# Impulsive Fractional Semilinear Integrodifferential Equations with Nonlocal Conditions

Authors: Xue Wang; Bo Zhu
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This paper is devoted to a class of impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions. Based on the semigroup theory and some fixed point theorems, the existence theory of PC-mild solutions is established under the condition of compact resolvent operator. Furthermore, the uniqueness of PC-mild solutions is proved in the case of the noncompact resolvent operator.


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## Body

## 1. Introduction

The fractional evolution equation has been applied to many fields, and scholars have obtained abundant research achievements [1–13]. Impulsive fractional integrodifferential equations can describe some phenomena which often occur in physics, geology, and economics, for instance, earthquake, the closing of the switch in the circuit, and so on. Many scholars are committed to this subject and have achieved plentiful results [1–7]. Based on the fact that nonlocal initial conditions are more effective than classical initial conditions in applied physics, the study of differential equations with nonlocal conditions has attracted more and more researchers’ attention [8–13].Ji and Li [14] studied the following impulsive differential evolution equations with nonlocal conditions:
(1)u′t=Aut+ft,ut,t∈0,b,t≠ti,Δut=ti=Iiuti,i=1,2,⋯,m,u0=gu,where A is the generator of a strongly continuous semigroup Tβt; sufficient conditions for the existence of mild solutions have been established by the Hausdorff measure of noncompactness and fixed point theorems.Zhu et al. [15] investigated the fractional semilinear integrodifferential equations of mixed type with nonlocal conditions:
(2)Dctβut=Atut+ft,ut,Gut,Sut,t∈0,T0,u0+gu=u0,where 0<β≤1, At is a closed linear operator with domain DA defined on a Banach space E; the existence and uniqueness of mild solutions have been established by k-set contraction and β-resolvent family.Gou and Li [16] studied the fractional impulsive integrodifferential equations in Banach space E; local and global existences of mild solutions have been proved by measure of noncompactness and Sadovskii’s fixed point theorem:
(3)Dctβut+Aut=ft,ut+∫0tqt−sgs,usds,t≥0,t≠tk,Δut=tk=Ikutk−,k=1,2,⋯,m,u0=u0∈E,where 0<β<1, A:DA⊂E⟶E is a closed linear operator and −A generates a uniformly bounded C0-semigroup Tt.Inspired by these contributions, we consider the following impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions:(4)Dctβxt−Atxt=ft,xt,Gxt,Hxt,t∈0,T,t≠tk,x0+wx=x0,Δxtk=Ikxtk,k=1,2,⋯,m,where Dctβ is the Caputo’s fractional derivative of order β, β∈0,1, At is a closed linear operator with domain DA defined on a Banach space E, and two integral operators G and H are defined by
(5)Gxt=∫0tgt,s,xsds,Hxt=∫0Tht,s,xsds,g:B×E⟶E, h:B0×E⟶E are continuous and nonlinear functions, B=t,s∣0≤s≤t≤T, B0=t,s∣0≤t,s≤T, f and ω are to be specified later, Ik:E⟶Ek=1,2,⋯,m are continuous impulsive functions, the prefixed numbers tkk=1,2,⋯,m satisfy 0=t0<t1<t2<⋯<tm<tm+1=T, xtk=xtk−, and xtk−=limh⟶0−xtk+h represent the left limit of xt at t=tk.In this paper, we demonstrate the existence of PC-mild solutions for problem (4) via the theory of semigroup and fixed point theorem under the condition of compact resolvent operator. Meanwhile, the uniqueness of PC-mild solutions is proved in the case of noncompact resolvent operator. The kernels g and h of the integral operators G and H are nonlinear functions; the function ω of the nonlocal conditions is noncompact. In addition, the closed linear operator At is dependent on t. The rest of this paper is organized as follows. In Section 2, some basic definitions and lemmas are collected that will be needed throughout the remaining sections. The existence and uniqueness of PC-mild solutions are shown in Section 3 via the theories of resolvent operators and various fixed point theorems. Finally, the summary of our results comes in Section 4.


## 2. Preliminaries

LetE,· be a Banach space, J=0,T and 0<T<∞. The collection of all continuous functions from J into E, denoted CJ,E, is a Banach space equipped with the norm xC=maxxt,t∈J for x∈CJ,E. Let PCJ,E=x∣x:J⟶E:x∈Ctk,tk+1,E,andthereexistxtk−andxtk+withxtk=xtk−,k=1,⋯,m endowed with the PC-norm xPC=supxt,t∈J, J0=0,t1, J1=t1,t2,⋯,Jm=tm,T.Lemma 1 (nonlinear alternative for single-valued maps).
LetE be a Banach space, C⊂E be a closed convex set, V be an open subset of C, and 0∈V. Suppose that Q:V¯⟶C is completely continuous, then either
(i)
Q has a fixed point in V¯ or(ii)
there is au∈∂V and λ∈0,1 with u=λQuLemma 2 (see [17]).
Let0<η<1, γ>0, denote
(6)Sn=ηn+Cn1ηn−1γ+Cn2ηn−22!γ2+⋯+γnn!,n∈N,where Cnk=n!/k!n−k!. Then, for any fixed constant 0<ξ<1 and any real number s>1, we get
(7)Sn≤Oξnn+o1ns=o1ns,asn⟶∞.Definition 3 (see [18, 19]).
The Caputo fractional derivative of orderβ of a function f:0,∞⟶R is defined as
(8)Dctβft=1Γn−β∫0tt−sn−β−1fnsds,where n−1<β<n, n∈N, Γ· denotes the Gamma function. The Laplace transform of the Caputo fractional derivative of order β is given as
(9)LDctβfts=sβLfs−∑j=1n−1sβ−j−1xj0,n−1<β≤n,where Lfs=∫0∞e−stftdt is the Laplace transform of the function ft.Definition 4 (see [20, 21]).
LetAt be a closed and linear operator with domain DA defined on a Banach space E and β>0. Let ρAt be the resolvent set of At; At is called the generator of a β-resolvent family if there exist ω≥0 and a strongly continuous function Uβ:ℝ+2⟶BE such that λβ:Reλ>ω⊂ρA and
(10)λβI−As−1x=∫0∞e−λt−sUβt,sxdt,Reλ>ω,x∈E.
In this case,Uβt,s is called the β-resolvent family generated by At.Lemma 5 (see [21, 22]).
Uβt,s satisfies the following properties:
(i)
Uβs,s=I, Uβt,s=Uβt,rUβr,s, for 0≤s≤r≤t≤a(ii)
t,s⟶Uβt,s is strongly continuous for 0≤s≤t≤a(iii)
IfUβt,s is compact for t,s>0, then the Uβt,s is continuous in the uniform operator topologyDefinition 6.
A functionx∈PCJ,E is said to be a PC-mild solution of problem (4) if xt satisfies the integral equation:
(11)xt=Uβt,0x0−ωx+∫0tUβt,sfs,xs,Gxs,Hxsds+∑0<tk<tUβt,tkIkxtk,t∈J.


## 3. Existence and Uniqueness of Mild Solution

Theorem 7.
Assume that the conditionsH1-H3 hold true and the resolvent operator Uβt,st,s>0 is compact.
H1 The function f:J×E×E×E⟶E is continuous, and there exist nonnegative Lebesgue integrable functions a, li∈LJ,R+i=1,2,3, for every t∈J, xi∈E, such that
(12)ft,x1,x2,x3≤at+l1tx1+l2tx2+l3tx3.
H2 There exist nonnegative Lebesgue integrable functions b, c, li∈LJ,ℝ+i=4,5, for all x∈E, such that
(13)gt,s,x≤bt+l4tx,t,s∈B,ht,s,x≤ct+l5tx,t,s∈B0.
H3 The functions ω:PCJ,E⟶E and Ik:E⟶E are continuous, and there exist constants dω, eω, dk, ek>0, such that
(14)ωx≤dωxPC+eω,x∈PCJ,E,Ikx≤dkx+ek,x∈E,k=1,2⋯m.
Then, problem (4) has at least one PC-mild solution in PCJ,E.Proof.
Let us consider the operatorQ:PCJ,E⟶PCJ,E as follows:
(15)Qxt=Uβt,0x0−ωx+∫0tUβt,sfs,xs,Gxs,Hxsds+∑0<tk<tUβt,tkIkxtk.
It is easy to see that the operatorQ is well defined in PCJ,E.
At first, we claim thatQ:PCJ,E⟶PCJ,E is a continuous operator. Let xn0∞⊂PCJ,E be a sequence such that xn⟶xn⟶∞ in PCJ,E. Since for all t∈J,
(16)Qxnt−Qxt≤Uβt,0ωxn−ωx+∫0tUβt,sfs,xns,Gxns,Hxns−fs,xs,Gxs,Hxsds+∑k=1mUβt,tkIkxntk−Ikxtk≤Mωxn−ωx+M∫0tfs,xns,Gxns,Hxns−fs,xs,Gxs,Hxsds+∑k=1mMIkxntk−Ikxtk,where M=max0≤s≤t≤TUβt,s. Using the fact that f:J×E×E×E⟶E, ω:PCJ,E⟶E, and Ik:E⟶Ek=1,2⋯m are continuous, we obtain
(17)Qxn−QxPC⟶0,asn⟶∞.
Therefore,Q:PCJ,E⟶PCJ,E is continuous.
Furthermore, for anyR>0, we prove that QTR is equicontinuous in Jkk=0,1,2⋯m. For all x∈TR=x∈PCJ,E:xPC≤R and τ1,τ2∈Jkτ1≤τ2, by the condition H3, we have
(18)Qxτ2−Qxτ1≤Uβτ2,0−Uβτ1,0x0−ωx+∫τ1τ2Uβτ2,sfs,xs,Gxs,Hxsds+∫0τ1Uβτ2,s−Uβτ1,sfs,xs,Gxs,Hxsds+∑0<tk<τ2Uβτ2,tkIkxtk−∑0<tk<τ1Uβτ1,tkIkxtk≤Uβτ2,0−Uβτ1,0x0+dωR+eω+∫τ1τ2Mfs,xs,Gxs,Hxsds+sups∈JkUβτ2,s−Uβτ1,s∫0τ1fs,xs,Gxs,Hxsds+∑k=1mUβτ2,tk−Uβτ1,tkIkxtk≕I1+I2+I3+I4.
For allx∈TR, s∈J, we get ∥xs∥≤R, by the condition H2,
(19)Gxs≤∫0sgs,ν,xνdν≤∫0sbν+l4νxνdν≤∫0Tbνdν+R∫0Tl4νdν,meanwhile,
(20)Hxs≤∫0Ths,ν,xνdν≤∫0Tcν+l5νxνdν≤∫0Tcνdν+R∫0Tl5νdν.
According to the conditionH1 and the above inequalities, for all s∈J, we get
(21)fs,xs,Gxs,Hxs≤as+l1sxs+l2sGxs+l3sHxs≤as+l1sR+l2s∫0Tbνdν+R∫0Tl4νdν+l3s∫0Tcνdν+R∫0Tl5νdν≤a1s+b1sR,where
(22)a1s=as+l2s∫0Tbνdν+l3s∫0Tcνdν,b1s=l1s+l2s∫0Tl4νdν+l3s∫0Tl5νdν.
Obviously,a1s and b1s are nonnegative Lebesgue integrable functions, then
(23)I2=∫τ1τ2Mfs,xs,Gxs,Hxsds≤M∫τ1τ2a1s+b1sRds,I3=sups∈JkUβτ2,s−Uβτ1,s∫0τ1fs,xs,Gxs,Hxsds≤sups∈JkUβτ2,s−Uβτ1,s∫0τ1a1s+b1sRds,I4=∑k=1mUβτ2,tk−Uβτ1,tkIkxtk≤∑k=1mUβτ2,tk−Uβτ1,tkdkR+ek.
In view of Lemma5, the compactness of the resolvent operator Uβt,st,s>0 implies the continuity in the uniform operator topology. As a result, from the above inequalities, we deduce that Qxτ2−Qxτ1⟶0 independently of x∈TR as τ2−τ1⟶0. That is, QTR is equicontinuous in Jkk=0,1,2⋯m.
In the end, we demonstrate thatQTR⊂PCJ,E is precompact.
For anyt0<t≤T, 0<ε<t, and x∈TR, the operator Qεx is defined by
(24)Qεxt=Uβt,0x0−ωx+∫0t−εUβt,sfs,xs,Gxs,Hxsds+∑0<tk<tUβt,tkIkxtk,t∈J.
SinceUβt,s is compact resolvent operator, the set Yεt=Qεxt:x∈TR is relatively compact in E for every ε (0<ε<t).
Moreover, for anyx∈TR, t∈J, one can show that
(25)Qxt−Qεxt=∫t−εtUβt,sfs,xs,Gxs,Hxsds≤M∫t−εtfs,xs,Gxs,Hxsds≤M∫t−εta1s+b1sRds.
Thus,Yt=Qxt:x∈TR is totally bounded. Hence, Yt is relatively compact in E, and so, with the help of the Arzelà-Ascoli theorem, Q:PCJ,E⟶PCJ,E is completely continuous.
For0<λ<1, let x=λQx, we get
(26)xt=λUβt,0x0−ωx+λ∫0tUβt,sfs,xs,Gxs,Hxsds+λ∑0<tk<tUβt,tkIkxtk.
Then, using the conditionsH1-H3, it follows that
(27)xt≤Mx0−ωx+M∫0tfs,xs,Gxs,Hxsds+∑k=1mMIkxtk≤Mx0+dωR+eω+M∫0ta1s+b1sRds+∑k=1mMdkR+ek≤Mx0+eω+M∫0Ta1sds+∑k=1mMek+Mdω+M∫0Tb1sds+∑k=1mMdkR≕ρ.
That is,xt≤ρ for t∈J, then there exists a constant ρ1>ρ such that xPC≠ρ1. Let V=x∈PCJ,E:xPC<ρ1, obviously, there is no x∈∂V such that x=λQx for 0<λ<1. Therefore, thanks to Lemma 1, one gets that Q has at least one fixed point x in V, which is a PC-mild solution of problem (4). This completes the proof. ☐Remark 8.
Theorem7 is proved under the condition that Uβt,s is compact for t,s>0 and the functions f,g,h meet corresponding conditions; in the case that the resolvent operator Uβt,s is noncompact, we would obtain Theorem 9 and Theorem 10.Theorem 9.
Suppose that the conditionsH4-H6 are satisfied, M=max0≤s<t≤TUβt,s, and MLω+∫0TL1sds+∫0TL2sds∫0TL4vdv+∫0TL3sds∫0TL5νdν+∑k=1mLIk<1.
H4 The function f:J×E×E×E⟶E is continuous, and there exist nonnegative Lebesgue integrable functions Li∈LJ,ℝ+i=1,2,3, for any ui, vi∈E, t∈J, such that
(28)ft,u1,u2,u3−ft,v1,v2,v3≤L1tu1−v1+L2tu2−v2+L3tu3−v3.
H5 There exist nonnegative Lebesgue integrable functions L4,L5∈LJ,ℝ+, for each u, v∈E, such that
(29)gt,s,u−gt,s,v≤L4tu−v,t,s∈B,ht,s,u−ht,s,v≤L5tu−v,t,s∈B0.
H6 The functions Ik:E⟶E and ω:PCJ,E⟶E are continuous, and there exist nonnegative constants Lω,LIk>0, such that
(30)ωu−ωv≤Lωu−vPC,u,v∈PCJ,E,Iku−Ikv≤LIku−v,u,v∈E,k=1,2⋯m.
Then, problem (4) has a unique PC-mild solution x⋆ in PCJ,E.Proof.
It follows from the conditionsH4-H6, for any u,v∈PCJ,E, t∈J, one can derive
(31)Qut−Qvt≤Uβt,0ωu−ωv+∑0<tk<tUβt,tkIkutk−Ikvtk+∫0tUβt,sfs,us,Gus,Hus−fs,vs,Gvs,Hvsds≤Mωu−ωv+∑k=1mMIkutk−Ikvtk+M∫0tfs,us,Gus,Hus−fs,vs,Gvs,Hvsds≤MLωu−vPC+∑k=1mMLIkutk−vtk+M∫0tL1sus−vs+L2sGus−Gvs+L3sHus−Hvsds≤MLωu−vPC+∑k=1mMLIku−vPC+M∫0tL1sdsu−vPC+M∫0tL2s∫0sL4νdνds+M∫0tL3s∫0TL5νdνdsu−vPC≤MLω+∑k=1mLIk+∫0TL1sds+∫0TL2sds∫0TL4νdν+∫0TL3sds∫0TL5νdνu−vPC.
Based on the assumption, we haveQu−QvPC<u−vPC, which means that the operator Q is a contraction mapping. Hence, the operator Q has a unique fixed point x⋆∈PCJ,E, which implies that problem (4) has a unique PC-mild solution. This completes the proof. ☐Theorem 10.
Assume that the conditionsH7 and H8 hold, ω≡0, M=max0≤s<t≤TUβt,s, a=∑k=1mMLIk<1.
H7 The function f:J×E×E×E⟶E is continuous, and there exist nonnegative Lebesgue integrable functions L′1,L′2∈LJ,ℝ+, for all ui, vi∈E, t∈J, such that
(32)ft,u1,u2,u3−ft,v1,v2,v3≤L′1tu1−v1+L′2tu2−v2.
H8 There exist constants Lg,LIk>0, for all u,v∈E, satisfying
(33)gt,s,u−gt,s,v≤Lgu−v,t,s∈B,Iku−Ikv≤LIku−v,k=1,2⋯m.
Then, problem (4) has a unique PC-mild solution x⋆ in PCJ,E. For all t∈J, x0∈PCJ,E, iterative sequence xnt are defined by
(34)xnt=Uβt,0x0+∫0tUβt,sfs,xn−1s,Gxn−1s,Hxn−1sds+∑0<tk<tUβt,tkIkxn−1tk,n=1,2,⋯,uniformly converge to the unique PC-mild solution x⋆t in t∈J, and for any s>0,
(35)xn−x⋆PC=o1ns,asn⟶∞.Proof.
Combining the conditionsH7 and H8, for all t∈J, u,v∈PCJ,E, we get
(36)Qut−Qvt≤M∫0tfs,us,Gus,Hus−fs,vs,Gvs,Hvsds+∑0<tk<tMIkutk−Ikvtk≤M∫0tL1′s+L2′sLgsus−vsds+∑k=1mMLIkutk−vtk≤∫0tLsus−vsds+∑k=1mMLIku−vPC≤∫0tLsds+∑k=1mMLIku−vPC≤∫0tLsds+au−vPC,where Ls=ML1′s+L2′sLgT. It is easy to see that L∈LJ,ℝ+. Notice that a=∑k=1mMLIk<1, then there is ε>0 such that 0<b=ε+a<1. For the above ε>0, there exists a continuous function ϕs such that
(37)∫0TLs−ϕsds<ε.
Consequently,(38)Qut−Qvt≤∫0tLs−ϕsds+∫0tϕsdsu−vPC+au−vPC≤ε+Φtu−vPC+au−vPC≤b+Φtu−vPC=C10b+C11Φt1!u−vPC,where Φ=maxϕs:s∈J. We next prove the following inequalities, for every positive integer n and t∈J,
(39)Qnut−Qnvt≤Cn0bn+Cn1bn−1Φt1!+Cn2bn−2Φt22!+⋯+Cnnb0Φtnn!u−vPC,where Cnm=n!/m!n−m!.
Assume that, for any positive integerk, we have
(40)Qkut−Qkvt≤Ck0bk+Ck1bk−1Φt1!+Ck2bk−2Φt22!+⋯+Ckkb0Φtkk!u−vPC.
By the formulaCk+1m=Ckm+Ckm−1, for all t∈J,
(41)Qk+1ut−Qk+1vt=QQkut−QQkvt≤M∫0tfs,Qkus,GQkus,HQkus−fs,Qkvs,GQkvs,HQkvsds+∑0<tk<tMIkQkutk−IkQkvtk≤M∫0tL1′s+L2′sLgsQkus−Qkvsds+∑k=1mMLIkQkutk−Qkvtk≤∫0tLsQkus−Qkvsds+∑k=1mMLIkQkutk−Qkvtk≤∫0tLs−ϕsCk0bk+Ck1bk−1Φs1!+Ck2bk−2Φs22!+⋯+Ckkb0Φskk!dsu−vPC+∫0tϕsCk0bk+Ck1bk−1Φs1!+Ck2bk−2Φs22!+⋯+Ckkb0Φskk!dsu−vPC+∑k=1mMLIkCk0bk+Ck1bk−1Φtk1!+Ck2bk−2Φtk22!+⋯+Ckkb0Φtkkk!u−vPC≤εCk0bk+Ck1bk−1Φt1!+Ck2bk−2Φt22!+⋯+Ckkb0Φtkk!u−vPC+Φ∫0tCk0bk+Ck1bk−1Φs1!+Ck2bk−2Φs22!+⋯+Ckkb0Φskk!dsu−vPC+aCk0bk+Ck1bk−1Φt1!+Ck2bk−2Φt22!+⋯+Ckkb0Φtkk!u−vPC≤bCk0bk+Ck1bk−1Φt1!+Ck2bk−2Φt22!+⋯+Ckkb0Φtkk!u−vPC+Φ∫0tCk0bk+Ck1bk−1Φs1!+Ck2bk−2Φs22!+⋯+Ckkb0Φskk!dsu−vPC≤Ck+10bk+1+Ck+11bkΦt1!+Ck+12bk−1Φt22!+⋯+Ck+1k+1b0Φtk+1k+1!u−vPC.
By mathematical induction, for every positive integern, we obtain
(42)Qnu−QnvPC≤Cn0bn+Cn1bn−1Φt1!+Cn2bn−2Φt22!+⋯+Cnnb0Φtnn!u−vPC.
Using Lemma2, it follows that
(43)Qnu−QnvPC≤o1nsu−vPC,asn⟶∞.
Thus, for any fixed constants>1, we can find a positive integer n0 such that, for any u,v∈PCJ,E and n>n0, we have
(44)Qnu−QnvPC≤1nsu−vPC.
Applying the general Banach contraction mapping principle, we deduce that the operatorQ has a unique fixed point x⋆ in PCJ,E, which means that problem (4) has a unique PC-mild solution x⋆ in PCJ,E. This completes the proof. ☐


## 4. Conclusion

In this paper, we demonstrate the existence theory of PC-mild solutions for the impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions (4) via the theory of semigroup and fixed point theorem under the condition of compact resolvent operator. Meanwhile, the uniqueness of PC-mild solutions is proved under the condition of noncompact resolvent operator. The kernels g and h of the integral operators G and H are nonlinear functions, and the function ω of the nonlocal initial conditions is noncompact. In addition, the closed linear operator At is dependent on t. As a consequence, our main theorems improve and generalize many existing results on this topic.



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