# Existence of Solutions for Fractional Impulsive Integrodifferential Equations in Banach Spaces

Authors: Haide Gou; Baolin Li
Journal: International Journal of Differential Equations
 (2016)

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## Abstract

We investigate the existence of solutions for a class of impulsive fractional evolution equations with nonlocal conditions in Banach space by using some fixed point theorems combined with the technique of measure of noncompactness. Our results improve and generalize some known results corresponding to those obtained by others. Finally, two applications are given to illustrate that our results are valuable.


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## Body

## 1. Introduction

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, and polymer rheology and have been emerging as an important area of investigation in the last few decades; see [1–4]. However, the theory of impulsive fractional evolution equations was still in the initial stages and many aspects of this theory need to be explored.The theory of impulsive differential equations is a new and important branch of differential equation theory, which has an extensive physical, population dynamics, ecology, chemical, biological systems, and engineering background. Therefore, it has been an object of intensive investigation in recent years; some basic results on impulsive differential equations have been obtained and applications to different areas have been considered by many authors; see [4–8].The study of nonlocal Cauchy problem for abstract evolution differential equations has been initiated by Byszewski [9]. The existence of solutions for fractional abstract differential equations with nonlocal initial condition was recently investigated by N’Guérékata [10] and Balachandran and Park [11]. In [12], Balachandran et al. were concerned with the existence of solutions of first-order nonlinear impulsive fractional integrodifferential equations in Banach spaces:(1)Dctαut=Aut+ft,ut,∫0tht,s,usds,t∈0,T,Δut=tk=Ikutk-,k=1,…,m,u0+gu=u0;the results are obtained by using fixed point principles.Shu and Wang [4] studied the existence of mild solutions for fractional differential equation with nonlocal conditions in a Banach space E:(2)Dctαut=Aut+ft,ut+∫0tqt-sgs,usds,t∈0,T,u0+mu=u0∈E,u′0+nu=u1∈E.By using the contraction mapping principle and Krasnoselskii’s fixed point theorem, they obtained the existence of solutions for the equation.In [13], Gou and Li investigated local and global existence of mild solution for an impulsive fractional functional integrodifferential equation with noncompact semigroup in Banach spaces E:(3)Dctαut+Aut=ft,ut+∫0tqt-sgs,usds,t≥0,t≠tk,Δut=tk=Ikutk-,k=1,…,m,u0=u0∈E,and they establish a general framework to find the mild solutions for impulsive fractional integrodifferential equations, which will provide an effective way to deal with such problems.Motivated by this consideration, we investigate the existence of solutions for a class of impulsive fractional evolution equations with nonlocal conditions in Banach spaceE: (4)Dctαut+Aut=ft,ut,Fut,Gut,t∈J,t≠tk,Δut=tk=Ikutk-,k=1,2,…,m,u0=gu+u0,by using some fixed point theorems combined with the technique of measure of noncompactness, where Dctα is the Caputo fractional derivative of order α∈(1,2], A:D(A)⊂E→E is a closed linear operator, and -A generates a C0-semigroup T(t)(t≥0) in E; f:J×E×E×E→E, J=[0,a], where a>0 is a constant, 0=t0<t1<t2<⋯<tm<tm+1=a; Ik:E→E is an impulsive function, and k=1,2,…,m; and x0∈E, p1,p2 is an E-valued function to be given later and(5)Fut=∫0tkt,sp1s,usds,k∈CD,R+,Gut=∫0aht,sp2s,usds,h∈CD0,R+,D=t,s∈R2:0≤s≤t≤a, D0=t,s∈R2:0≤t,s≤a, pi∈C(J×E,E), i=1,2, and Δut=tk denotes the jump of u(t) at t=tk; that is, Δut=tk=u(tk+)-u(tk-), u(tk+) and u(tk-) represent the right and left limits of u(t) at t=tk, respectively.The paper is organized as follows. In Section2, we recall some concepts and facts about the Kuratowski measure of noncompactness and some fixed point theorems. In Section 3, we obtain the existence solutions of problem (4). In Section 4, we give two examples to illustrate our results.


## 2. Preliminaries

In this section, we briefly recall some definitions and the fixed point theorems which will be used in the sequel. Throughout this paper, letE be a Banach space; we assume that A:D(A)⊂E→E is a closed linear operator and -A generates a uniformly bounded C0-semigroup T(t)(t>0) on a Banach space (E,·) and M=supt∈[0,+∞)⁡T(t)B(E), where B(E) stands for the Banach space of all linear and bounded operators in E.C ( J , E ) denote the Banach space of all continuous E-value functions on interval J and PCJ,E={u:J→E:u∈Ctk-1,tk,E,k=1,2,…,m, and there exist utk- and utk+, k=1,2,…,m with utk-=utk}. Obviously, PCJ,E is a Banach space with the supnorm u=supt∈Jut.Definition 1.
The fractional integral of orderα>0 with the lower limit zero for a function u is defined as(6)Itαut=1Γα∫0tt-sα-1usds,t>0,where Γ· is the Gamma function.Definition 2.
The Caputo fractional derivative of orderα>0 with the lower limit zero for a function u is defined as(7)Dctαut=1Γn-α∫0tt-sn-α-1unsds,t>0,0≤n-1<α<n,where the function u(t) has absolutely continuous derivatives up to order n-1.Remark 3.
Ifu is an abstract function with values in E, then the integrals which appear in Definitions 1 and 2 are taken in Bochner’s sense.Definition 4.
By a mild solution of the initial value problem(8)Dctαut+Aut=ft,ut,t≥0,u0=u0,on [0,∞), we mean that a continuous function u defined from [0,∞) into E satisfying (9)ut=Tαtu0+∫0tt-sα-1Sαt-sfs,usds,t∈0,∞,where(10)Tαt=∫0∞θαsTtαsds,Sαt=α∫0∞sθαsTtαsds,θαs=1πα∑n=1∞-sn-1Γnα+1n!sin⁡nπα,s∈0,∞are the functions of Wright type defined on (0,∞) which satisfies (11)∫0∞θαsds=1,θαs≥0,s∈0,∞,∫0∞svθαsds=Γ1+vΓ1+αv,v∈0,1.The following lemma can be found in [14, 15].Lemma 5.
The operatorsTαt and Sαtt≥0 have the following properties:(i)
For any fixedt≥0,Tα(t) and Sα(t) are linear and bounded operators; that is, for any u∈E,(12)Tαtu≤Mu,Sαtu≤αMΓα+1u=MΓαu.(ii)
For everyu∈E, t→Tα(t)u and t→Sα(t)u are continuous functions from [0,∞) into E.(iii)
The operatorsTα(t)(t≥0) and Sα(t)(t≥0) are strongly continuous, which means that, for  ∀u∈E and 0≤t′<t′′≤a, one has (13)Tαt′′u-Tαt′u⟶0,Sαt′′u-Sαt′u⟶0,ast′′-t′⟶0.(iv)
If the semigroupT(t) is continuous by operator norm for every t>0, then Tα(t) and Sα(t) are continuous in (0,+∞) by the operator norm.Letβ· denote the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see [6].The following lemmas are to be used in proving our main results.Lemma 6 (see [16]).
LetE be a Banach space, and let D⊂C(J,E) be equicontinuous and bounded; then β(D(t)) is continuous on J, and β(D)=maxt∈J⁡β(D(t)).Lemma 7 (see [7]).
LetD⊂E be bounded. Then there exists a countable set D0⊂D, such that β(D)≤2β(D0).Lemma 8 (see [17]).
LetE be a Banach space, and let D=un⊂CI,X be a bounded and countable set. Then βDt is the Lebesgue integral on X, and(14)β∫Iuntdt∣n∈N≤2∫IβDtdt.In the following, we introduce the definition of(γ,ψ,p)-contractive mapping.Definition 9 (see [18]).
Let(E,d) be a metric space with w distance p and f:E→E a given mapping. We say that f is an (γ,ψ,p)-contractive mapping if there exist two functions γ:E×E→[0,∞) and ψ∈Ψ such that (15)γx,ypfx,fy≤ψpx,yfor all x,y∈E.We will show some fixed point theorems about condensing operator and(γ,ψ,p)-contractive mapping, which play a key role in the proof of our main results.Lemma 10 (see [19]).
LetE be a Banach space. Assume that D⊂E is a bounded closed and convex set on E, Q:D→D is condensing. Then Q has at least one fixed point in D.Lemma 11 (see [18]).
Letp be a w distance on a complete metric space (E,d) and let f:E→E be an (γ,ψ,p)-contractive mapping. Suppose that the following conditions hold:(i)
f is an γ-admissible mapping;(ii)
there exists a pointx0∈E such that γ(x0,fx0)≥1;(iii)
eitherf is continuous or, for any sequence {xn} in E, if γ(xn,xn+1)≥1 for all n∈N and xn→x∈E as n→∞, then γ(xn,x)≥1 for all n∈N. Then there exists a point u∈E such that fu=u. Moreover, if γ(u,u)≥1, then p(u,u)=0.


## 3. Main Results

In this section, we will establish the existence theorems of solutions for the nonlocal problem (4). For convenience, we give some notations.ForB⊂C(J,E), let B(t)={u(t):u∈B} and denote BR={u∈E:u≤R}. Let k0=max⁡{k(t,s):(t,s)∈D}, h0=max⁡{h(t,s):(t,s)∈D0}. First of all, let us start by defining what we mean by a solution of problem (4).Definition 12.
A functionu∈PC(J,E) is said to be a mild solution of problem (4) if u satisfies the equation(16)ut=Tαtgu+u0+∑0<tk<t∫tk-1tktk-sα-1Sαt-sfs,us,Fus,Gusds+∫tktt-sα-1Sαt-sfs,us,Fus,Gusds+∑0<tk<tTαt-tkIkutk-.To prove our main results, we state the following basic assumptions of this paper.(H1)
There exists a positive constantKi,(i=1,2,3) such that (17)ft,ut,Fut,Gut-ft,vt,Fvt,Gvt≤K1u-v+K2Fu-Fv+K3Gu-Gvand forpi∈C(J×E,E), t∈J,u,v∈BR, there exists a constant bi,ci>0 such that(18)pit,u-pit,v≤biu-v,pit,u≤ciu,i=1,2.(H2)
For anyR>0, there exist a Lebesgue-integrable function MR:I→R+ and nondecreasing continuous function Ω:0,∞→0,∞ such that (19)ft,x,y,z≤MRtΩRfor allt∈I, (x,y,z)∈BR×BR×BR.(H3)
g:E→E is a continuous and compact mapping; furthermore, there exists positive number N1,N2 such that g(u)-g(v)≤N1 and g(u)≤N2 for any u,v∈BR.(H4)
The functionsIk:E→E are continuous and there exists a constant μ>0 and ρ>0 such that (20)Ikt,u-Ikt,v≤μu-v,Iku≤ρu,∀u,v∈BR,k=1,2,…,m.(H5)
There exist constantsLi>0,(i=1,2,3) such that for any bounded and equicontinuous sets Di⊂C(J,E),(i=1,2,3) and t∈J,(21)βft,D1,D2,D3≤∑i=13LiβDi,βkt,sp1s,D4≤L4βD4,βht,sp2s,D5≤L5βD5.Lemma 13 (see [13]).
Letf,g:J×E→E be a continuous function and let -A be the generator of a C0-semigroup (T(t))(t≥0). If u∈PC(J,E) is a mild solution of (4) in the sense of Definition 12, then for any t∈(tk-1,tk],k=1,…,m,(22)ut=Tαtgu+u0+∑0<tk<t∫tk-1tktk-sα-1Sαt-sfs,us,Fus,Gusds+∫tktt-sα-1Sαt-sfs,us,Fus,Gusds+∑0<tk<tTαt-tkIkutk-is a solution of (4). In other words u(t) is a mild solution of (4).Theorem 14.
LetE be a Banach space, let A:D(A)⊂E→E be a closed linear operator, and -A generates an equicontinuous C0-semigroup T(t)(t≥0) of uniformly bounded operators in E. Suppose that the conditions (H1)–(H5) are satisfied. Then for every u0∈PC(J,E) there exists a τ1=τ1(u0), 0<τ1<a such that problem (4) has a solution u∈PC0,τ1,E.Proof.
Since we are interested here only in local solutions, we may assume thata<∞. By using our assumption (H1)–(H4), let t′>0,R=M(2u0+1)>0 be such that BR(u0)={u:u-u0≤R}, for 0≤t≤t′ and u∈BR(u0), and let us choose(23)τ1=min⁡t′,a,Γα+1x0+1-N2-mρ1m+1MRΩM2u0+11/α,Γα+12Mm+1L1+ak0L2L4+ah0L3L51/α.Set Ω=u∈PC0,τ1,E:ut≤R,t∈0,τ1; then Ω is a closed ball in PC0,τ1,E with center θ and radius R. Consider the operator Q:Ω→PC0,τ1,E defined by (24)Qut=Tαtgu+u0+∑0<tk<t∫tk-1tktk-sα-1Sαt-sfs,us,Fus,Gusds+∫tktt-sα-1Sαt-sfs,us,Fus,Gusds+∑0<tk<tTαt-tkIkutk-.It is easy to see that the fixed points of Q are the solutions of the nonlocal problem (4); we shall prove that Q has a fixed point by using Lemma 10. For any u∈Ω and t∈[0,τ1], by Lemma 5(i), we have (25)Fus=∫0tkt,sp1s,usds≤k0∫0ap1s,usds≤Rc1k0≤R,where c1=k0-1. So Tx∈Ω. Similarly, we prove Gu∈Ω.
And by (19), we have(26)Qut≤Tαtgu+u0+MMRΩRΓα∑0<tk<t∫tk-1tktk-sα-1ds+MMRΩRΓα∫tktt-sα-1ds+M∑0<tk<tIkutk-≤MN2+u0+MMRΩRm+1τ1αΓα+1+Mmρ≤R.Therefore, Qu∈Ω. Now we show that Q is continuous from Ω into Ω. To show this, we first observe that since f is continuous in [0,a]×E, it follows that for any ϵ>0 and for a fixed u∈BR(u0) there exists δ1(u),δ2(u)>0 such that for any v∈BR(u0) and let δ(u)=min⁡δ1u,δ2u. Then for any v∈Ω, u-v<δu and choose (27)Mm+1K1+K2ak0b1+K3ah0b2τ1αΓα+1+Mmμ+N1<ϵδu.Then we have (28)Qut-Qvt≤Tαtgu-gv+∑0<tk<t∫tk-1tktk-sα-1Sαt-s·fs,us,Fus,Gus-fs,vs,Fvs,Gvsds+∫tktt-sα-1Sαt-s·fs,us,Fus,Gus-fs,vs,Fvs,Gvsds+∑0<tk<tTαt-tkIkutk--Ikvtk-≤Mm+1K1+K2ak0b1+K3ah0b2τ1αΓα+1+Mmμ+N1u-v≤ϵ.Thus, we proved that Q:Ω→Ω is a continuous operator.
Now, we demonstrate that the operatorQ:Ω→Ω is equicontinuous. For any u∈Ω and 0≤t1<t2≤τ1, we get that (29)Qut2-Qut1=Tαt2gu+u0-Tαt1gu+u0+∑0<tk<t∫tk-1tktk-sα-1Sαt2-s-Sαt1-s×fs,us,Fus,Gusds+∫t1t2t2-sα-1Sαt2-sfs,us,Fus,Gusds+∫tkt1t2-sα-1-t1-sα-1Sαt2-sfs,us,Fus,Gusds+∫tkt1t1-sα-1Sαt2-s-Sαt1-sfs,us,Fus,Gusds+∑0<tk<tTαt2-tk-Tαt1-tk×Ikutk-=I1+I2+I3+I4+I5+I6,where(30)I1=Tαt2gu+u0-Tαt1gu+u0,I2=∑0<tk<t∫tk-1tktk-sα-1Sαt2-s-Sαt1-s×fs,us,Fus,Gusds,I3=∫t1t2t2-sα-1Sαt2-sfs,us,Fus,Gusds,I4=∫tkt1t2-sα-1-t1-sα-1Sαt2-sfs,us,Fus,Gusds,I5=∫tkt1t1-sα-1Sαt2-s-Sαt1-sfs,us,Fus,Gusds,I6=∑0<tk<tTαt2-tk-Tαt1-tk×Ikutk-.Here we calculate (31)Qut2-Qut1≤∑i=16Ii.Therefore, we inspect that Ii tend to 0, when t2-t1→0, i=1,2,…,6.
ForI1, by Lemma 5(iii), I1→0 as t2-t1→0.
ForI2, by Lemma 5(iii), we have (32)I2≤∑0<tk<t∫tk-1tktk-sα-1Sαt2-s-Sαt1-s·MRΩRds⟶0,t2-t1⟶0.
ForI3, by Lemma 5(i), we have (33)I3≤MMRΩRΓα∫t1t2t2-sα-1ds≤MMRΩRΓα+1t2-t1α⟶0,t2-t1⟶0.
ForI4, by Lemma 5(i), we have (34)I4≤MMRΩRΓα∫tkt1t2-sα-1-t1-sα-1ds≤MMRΩRΓα+1t1-t2α-tk-t2α+tk-t1α⟶0,t2-t1⟶0.
ForI5, by Lemma 5(iii), we have (35)I5≤∫tkt1t1-sα-1Sαt2-s-Sαt1-s·MRΩRds⟶0,t2-t1⟶0.
ForI6, by Lemma 5(iii), we have (36)I6≤mρTαt2-tk-Tαt1-tk⟶0,t2-t1⟶0.In conclusion, (Qu)(t2)-(Qu)(t1) tends to 0 as t2-t1→0, which implies that Q(Ω) is equicontinuous.
LetB=co¯Q(Ω). Then it is easy to verify that Q maps B into itself and B⊂PC(J,E) is equicontinuous. Now, we prove that Q:B→B is a condensing operator. For any D⊂B, by Lemma 7, there exists a countable set D0=un⊂D, such that (37)βQD≤2βQD0.By the equicontinuity of B, we know that D0⊂B is also equicontinuous.
By the fact that(38)∫0ausds∈aco¯us∣s∈J,u∈CJ,E,we have (39)β∫0tkt,sp1s,usds∣u∈B,t∈J≤ak0L4βut∣u∈B,t∈J,β∫0tht,sp2s,usds∣u∈B,t∈J≤ah0L5βut∣u∈B,t∈J.
Thus, by (H1), (H2), (39), and Lemma 13, we have (40)βQD0t=βTαtgun+u0+∑0<tk<t∫tk-1tktk-sα-1Sαt-sfs,uns,Funs,Gunsds+∫tktt-sα-1Sαt-sfs,uns,Funs,Gunsds+∑0<tk<tTαt-tkIkuntk-≤2Mm+1Γα∫0tt-sα-1αSαt-sfs,uns,Funs,Gunsds≤2Mm+1L1+ak0L2L4+ah0L3L5τ1αΓα+1βD.Since Q(D0)⊂B is bounded and equicontinuous, we know from Lemma 6 that (41)βQD0=maxt∈I⁡βQD0t.Therefore, from (37), (40), and (41), we know that(42)βQD≤2Mm+1L1+ak0L2L4+ah0L3L5τ1αΓα+1·βD≤βD.Thus, Q:B→B is a condensing operator. It follows from Lemma 10 that Q has at least one fixed point u(t0)∈B, which is just a solution of problem (4) on the interval [0,τ1]. This completes the proof.Corollary 15.
LetE be a Banach space. A:D(A)⊂E→E be a closed linear operator and -A generates an equicontinuous C0-semigroup T(t)(t≥0) of uniformly bounded operators in E. Suppose that the conditions (H1)–(H5) are satisfied. Then for every u0∈PC(J,E) there exists a τ1=τ1(u0),0<τ1<a such that the nonlocal problem(43)Dctαut+Aut=ft,ut,Fut,Gut,t∈J,t≠tk,Δut=tk=Ikutk-,k=1,2,…,m,u0=u0+∑i=1mciutihas a solution u∈PC([0,τ1],E).Proof.
Let the functiong(u)=∑i=1mciu(ti); by the similar way one can easily verify the conditions (H1)–(H5) by properly choosing ci. Hence, by Theorem 14, problem (43) has a solution.Theorem 16.
Letξ:E×E→R+ be a given function. Assume that the following conditions hold:(A)
there existsψ∈Ψ such that (44)fs,us,Fus,Gus-fs,vs,Fvs,Gvs≤Γα+13Mm+1aαψu-v,gu-gv≤13Mψu-v,Ikutk--Ikvtk-≤13Mmψu-v,for allt∈J and for all a,b∈E with ξa,b≥0;(B)
there existsu0∈PC(J,E) such that ξ(u0(t), Qu0(t))≥0 for all t∈J, where a mapping Q:PC(J,E)→PC(J,E) is defined by(45)Qut=Tαtgu+u0+∑0<tk<t∫tk-1tktk-sα-1Sαt-sfs,us,Fus,Gusds+∫tktt-sα-1Sαt-sfs,us,Fus,Gusds+∑0<tk<tTαt-tkIkutk-;(C)
for eacht∈J, and u,v∈PC(J,E),ξ(u(t),v(t))≥0 implies that ξQut,Qvt≥0;(D)
for eacht∈J, if un is a sequence in PCJ,E such that un→u in PC(J,E) and ξ(un(t),un+1(t))≥0 for all n∈N, then (46)ξunt,ut≥0for alln∈N.Proof.
First of all, letE=PC(J,E). It is easy to see that u∈E is a solution of (4) if and only if u∈E is a solution of the integral equation(47)ut=Tαtgu+u0+∑0<tk<t∫tk-1tktk-sα-1Sαt-sfs,us,Fus,Gusds+∫tktt-sα-1Sαt-sfs,us,Fus,Gusds+∑0<tk<tTαt-tkIkutk-.Then problem (4) is equivalent to finding u∗∈E which is a fixed point of Q.
Now, letu,v∈E such that ξ(u(t),v(t))≥0 for all t∈J. By condition (A), we have (48)Qut-Qvt=Tαtgu+u0+∑0<tk<t∫tk-1tktk-sα-1Sαt-sfs,us,Fus,Gusds+∫tktt-sα-1Sαt-sfs,us,Fus,Gusds+∑0<tk<tTαt-tkIkutk--Tαtgv+v0-∑0<tk<t∫tk-1tktk-sα-1Sαt-sfs,vs,Fvs,Gvsds-∫tktt-sα-1Sαt-sfs,vs,Fvs,Gvsds-∑0<tk<tTαt-tkIkvtk-≤Mgu-gv+Mm+1Γα∫0tt-sα-1fs,us,Fus,Gus-fs,vs,Fvs,Gvs+MmIkutk--Ikvtk-≤13ψu-v+13ψu-v+13ψu-v=ψu-v.This implies that for each u,v∈E with ξ(u(t),v(t))≥0 for all t∈J, we obtain that (49)Qxt-Qyt≤ψu-vfor all u,v∈E. Now, we define the function γ:E×E→[0,∞) by (50)γu,v=1ifξut,vt≥0∀t∈J,0otherwise,and also we define the w-distance p on E by p(u,v)=u-v. From (49), we have (51)γu,vpQu,Qv≤ψpu,vfor all u,v∈E. This implies that Q is an (γ,ψ,p)-contractive mapping. From condition (B), there exists u0∈E such that γ(u0,Qu0)≥1. Next, by using condition (C), the following assertions hold for all u,v∈E: (52)γu,v≥1⟹ξut,vt≥0⟹ξQut,Qvt≥0⟹γQu,Qv≥1and hence Q is an γ-admissible mapping. Finally, from condition (D), we get that condition (iii) of Lemma 11 holds. Therefore, by Lemma 11, we find x∗∈E such that x∗=Qx∗ and so x∗ is a solution of problem (4), which completes the proof.


## 4. Applications

In this section, we present two examples, which illustrate the applicability of our main results.Throughout this section, we letE=L20,π and consider the operator A:D(A)⊆E→E defined by (53)DA=u∈E:∂u∂x,∂2u∂x2∈E,u0=uπ=0,Au=-∂2∂x2u.It is well known that -A generates a uniformly bounded C0 semigroup  (T(t))t>0 in E.Example 1.
We consider the following impulsive fractional differential equation:(54)∂α∂tαux,t-∂2∂x2ux,t=1t+52ux,t1+ux,t+15∫0te-s-t5ux,sds+15∫0te-s-t/25ux,sds,u0,t=uπ,t,t≥0,Δut=1/2=sin⁡17ux,12-,ux,0=u0x,where t∈0,1, x∈0,π, t≠1/2, ∂α/∂tα is the Caputo fractional-order partial derivative of order α,0<α≤1,u0∈L20,π. Take J:=[0,1] and so a=1.Let(55)p1t,ut,x=p2t,ut,x=ut,x,Fu=∫0te-s-t5ux,sds,Gu=∫0te-s-t/25ux,sds,ft,u,Fu,Gu=1t+52ux,t1+ux,t+Fu+Gu,Iku=sin⁡17ux,t;then the impulsive fractional differential equation (54) can be transformed into the abstract form of problem (4).Next, letu,v∈C(J,E), we calculate(56)Fu-Fv=∫0te-s-t5ux,sds-∫0te-s-t5vx,sds≤25u-v,Gu-Gv=∫0te-s-t/25ux,sds-∫0te-s-t/25vx,sds≤25u-v,ft,u,Fu,Gu-ft,v,Fv,Gv=1t+52u1+u-v1+v+225Fu-Fv+225Gu-Gv≤125u-v+225Fu-Fv+225Gu-Gv,ft,u,Fu,Gu=1t+52ux,t1+ux,t+15∫0te-s-t5ux,sds+15∫0te-s-t/25ux,sds≤125u+225u+225u=15u,p1t,u-p1t,v≤2u-v,p2t,u-p2t,v≤2u-v,p1t,u≤2u,p2t,u≤2u;hence the condition (H1), (H2) holds with K1=1/25, K2=K3=2/25, MR(t)=1/5,Ω(r)=r, b1=b2=2, and c1=c2=2. Then we have (57)Iku-Ikv=sin⁡17u-sin⁡17v≤17u-v,Iku=sin⁡17u≤17u;for any bounded set X,Y,Z∈C(J,E),I⊂J, we have (58)βkI,Ip1I,Y≤25βY,βhI,Ip1I,Z≤25βZ,βft,X,Y,Z≤125βX+225βY+225βZ.Hence the condition (H4), (H5) holds with μ=ρ=1/7, L1=1/25, L2=L3=2/25, and L4=L5=2/5. Then, by Theorem 14, problem (54) has a solution on [0,1].Example 2.
Consider the following impulsive fractional differential equation:(59)∂α∂tαux,t-∂2∂x2ux,t=t25ux,t1+ux,t+15∫0te-s-t5ux,sds+15∫0te-s-t/25ux,sds,u0,t=uπ,t,t≥0,Δut=1/2=ux,1/2-20+ux,1/2-,ux,0=u0x+∑i=1mηiux,12,where t∈0,1, x∈[0,π], t≠1/2, and ∂α/∂tα is the Caputo fractional-order partial derivative of order α, 0<α≤1, u0∈L2[0,π], 0<η1<η2<⋯<1, and ηi is given positive constants with ∑i=1mηi<2/15. Take J≔[0,1] and so a=1.Let(60)Fu=∫0te-s-t5ux,sds,Gu=∫0te-s-t/25ux,sds,ft,u,Fu,Gu=t25ux,t1+ux,t+Fu+Gu,Iku=ux,1/2-20+ux,1/2-,gu=∑i=1mηiux,12;then the impulsive fractional differential equation (59) can be transformed into the abstract form of problem (43).Next, lettingu,v∈C(J,E), we calculate (61)Fu-Fv=∫0te-s-t5ux,sds-∫0te-s-t5vx,sds≤25u-v,Gu-Gv=∫0te-s-t/25ux,sds-∫0te-s-t/25vx,sds≤25u-v,ft,u,Fu,Gu-ft,v,Fv,Gv=t25u1+u-v1+v+225Fu-Fv+225Gu-Gv≤125u-v+225Fu-Fv+225Gu-Gv,ft,u,Fu,Gu=t25ux,t1+ux,t+15∫0te-s-t5ux,sds+15∫0te-s-t/25ux,sds≤125u+225u+225u=15u,p1t,u-p1t,v≤2u-v,p2t,u-p2t,v≤2u-v,p1t,u≤2u,p2t,u≤2u;hence the condition (H1), (H2) holds with K1=1/25, K2=K3=2/25, MR(t)=1/5, and Ω(r)=r. Then we have (62)Iku-Ikv=u20+u-v20+v≤120u-v,gu-gv=∑i=1mηiu-∑i=1mηiv≤215u-v,Iku=u20+u≤120u,gu=∑i=1mηiu≤215u;for any bounded set X,Y,Z∈C(J,E),I⊂J, we have (63)βkI,Ip1I,Y≤25βY,βhI,Ip1I,Z≤25βZ,βft,X,Y,Z≤125βX+225βY+225βZ.Hence the condition (H3), (H4) holds with μ=ρ=1/7, L1=1/25, L2=L3=2/25, L4=L5=2/5, and N1=N2=2/15. Therefore, by Corollary 15, problem (59) has a solution on [0,1].


## 5. Conclusions

In this paper, we studied the existence of solutions for a class of impulsive fractional evolution equations with nonlocal conditions in Banach space by using some fixed point theorems combined with the technique of measure of noncompactness. Our results improve and generalize some known results corresponding to those obtained by others.



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*Source: 5648798-2016-12-06.xml*