# Existence Theorems for Fractional Semilinear Integrodifferential Equations with Noninstantaneous Impulses and Delay

Authors: Bo Zhu; Minhui Zhu
Journal: Journal of Function Spaces
 (2020)

---

## Abstract

In this paper, we consider a class of fractional semilinear integrodifferential equations with noninstantaneous impulses and delay. By the semigroup theory and fixed point theorems, we establish various theorems for the existence of mild solutions for the problem. An example involving partial differential equations with noninstantaneous impulses is given to show the application of our main results.


---

## Body

## 1. Introduction

Over the last couple of decades, fractional differential equations have been applied successfully to model many phenomena in physics, engineering, chemistry, financial, biology, etc. Consequently, the subject of fractional differential equations has attracted more and more attention worldwide and for more details, see for example [1–19] and the references therein.Meanwhile, differential equations with impulsive effects have been used widely as mathematical models for the study of many phenomena in physical, biology, optimal control model of economics, etc. Much attention has been paid to the existence of solutions for the differential equations with impulses in abstract space. For details, see [7, 20–28].In [26], the author initially studied the differential equations with noninstantaneous impulsive effects as follows:(1)u′t=Aut+ft,ut,t∈sk,tk+1,k=0,1,⋯,N,ut=gkt,ut,t∈tk,sk,k=1,2,⋯,N,u0=u0,where A:DA⊂E→E is the generator of a C0-semigroup of bounded operators Ttt≥0 defined on a Banach space E.In [20], the author studied the following integer order integrodifferential equations with instantaneous impulses in a Banach space E:(2)u′t=Aut+ft,uτ1t,uτ2t,⋯,uτnt,∫0tht,s,uτn+1sds,t∈0,b,t≠tk,k=1,2,⋯,m,u0+gu=u0,Δut=tk=Ikutk,k=1,2,⋯,m,where for any t≥0, the linear operator A is the infinitesimal generator of a compact, analytic semigroup, and the nonlinear term is Lipschitz continuous. The existence of mild solutions has been proved.In this paper, we investigate the following fractional semilinear integrodifferential equations with noninstantaneous impulses and delay:(3)cDtβut=Atut+ft,uτ1t,uτ2t,⋯,uτnt,∫0tht,s,uτn+1sds,t∈sk,tk+1,k=0,1,⋯,N,ut=lkt,ut,t∈tk,sk,k=1,2,⋯,N,u0+gu=u0,where β∈0,1, cDtβ is the Caputo’s fractional derivative of order β, At is a closed and linear operator with domain DA defined on a Banach space E, and the fixed points si and ti satisfying 0=s0<t1≤s1<t2≤⋯<tN≤sN<tN+1=T0 are prefixed numbers. f,lk,h,gk=1,2,⋯,N and τii=1,2,⋯,n+1 are to be specified later.Inspired by the results mentioned above, by the semigroup theory and fixed point theorems, we consider the existence of mild solutions for the fractional semilinear integrodifferential equations with noninstantaneous impulses and delay (3). In [7], the authors discussed the existence of solutions for the fractional ordinary differential equation with a generalized impulsive term. In [20–23], the authors discussed the integer or fractional differential equations with instantaneous impulses and the linear operator A is independent of t. In [26, 29, 30], the authors discussed the integer-order differential equations with noninstantaneous impulses and the linear operator A is independent of t. In [31–33], the authors discussed the fractional differential equations with noninstantaneous impulses and the linear operator A is also independent of t. In this paper, we consider the fractional semilinear integrodifferential equations with noninstantaneous impulses and delay, and the linear operator At is assumed to be dependent on t. Therefore, the mentioned results above are special cases of the problem investigated in this paper. Our results improve and generalize the results in References [7, 20–23, 26, 29–33].The rest of this paper is organized as follows. In Section2, we present the basic notation and preliminary results. In Section 3, we prove the existence of mild solutions. In Section 4, we give an illustrative example, followed by the conclusion of this paper in Section 5.


## 2. Preliminaries

LetE,· be a Banach space, J=0,T0, CJ,E=u:J⟶Eiscontinuous, PCJ,E=u:J⟶E:u∈Csk,tk+1,Eandthereexistutk−andutk+withutk−=utk,k=1,2,⋯,N with the PC-norm ∥u∥PC=sup∥ut∥:t∈J.Definition 1 [34].
The Riemann-Liouville fractional integral of orderα>0 of a function f:0,∞→R is given by(4)I0+αft=1Γα∫0tt−sα−1fsds,provided that the right-hand side is pointwise defined on 0,∞.Definition 2 [34].
The Caputo fractional derivative of orderα>0 of a function f:0,∞⟶R is given by(5)cDtαft=1Γn−α∫0tt−sn−α−1fnsds,where Γ· denotes the Gamma function, α is a fractional number, n=α+1, provided that the right-hand side is pointwise defined on R+.Definition 3 [35, 36].
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, we call At the generator of an β-resolvent family if there exist ω≥0 and a strongly continuous function Uβ:R+2⟶BE such that λβ:Reλ>ω⊂ρA and(6)λβI−As−1u=∫0∞e−λt−sUβt,sudt,Reλ>ω,u∈E.In this case,Uβt,s is called the β-resolvent family generated by At.Definition 4.
A functionu∈PCJ,E is called a mild solution of the problem (3), if u0+gu=u0, and(7)ut=Uβt,0u0−gu+∫0tUβt,sfs,uτ1s,uτ2s,⋯,uτns,∫0shs,σ,uτn+1σdσds,t∈0,t1,lkt,ut,t∈tk,sk,k=1,2,⋯,N,Uβt,sklksk,usk+∫sktUβt,sfs,uτ1s,uτ2s,⋯,uτns,∫0shs,σ,uτn+1σdσds,t∈sk,tk+1,k=1,2,⋯,N.Lemma 5.
(Sadovskii fixed point theorem). LetB be a convex, closed, and bounded subset of a Banach space E and Q:B⟶B be a condensing map. Then, Q has one fixed point in B.


## 3. Main Results

For convenience in presentation, we give here the basic assumptions to be used later throughout the paper.(H1) The function f:J×En+1⟶E is continuous, and for all r>0, there exist nonnegative Lebesgue integrable functions Li∈L1J,R+i=1,2,⋯,n+1 such that for any t∈J,xi,yi∈Tr=x∈E:∥x∥≤ri=1,2,⋯,n+1, we have(8)∥ft,x1,x2,⋯,xn+1−ft,y1,y2,⋯,yn+1∥≤∑i=1n+1Lit∥xi−yi∥.(H2) The function h:J×J×E⟶E is continuous, and for all r>0, there exists a nonnegative Lebesgue integrable function Lht,s∈L1J×J,R+ such that for any t∈J,x,y∈Tr, we have(9)∥ht,s,x−ht,s,y∥≤Lht,s∥x−y∥.(H3) τi:J⟶J are continuous such that 0≤τit≤ti=1,2,⋯,n+1 for all t∈J.(H4) The functions lk:J×E⟶E and g:PCJ,E⟶E are continuous, and for all r>0, there exist nonnegative constants Llk,Lg such that for any t∈J,x,y∈Tr,u,v∈PCJ,E, we have(10)∥lkt,x−lkt,y∥≤Llk∥x−y∥,k=1,2,⋯,N,∥gu−gv∥≤Lg∥u−v∥PC.(H4’) There is a function ϕk:J⟶R+k=1,2,3,⋯,N such that ∥lkt,x∥≤ϕkt for every t∈tk,sk and for every x∈Tr. Let Mk≔supt∈tk,skϕkt<∞,k=1,2,3,⋯,N.(H5) The function f:J×En+1⟶E satisfies the Carathéodory condition, i.e., f·,x1,x2,⋯,xn+1 is measurable for all xi∈E, and ft,·,·,⋯,· is continuous for a.e. t∈J, and for all r>0, there exists mfi∈L1J,R+i=1,2,⋯,n+1 such that ∥ft,x1,x2,⋯,xn+1∥≤∑i=1n+1mfit∥xi∥ for t∈J,xi∈Tr.(H6) The function h:J×J×E→E is continuous, and for all r>0, there exists mh∈L1J×J,R+ such that ∥ht,s,x∥≤mht,s∥x∥ for s,t∈J,x∈Tr.Theorem 6.
Assumeρ=maxΔ,MΔ+N1+N2<1 and conditions (H1)–(H4) hold, where M=max0≤s<t≤T0∥Uβt,s∥<+∞, Δ=maxLg,Llk,k=1,2,⋯,N,(11)N1=max∫0t1∑i=1nLisds,∫sktk+1∑i=1nLisds,k=1,2,⋯,N,N2=max∫0t1Ln+1s∫0t1Lhs,σdσds,∫sktk+1Ln+1s∫0tk+1Lhs,σdσds,k=1,2,⋯,N.Then, there exists a unique mild solutionu∗t of the problem (3) on PCJ,E and ∥u∗∥PC≤r, where(12)r≥maxlkt,01−Llk,M∥g0∥PC+∫0t1fs,0,0,⋯,0ds+∫0t1∫0shs,σ,0dσds1−MLg+∫0t1∑i=1nLisds+∫0t1Ln+1s∫0sLhs,σdσds,Mlksk,0+∫sktk+1fs,0,0,⋯,0ds+∫sktk+1∫0shs,σ,0dσds1−MLlk+∫sktk+1∑i=1nLisds+∫sktk+1Ln+1s∫0sLhs,σdσds,k=1,2,3,⋯,N.Proof.
Define an operatorQ:PCJ.E⟶PCJ.E by(13)Qut=Uβt,0u0−gu+∫0tUβt,sfs,uτ1s,⋯,uτns,∫0shs,σ,uτn+1σdσds,t∈0,t1,lkt,ut,t∈tk,sk,k=1,2,⋯,N,Uβt,sklksk,usk+∫sktUβt,sfs,uτ1s,⋯,uτns,∫0shs,σ,uτn+1σdσds,t∈sk,tk+1,k=1,2,⋯,N.
LetBr=u∈PCJ,E:∥u∥PC≤r. We will prove QBr⊂Br. Let u∈Br, for t∈0,t1, we have(14)Qut≤MLguPC+g0PC+M∫0t∑i=1nLisuτis+fs,0,0,⋯,0ds+M∫0tLn+1s∫0sLhs,σuτn+1σdσ+∫0shs,σ,0dσds≤Mg0PC+∫0t1fs,0,0,⋯,0ds+∫0t1∫0shs,σ,0dσds+MLg+∫0t1∑i=1nLisds+∫0t1Ln+1s∫0sLhs,σdσdsr≤r.
For allt∈sk,tk+1k=1,2,⋯,N, we have(15)Qut≤MLlkr+lksk,0+M∫sktk+1r∑i=1nLis+fs,0,0,⋯,0ds+M∫sktk+1rLn+1s∫0sLhs,σdσ+∫0shs,σ,0dσds≤Mlksk,0+∫sktk+1fs,0,0,⋯,0ds+∫sktk+1∫0shs,σ,0dσds+MLlk+∫sktk+1∑i=1nLisds+∫sktk+1Ln+1s∫0sLhs,σdσdsr≤r.
For allt∈tk,skk=1,2,⋯,N, we have(16)∥Qut∥≤Llkr+∥lkt,0∥≤r.
From the above inequalities, we have thatQu∈Br. Next, we prove that Q is a contraction map on Br. For all u,v∈PCJ,E and t∈0,t1, we have(17)∥Qut−Qvt∥≤∥Uβt,0gu−gv∥+∫0tt−sα−1∥Uβt,s∥×fs,uτ1s,uτ2s,⋯,uτns,∫0shs,σ,uτn+1σdσ−fs,vτ1s,vτ2s,⋯,vτns,∫0shs,σ,vτn+1σdσds≤MLg∥u−v∥PC+M∫0t∑i=1nLis∥uτis−vτis∥ds+M∫0tLn+1s∫0shs,σ,uτn+1σdσ−∫0shs,σ,vτn+1σdσds≤MLg∥u−v∥PC+M∫0t∑i=1nLisds∥u−v∥PC+M∫0tLn+1s∫0sLhs,σ∥uτn+1σ−vτn+1σ∥dσds≤MLg∥u−v∥PC+M∫0t∑i=1nLisds∥u−v∥PC+M∫0tLn+1s∫0sLhs,σdσds∥u−v∥PC≤MLg+∫0t1∑i=1nLisds+∫0t1Ln+1s∫0t1Lhs,σdσds∥u−v∥PC≤MLg+N1+N2∥u−v∥PC.
For allt∈sk,tk+1k=1,2,⋯,N, we have(18)∥Qut−Qvt∥≤∥Uβt,sklksk,usk−lksk,vsk∥+M∫sktfs,uτ1s,uτ2s,⋯,uτns,∫0shs,σ,uτn+1σdσ−fs,vτ1s,vτ2s,⋯,vτns,∫0shs,σ,vτn+1σdσds≤MLlk∥u−v∥PC+M∫skt∑i=1nLis∥uτis−vτis∥ds+M∫sktLn+1s∫0shs,σ,uτn+1σdσ−∫0shs,σ,vτn+1σdσds≤MLlk∥u−v∥PC+M∫skt∑i=1nLisds∥u−v∥PC+M∫sktLn+1s∫0sLhs,σ∥uτn+1σ−vτn+1σ∥dσds≤MLlk∥u−v∥PC+M∫sktk+1∑i=1nLisds∥u−v∥PC+M∫sktk+1Ln+1s∫0tk+1Lhs,σdσds∥u−v∥PC≤MLlk+N1+N2∥u−v∥PC.
For allt∈tk,skk=1,2,⋯,N, we have(19)∥Qut−Qvt∥=∥lkt,ut−lkt,vt∥≤Llk∥u−v∥PC.
From the above results, for allu,v∈PCJ,E, we have(20)∥Qu−Qv∥PC≤ρ∥u−v∥PC.
Therefore,Q:Br→Br is a contraction map and there exists a unique mild solution u∗ of the problem (3) in Br and ∥u∗∥PC≤r.Theorem 7.
Assume that the conditions (H3), (H4), (H4’), (H5), and (H6) hold, Uβt,s is compact for t,s>0 and(21)ϱ≔maxLlk,MLlk,MLg+∫0t1∑i=1nmfisds+∫0t1mfn+1s∫0t1mhs,σdσds,MMk+∫sktk+1∑i=1nmfisds+∫sktk+1mfn+1s∫0tk+1mhs,σdσds,k=1,2,⋯,N<1.Then, the problem (3) has a mild solution u∗∈PCJ,E and ∥u∗∥PC≤r, where(22)r≥maxMk,M∥u0∥+∥g0∥1−MLg+∫0t1∑i=1nmfisds+∫0t1mfn+1s∫0t1mhs,σdσds,MMk1−M∫sktk+1∑i=1nmfisds+∫sktk+1mfn+1s∫0tk+1mhs,σdσds,k=1,2,3,⋯,N.Proof.
We introduce the decompositionQ=Q1+Q2, where Qj:PCJ,E→PCJ,Ej=1,2 are defined by(23)Q1ut=Uβt,0u0−gu,t∈0,t1,lkt,ut,t∈tk,sk,k=1,2,⋯,N,Uβt,sklksk,usk,t∈sk,tk+1,k=1,2,⋯,N,Q2ut=∫0tUβt,sfs,uτ1s,⋯,uτns,∫0shs,σ,uτn+1σdσds,t∈0,t1,0,t∈tk,sk,k=1,2,⋯,N,∫sktUβt,sfs,uτ1s,⋯,uτns,∫0shs,σ,uτn+1σdσds,t∈sk,tk+1,k=1,2,⋯,N.
LetBr=u∈PCJ,E:∥u∥PC≤r, we then prove that the operator Q=Q1+Q2 is a condensing map on Br. It is easy to see that Br is a closed, bounded, and convex subset of PCJ,E.Step 1.
We proveQBr⊂Br. Let u∈Br. For t∈0,t1, we have(24)∥Qut∥≤∥Uβt,0u0−gu∥+∫0t∥Uβt,s∥×fs,uτ1s,uτ2s,⋯,uτns,∫0shs,σ,uτn+1σds≤M∥u0∥+∥g0∥+Lg∥u∥PC+M∫0t∑i=1nmfis∥uτis∥ds+M∫0tmfn+1s∫0shs,σ,uτn+1σdσds≤M∥u0∥+∥g0∥+Lgr+M∫0t∑i=1nmfisds∥u∥PC+M∫0tmfn+1s∫0smhs,σdσds∥u∥PC≤M∥u0∥+∥g0∥+Lgr+M∫0t1∑i=1nmfisds+∫0t1mfn+1s∫0t1mhs,σdσdsr≤r.For allt∈tk,skk=1,2,⋯,N, we have(25)∥Qut∥=∥Q1ut∥=∥lkt,ut∥≤Mk≤r.For allt∈sk,tk+1k=1,2,⋯,N, we have(26)∥Qut∥≤∥Uβt,sklksk,usk∥+∫sktUβt,sfs,uτ1s,uτ2s,⋯,uτns,∫0shs,σ,uτn+1σdσds≤MMk+M∫skt∑i=1nmfis∥uτis∥ds+M∫sktmfn+1s∫0shs,σ,uτn+1σdσds≤MMk+M∫skt∑i=1nmfisds∥u∥PC+M∫sitmfn+1s∫0smhs,σ∥uτn+1σ∥dσds≤MMk+M∫sktk+1∑i=1nmfisds∥u∥PC+M∫sktk+1mfn+1s∫0tk+1mhs,σdσds∥u∥PC≤MMk+M∫sktk+1∑i=1nmfisds+∫sktk+1mfn+1s∫0tk+1mhs,σdσdsr≤r.Hence,QBr⊂Br.Step 2.
We prove thatQ1 is a contraction on Br. For any u,v∈Br and t∈0,t1, we have(27)∥Q1ut−Q1vt∥≤∥Uβt,0gu−gv∥≤MLg∥u−v∥PC.
For allt∈tk,skk=1,2,⋯,N, we have(28)∥Q1ut−Q1vt∥≤Llk∥u−v∥PC.
For allt∈sk,tk+1k=1,2,⋯,N, we have(29)∥Q1ut−Q1vt∥≤∥Uβt,sklksk,usk−lksk,vsk∥≤MLlk∥u−v∥PC.Thus, sinceϱ<1, we get that Q1 is a contraction on Br.Step 3.
We prove thatQ2 is completely continuous on Br. Firstly, we prove that the operator Q2 is continuous. Let um0∞⊂Br, and um→u∈Br. For any t∈0,t1, by H3, we have(30)∥umτit−uτit∥≤∥um−u∥PC→0m→∞,i=1,2,⋯,n+1.
By (H5),(31)ft,umτ1t,umτ2t,⋯,umτnt,∫0shs,σ,umτn+1σdσ→ft,uτ1t,uτ2t,⋯,uτnt,∫0shs,σ,uτn+1σdσ,m→∞,a.e.t∈0,t1,ft,umτ1t,⋯,umτnt,∫0shs,σ,umτn+1σdσ−ft,uτ1t,uτ2t,⋯,uτnt,∫0shs,σ,uτn+1σdσ≤2r∑i=1nmfit+mfn+1t∫0tmht,sds,a.e.t∈0,t1.
By the Lebesgue dominated convergence theorem, we have(32)∥Q2umt−Q2ut∥≤M∫0tft,umτ1t,⋯,umτnt,∫0shs,σ,umτn+1σdσ−ft,uτ1t,uτ2t,⋯,uτnt,∫0shs,σ,uτn+1σdσds→0,
asm→∞. Therefore, ∥Q2um−Q2u∥PC→0, as m→∞. For t∈sk,tk+1k=1,2,⋯,N, the proof is similar to that for t∈0,t1. Hence, Q2 is continuous. By the proof of Step 1, for any t∈J, it is easy to see that Q2 maps a bounded set into a bounded set in Br.Secondly, we prove thatQ2Br is equicontinuous.Case 1.
For0,t1,0≤e1<e2≤t1,u∈Br,(33)∥Q2ue2−Q2ue1∥≤∫0e1Uβe2,s−Uβe1,s×fs,uτ1s,uτ2s,⋯,uτns,∫0shs,σ,uτn+1σdσds+∫e1e2Uβe2,s×fs,uτ1s,uτ2s,⋯,uτns,∫0shs,σ,uτn+1σdσds≔I1+I2.
By (H5), we have(34)I1≤∫0t1∑i=1nmfisds+∫0t1mfn+1s∫0t1mhs,σdσdsrsups∈0,t1∥Uβe2,s−Uβe1,s∥,I2≤M∫e1e2∑i=1nmfisds+∫e1e2mfn+1s∫0smhs,σdσdsr.Further, the compactness ofUβt,s for t,s>0 implies that the Uβt,s for t,s>0 is continuous in the sense of uniformly operator topology. Thus, I1→0 as e2→e1. It is easy to see that I2→0 as e2−e1→0 since the functions mfi∈L1J,R+i=1,2,⋯,n+1 and mht,s∈L1J×J,R+. Therefore, ∥Q2ue2−Q2ue1∥→0 as e2−e1→0.Case 2.
Forsk,tk+1k=1,2,⋯,N,sk≤e1<e2≤tk+1,u∈Br,(35)∥Q2ue2−Q2ue1∥≤∫ske1Uβe2,s−Uβe1,s×fs,uτ1s,uτ2s,⋯,uτns,∫0shs,σ,uτn+1σdσds+∫e1e2Uβe2,s×fs,uτ1s,uτ2s,⋯,uτns,∫0shs,σ,uτn+1σdσds≔I3+I4.Similar to the proof for Case1, we get ∥Q2ue2−Q2ue1∥→0 as e2−e1→0. Therefore, Q2Br is equicontinuous.Finally, we prove thatQ2Br is precompact. For every fixed t0<t≤t1 and 0<ε<t, let u∈Br and define(36)Q2,εut=∫0t−εUβt,sfs,uτ1s,uτ2s,⋯,uτns,∫0shs,σ,uτn+1σdσds.Due to the compactness ofUt,s for t,s>0, the set Yεt=Q2,εut:u∈Br is relatively compact in E for any ε0<ε<t. For any u∈Br, it follows from(37)∥Q2ut−Q2,εut∥≤∫t−εtUβt,sfs,uτ1s,⋯,uτns,∫0shs,σ,uτn+1σdσds≤M∫t−εt∑i=1nmfisds+∥mh∥∫t−εtmfn+1s∫0t1mhs,σdσdsr,thatYt=Q2ut:u∈Br is totally bounded. Thus, Yt is relatively compact in E. Similarly, we can prove that Q2Brt⊂E,t∈sk,tk+1k=1,2,⋯,N is precompact. Using the Arzelà-Ascoli theorem, we have that Q2:PCJ,E→PCJ,E is completely continuous. Hence, the operator Q=Q1+Q2 is a condensing operator on Br. By Lemma 5, Q has a fixed point u∗ in Br, which is a mild solution of the problem (3).Remark 8.
In this paper, we consider the fractional semilinear integrodifferential equations with noninstantaneous impulses and delay, and the linear operatorAt is assumed to be dependent on t. In conditions (H1) and (H2), Li∈L1J,R+i=1,2,⋯,n+1 and Lht,s∈L1J×J,R+ are Lebesgue integrable functions, not constants. The mentioned results above are special cases of the problem investigated in this paper.


## 4. An Application

In order to show the application of the main results of this paper, we consider the following problem:(38)cDtαux,t=t∂2∂x2ux,t+t1+t2ux,sint+11+t2∫0tux,sinsds1/2,t,x∈∪i=1Nsi,ti+1×0,π,ux,t=Git,ux,t,x∈0,π,t∈ti,si,u0,t=uπ,t=0,0≤t≤1,ux,0+13ucosu=u0x,0≤x≤π,where 0=s0<t1≤s1<⋯<tN≤sN<tN+1=1,0<α≤1, u0x∈E=L20,π and the operator A:DA⊂E→E is defined by Atz=t∂2u/∂x2 with(39)DA=u∈E:u′′∈E,u0=uπ=0.It is well known thatAt generates a β-resolvent family on E.Let(40)ut=u·,t,ft,x,y=t1+t2xsint+11+t2y1/2,lit,u=Git,u·,t,gu=13ucosu.Then, (38) takes the abstract form (3). Assume that all the conditions of Theorems 6 and 7 are satisfied, then the problem (38) has a mild solution on 0,1.


## 5. Conclusions

This paper investigates the existence of mild solutions for a class of fractional semilinear integrodifferential equations with noninstantaneous impulses and delay by the fixed point theorem and the semigroup theory. In [7, 20–23, 26, 29–33], the linear operator A is independent of t, while in this paper, the linear operator At is allowed to be dependent on t. Therefore, the differential equations in [7, 20–23, 26, 29–33] are special cases of the problem investigated in this paper. Hence, our results improve and generalize the results of References [7, 20–23, 26, 29–33].



---

*Source: 2914269-2020-08-04.xml*