# Instantaneous and Noninstantaneous Impulsive Integrodifferential Equations in Banach Spaces

Authors: Saïd Abbas; Mouffak Benchohra; Gaston M. N'Guérékata
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

This paper deals with some existence of mild solutions for two classes of impulsive integrodifferential equations in Banach spaces. Our results are based on the fixed point theory and the concept of measure of noncompactness with the help of the resolvent operator. Two illustrative examples are given in the last section.


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## Body

## 1. Introduction

The existence of mild solutions is developed in [1, 2] for some semilinear functional differential equations. There has been a significant development in functional evolution equations in recent years (see the monographs [3–5], the papers [6–12], and the references therein).The study of an abstract nonlocal Cauchy problem was initiated by Byszewski [13] in 1991. Evolution equations with nonlocal initial conditions were motivated by physical problems. As a matter of fact, it is demonstrated that the evolution equations with nonlocal initial conditions have better effects in applications than the classical Cauchy problems. For example, it was pointed in [14, 15] that the nonlocal problems are used to represent mathematical models for evolution of various phenomena, such as nonlocal neural networks, nonlocal pharmacokinetics, nonlocal pollution, and nonlocal combustion. Due to nonlocal problems having a wide range of applications in real-world applications, evolution equations with nonlocal initial conditions were studied by many authors. Xue [16] studied the existence of mild solutions for semilinear differential equations with nonlocal initial conditions in separable Banach spaces. Xue discussed the semilinear nonlocal differential equations when the semigroup Tt generated by the coefficient operator is compact and the nonlocal term g is not compact. Fan and Li [4] discussed the existence for impulsive semilinear differential equations with nonlocal conditions by using Sadovskii’s fixed point theorem and Schauder’s fixed point theorem.Recently, several researchers obtained other results by application of the technique of measure of noncompactness (see [17–19] and the references therein).Impulsive differential equations have become more important in recent years in some mathematical models of real phenomena, especially in biological or medical domains, and in control theory (see, for example, the monographs [20–22] and the papers [23–25]). In this paper, we first discuss the existence of mild solutions for the following nonlocal problem of impulsive integrodifferential equations
(1)u′t=Aut+∫0tYt−susds+ft,ut;t∈Ik,k=0,⋯,m,utk+=utk−+Lkutk−;k=1,⋯,m,u0+gu=u0∈E,where I0=0,t1,Ik≔tk,tk+1;k=1,⋯,m,0=t0<t1<⋯<tm<tm+1=T,f:Ik×E⟶E;k=1,⋯,m,Lk:E⟶E;k=1,⋯,m,g:PC⟶E are given functions, the set PC is given later, E is a real (or complex) Banach space with norm ∥·∥, u′t≔du/dt,A:DA⊂E⟶E generates a C0-semigroup on the Banach space E, and Yt is a closed linear operator on E with DA⊂DY.In [26–29] the authors initially offered to study some classes of impulsive differential equations with noninstantaneous impulses. Motivated by the above papers, we next discuss the existence of mild solutions for the following nonlocal problem of noninstantaneous impulsive integrodifferential equations:
(2)u′t=Aut+∫0tYt−susds+ft,ut;t∈Ik,k=0,⋯,m,ut=gkt,utk−;t∈Jk,k=1,⋯,m,usk+gu=uk∈E;k=0,⋯,m,where I0≔0,t1,Jk≔tk,sk,Ik≔sk,tk+1;k=1,⋯,m,f:Ik×E⟶E,gk:Jk×E⟶E are given functions such that gkt,utk−t=sk=uk∈E;k=1,⋯,m,g:PC⟶E is a given function, the set PC is given later, and 0=s0<t1≤s1<t2≤s2<⋯≤sm−1<tm≤sm<tm+1=T.


## 2. Preliminaries

ByBE, we denote the space of the bounded linear operator from E into itself. Let CI≔CI,E be the Banach space of continuous functions from I≔0,T into E. Let L∞I be the Banach space of measurable functions v:I⟶ℝ that are essentially bounded and equipped with the norm
(3)∥v∥L∞=infc>0:∣vt∣≤c,a.e.t∈I.Consider the Banach space(4)PC=u:I⟶E:u∈CIk;k=0,⋯,m,andthereexistutk−andutk+;k=1,⋯,m,withutk−=utkwith the norm
(5)∥u∥PC=supt∈I∥ut∥.A semigroup of bounded linear operatorsTt is uniformly continuous if
(6)limt→0∥Tt−I∥E=0.Here,I denotes the identity operator in E.We note that if a semigroupTt is of class C0, then it satisfies the growth condition∥Tt∥BE≤Meβt, for 0≤t<∞ with some constants M>0 and β≥0.If, in particular,M=1 and β=0, i.e, ∥Tt∥BE≤1, for t≥0, then the semigroup Tt is called a contraction semigroup. For more details on strongly continuous operators, we refer the reader to the books [30, 31].LetMX denote the class of all bounded subsets of a metric space X.Definition 1.
LetXbe a complete metric space. A mapμ:MX⟶0,∞is called a measure of noncompactness onXif it satisfies the following properties for allB,B1,B2∈MX.
(a)
μB=0 if and only if B is precompact (regularity)
(b)
μB=μB¯ (invariance under closure)
(c)
μB1∪B2=maxμB1,μB2 (semiadditivity)Definition 2 [32].
LetXbe a Banach space and letΩXbe the family of bounded subsets ofX. The Kuratowski measure of noncompactness is the mapμ:ΩX⟶0,∞defined by
(7)μM=infε>0:M⊂∪j=1mMj,diamMj≤ε,where M∈ΩX.For our purpose, we will need the following fixed point theorem:Theorem 3.
(Monch’s fixed point theorem [33]). Let Dbe a bounded, closed, and convex subset of a Banach space such that0∈Dand letNbe a continuous mapping ofDinto itself. If the implication
(8)V=conv¯NVorV=NV∪0⇒V¯iscompact,holds for every subset V of D, then N has a fixed point.


## 3. Mild Solutions with Instantaneous Impulses

In this section, we are concerned with the existence results of the problem (1).Definition 4 [10].
A resolvent operator for the Cauchy problem
(9)u′t=Aut+∫0tYt−susds;t∈0,∞,u0=u0∈E,is a bounded linear operator-valued function Rt∈BE;t≥0, verifying the following conditions:
(i)
R0=I (the identity map of E) and ∥Rt∥≤Neνt for some constants N>0, and ν∈ℝ
(ii)
For eachu∈E,Rtu is strongly continuous for t≥0
(iii)
Rt is bounded for t≥0. For u∈DA,R·u∈Cℝ+,DA∩C1ℝ+,Eand R′tu=ARtu+∫0tYt−sRsuds=RtAu+∫0tRt−sYsuds;t∈0,∞Let us introduce the following hypotheses:R1 The operator A is the infinitesimal generator of a uniformly continuous semigroup Stt≥0R2 For all t≥0,Yt is a closed linear operator from DA to E and Yt∈BE. For any u∈E, the map t↦Ytu is bounded differentiable and the derivative t↦Y′tu is bounded uniformly continuous on ℝ+.Theorem 5 [10, 34].
Assume thatR1 and R2 hold. Then, there exists a unique uniformly continuous resolvent operator for the Cauchy problem (9).Definition 6 [34].
By a mild solution of the problem (1), we mean a function u∈PC that satisfies
(10)ut=Rtu0−gu+∫0tRt−sfs,usds+∑0<ti<tRt−tiLiuti;t∈I.The following hypotheses will be used in the sequel.H1 The function t↦ft,u is measurable on I for each u∈E, and the function u↦ft,u is continuous on E for a.e. t∈Ik,H2 There exists a function p∈L∞I, such that
(11)∥ft,u∥≤pt1+∥u∥;fora.e.t∈Ik,andeachu∈EH3 There exist positive constants q∗,lk∗;k=0,⋯,m, such that
(12)∥gu∥≤q∗1+∥u∥PC;foreachu∈PC,and
(13)∥Lku∥≤lk∗1+∥u∥;k=0,⋯,m,fora.e.t∈I,andeachu∈E,H4 For each bounded set B⊂E, we have
(14)μft,B≤ptμB,μLkB≤lk∗μB;k=0,⋯,m,and for each bounded set B1⊂PC, we have
(15)μgB1≤q∗supt∈IμB1t,where B1t=ut:u∈B1;t∈I.Set(16)p∗=∥p∥L∞,andM=supt∈I∥Rt∥BE.Theorem 7.
Assume that the hypothesesR1,R2,H1–H4 hold. If
(17)ℓ≔Mq∗+Tp∗+∑k=0mlk∗<1,then problem (1) has at least one mild solution defined on I.Proof.
Transform problem (1) into a fixed point problem. Consider the operator N:PC⟶PC defined by
(18)Nut=Rtu0−gu+∫0tRt−sfs,usds+∑0<ti<tRt−tiLiuti;t∈I.
Letρ>0, such that
(19)ρ≥M∥u0∥+q∗+Tp∗+∑k=0mlk∗1−Mq∗+Tp∗+∑k=0mlk∗,and consider the ball Bρ≔B0,ρ=w∈PC:∥w∥PC≤ρ.
For anyu∈Bρ and each t∈I, we have
(20)∥Nut∥≤∥Rt∥BE∥u0∥+∥gu∥+∫0t∥Rt−s∥BE∥fs,us∥ds+∑0<ti<t∥Rt−ti∥BE∥Liuti∥≤M∥u0∥+1+ρq∗+Tp∗+∑k=0mlk∗:≤ρ.
Thus,(21)∥Nu∥PC≤ρ.This proves thatN transforms the ball Bρ into itself. We shall show that the operator N:Bρ⟶Bρ satisfies all the assumptions of Theorem 3. The proof will be given in three steps.Step 1.
N:Bρ⟶Bρis continuous.
Letunn∈ℕ be a sequence such that un⟶u as n⟶∞ in Bρ. Then, for each t∈I, we have
(22)Nunt−Nut≤∥Rt∥BE∥gun−gu∥+∫0t∥Rt−s∥BE∥fs,uns−fs,us∥ds+∑0<ti<t∥Rt−ti∥BE∥Liunti−Liuti∥≤M∥gun−gu∥+M∫0T∥fs,uns−fs,us∥ds+M∑0<ti<t∥Liunti−Liuti∥.
Sinceun⟶u as n⟶∞ and f,g,Li are continuous, the Lebesgue-dominated convergence theorem implies that
(23)∥Nun−Nu∥PC⟶0asn⟶∞.Step 2.
NBρis bounded and equicontinuous.
SinceNBρ⊂Bρ and Bρ is bounded, then NBρ is bounded.
Next, lett,τ∈I, τ<t and let u∈Bρ. Thus, we have
(24)Nut−Nuτ≤∥Rt−Rτ∥BE∥u0∥+∥gu∥+∫0τ∥Rt−s−Rτ−s∥BE∥fs,us∥ds+∫τt∥Rt−s∥BE∥fs,us∥ds+∑0<ti<t∥Rt−ti−Rτ−ti∥BE∥Liuti∥.
Hence, we get(25)Nut−Nuτ≤∥u0∥+q∗1+ρ∥Rt−Rτ∥BE+p∗1+ρ∫0τ∥Rt−s−Rτ−s∥BEds+Mp∗1+ρt−τ+∑0<ti<tli∗1+ρ∥Rt−ti−Rτ−ti∥BE.
As the resolvent operatorR· is uniformly continuous, the right-hand side of the above inequality tends to zero as τ⟶t.Step 3.
The implication (8) holds.Now letV be a subset of Bρ such that V⊂NV¯∪0. V is bounded and equicontinuous, and therefore, the function t⟶vt=μVt is continuous on I. By H3 and the properties of the measure μ, for each t∈I, we have
(26)vt≤μNVt∪0≤μNVt≤∥Rt∥BEq∗supt∈IμVt+∫0t∥Rt−s∥BEpsμVsds+∑k=0m∥Rt−tk∥BElktμVt≤Mq∗∥v∥∞+Mp∗∫0tvsds+M∑k=0mlk∗vt≤Mq∗+Tp∗+∑k=0mlk∗∥v∥∞.Hence,(27)∥v∥∞≤ℓ∥v∥∞.From (17), we get ∥v∥∞=0, that is, vt=μVt=0, for each t∈I, and then Vt is relatively compact in E. In view of the Ascoli-Arzelà theorem, V is relatively compact in Bρ. Applying now Theorem 3, we conclude that N has a fixed point which is a mild solution of our problem (1).


## 4. Mild Solutions with Noninstantaneous Impulses

In this section, we are concerned with the existence results of problem (2). Denote by
(28)PC=y:I⟶E:y∈C0,t1∪tk,sk∪sk,tk+1,E,k=1,⋯,m,and there exist ytk−,ytk+,ysk−, and ysk+,k=1,⋯,m with ytk−=ytk and ysk−=ysk}, the Banach space equipped with the standard supremum norm.Definition 8 [34].
By a mild solution of problem (2), we mean a function u∈PC that satisfies
(29)ut=Rtuk−gu+∫sktRt−sfs,usds;t∈Ik,k=0,⋯,m,ut=gkt,utk−;t∈Jk,k=1,⋯,m.The following hypotheses will be used in the sequel:H01 The functions t↦ft,u and t↦gkt,u are measurable on Ik and Jk, respectively, for each u∈E, and the functions u↦ft,u and u↦gkt,u are continuous on E for a.e. t in Ik and Jk, respectively.H02 There exist functions p,lk∈L∞I;k=0,⋯,m, such that
(30)∥ft,u∥≤pt1+∥u∥;fora.e.t∈Ik,andeachu∈E,(31)∥gkt,u∥≤lkt1+∥u∥;k=1,⋯,m,fora.e.t∈Jk,andeachu∈E.H03 There exists a positive constant q∗, such that
(32)∥gu∥≤q∗1+∥u∥PC;fora.e.t∈I,andeachu∈PC.H04 For each bounded set B⊂E and for each t∈I, we have
(33)μft,B≤ptμB,μgkt,B≤lktμB;k=0,⋯,mand for each bounded set B0⊂PC, we have
(34)μgB0≤q∗supt∈IμB0t,where B0t=ut:u∈B0;t∈I.Set(35)p∗=∥p∥L∞,l∗=maxk=0,⋯,m∥lk∥L∞,M=supt∈I∥Rt∥BE.Theorem 9.
Assume that the hypothesesR1,R2,H01–H04hold. If
(36)ℓ≔maxl∗,Mq∗+Tp∗<1,then problem (2) has at least one mild solution defined on I.Proof.
Transform problem (2) into a fixed point problem. Consider the operator N:PC⟶PC defined by
(37)Nut=Rtuk−gu+∫sktRt−sfs,usds;t∈Ik,k=0,⋯,m,Nut=gkt,utk−;t∈Jk,k=1,⋯,m.
LetL>0, such that
(38)L≥M∥uk∥+q∗+Tp∗1−Mq∗+Tp∗.
For anyu∈PC and each t∈Ik, we have
(39)∥Nut∥≤∥Rt∥BE∥uk∥+∥gu∥+∫skt∥Rt−s∥BE∥fs,us∥ds≤M∥uk∥+1+Lq∗+Tp∗≤L.
Thus,(40)∥Nu∥PC≤L.
Next, for eacht∈Jk;k=1,⋯,m, it is clear that
(41)∥Nut∥E≤l∗.
Hence,(42)∥Nu∥PC≤maxL,l∗≔ρ.
This proves thatN transforms the ball Bρ≔w∈PC:∥w∥PC≤ρ into itself.
We shall show that the operatorN:Bρ⟶Bρ satisfies all the assumptions of Theorem 3. The proof will be given in three steps.Step 1.
N:Bρ⟶Bρis continuous.
Letunn∈ℕ be a sequence such that un⟶u as n⟶∞ in Bρ. Then, for each t∈Jk;k=1,⋯,m, we have
(43)∥Nunt−Nut∥≤∥Rt∥BE∥gkt,untk−−gkt,utk−∥,and for each t∈Ik;k=0,⋯,m, we have
(44)∥Nunt−Nunt∥≤∥Rt∥BE∥gun−gu∥+∫skt∥Rt−s∥BE∥fs,uns−fs,us∥ds≤M∥gun−gu∥+M∫skT∥fs,uns−fs,us∥ds.
Sinceun⟶u as n⟶∞ and f,g,gk are continuous, the Lebesgue-dominated convergence theorem implies that
(45)∥Nun−Nu∥PC⟶0asn⟶∞.Step 2.
NBρis bounded and equicontinuous.
SinceNBR⊂Bρ and Bρ is bounded, then NBρ is bounded.
Next, lett,τ∈Ik, τ<t and let u∈Bρ. Thus, we have
(46)∥Nut−Nuτ∥≤∥Rt−Rτ∥BE∥u0∥+∥gu∥+∫0τ∥Rt−s−Rτ−s∥BE∥fs,us∥ds+∫τt∥Rt−s∥BE∥fs,us∥ds.
Hence, we get(47)∥Nut−Nuτ∥≤∥u0∥+q∗1+ρ∥Rt−Rτ∥BE+p∗1+ρ∫0τ∥Rt−s−Rτ−s∥BEds.
Asτ⟶t, the right-hand side of the above inequality tends to zero.Step 3.
The implication (8) holds.
Now letV be a subset of BR such that V⊂NV¯∪0. V is bounded and equicontinuous, and therefore, the function t⟶vt=μVt is continuous on I. By H03 and the properties of the measure μ, for each t∈Ik, we have
(48)vt≤μNVt∪0≤μNVt≤l∗∥v∥∞≤ℓ∥v∥∞.
Next, for eacht∈Ik, we have
(49)vt≤μNVt∪0≤μNVt≤∥Rt∥BEq∗supt∈IμVt+∫0t∥Rt−s∥BEpsμVsds≤Mq∗∥v∥∞+Mp∗∫0tvsds≤Mq∗+Tp∗∥v∥∞≤ℓ∥v∥∞.
Thus, for eacht∈I, we get
(50)vt≤ℓ∥v∥∞.
Hence,(51)∥v∥∞≤ℓ∥v∥∞.
From (36), we get ∥v∥∞=0; that is, vt=μVt=0, for each t∈I, and then Vt is relatively compact in E. In view of the Ascoli-Arzelà theorem, V is relatively compact in Bρ. Applying now Theorem 3, we conclude that N has a fixed point which is a mild solution of problem (2).


## 5. Examples

Let(52)H≔L20,π=u:0,π⟶ℝ:∫0πux2dx<∞,be the Hilbert space with the scalar product <u,v>=∫0πuxvxdx. It is known that H is a Banach space with the norm
(53)∥u∥2=∫0πux2dx1/2.Example 1.
Consider the following problem of impulsive integrodifferential equations(54)∂∂tzt,x=∂2∂x2zt,x+Qt,zt,x+∫0tbt−s∂2∂x2zs,xds;t∈0,1∪1,2,x∈0,π,z1+,x=z1−,x+L1z1−,x;x∈0,π,zt,0=zt,π=0;t∈0,1∪1,2,z0,x+gz=1+x2;x∈0,π,z∈PC,where t∈I=0,2,PC≔PC0,2,H,
(55)Qt,zt,x=ct21+∥z∥2e−7+1et+x+51+zt,x;t∈0,1∪1,2,L1z1−,x=z1−,x3e41+∥z1−,x∥2,and
(56)gz=∫0πKx,ye−y1+∥z∥PCdy,with ∫0π∫0πK2x,ydxdy<∞.
We define the strongly elliptic operatorA:DA⊂H⟶H by
(57)Au=Ax,Du=∑∣μ∣≤2maμxDμu,where aμ∈C2m0,π and DA=H2m0,π∩H0m0,π.
It is well known (see [31]) that A generates a uniformly continuous semigroup Tt;t≥0 in the Hilbert space H.
Forx∈0,π, we have
(58)utx=zt,x;t∈0,1∪1,2,ft,utx=Qt,zt,x;t∈0,1∪1,2,Yt=btA,u0x=1+x2;x∈0,π.
Thus, under the above definitions off, u0, and A, system (54) can be represented by problem (1). Furthermore, more appropriate conditions on Q ensure the hypotheses R1,R2,H1−H5. Consequently, Theorem 7 implies that problem (54) has at least one mild solution on 0,2.Example 2.
Consider now the following problem of impulsive integrodifferential equations(59)∂∂tzt,x=∂2∂x2zt,x+Qt,zt,x+∫0tbt−s∂2∂x2zs,xds;t∈0,1∪2,3,x∈0,π,zt,x=g1t,z1−,x;t∈1,2,x∈0,π,zt,0=zt,π=0;t∈0,1∪2,3,z0,x+gz=1+ex;x∈0,π,z2,x+gz=2+ex;x∈0,π,where t∈0,3,PC≔PC0,3,H,
(60)Qt,zt,x=ct21+∥z∥2e−7+1et+x+51+zt,x;t∈0,1∪2,3,g1t,z1−,x=z1−,x3e41+∥z1−,x∥2;t∈1,2,x∈0,π,(61)gz=∫0πKx,ye−y1+∥z∥PCdy,with ∫0π∫0πK2x,ydxdy<∞.
Again, as the above example, simple computations show that all conditions of Theorem 9 are satisfied. It follows that problem (59) has at least one mild solution on 0,3.



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