈J.Moreover, aCarathéodory function F is called L1-Carathéodory if(iii) for eachα>0, there exists φα∈L1([0,1],R+) such that (9)Ft,x=sup⁡v:v∈Ft,x≤φαtfor all x≤α for a.e.t∈[0,1].For eachy∈C(J,R), define that the set of selections for F by (10)SF,y=v∈L1J,R:vt∈Ft,yta.e.t∈Jis nonempty.Lemma 2 (see [13]).
LetX be a Banach space. Let F:J×R→Pcp,c(X) be an L1-Carathéodory multivalued map, and let Θ be a linear continuous mapping from L1(J,X) to C(J,X). Then the operator (11)Θ∘SF:CJ,X→Pcp,cXCJ,Xand (12)x↦Θ∘SFx=ΘSF,x,yis a closed graph operator in C(J,X)×C(J,X).For more details, please refer to [13–15].Definition 3.
A functionu(t)∈PC(J,R) is called a solution of (1) if there exists a function f∈L1(J,R) with f(t)∈F(t,u(t)),a.e.t∈J such that cD0,tα(cD0,tβu(t))+λu(t)=f(t,u(t)),a.e.t∈J, Δu(tk)=u(tk+)-u(tk-)=Ik(u(tk)),t=tk,k=1,2,…n, and au0+bu1=∫01gs,usds,[cD0,tβu(t)]t=tk=ck,k=0,1,…,n.Next, we present the following necessary basic knowledge of fractional calculus theory which is used in the later paper.Definition 4 (see [4]).
The Riemann-Liouville fractional integral of orderα>0 of a function f:[0,+∞)→R is given by (13)Itαft=1Γα∫att-sα-1fsds,a>0,provided that the right-hand side is pointwise defined on [0,+∞), where Γ(·) is the gamma function.Definition 5 (see [4]).
The Riemann-Liouville fractional derivative of orderα>0 of a function f:[0,+∞)→R is given by (14)lDtαft=1Γn-αdndtn∫att-sn-α-1fsds,t>0,where n-1<α≤n, provided that the right-hand side is pointwise defined on [0,+∞).Definition 6 (see [4]).
The Caputo fractional derivative of orderα>0 of a function f:[0,+∞)→R is given by (15)cDtαft=lDtαft-∑k=0n-1tkk!fk0,t>0,where n-1<α≤n, provided that the right-hand side is pointwise defined on [0,+∞).Definition 7 (see [10]).
FunctionsEα(z) and Eα,β(z) are called classical and generalized Mittag-Leffler functions, respectively, given by (16)Eαz=∑k=0∞zkΓαk+1,Eα,βz=∑k=0∞zkΓαk+β.Lemma 8 (see [10]).
Let0<α,β<1, and then functions Eα(z),Eα,α(z), and Eα,α+β are nonnegative and have the following properties.
(i) For anyλ>0 and t∈J,(17)Eα-tαλ≤1,Eα,α-tαλ≤1Γα,Eα,α+β-tαλ≤1Γα+β.Moreover,(18)Eα0=1,Eα,α0=1Γα,Eα,α+β0=1Γα+β.
(ii) For anyλ>0 and t1,t2∈J, when t2→t1, we have(19)Eα-t2αλ→Eα-t1αλ,Eα,α-t2αλ→Eα,α-t1αλ,Eα,α+β-t2αλ→Eα,α+β-t1αλ.
(iii) For anyλ>0 and t1,t2∈J and t1≤t2, we have(20)Eα-t1αλ≥Eα-t2αλ,Eα,α-t1αλ≥Eα,α-t2αλ,Eα,α+β-t1αλ≥Eα,α+β-t2αλ.Lemma 9 (see [16]).
Leta+bEα+β(-λ)≠0. For a given f∈L1(J,R) with f(t)∈F(t,u(t)),a.e.t∈J, then the boundary value problem (1) has a unique solution u(t)∈PC(J,R) which is defined by the following form: (21)ut=Eα+β-λtα+βa+bEα+β-λa∑i=1nIi-tiβEα+β,β+1-λtiα+βci-ci-1Eα+β-λtiα+β-b∫011-sα+β-1Eα+β,α+β-λ1-sα+βfsds-bcnEα+β,β+1-λ+∫01gs,usds-Eα+β-λtα+β×∑j=k+1nIj-tjβEα+β,β+1-λtjα+βcj-cj-1Eα+β-λtjα+β+∫0tt-sα+β-1Eα+β,α+β-λt-sα+βfsds+cktβEα+β,β+1-λtα+β∀t∈tk,tk+1,k=0,1,…,n.
Finally, we give the following lemma which is greatly important in the proof of our main results.Lemma 10 (see [17, Bohnenblust-Karlin]).
LetX be a Banach space, D a nonempty subset of X, which is bounded, closed, and convex. Suppose G:D→2X∖0 is u.s.c. with closed, convex values, and such that G(D)⊂D and G(D)¯ are compact. Then G has a fixed point.


## 3. Main Results

In order to begin our main results, we also need the following conditions:(H1)
There exists0<q<α+β<1, and a real function mr(t)∈L1/q(J,R+) such that (22)Ft,u=sup⁡f:ft∈Ft,u≤mrt,∀u≤rfora.e.t∈J,for eachr>0.(H2)
g(t,0)=0 and there exists L>0 such that (23)gt,u-gt,v≤Lu-vforu,v∈R and t∈[0,1], where L satisfies L<a in which a is defined in (1).For convenience, we denote(24)Ω=∑i=1nIi+tiβEα+β,β+1-λtiα+βci-ci-1Eα+β-λtiα+β.Theorem 11.
Suppose that (H1) and (H2) hold; then system (1) has at least one solution on J.Proof.
We transform problem (1) into a fixed point problem. Consider the operator N:C(J,R)→PC(J,R) defined by(25)Nu=ht∈PCJ,R:ht=Eα+β-λtα+βa+bEα+β-λa∑i=1nIi-tiβEα+β,β+1-λtiα+βci-ci-1Eα+β-λtiα+β-b∫011-sα+β-1Eα+β,α+β-λ1-sα+βfsds-bcnEα+β,β+1-λ+∫01gs,usds-Eα+β-λtα+β×∑j=k+1nIj-tjβEα+β,β+1-λtjα+βcj-cj-1Eα+β-λtjα+β+∫0tt-sα+β-1Eα+β,α+β-λt-sα+βfsds+cktβEα+β,β+1-λtα+β∀t∈tk,tk+1,k=0,1,…,n.for f∈SF,u.
Next we shall show thatN satisfies all the assumptions of Lemma 10; that is to say, N has a fixed point which is a solution of problem (1). For the sake of convenience, we subdivide the proof into several steps.
Step 1 (N(u) is convex for each u∈PC(J,R)). In fact, assume h1,h2∈N(u), then there exist f1,f2∈SF,u such that, for each t∈J, we have(26)hit=Eα+β-λtα+βa+bEα+β-λa∑i=1nIi-tiβEα+β,β+1-λtiα+βci-ci-1Eα+β-λtiα+β-b∫011-sα+β-1Eα+β,α+β-λ1-sα+βfisds-bcnEα+β,β+1-λ+∫01gs,usds-Eα+β-λtα+β×∑j=k+1nIj-tjβEα+β,β+1-λtjα+βcj-cj-1Eα+β-λtjα+β+∫0tt-sα+β-1Eα+β,α+β-λt-sα+βfisds+cktβEα+β,β+1-λtα+β,i=1,2.Let 0≤χ≤1. Then, for each t∈J, we have(27)χh1+1-χh2t=Eα+β-λtα+βa+bEα+β-λa∑i=1nIi-tiβEα+β,β+1-λtiα+βci-ci-1Eα+β-λtiα+β-b∫011-sα+β-1Eα+β,α+β-λ1-sα+βχf1s+1-χf2sds-bcnEα+β,β+1-λ+∫01gs,usds-Eα+β-λtα+β×∑j=k+1nIj-tjβEα+β,β+1-λtjα+βcj-cj-1Eα+β-λtjα+β+∫0tt-sα+β-1Eα+β,α+β-λt-sα+βχf1s+1-χf2sds+cktβEα+β,β+1-λtα+β.Since SF,u is convex (F has convex values), so it follows that χh1+(1-χ)h2∈N(u).
Step 2. Let Br=u∈PCJ,R:u≤r, where(28)aa-La+bΩa+bcaΓ1+β+cΓ1+β+a+bmrL1/qaΓα+β1-qα+β-q1-q≤r.Then Br is a bounded closed convex set in PC(J,R). Thus we need to verify N(Br)⊆Br. In fact, from Lemma 8, (H1), and (H2), for each u∈Br,t∈Jk,k=0,1,…,n, we have(29)
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+
β
1
-
q
α
+
β
-
q
1
-
q
·
t
α
+
β
-
q
≤
a
+
b
Ω
a
+
b
c
a
Γ
1
+
β
+
c
Γ
1
+
β
+
L
r
a
+
a
+
b
m
r
L
1
/
q
a
Γ
α
+
β
1
-
q
α
+
β
-
q
1
-
q
.From (28), we have N(Br)⊆Br.
Step 3 (N(Br) is equicontinuous). Let Δ=J×Br,δ1,δ2∈J,δ1<δ2. For convenience, we also let sup(t,u)∈Δft,u≔Υ,(30)Φ=1a+bEα+βa∑i=1nIi-tiβEα+β,β+1-λtiα+βci-ci-1Eα+β-λtiα+β-bcnEα+β,β+1-λ+∫01gs,usdsand(31)Ψ=∑j=k+1nIj-tjβEα+β,β+1-λtjα+βcj-cj-1Eα+β-λtjα+β,and then we have(32)
N
u
δ
1
-
N
u
δ
2
=
E
α
+
β
-
λ
δ
1
α
+
β
-
E
α
+
β
-
λ
δ
2
α
+
β
Φ
-
E
α
+
β
-
λ
δ
1
α
+
β
-
E
α
+
β
-
λ
δ
2
α
+
β
Ψ
+
∫
0
δ
1
δ
1
-
s
α
+
β
-
1
E
α
+
β
,
α
+
β
-
λ
δ
1
-
s
α
+
β
f
s
d
s
-
∫
0
δ
2
δ
2
-
s
α
+
β
-
1
E
α
+
β
,
α
+
β
-
λ
δ
2
-
s
α
+
β
f
s
d
s
+
c
k
δ
1
β
E
α
+
β
,
β
+
1
-
λ
δ
1
α
+
β
-
c
k
δ
2
β
E
α
+
β
,
β
+
1
-
λ
δ
2
α
+
β
≤
E
α
+
β
-
λ
δ
1
α
+
β
-
E
α
+
β
-
λ
δ
2
α
+
β
Φ
-
Ψ
+
∫
0
δ
1
δ
1
-
s
α
+
β
-
1
E
α
+
β
,
α
+
β
-
λ
δ
1
-
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α
+
β
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δ
2
-
s
α
+
β
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1
E
α
+
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,
α
+
β
-
λ
δ
2
-
s
α
+
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f
s
d
s
-
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δ
1
δ
2
δ
2
-
s
α
+
β
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1
E
α
+
β
,
α
+
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-
λ
δ
2
-
s
α
+
β
f
s
d
s
+
c
k
δ
1
β
E
α
+
β
,
β
+
1
-
λ
δ
1
α
+
β
-
c
k
δ
2
β
E
α
+
β
,
β
+
1
-
λ
δ
2
α
+
β
≤
E
α
+
β
-
λ
δ
1
α
+
β
-
E
α
+
β
-
λ
δ
2
α
+
β
Φ
-
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+
∫
0
δ
1
δ
1
-
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α
+
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1
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δ
2
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α
+
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1
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α
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+
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1
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f
s
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+
∫
0
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+
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1
E
α
+
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,
α
+
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-
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δ
1
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+
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+
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δ
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f
s
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+
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δ
1
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2
δ
2
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α
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1
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α
+
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,
α
+
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1
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1
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1
α
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c
k
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1
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δ
1
α
+
β
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E
α
+
β
,
β
+
1
-
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δ
2
α
+
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,
≤
E
α
+
β
-
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δ
1
α
+
β
-
E
α
+
β
-
λ
δ
2
α
+
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-
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+
Υ
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α
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∫
0
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1
δ
1
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α
+
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1
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2
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1
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1
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,
α
+
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δ
2
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α
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f
s
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s
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Υ
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∫
δ
1
δ
2
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2
-
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1
d
s
+
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k
δ
1
β
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2
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α
+
β
,
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+
1
-
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δ
1
α
+
β
+
c
k
δ
2
β
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α
+
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,
β
+
1
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δ
1
α
+
β
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α
+
β
,
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+
1
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δ
2
α
+
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≤
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α
+
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δ
1
α
+
β
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E
α
+
β
-
λ
δ
2
α
+
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-
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+
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2
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1
α
+
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2
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δ
1
α
+
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+
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δ
2
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1
α
+
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α
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Γ
α
+
β
+
∫
0
δ
1
δ
2
-
s
α
+
β
-
1
E
α
+
β
,
α
+
β
-
λ
δ
1
-
s
α
+
β
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α
+
β
,
α
+
β
-
λ
δ
2
-
s
α
+
β
f
s
d
s
+
c
k
δ
1
β
-
δ
2
β
E
α
+
β
,
β
+
1
-
λ
δ
1
α
+
β
+
c
k
δ
2
β
E
α
+
β
,
β
+
1
-
λ
δ
1
α
+
β
-
E
α
+
β
,
β
+
1
-
λ
δ
2
α
+
β
,and from Lemma 8, we clearly see the right hand side of the above inequality tends to zero as δ1→δ2. This implies that N is equicontinuous on J. As a consequence of Steps 1–3 together with the Ascoli-Arzela theorem, we can conclude that N is a compact valued map.
Step 4 (N has a closed graph). Let un→u∗,hn∈N(un) and hn→h∗. Then we need to verify h∗∈N(u∗). hn∈N(un) implies that there exists fn∈SF,un such that for each t∈J we have(33)hnt=Eα+β-λtα+βa+bEα+β-λa∑i=1nIi-tiβEα+β,β+1-λtiα+βci-ci-1Eα+β-λtiα+β-b∫011-sα+β-1Eα+β,α+β-λ1-sα+βfnsds-bcnEα+β,β+1-λ+∫01gs,usds-Eα+β-λtα+β×∑j=k+1nIj-tjβEα+β,β+1-λtjα+βcj-cj-1Eα+β-λtjα+β+∫0tt-sα+β-1Eα+β,α+β-λt-sα+βfnsds+cktβEα+β,β+1-λtα+β,and thus we must verify that there exists f∗∈SF,u∗ such that for each t∈J we have(34)h∗t=Eα+β-λtα+βa+bEα+β-λa∑i=1nIi-tiβEα+β,β+1-λtiα+βci-ci-1Eα+β-λtiα+β-b∫011-sα+β-1Eα+β,α+β-λ1-sα+βf∗sds-bcnEα+β,β+1-λ+∫01gs,usds-Eα+β-λtα+β×∑j=k+1nIj-tjβEα+β,β+1-λtjα+βcj-cj-1Eα+β-λtjα+β+∫0tt-sα+β-1Eα+β,α+β-λt-sα+βf∗sds+cktβEα+β,β+1-λtα+β.
Consider the continuous linear operator(35)Θ:L1J,R→CJ,R,(36)f↦Θft=∫0tt-sα+β-1Eα+β,α+β-λt-sα+βfsds-bEα+β-λtα+βa+bEα+β-λ∫011-sα+β-1Eα+β,α+β-λ1-sα+βfsds;then,(37)hnt-Eα+β-λtα+βΦ-Ψ-cktβEα+β,β+1-λtα+β-h∗t-Eα+β-λtα+βΦ-Ψ-cktβEα+β,β+1-λtα+β→0asn→∞.By Lemma 2, we know Θ∘SF is a closed graph operator.
Also from the definition ofΘ we have (38)hnt-Eα+β-λtα+βΦ-Ψ-cktβEα+β,β+1-λtα+β∈ΘSF,un.Since un→u∗, Lemma 2 implies that (39)h∗t-Eα+β-λtα+βΦ-Ψ-cktβEα+β,β+1-λtα+β=∫0tt-sα+β-1Eα+β,α+β-λt-sα+βf∗sds-bEα+β-λtα+βa+bEα+β-λ∫011-sα+β-1Eα+β,α+β-λ1-sα+βf∗sdsfor some f∗∈SF,u∗.
Therefore,N is a compact multivalued map, u.s.c. with convex closed values. By Lemma 10, we have that N has a fixed point u(t) which is a solution of problem (1).Corollary 12 ..
Assume that (H2) and (H3) hold.
(H3) There exist continuous and bounded functionsτ1(t),τ2(t)∈L1(J,R+),σ∈[0,1] such that (40)Ft,u≤τ1t+τ2tuσ;then problem (1) has at least a solution on J.Proof.
The proof is the same as Theorem11 which we can take as m(t)=τ1(t)+τ2(t)uσ.Remark 13.
If we letf(t,u)∈Ft,u and g(t,u) be a constant function, then the above Corollary 12 improves Theorem 3.1 in [12].Remark 14.
Note that ifγ=0 and γ=1, we have D0ctγu(t)=u(t) and D0ctγu(t)=u′(t), respectively. Thus, in this paper, let α=1,β=0,λ=0,ck=0; our system (1) reduces to [18], so our problem (1) gives generalization of [18].Remark 15 ..
Ifβ=0,λ=0, the boundary value condition becomes u(0)=u0, and our system (1) reduces to [16, 19]. If α=1,β=0, the boundary value condition becomes u(0)-u(T)=μ, and our system (1) reduces to [20]. Thus, our problem (1) gives generalizations of [16, 19, 20].


## 4. An Example

In this part, we will give corresponding example to illustrate the main results in our paper.Example 1.
Consider the following system:(41)cD0,tαcD0,tβut+λut∈Ft,ut,a.e.t∈J=0,1∖15Δu15=I1u15au0+bu1=∫01gs,uds,cD0,tβutt=0=c0,cD0,tβutt=1/5=c1,where 0<α+β<1,λ>0,a=4,b=1, and let F(t,u(t))=[sin⁡t/et(cos⁡u(t)+1),sin⁡t/et(cos⁡u(t)+3)] and g(t,u(t))=cos⁡t/etu(t)/1+u(t),(t,u)∈[0,1]×[0,+∞). Then we let mr(t)=4sin⁡t/et and L=1, and we have (42)gt,u-gt,v≤u-v;then (H1) and (H2) of Theorem 11 all hold. Hence, system (41) has at least one solution on J.



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*Source: 1323176-2019-02-18.xml*