# On Approximate Controllability of Second-Order Neutral Partial Stochastic Functional Integrodifferential Inclusions with Infinite Delay and Impulsive Effects

Authors: Zuomao Yan
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We discuss the approximate controllability
of second-order impulsive neutral partial stochastic functional
integrodifferential inclusions with infinite delay under the
assumptions that the corresponding linear system is approximately
controllable. Using the fixed point strategy, stochastic analysis,
and properties of the cosine family of bounded linear operators
combined with approximation techniques, a new set of sufficient
conditions for approximate controllability of the second-order
impulsive partial stochastic integrodifferential systems are
formulated and proved. The results in this paper are generalization
and continuation of the recent results on this issue. An example is
provided to show the application of our result.


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## Body

## 1. Introduction

Impulsive effects exist widely in many evolution processes in which states are changed abruptly at certain moments of time, involving fields such as physics, chemical technology, population dynamics, biotechnology, and economics; see [1–4] and the references therein. However, in addition to impulsive effects, stochastic effects likewise exist in real systems. A lot of dynamical systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, and other areas of science. Therefore, it is necessary and important to consider the impulsive stochastic dynamical systems. Particularly, the authors in [5–7] studied the existence of mild solutions for a class of abstract impulsive neutral stochastic functional differential and integrodifferential equations with infinite delay in Hilbert spaces.The concept of controllability leads to some very important conclusions regarding the behavior of linear and nonlinear dynamical systems. In the case of infinite-dimensional systems, two basic concepts of controllability can be distinguished. There are exact and approximate controllability. However, the concept of exact controllability is usually too strong [8]. Therefore, approximate controllability problems for deterministic and stochastic dynamical systems in infinite dimensional spaces are well developed using different kind of approaches (see [9, 10]). Stochastic control theory is a stochastic generalization of classic control theory. So significant progress has been made in the approximate controllability of linear and nonlinear stochastic systems in Banach spaces (see, e.g., [9–12]). Several papers [13–16] have appeared on the approximate controllability of nonlinear impulsive stochastic differential systems in Hilbert spaces.In many cases, it is advantageous to treat the second-order stochastic differential equations directly rather than to convert them to first-order systems. The second-order stochastic differential equations are the right model in continuous time to account for integrated processes that can be made stationary. Recently, based on the fixed point theory, the existence and approximate controllability of mild solutions for various second-order stochastic partial differential equations and impulsive stochastic partial differential equations have been extensively studied. For example, Ren and Sun [17], Cui and Yan [18], and Mahmudov and McKibben [19] proved the approximate controllability of second-order neutral stochastic evolution differential equations. Muthukumar and Balasubramaniam [20] established sufficient conditions for the approximate controllability of a class of second-order nonlinear stochastic functional differential equations of McKean-Vlasov type. Balasubramaniam and Muthukumar in [21] discussed the approximate controllability of second-order neutral stochastic distributed implicit functional differential equations with infinite delay. Sakthivel et al. in [22] studied the approximate controllability of second-order impulsive stochastic differential equations. On the other hand, many systems arising from realistic models can be described as partial stochastic differential or integrodifferential inclusions (see [23–27] and references therein), so it is natural to extend the concept controllability of mild solution for second-order impulsive stochastic evolution equations to second-order impulsive systems represented by stochastic partial differential or integrodifferential inclusions. In this paper, we consider the approximate controllability of the following second-order impulsive neutral partial stochastic functional integrodifferential inclusions with infinite delay in Hilbert spaces of the form(1)dx′t-gt,xt,x′t∈Axt+Butdt+Ft,xt,x′t,∫0tht,s,xs,x′sdsdwt,t∈J,t≠tk,Δxtk=Ikxtk,x′tk,k=1,…,m,Δx′tk=Jkxtk,x′tk,k=1,…,m,x0=φ∈B,x′0=ϕ,where the state x(·) takes values in a separable real Hilbert space H with inner product 〈·,·〉H and norm ∥·∥H. The operator A:D(A)→H is the infinitesimal generator of a strongly continuous cosine family on H. The control function u∈Lp(J,U), a Hilbert space of admissible control functions, p≥2 is an integer, and B is a bounded linear operator from a Banach space U to H. Let K be another separable Hilbert space with inner product 〈·,·〉K and norm ∥·∥K. Suppose that {w(t):t≥0} is a given K-valued Wiener process with a covariance operator Q>0 defined on a complete probability space (Ω,F,P) equipped with a normal filtration {Ft}t≥0, which is generated by the Wiener process w. The time history xt:(-∞,0]→H given by xt(θ)=x(t+θ) belongs to some abstract phase space B defined axiomatically; F, g, h, Ik, Jk(k=1,…,m) are given functions to be specified later. Moreover, let 0<t1<⋯<tm<b be prefixed points and the symbol Δx(tk)=x(tk+)-x(tk-), where x(tk-) and x(tk+) represent the right and left limits of x(t) at t=tk, respectively. The initial data {φ(t):-∞<t≤0} is an F0-adapted, B-valued random variable independent of the Wiener process w with finite second moment.To the best of the author’s knowledge, there are no results about the existence and approximate controllability of mild solutions for second-order impulsive second-order neutral partial stochastic functional integrodifferential inclusions with infinite delay, which is expressed in the form of (1). In order to fill this gap, this paper studies this interesting problem. We derive the sufficient conditions for the approximate controllability of system (1) by using the fixed point theorem for multivalued mapping due to Dhage [28] with stochastic analysis and properties of the cosine family of bounded linear operators combined with approximation techniques. The obtained result can be seen as a contribution to this emerging field. Moreover, the operators Ik, Jk(k=1,…,m) are continuous but without imposing completely continuous and Lipschitz condition. The results shown are also new for deterministic second-order systems with impulsive effects.The rest of this paper is organized as follows. In Section2, we introduce some notations and necessary preliminaries. Section 3 verifies the existence of solutions for impulsive stochastic control system (1). In Section 4 we establish the approximate controllability of impulsive stochastic control system (1). Finally in Section 5, an example is given to illustrate our results.


## 2. Preliminaries

Let(Ω,F,P) be a complete probability space equipped with a normal filtration Ft,t∈[0,b]. Let H and K be the separable Hilbert spaces and let w be a Q-Weiner process on (Ω,Fb,P) with the covariance operator Q such that trQ<∞. We assume that there exists a complete orthonormal system {en}n=1∞ in K, a bounded sequence of nonnegative real numbers {λn}n=1∞ such that Qei=λiei, and a sequence βi of independent Brownian motions such that (2)wt,e=∑n=1∞λnen,eβnt,e∈K,t∈J,and Ft=Ftw, where Ftw is the σ-algebra generated by {w(s):0≤s≤t}. Let L02=L2(Q1/2K;H) be the space of all Hilbert-Schmidt operators from Q1/2K to H with the inner product 〈ψ,θ〉L02=Tr(ψQθ∗). Let Lp(Fb,H) be the Banach space of all Fb-measurable pth power integrable random variables with values in the Hilbert space H. Let C([0,b];Lp(F,H)) be the Banach space of continuous maps from [0,b] into Lp(F,H) satisfying the condition supt∈JExtHp<∞.We use the notations thatP(H) is the family of all subsets of H. Let us introduce the following notations: (3)PclH=x∈PH:xis closed,PbdH=x∈PH:xis bounded,PcvH=x∈PH:xis convex,PcpH=x∈PH:xis compact.Consider Hd:P(H)×P(H)→R+∪{∞} given by (4)HdA~,B~=max⁡supa~∈A~⁡da~,B~,supb~∈B~⁡dA~,a~,where dA~,b~=infa~∈A~da~,b~ and d(a~,B~)=infb~∈B~d(a~,b~). Then (Pbd,cl(H),Hd) is a metric space and (Pcl(H),Hd) is a generalized metric space.In what follows, we briefly introduce some facts on multivalued analysis. For more details, one can see [29, 30].A multivalued mapΦ:H→P(H) is convex (closed) valued if G(H) is convex (closed) for all x∈H. Φ is bounded on bounded sets if Φ(D)=⋃x∈D‍Φ(x) is bounded in H for any bounded set D of H; that is, supx∈D{sup{yH:y∈Φ(x)}}<∞.Φ is called upper semicontinuous (u.s.c., in short) on H, if, for any x∈H, the set Φ(x) is a nonempty, closed subset of H and if, for each open set B of H containing Φ(x), there exists an open neighborhood N of x such that Φ(N)⊆B.Φ is said to be completely continuous if Φ(D) is relatively compact for every bounded subset D of H. If the multivalued map Φ is completely continuous with nonempty compact values, then Φ is u.s.c. if and only if Φ has a closed graph; that is, xn→x∗, yn→y∗, and yn∈Φ(xn) imply y∗∈Φ(x∗).Φ is said to be completely continuous if Φ(D) is relatively compact, for every bounded subset D⊆H.A multivalued mapΦ:J→Pcl is said to be measurable if, for each x∈H, the function Y:J→R+ defined by Y(t)=d(x,Φ(t))=inf{d(x,z):z∈Φ(t)} is measurable.In this paper,A is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators (C(t))t∈R on H. The corresponding strongly continuous sine family {S(t):t∈R}⊂L(H) is defined by S(t)x=∫0t‍Csxds, t∈R,x∈H. The generator A:H→H of {C(t):t∈R} is given by Ax=(d2/dt2)C(t)x|t=0 for all x∈D(A)={x∈H:C(·)x∈C2(R,H)}. It is well known that the infinitesimal generator A is a closed densely defined operator on H. As usual we denote by [D(A)] the domain of operator A endowed with the graph norm xA=xH+AxH,x∈D(A). Moreover, the notation E stands for the space formed by the vectors x∈H for which C(·)x is of class C1 on R. It was proved by Kisyński [31] that E endowed with the norm xE=xH+sup0≤t≤1AS(t)xH, x∈E, is a Banach space. Such cosine and corresponding sine families and their generators satisfy that the following properties.Lemma 1 (see [32]).
Suppose thatA is the infinitesimal generator of a cosine family of operators {C(t):t∈R}. Then, the following hold: (a)
There existM~1≥1 and α≥0 such that C(t)H≤M~1eαt and hence S(t)H≤M~1eαt.(b)
ConsiderA∫sr‍S(u)xdu=[C(r)-C(u)]x for all 0≤s≤r<∞.(c)
There existsM~2≥1 such that S(s)-S(r)H≤M~2∫rs‍eαθdθ for all 0≤r≤s<∞.The existence of solutions of the second-order linear abstract Cauchy problem(5)x′′t=Axt+Ft,t∈J,x0=z0,x′0=z1,where h:J→H is an integrable function, has been discussed in [33]. Similarly, the existence of solutions for semilinear second-order abstract Cauchy problem has been treated in [32]. We only mention here that the function x(·) given by (6)xt=Ctz0+Stz1+∫0tSt-sFsds,t∈J,is called a mild solution of (5) and if z1∈E, the function x(·) is continuously differentiable and (7)x′t=AStz0+Ctz1+∫0tCt-sFsds,t∈J.A functionx:[μ,τ]→H is said to be normalized piecewise continuous function on [μ,τ] if x is piecewise continuous and left continuous on (μ,τ]. We denote by PC([μ,τ],H) the space formed by the normalized piecewise continuous, Ft-adapted measurable processes from [μ,τ] into H. In particular, we introduce the space PC formed by all Ft-adapted measurable, H-valued stochastic processes {x(t):t∈[0,b]} such that x is continuous at t≠tk,x(tk)=x(tk-) and x(tk+) exists for k=1,2,…,m. Similarly, PC1 formed by all Ft-adapted measurable, H-valued stochastic processes {x′(t):t∈[0,b]} such that x′ is continuous at t≠tk, x′(tk)=x′(tk-), and x′(tk+) exists for k=1,2,…,m. In this paper, we always assume that PC is endowed with the norm xPC=(sup0≤t≤bEx(t)Hp)1/p. Then (PC,·PC) is a Banach space. Next, for x∈PC1, we represent by x′(t) the right derivative at t∈(0,b] and by x′(0) the right derivative at zero. It is easy to see that PC1 is provided with the norm xPC1=xPC+x′PC being a Banach space.In this paper, we assume that the phase space(B,·B) is a seminormed linear space of F0-measurable functions mapping (-∞,0] into H and satisfying the following fundamental axioms due to Hale and Kato (see, e.g., [34]).(A)
Ifx:(-∞,σ+b]→H,b>0, such that x|[σ,σ+b]∈PC1([σ,σ+b],H) and xσ∈B, then for every t∈[σ,σ+b] the following conditions hold:
(i)
xt is in B;(ii)
x(t)H≤H~xtB;(iii)
xtB≤K(t-σ)sup{x(s)H:σ≤s≤t}+M(t-σ)xσB, where H~≥0 is a constant; K,M:[0,∞)→[1,∞),K is continuous and M is locally bounded, and H~, K, and M are independent of x(·).(B)
For the functionx(·) in (A), the function t→xt is continuous from [σ,σ+b] into B.(C)
The spaceB is complete.Example 2.
The phase spacePCr×Lp(h~,H). Let r≥0, 1≤p<+∞ and let h~:(-∞,-r]→R be a nonnegative measurable function which satisfies conditions (h-5) and (h-6) in the terminology of Hino et al. [35]. Briefly, this means that h~ is locally integrable and there is a nonnegative, locally bounded function γ on (-∞,0] such that h~(ξ+τ)≤γ(ξ)h~(τ) for all ξ≤0 and ξ∈(-∞,-r)∖Nξ, where Nξ⊆(-∞,-r) is a set whose Lebesgue measure is zero. We denote by PCr×Lp(h~,H) the set consisting of all classes of functions φ:(-∞,0]→H such that φ|[-r,0]∈PC([-r,0],H),φ(·) is Lebesgue measurable on (-∞,-r), and h~φp is Lebesgue integrable on (-∞,-r). The seminorm is given by (8)φB=sup-r≤τ≤0⁡φτ+∫-∞-rh~τφpdτ1/p.The space B=PCr×Lp(h~,H) satisfies axioms (A)–(C). Moreover, when r=0 and p=2, we can take H~=1, M(t)=γ(-t)1/2, and K(t)=1+(∫-t0‍h~(τ)dτ)1/2, for t≥0 (see [35, Theorem 1.3.8] for details).Remark 3 (see [4]).
In retarded functional differential equations without impulses, the axioms of the abstract phase spaceB include the continuity of the function t→xt. Due to the impulsive effect, this property is not satisfied in impulsive delay systems and, for this reason, has been eliminated in our abstract description of B.Remark 4.
In the rest of this paperMb and Kb are the constants defined by Mb=supt∈JM(t) and Kb=supt∈JK(t).Forx∈PC, we denote by x^k, k=0,1,…,m, the unique continuous function x^k∈C([tk,tk+1];H) such that (9)x^kt≔xtfort∈tk,tk+1,xtk+fort=tk.Moreover, for B⊆PC we denote by B^k, k=0,1,…,m, the set B^k={x^k:x∈B}. The notation Br(x,H) stands for the closed ball with center at x and radius r>0 in H.Lemma 5.
A setB⊆PC is relatively compact in PC if and only if the set B^k is relatively compact in C([tk,tk+1];L2(Ω,H)), for every k=0,1,…,m.Furthermore, we need the following result.Lemma 6 (see [36]).
Letν:J→H be an integrable function such that ν∈PC. Then the function v(t)=∫0t‍C(t-s)ν(s)ds belongs to PC1, the function s→AS(t-s)ν(s) is integrable on 0,t,t∈J, and v′t=νt+A∫0t‍St-sνsds=νt+∫0t‍ASt-sνsds, t∈J.Letxb(x0;u) be the state value of system (1) at terminal time b corresponding to the control u and the initial value x0=φ(t)∈B. Introduce the set (10)Bb,x0=xbx0;u0:u·∈LpJ,U,which is called the reachable set of system (1) at terminal time b, and its closure in H is denoted by B(b,x0)¯.Now we give the definitions of mild solutions and approximate controllability for system (1).Definition 7.
AnFt-adapted stochastic process x:(-∞,b]→H is called a mild solution of system (1) if x0=φ∈B, x′(0)=ϕ,x|J∈PC1, and the impulsive conditions Δxtk=Ikxtk,x′tk and Δx′tk=Jkxtk,x′tk,k=1,…,m, are satisfied and (i)
xt is adapted to Ft,t≥0.(ii)
x(t)∈H has càdlàg paths on t∈J a.s. and, for each t∈J, x(t) satisfies the integral equation(11)xt=Ctφ0+Stϕ-g0,φ,ϕ+∫0tCt-sgs,xs,x′sds+∫0tSt-sBusds+∫0tSt-sfsdws+∑0<tk<tCt-tkIkxtk,x′tk+∑0<tk<tSt-tkJkxtk,x′tk,t∈J,where f∈SF,x={f∈Lp(K,H):f(t)∈F(t,xt,x′(t),∫0t‍h(t,s,xs,x′(s))ds)a.e.t∈J}.Definition 8.
System (1) is said to be approximately controllable on the interval J if B(b,x0)¯=H.It is convenient at this point to define operators(12)Γτb=∫τbSαb-sBB∗S∗b-sds,0≤τ<b,Γ0b=∫0bSb-sBB∗S∗b-sds,Ra,Γ0b=aI+Γ0b-1fora>0,where B∗ denotes the adjoint of B and S∗(t) is the adjoint of S(t). It is straightforward that the operator Γ0b is a linear bounded operator:(S1)
aRa,Γτb→0,0≤τ<s≤b, as a→0+ in the strong operator topology.Lemma 9.
Assumption (S1) holds if and only ifaRa,Γτb→0,0≤τ<s≤b, as a→0+ in the strong operator topology.The proof of this lemma is a straightforward adaptation of the proof of [9, Theorem 2].Lemma 10 (see [10]).
For anyx~b∈Lp(Fb,H) there exists ϕ~∈LFp(Ω;L2(0,b;L20)) such that x~b=Ex~b+∫0b‍ϕ~(s)dw(s).Now for anya>0 and x~b∈Lp(Fb,H) we define the control function(13)uxat=B∗S∗b-taI+Γ0b-1Ex~b+∫0bϕ~sdws-Cbφ0-Sbϕ-g0,φ,ϕ-B∗S∗b-t·∫0baI+Γsb-1Cb-sgs,xs,x′sds-B∗S∗b-t∫0baI+Γsb-1Sb-sfsdws-B∗S∗b-taI+Γsb-1·∑k=1mCb-tkIkxtk,x′tk-B∗S∗b-taI+Γsb-1∑k=1mSb-tkJkxtk,x′tk,where f∈SF,x={f∈Lp(K,H):f(t)∈F(t,xt,x′(t),∫0t‍h(t,s,xs,x′(s))ds)a.e.t∈J}.The next result is a consequence of the phase space axioms.Lemma 11.
Letx:(-∞,b]→H be an Ft-adapted measurable process such that F0-adapted process x0=φ(t)∈L20(Ω,B) and x|J∈PC1(J,H); then (14)xsB≤MbEφB+Kbsup0≤s≤b⁡ExsH.Lemma 12 (see [37]).
For anyp≥1 and for arbitrary L20(K,H)-valued predictable process ϕ(·), (15)sups∈0,t⁡E∫0sϕvdwvH2p≤p2p-1p∫0tEϕsL202p1/pdsp,t∈0,∞.The consideration of this paper is based on the following fixed point theorem due to Dhage [28].Lemma 13.
LetH be a Hilbert space, and let Φ1:H→Pcl,cv,bd(H) and Φ2:H→Pcp,cv(H) be two multivalued operators satisfying that(a)
Φ1 is a contraction(b)
Φ2 is completely continuous.Then either (i)
the operator inclusionx∈Φ1x+Φ1x has a solution or(ii)
the setG={x∈H:x∈λΦ1x+λΦ2x} is unbounded for λ∈(0,1).


## 3. Existence of Solutions for Impulsive Stochastic Control System

In this section, we prove the existence of solutions for impulsive stochastic control system (1). We make the following hypotheses:(H1)
A is the infinitesimal generator of a strongly continuous cosine family {C(t):t≥0} on H and the corresponding sine family {S(t):t≥0} satisfies the conditions C(t)H≤Me-αt, S(t)H≤Me-βt, and AS(t)H≤Me-γt, t≥0 for some constants M, α, β, and γ.(H2)
Ct,t>0, is compact.(H3)
The functiong:J×B×H→H is continuous and there exist Lg>0 such that(16)Egt1,ψ1,y1-gt2,ψ2,y2Hp≤Lgt1-t2+ψ1-ψ2Bp+Ey1-y2Hpfor allti∈J,ψi∈B,yi∈H,i=1,2, and (17)Egt,ψ,yHp≤LgψBp+EyHp+1,t∈J,ψ∈B,y∈H,withLg{Mpbp-1(pα)-1+2p-1[1+M2pbp-1(pγ)-1]}(2p-1Kbp+1)<1.(H4)
The functionh:J×J×B×H→H satisfies the following conditions:
(i)
For each(t,s)∈J×J the function h(t,s,·,·):B×H→H is continuous and, for each (ψ,y)∈B×H, the function h(·,·,ψ,y):J×J→H is strongly measurable.(ii)
There exists a continuous functionmh:J×J→[0,∞), such that (18)Eht,s,ψ,yHp≤mht,sΘhψBp+yHpfor a.e.t,s∈J,(ψ,y)∈B×H, where Θh:[0,∞)→(0,∞) is a continuous nondecreasing function.(H5)
The multivalued mapF:J×B×H×H→Pbd,cl,cv(L(K,H)) satisfies the following conditions:
(i)
For eacht∈J, the function F(t,·,·,·):B×H×H→Pbd,cl,cv(L(K,H)) is u.s.c. and, for each (ψ,y,z)∈B×H×H, the function F(·,·,·) is measurable and the set (19)SF,ψ=f∈LpJ,LK,H:ft∈Ft,ψt,ψ′t,∫0tht,s,ψs,ψ′sdsfor a.e.t∈Jis nonempty.(ii)
There exist a continuous functionmf:J→[0,∞) and a continuous nondecreasing function Θf:[0,∞)→(0,∞) such that (20)Ft,ψ,y,zHp≤mftΘfψBp+EyHp+EzHp,a.e.t∈J,ψ∈B,y,z∈H, with(21)∫1∞11+s+Θhs+Θfsds=∞,where(22)Ft,ψ,y,zHp=sup⁡EfHp:f∈Ft,ψ,y,z.(H6)
The functionsIk,Jk:B×H→H are continuous and there are constants ck, dk, k=1,2,…,m, j=1,2,3,4 such that (23)0≤limsupψBp+yHp→∞⁡EIkψ,yHpψBp+EyHp≤ck,0≤limsupψBp+EyHp→∞⁡EJkψ,yHpψBp+EyHp≤dkfor everyψ∈B,y∈H.Lemma 14 (see [38]).
LetJ be a compact interval and let H be a Hilbert space. Let F be a multivalued map satisfying (H5) (i) and let Γ be a linear continuous operator from Lp(J,H) to C(J,H). Then the operator Γ∘SF:C(J,H)→Pcp,cv(C(J,H)) is a closed graph in C(J,H)×C(J,H).Remark 15.
In what follows, we setM1=BH, M∗=Mmax⁡{1,e-αb,e-βb,e-γb}, and N∗=max⁡{1,eαb,eβb,eγb}.Theorem 16.
If assumptions (H1)–(H6) are satisfied, further, suppose that, for alla>0, system (1) has at least one mild solution on J, provided that(24)12p-1Lg+10p-1Kbp+12p-1mp-1M2p∑k=1mck+dkN∗p<1.Proof.
LetBb0={x:(-∞,b]→H;x0=0,x|J∈PC1,x′(0)=g(0,φ,ϕ)} endowed with the norm of PC1. Thus (Bb0,·PC1) is Banach space. Now we can define the multivalued map Φ:Bb0→P(Bb0) by Φx the set of ρ∈Bb0 such that (25)ρt=0,t∈-∞,0,∫0tCt-sgs,x¯s,x¯′sds+∫0tSt-sBux¯sds+∫0tSt-sfsdws+∑0<tk<tCt-tkIkx¯tk,x¯′tk+∑0<tk<tSt-tkJkx¯tk,x¯′tk,t∈J,where(26)uxat=B∗S∗b-taI+Γ0b-1Ex~b+∫0bϕ~sdws-Cbφ0-Sbϕ-g0,φ,ϕ-B∗S∗b-t·∫0baI+Γsb-1Cb-sgs,x¯s,x¯′sds-B∗S∗b-t∫0baI+Γsb-1Sb-sfsdws-B∗S∗b-taI+Γsb-1·∑k=1mCb-tkIkx¯tk,x¯′tk-B∗S∗b-taI+Γsb-1∑k=1mSb-tkJkx¯tk,x¯′tk,and f∈SF,x¯={f∈Lp(L(K,H)):f(t)∈F(t,x¯t,x¯′(t),∫0th(t,s,x¯s,x¯′(s))ds)a.e.t∈J}, and x¯:(-∞,0]→H is such that x¯0=φ and x¯=x on J. In what follows, we aim to show that the operator Φ has a fixed point, which is a solution of problem (1).
Let{σn:n∈N} be a decreasing sequence in (0,t1)⊂(0,b) such that limn→∞σn=0. To prove the above theorem, we consider the following problem:(27)dx′t-gt,xt,x′t∈Axt+Butdt+Ft,xt,x′t,∫0tht,s,xs,x′sdsdwt,t∈J,t≠tk,Δxtk=CσnIkxtk,x′tk,k=1,…,m,Δx′tk=SσnJkxtk,x′tk,k=1,…,m,x0=φ∈B,x′0=ϕ.We will show that the problem has at least one mild solution xn∈Bb0.
For fixedn∈N, set the multivalued map Φn:Bb0→P(Bb0) by Φnx the set of ρn∈Bb0 such that (28)ρnt=0,t∈-∞,0,∫0tCt-sgs,x¯s,x¯′sds+∫0tSt-sBux¯asds+∫0tSt-sfsdws+∑0<tk<tCt-tkCσnIkx¯tk,x¯′tk+∑0<tk<tSt-tkSσnJkx¯tk,x¯′tk,t∈J,where (29)ux¯at=B∗S∗b-taI+Γ0b-1Ex~b+∫0bϕ~sdws-Cbφ0-Sbϕ-g0,φ,ϕ-B∗S∗b-t·∫0baI+Γsb-1Cb-sgs,x¯s,x¯′sds-B∗S∗b-t∫0baI+Γsb-1Sb-sfsdws-B∗S∗b-taI+Γsb-1·∑k=1mCb-tkCσnIkx¯tk,x¯′tk-B∗Sα∗b-taI+Γsb-1∑k=1mSb-tkSσnJkx¯tk,x¯′tk,and f∈SF,x¯. It is easy to see that the fixed point of Φn is a mild solution of the Cauchy problem (27).
Letφ¯:(-∞,0)→H be the extension of (-∞,0] such that φ¯(θ)=φ(0)=0 on J. Now, we consider the following multivalued operators Λn and Υn defined by (30)Λnxt=0,t∈-∞,0,∫0tCt-sgs,x¯s,x¯′sds,t∈J,Υnxt=0,t∈-∞,0,∫0tSt-sBux¯sds+∫0tSt-sfsdws+∑0<tk<tCt-tkCσnIkx¯tk,x¯′tk+∑0<tk<tSt-tkSσnJkx¯tk,x¯′tkt∈J.It is clear that Φn=Λn+Υn. The problem of finding mild solutions of (27) is reduced to find the solutions of the operator inclusion x∈Λn(x)+Υn(x). In what follows, we show that operators Λn and Υn satisfy the conditions of Lemma 13.
Step 1.
Λ
n is a contraction on Bb0.
Lett∈J and x∗,x∗∗∈Bb0. From (H3) and Lemma 11, we have(31)EΛnx∗t-Λnx∗∗tHp=E∫0tCt-s·gs,x∗¯t,x∗¯′s-gs,x∗∗¯t,x∗∗¯′sdsHp≤MpE∫0te-αt-sgs,x∗¯t,x∗¯′s-gs,x∗∗¯t,x∗∗¯′sHdsp≤MpLgbp-1∫0te-pαt-sx∗¯s-x∗∗¯sBp+Ex∗¯′s-x∗∗¯′sHpds≤MpLgbp-1∫0te-pαt-sds2p-1Kbpsups∈0,b⁡Ex∗¯s-x∗∗¯sHp+sups∈0,b⁡Ex∗¯′s-x∗∗¯′sHp≤MpLgbp-1pα-12p-1Kbpsups∈0,b⁡Ex∗s-x∗∗sHp+sups∈0,b⁡Ex∗′s-x∗∗′sHpsincex¯=xonJ≤MpLgbp-1pα-12p-1Kbp+1x∗-x∗∗PC1p.By Lemma 6, we have (32)Λn′t=gt,x¯t,x¯′t+∫0tASt-sgs,x¯s,x¯′sds,t∈J.Similarly, for any t∈[0,b], we have (33)EΛn′x∗t-Λn′x∗∗tHp≤2p-1Egt,x∗¯t,x∗¯′t-gt,x∗∗¯t,x∗∗¯′tHp+2p-1E∫0tASt-sgs,x∗¯s,x∗¯′s-gs,x∗∗¯s,x∗∗¯′sdsHp≤2p-1x∗¯t-x∗∗¯tBp+Ex∗¯′t-x∗∗¯′tHp+2p-1MpLgbp-1∫0te-pγt-sx∗¯s-x∗∗¯sBp+Ex∗¯′s-x∗∗¯′sHpds≤2p-1Lg2p-1Kbpsupt∈0,b⁡Ex∗¯t-x∗∗¯tHp+supt∈0,b⁡Ex∗¯′t-x∗∗¯′tHp+2p-1MpLgbp-1∫0te-pγt-sds2p-1Kbp·sups∈0,b⁡Ex∗¯s-x∗∗¯sHp+sups∈0,b⁡Ex∗¯′s-x∗∗¯′sHp≤2p-1Lg+2p-1M2pLge-ασnbp-1pγ-12p-1Kbp·sups∈0,b⁡Ex∗s-x∗∗sHp+sups∈0,b⁡Ex∗′s-x∗∗′sHpsincex¯=xonJ≤2p-1Lg1+Mpbp-1pγ-12p-1Kbp+1x∗-x∗∗PC1p.Taking supremum over t, it follows that (34)Λnx∗-Λn∗∗PC1p≤L0x∗-x∗∗PC1p,where L0=Lg{Mpbp-1(pα)-1+2p-1[1+M2pbp-1(pγ)-1]}(2p-1Kbp+1)<1. Hence, Λn is a contraction on Bb0.
Step 2.
Υ
n has compact, convex values and it is completely continuous.
(
1
)
Υ
n
x is convex for each x∈Bb0.
In fact, ifρ~n1, ρ~n2 belong to Υnx, then there exist f1,f2∈SF,x¯ such that (35)ρ~nit=∫0tSt-sBB∗S∗b-saI+Γ0b-1Ex~b+∫0bϕ~ηdwη-Cbφ0-Sbϕ-g0,φ,ϕ-B∗S∗b-s·∫0baI+Γηb-1Cb-ηgη,x¯η,x¯′ηdη-B∗S∗b-s·∫0baI+Γηb-1Sb-ηfiηdwη-B∗S∗b-saI+Γηb-1·∑k=1mCb-tkCσnIkx¯tk,x¯tk-B∗S∗b-saI+Γηb-1·∑k=1mSb-tkSσnJkx¯tk,x¯′tkds+∫0tSt-sfisdws+∑0<tk<tCt-tk·CσnIkx¯tk,x¯′tk+∑0<tk<tSt-tkSσn·Jkx¯tk,x¯′tk,t∈0,b,i=1,2.Let 0≤λ≤1. For each t∈[0,b] we have (36)λρ~n1+1-λρ~n2t=∫0tSt-sBB∗S∗b-s·aI+Γ0b-1Ex~b+∫0bϕ~ηdwη-Cbφ0-Sbϕ-g0,φ,ϕ-B∗Sα∗b-s·∫0baI+Γηb-1Cb-ηgη,x¯η,x¯′ηdη-B∗Sα∗b-s∫0baI+Γηb-1Sb-η·λf1η+1-λf2ηdwη-B∗S∗b-saI+Γηb-1∑k=1mCb-tkCσn·Ikx¯tk,x¯tk-B∗S∗b-saI+Γηb-1·∑k=1mSb-tkSσnJkx¯tk,x¯′tkds+∫0tSt-sλf1s+1-λf2sdws+∑0<tk<tCt-tkCσnIkx¯tk,x¯′tk+∑0<tk<tSt-tkSσnJkx¯tk,x¯′tk.Since SF,x¯ is convex (because F has convex values) we have (λρ~n1+(1-λ)ρ~n2)∈Υnx.
2
Υ
n maps bounded sets into bounded sets in Bb0.
Indeed, it is enough to show that there exists a positive constantL such that, for each ρ~n∈Υnx,x∈Br(0,Bb0)={x∈Bb0:xPC1p≤r}, one has ρ~nPC1p≤L. If ρ~n∈Υnx, then there exists f∈SF,x¯ such that, for each t∈J,(37)ρ~nt=∫0tSt-sBux¯asds+∫0tSt-sfsdws+∑0<tk<tCt-tkCσnIkx¯tk,x¯′tk+∑0<tk<tSt-tkSσnJkx¯tk,x¯′tk.However, on the other hand, from the condition (H6), we conclude that there exist positive constants ϵkk=1,…,m,γ1 such that, for all ψBp+yHp>γ1,(38)EIkψ,yHp≤ck+ϵkψBp+EyHp,EJkψ,yHp≤dk+ϵkψBp+EyHp,12p-1Lg+10p-1Kbp+12p-1·mp-1M2p∑k=1mck+dk+2ϵkN∗p<1.Let (39)F1=ψ:ψBp+EyHp≤γ1,F2=ψ:ψBp+EyHp>γ1,C1=max⁡Ikψ,yHp,x∈F1.Therefore(40)EIkψ,yHp≤C1+ck+ϵkψBp+EyHp,(41)EJkψ,yHp≤C1+dk+ϵkψBp+EyHp.If x∈Br(0,Bb0), from Lemma 11, it follows that (42)x¯sBp≤2p-1MbφBp+2p-1Kbpr≔r∗.By (H1)–(H5) and (40)-(41), from (37) we have for t∈J(43)Eρ~nHp≤4p-1E∫0tSt-sBB∗S∗b-saI+Γ0b-1Ex~b+∫0bϕ~ηdwη-Cbφ0-Sbϕ-g0,φ,ϕ-B∗S∗b-s∫0baI+Γηb-1Cb-ηgη,x¯η,x¯′ηdη-B∗S∗b-s∫0baI+Γηb-1Sb-ηfiηdwη-B∗S∗b-saI+Γηb-1∑k=1mCb-tkCσnIkx¯tk,x¯tk-B∗S∗b-saI+Γηb-1∑k=1mSb-tkSσnJkx¯tk,x¯′tkdsHp+4p-1E∫0tSt-sfsdwsHp+4p-1E∑0<tk<tCt-tkCσnIkx¯tk,x¯′tkHp+4p-1E∑0<tk<tSt-tkSσnJkx¯tk,x¯′tkHp≤20p-1Mpe-pβt·1apMpe-pβbM12pbp-1∫0te2pβs4p-1Ex~bHp+Cp∫0bEϕ~ηHp2/pdηp/2+Mpe-pαbEφ0Hp+2p-1ϕHp+g0,φ,ϕHp+Mpbp-1Lg∫0be-pαb-ηx¯ηBp+Ex¯′ηHp+1dη+CpMp∫0be-pβb-ηEfηHp2/pdηp/2+mp-1M2p∑k=1me-pαb-tke-pασnEIkx¯tk,x¯′tkHp+mp-1M2p∑k=1me-pβb-tke-pβσnEJkx¯tk,x¯′tkHpds+4p-1CpMp∫0te-pβt-sEfsHp2/pdsp/2+4mp-1M2p∑k=1me-pαt-tke-pασnEIkx¯tk,x¯′tkHp+4mp-1M2p∑k=1me-pβt-tke-pβσnEJkx¯tk,x¯′tkHp≤20p-1Mpe-pβt1apMpe-pβbM12pbp-1∫0te2pβs4p-1Ex~bHp+Cpbp/2-1∫0bEϕ~ηHpdη+Mpe-pαbEφ0Hp+2p-1ϕHp+g0,φ,ϕHp+Mpbp-1Lg∫0be-pαt-ηx¯ηBp+Ex¯′ηHp+1dη+CpMpbp/2-1∫0be-pβb-ηmfηΘfx¯ηBp+Ex¯′sHp+bp-1∫0ηmhη,νΘhx¯νBp+Ex¯′νHpdνdη+mp-1M2p∑k=1me-pαb-tke-pασnC1+ck+ϵkx¯tkBp+Ex¯′tkHp+mp-1M2p∑k=1me-pβb-tke-pβσnC1+dk+ϵkx¯tkBp+Ex¯′tkHpds+4p-1CpMpbp/2-1∫0te-pβt-smfs·Θfx¯sBp+Ex¯′sHp+bp-1∫0smhs,τΘhx¯τBp+Ex¯′τHpdτds+4mp-1·M2p∑k=1me-pαt-tke-pασnC1+ck+ϵkx¯tkBp+Ex¯′tkHp+4mp-1M2p∑k=1me-pβt-tke-pβσnC1+dk+ϵk·x¯tkBp+Ex¯′tkHp≤20p-1M∗p1apMpe-pβbM12pbpN∗2p4p-1Ex~bHp+Cpbp/2-1∫0bEϕ~ηHpdη+Mpe-pαbEφ0Hp+2p-1ϕHp+g0,φ,ϕHp+Mpbp-1Lgr∗+r+1∫0be-αpt-ηdη+CpMpbp/2-1Θfr∗∗∫0be-pβb-ηmfηdη+mp-1M2pe-pαbe-pασnN∗p∑k=1mC1+ck+ϵkr∗+r+mp-1M2pe-pβbe-pβσnN∗p∑k=1mC1+dk+ϵkr∗+r+4p-1CpMpbp/2-1Θfr∗∗∫0be-pβb-smfsds+4mp-1M∗pMpe-pασnN∗p∑k=1mC1+ck+ϵkr∗+r+4mp-1M∗pMpe-pβσnN∗p∑k=1mC1+dk+ϵkr∗+r≔L1,where r∗∗=r∗+r+bp-1∫0bmh(s,s)Θh(r)ds, Cp=(p(p-1)/2)p/2. By Lemma 6, we have (44)ρ~n′t=∫0tCt-sBux¯asds+λ∫0tCt-sfsdws+∑0<tk<tASt-tkCσnIkx¯tk,x¯′tk+∑0<tk<tCt-tkSσnJkx¯tk,x¯′tk,t∈J.Similarly, for any t∈J, we have (45)Eρ~n′tHp≤20p-1M∗p1ap·Mpe-pαbM12pbp-1N∗2p4p-1Ex~bHp+Cpbp/2-1∫0bEϕ~ηHpdη+Mpe-pαbEφ0Hp+2p-1ϕHp+g0,φ,ϕHp+Mpbp-1Lgr∗+r+1∫0be-pαb-ηdη+CpMpbp/2-1Θfr∗∗·∫0be-pβb-ηmfηdη+mp-1M2pe-pγbe-pασnN∗p∑k=1mC1+ck+ϵkr∗+r+mp-1M2pe-pαbe-pβσnN∗p∑k=1mC1+dk+ϵkr∗+r+4p-1CpMpbp/2-1Θfr∗∗∫0be-pαb-smfsds+4mp-1M∗pMpe-pασnN∗p∑k=1mC1+ck+ϵkr∗+r+4mp-1M∗pMpe-pβσnN∗p∑k=1mC1+dk+ϵkr∗+r≔L2.Take L=L1+L2. Then for each ρ~n∈Υnx, we have ρ~PC1p≤L.
3
Υ
n is a compact multivalued map.
To this end, we decomposeΥn by Υn=Γn1+Γn2, where the map Γn1:Br(0,Bb0)→P(Bb0) is defined by Γn1x and the set γ~n1∈Bb0 such that (46)γ~n1tt=∫0tSt-sBux¯asds+∫0tSt-sfsdws,and the map Γn2:Br(0,Bb0)→P(Bb0) is defined by Γn2x, and the set γ~n2∈Bb0 such that (47)γ~n2t=∑0<tk<tCt-tkCσnIkx¯tk,x¯′tk+∑0<tk<tSt-tkSσnJkx¯tk,x¯′tk.
First,Γn1(Br(0,Bb0)) is a compact multivalued map. We begin by showing that Γn1(Br(0,Bb0)) is equicontinuous. Let 0<τ1<τ2≤b. Then, we have, for each x∈Br(0,Bb0),(48)Eγ~n1τ2-γ~n1τ1Hp≤6p-1E∫0τ1-εSτ2-s-Sτ1-sBux¯sdsHp+6p-1E∫τ1-ετ1Sτ2-s-Sτ1-s·Bux¯sdsHp+6p-1E∫τ1τ2Sτ2-sBux¯sdsHp+6p-1E∫0τ1-εSτ2-s-Sτ1-sfsdwsHp+6p-1E∫τ1-ετ1Sτ2-s-Sτ1-sfsdwsHp+6p-1E∫τ1τ2Sτ2-sfsdwsHp≤6p-1E∫0τ1-εSτ2-s-Sτ1-sHBux¯sHdsp+6p-1E∫τ1-ετ1Sτ2-s-Sτ1-sHBux¯sHdsp+6p-1E∫τ1τ2Sτ2-sHBux¯sHdsp+6p-1Cp∫0τ1-εSτ2-s-Sτ1-sHpEfsHp2/pdsp/2+6p-1Cp∫τ1-ετ1Sτ2-s-Sτ1-sHpEfsHp2/pdsp/2+6p-1Cp∫τ1τ2Sτ2-sHpEfsHp2/pdsp/2≤6p-1bpM1p∫0τ1-εSτ2-s-Sτ1-sHpM2ds+12p-1MpM1p∫τ1-ετ1e-βτ1-sdsp-1∫τ1-ετ1e-βτ1-sM2ds+6p-1MpM1p∫τ1τ2e-βτ2-sdsp-1∫τ1τ2e-βτ2-sM2ds+6p-1Cp∫0τ1-εSτ2-s-Sτ1-sHpmfs·Θfx¯sBp+Ex¯′sHp+bp-1∫0smhs,τΘhx¯τBp+Ex¯′τHpdτ2/pdsp/2+12p-1CpMp∫τ1-ετ1e-pβτ1-smfs·Θfx¯sBp+Ex¯′sHp+bp-1∫0smhs,τΘhx¯τBp+Ex¯′τHpdτ2/pdsp/2+6p-1CpMp∫τ1τ2e-pβτ2-smfs·Θfx¯sBp+Ex¯′sHp+bp-1∫0smhs,τΘhx¯τBp+Ex¯′τHpdτ2/pdsp/2≤6p-1bpM1p∫0τ1-εSτ2-s-Sτ1-sHpM2ds+12p-1MpM1pβ1-p∫τ1-ετ1e-βτ1-sM2ds+6p-1MpM1pβ1-p∫τ1τ2e-βτ2-sM2ds+6p-1CpΘfr∗∗∫0τ1-εSτ2-s-Sτ1-sHpmfs2/pdsp/2+12p-1CpMpΘfr∗∗2βp-1p-21-p/2·∫τ1-ετ1e-βτ1-smfsds+6p-1CpMpΘfr∗∗2βp-1p-21-p/2∫τ1τ2e-βτ2-smfsds,where(49)Eux¯asHp≤5p-1Mpe-pβb1ap·M1pbpN∗p4p-1Ex~bHp+Cpbp/2-1∫0bEϕ~ηHpdη+Mpe-pαbEφ0Hp+2p-1ϕHp+g0,φ,ϕHp+Mpbp-1Lgr∗+r+1∫0be-pαb-ηdη+5p-1CpMpbp/2-1Θfr∗∗∫0be-pβb-ηmfηdη+mp-1M2pe-pαbe-pασnN∗p∑k=1mC1+ck+ϵkr∗+r+mp-1M2pe-pβbe-pβσnN∗p∑k=1mC1+dk+ϵkr∗+r≔M2.The fact of the compactness of S(t) for t>0 implies the continuity in the uniform operator topology. So as τ2-τ1→0, with ε being sufficiently small, the right-hand side of the above inequality is independent of x∈Br(0,Bb0) and tends to zero. The equicontinuities for the cases τ1<τ2≤0 or τ1≤0≤τ2≤b are very simple. Thus the set {Γn1x:x∈Br(0,Bb0)} is equicontinuous.
We now prove thatΓn1(Br(0,Bb0))(t)={γ~n1(t):γ~n1(t)∈Γn1(Br(0,Bb0))} is relatively compact for every t∈[0,b]. Let 0<t≤s≤b be fixed and let ε be a real number satisfying 0<ε<t. For x∈Br(0,Bb0), we define (50)γ~n,ε1t=∫0t-εSt-sBux¯asds+∫0t-εSt-sfsdws,where f∈SF,x¯. Using the compactness of S(t) for t>0, we deduce that the set Uε(t)={γ~n,ε1(t):x∈Br(0,Bb0)} is relatively compact in H for every ε, 0<ε<t. Moreover, for every x∈Br(0,Bb0) we have (51)Eγ~n1t-γ~n,ε1tHp≤2p-1E∫t-εtSt-sBux¯sdsHp+2p-1E∫t-εtSt-sfsdwsHp≤2p-1MpE∫t-εte-βt-sBux¯sHdsp+2p-1CpMp∫t-εte-pβt-sEfsHp2/pdsp/2≤2p-1MpM1p∫t-εte-βt-sdsp-1∫t-εte-βt-sM2ds+2p-1CpMp∫t-εte-pβt-smfs·Θfx¯sBp+Ex¯′sHp+bp-1∫0smhs,τΘhx¯τBp+Ex¯′τHpdτ2/pdsp/2≤2p-1MpM1pβ1-p∫t-εte-βt-sM2ds+2p-1CpMpΘfr∗∗2βp-1p-21-p/2∫t-εte-βt-smfsds.There are relatively compact sets arbitrarily close to the set W(t)={(Γn1x)(t):x∈Br(0,Bb0)}, and W(t) is a relatively compact in H. Hence, the Arzelá-Ascoli theorem shows that Γn1 is a compact multivalued map.
Secondly,Γn2(Br(0,Bb0)) is a compact multivalued map. We begin by showing that Γn2(Br(0,Bb0)) is equicontinuous. For each x∈Br0,Bb0, t∈(0,b) is fixed, t∈[ti,ti+1], and γ~n2∈Γn2x, such that(52)γ~n2t=∑0<tk<tCt-tkCσnIkx¯tk,x¯′tk+∑0<tk<tSt-tkSσnJkx¯tk,x¯′tk.Next, for τ1≤s<t≤τ2, ε>0, we have, using the property of compact operator, (53)Eγ~n2^iτ2-γ~n2^iτ1Hp≤2mp-1·∑k=1mECτ2-tk-Cτ1-tkCσn·Ikx¯tk,x¯′tkHp+2mp-1·∑k=1mESτ2-tk-Sτ1-tkSσn·Jkx¯tk,x¯′tkHp.As τ2-τ1→0, the right-hand side of the above inequality tends to zero independently of x due to the sets {C(σn)Ik(x¯tk,x¯′(tk)):x∈Br(0,Bb0)},{C(σn)Jk(x¯tk,x¯′(tk)):x∈Br(0,Bb0)} which are relatively compact in H and the strong continuity of C(·),S(·). So [γ~n2^]i,i=1,2,…,m, are equicontinuous.
Now we prove that[γ~n2^]i(t),i=1,2,…,m, is relatively compact for every t∈[0,b].
From the following relations(54)γ~n2^it=∑0<tk<tCt-tkCσnIkx¯tk,x¯′tk+∑0<tk<tSt-tkSσnJkx¯tk,x¯′tk∈2mp-1·∑k=1mCt-tkCσnIkBr∗0,B,Br0,H+2mp-1·∑k=1mSt-tkSσnJkBr∗0,B,Br0,H,we conclude that [γ~n2(Br(0,Bb0))^]i(t), i=1,2,…,m, is relatively compact for every t∈[ti,ti+1]. By Lemma 5, we infer that Γn2(Br(0,Bb0)) is relatively compact. Moreover, using the continuity of the operators C(t),S(t), for all t∈[0,b], we conclude that operator Γn2 is also a compact multivalued map.
4
Υ
n has a closed graph.
Letx(j)→x∗, ρ~n(j)∈Υnx(j), x(j)∈Br(0,Bb0), and ρ~n(j)→ρ~n∗. From axiom (A), it is easy to see that (x(j)¯)s→x∗¯s uniformly for s∈(-∞,b] as j→∞. We prove that ρ~n∗∈Υnx∗¯. Now ρ~n(j)∈Υnx(j)¯ means that there exists f(j)∈SF,x(j)¯ such that, for each t∈[0,b],(55)ρ~nj=∫0tSt-sBuxj¯asds+∫0tSt-sfjsdws+∑0<tk<tCt-tkCσnIkxj¯tk,xj¯′tk+∑0<tk<tSσnSt-tkJkxj¯tk,xj¯′tk,t∈J,where (56)uxj¯at=B∗S∗b-taI+Γ0b-1Ex~b+∫0bϕ~sdws-Cbφ0-Sb·ϕ-g0,φ,ϕ-B∗S∗b-t∫0baI+Γsb-1·Cb-sgs,xj¯s,xj¯′sds-B∗S∗b-t∫0baI+Γsb-1Sb-s·fjsdws-B∗S∗b-taI+Γsb-1·∑k=1mCb-tkCσnIkxj¯tk,xj¯′tk-B∗Sα∗b-taI+Γsb-1∑k=1mSb-tkSσn·Jkxj¯tk,xj¯′tk.We must prove that there exists f∗∈SF,x∗¯ such that, for each t∈[0,b],(57)ρ~n∗t=∫0tSt-sBux∗¯sds+∫0tSt-sf∗sdws+∑0<tk<tCt-tkCσnIkx∗¯tk,x∗¯′tk+∑0<tk<tSt-tkSσnJkx∗¯tk,x∗¯′tk,t∈J,where (58)ux∗¯at=B∗S∗b-taI+Γ0b-1Ex~b+∫0bϕ~sdws-Cbφ0-Sb·ϕ-g0,φ,ϕ-B∗S∗b-t∫0baI+Γsb-1·Cb-sgs,x∗¯s,x∗¯′sds-B∗S∗b-t∫0baI+Γsb-1Sb-sf∗sdws-B∗S∗b-taI+Γsb-1∑k=1mCb-tkCσn·Ikx∗¯tk,x∗¯′tk-B∗Sα∗b-taI+Γsb-1∑k=1mSb-tkSσnJkx∗¯tk,x∗¯′tk.Since g, Ik, Jk, k=1,2,…,m, are continuous, we obtain that (59)ρ~nj-∫0tSt-sBB∗S∗b-saI+Γ0b-1∫0baaA∫0basAAttti·Ex~b+∫0bϕ~ηdwη-Cbφ0ttti∫0bϕ~ηdw-Sbϕ-g0,φ,ϕ-B∗S∗b-sttti·∫0baI+Γηb-1Cb-ηgttti·η,xj¯η,xj¯′ηdηttti-B∗S∗b-saI+Γηb-1∑k=1mCb-tkCσnIkttti·xj¯tk,xj¯′tk-B∗Sα∗b-saI+Γηb-1ttti·∑k=1mSb-tkSσnJk∫0baaAxj¯tk,xj¯′tkdsttti-∑0<tk<tCt-tkIkxj¯tk,xj¯′tkttti-∑0<tk<tSt-tkJkxj¯tk,xj¯′tkttti-ρ~n∗t∫0baaAA-∫0tSt-sBttti·B∗S∗b-saI+Γ0b-1∫0baaAEx~b+∫0bϕ~ηdwηttti-Cbφ0∫0bϕ~ηdw-Sbϕ-g0,φ,ϕttti-B∗S∗b-s∫0baI+Γηb-1Cb-ηgttti·η,x∗¯η,x∗¯′ηdη-B∗S∗b-saI+Γηb-1ttti·∑k=1mCb-tkCσnIkx∗¯tk,x∗¯′tkttti-B∗Sα∗b-saI+Γηb-1∑k=1mSb-tkSσnJkttti∫0baaA·x∗¯tk,x∗¯′tkds-∑0<tk<tCt-tkIkttti·x∗¯tk,x∗¯′tk-∑0<tk<tSt-tkJkttti∫0baaAA·x∗¯tk,x∗¯′tkHp⟶0B∗Sα∗b-saI+Γηb-1B∗Sα∗aI+Γηb-1asj⟶∞.Consider the linear continuous operator Γ:Lp(J,H)→C(J,H),(60)Γft=∫0tSt-sfsdws+∫0tSt-s·BB∗S∗b-s·∫0baI+Γηb-1Sb-ηfηdηds.From  Lemma 14,  it follows that Γ∘SF is a closed graph operator. Also, from the definition of Γ, we have that, for every t∈[0,b],(61)ρ~nj-∫0tSt-sBB∗S∗b-saI+Γ0b-1Ex~b+∫0bϕ~ηdwη-Cbφ0-Sb·ϕ-g0,φ,ϕ-B∗S∗b-s∫0baI+Γηb-1·Cb-ηgη,xj¯η,xj¯′ηdη-B∗S∗b-saI+Γηb-1∑k=1mCb-tkCσn·Ikxj¯tk,xj¯′tk-B∗Sα∗b-saI+Γηb-1∑k=1mSb-tkSσn·Jkxj¯tk,xj¯′tkds-∑0<tk<tCt-tkIkxj¯tk,xj¯′tk-∑0<tk<tSt-tk·Jkxj¯tk,xj¯′tk∈ΓSF,xj¯.Since x(j)¯→x∗¯, for some f∗∈SF,x∗¯ it follows that, for every t∈[0,b], we have (62)ρ~n∗t-∫0tSt-sBB∗S∗b-saI+Γ0b-1Ex~b+∫0bϕ~ηdwη-Cbφ0-Sb·ϕ-g0,φ,ϕ-B∗S∗b-s∫0baI+Γsb-1·Cb-ηgη,x∗¯η,x∗¯′ηdη-B∗S∗b-saI+Γηb-1∑k=1mCb-tkCσn·Ikx∗¯tk,x∗¯′tk-B∗Sα∗b-saI+Γsb-1∑k=1mSb-tkSσn·Jkx∗¯tk,x∗¯′tkds-∑0<tk<tCt-tk·Ikx∗¯tk,x∗¯′tk-∑0<tk<tSt-tk·Jkx∗¯tk,x∗¯′tk=∫0tSt-sf∗sdws+∫0tSt-sBB∗S∗b-s·∫0baI+Γηb-1Sb-ηf∗ηdηds.Therefore, Υn is a completely continuous multivalued map, u.s.c. with convex closed, compact values.
Step 3. We will show that the set G={x∈Bb0:x∈λΛnx+λΥnxfor someλ∈(0,1)} is bounded on J.
Letx∈Bb0, and then there exists f∈SF,x¯ such that we have (63)xt=λ∫0tCt-sgs,x¯s,x¯′sds+λ∫0tSt-sBux¯asds+λ∫0tSt-sfsdws+λ∑0<tk<tCt-tkCσnIkx¯tk,x¯′tk+λ∑0<tk<tSt-tkSσnJkx¯tk,x¯′tk,t∈J.It also follows from Lemma 6 that (64)x′t=λgt,x¯t,x¯′t+λ∫0tASt-sgs,x¯s,x¯′sds+λ∫0tCt-sBux¯asds+λ∫0tCt-sfsdws+λ∑0<tk<tASt-tkCσnIkx¯tk,x¯′tk+λ∑0<tk<tCt-tkSσnJkx¯tk,x¯′tk,t∈J.This implies by (H1)–(H5) and (41) that for each t∈J we have (65)ExtHp≤5p-1E∫0tCt-sgs,x¯s,x¯′sdsHp+5p-1E∫0tSt-sBux¯asdsHp+5p-1E∫0tSt-s·fsdwsHp+5p-1E∑0<tk<tCt-tkCσnIkx¯tk,x¯′tkHp+5p-1E∑0<tk<tSt-tkSσn·Jkx¯tk,x¯′tkHp≤5p-1Mpbp-1Lg∫0te-pαt-sx¯sBp+Ex¯′sHp+1ds+20p-1Mpe-pβt1ap·Mpe-pβbM12pbp-1∫0te2pβs4p-1Ex~bHp+Cpbp/2-1∫0bEϕ~ηHpdη+Mpe-pαbEφ0Hp+2p-1ϕHp+g0,φ,ϕHp+Mpbp-1Lg∫0be-pαb-ηx¯ηBp+Ex¯′ηHp+1dη+CpMpbp/2-1∫0be-pβb-ηmfηΘfx¯ηBp+Ex¯′ηHp+bp-1∫0ηmhη,νΘhx¯νBp+Ex¯′νHpdνdη+mp-1M2p∑k=1me-pαb-tke-pασnC1+ck+ϵkx¯tkBp+Ex¯′tkHp+mp-1M2p∑k=1me-pβb-tke-pβσnC1+dk+ϵkx¯tkBp+Ex¯′tkHpds+5p-1CpMpbp/2-1∫0te-pβt-smfs·Θfx¯sBp+Ex¯′sHp+bp-1∫0smhs,τΘhx¯τBp+Ex¯′τHpdτds+5mp-1·M2p∑k=1me-pαt-tke-pασnC1+ck+ϵkx¯tkBp+Ex¯′tkHp+5mp-1M2p∑k=1me-pβt-tke-pβσnC1+dk+ϵk·x¯tkBp+Ex¯′tkHp.Similarly, for any t∈J, we have (66)Ex′tHp≤6p-1Lgx¯tBp+Ex¯′tHp+1+6p-1Mpbp-1Lg∫0te-pγt-sx¯sBp+Ex¯′sHp+1ds+20p-1Mpe-pαt1apMpe-pαbM12pbp-1∫0te2pαs4p-1Ex~bHp+Cpbp/2-1∫0bEϕ~ηHpdη+Mpe-pαbEφ0Hp+2p-1ϕHp+g0,φ,ϕHp+Mpbp-1Lg∫0be-pαb-ηx¯ηBp+Ex¯′ηHp+1dη+CpMpbp/2-1∫0be-pβb-ηmfηΘfx¯ηBp+Ex¯′ηHp+bp-1∫0ηmhη,νΘhx¯νBp+Ex¯′νHpdνdη+mp-1M2p∑k=1me-pαb-tke-pασnC1+ck+ϵkx¯tkBp+Ex¯′tkHp+mp-1M2p∑k=1me-pβb-tke-pβσnC1+dk+ϵkx¯tkBp+Ex¯′tkHpds+5p-1CpMpbp/2-1∫0te-pαt-smfs·Θfx¯sBp+Ex¯′sHp+bp-1∫0smhs,τΘhx¯τBp+Ex¯′τHpdτds+5mp-1M2p∑k=1me-γpt-tkC1+ck+ϵkx¯tkBp+Ex¯′tkHp+5mp-1M2p∑k=1me-αpt-tkC1+dk+ϵkx¯tkBp+Ex¯′tkHp.By Lemma 11, it follows that (67)sup⁡x¯sBp:0≤s≤t≤2p-1MbEφBp+2p-1Kbpsup⁡ExsHp:0≤s≤t.Consider the function defined by (68)μt=2p-1MbEφBp+2p-1Kbpsup⁡ExsHp:0≤s≤t+2p-1sup⁡Ex¯′sHp:0≤s≤t,0≤t≤b,where (69)x¯sBp+Ex¯′sHp≤sup⁡x¯sBp:0≤s≤t+sup⁡Ex¯′sHp:0≤s≤t.For each t∈[0,b], we have (70)ExtHp≤5p-1Mpbp-1Lge-pαt∫0tepαsμsds+20p-1Mpe-pβt1ap·Mpe-pβbM12pbp-1∫0te2pβs4p-1Ex~bHp+Cpbp/2-1∫0bEϕ~ηHpdη+Mpe-pαbEφ0Hp+2p-1ϕHp+g0,φ,ϕHp+Mpbp-1Lg∫0be-pαb-ημη+1dη+CpMpbp/2-1∫0be-pβb-ηmfη·Θfμη+bp-1∫0ηmhη,νΘhμνdνdη+mp-1M2p∑k=1me-pαb-tke-pασnC1+ck+ϵkμη+mp-1M2p∑k=1me-pβb-tke-pασnC1+dk+ϵkμηds+5p-1CpMpbp/2-1∫0te-pαt-smfsΘfμs+bp-1∫0smhs,τΘhμτdτds+5mp-1·M2p∑k=1me-pαt-tke-pασnC1+ck+ϵkμs+5mp-1·M2p∑k=1me-pβt-tke-pασnC1+dk+ϵkμs,Ex′tHp≤6p-1Lgμt+1+6p-1Mpbp-1Lge-pγt∫0tepγsμsds+24p-1Mpe-pαt1ap·Mpe-pαbM12pbp-1∫0te2pαs4p-1Ex~bHp+Cpbp/2-1∫0bEϕ~ηHpdη+Mpe-pαbEφ0Hp+2p-1ϕHp+g0,φ,ϕHp+Mpbp-1Lg∫0be-pαb-ημη+1dη+CpMpbp/2-1∫0be-pβb-ηmfη·Θfμη+bp-1∫0ηmhη,νΘhμνdνdη+mp-1M2p∑k=1me-pαb-tke-pασnC1+ck+ϵkμη+mp-1M2p∑k=1me-pβb-tke-pασnC1+dk+ϵkμηds+6p-1CpMpbp/2-1∫0te-pαt-smfsΘfμs+bp-1∫0smhs,τΘhμτdτds+6mp-1·M2p∑k=1me-pγt-tke-pασnC1+ck+ϵkμs+6mp-1·M2p∑k=1me-pαt-tke-pασnC1+dk+ϵkμs.Since limn→∞σn=0, it follows that (71)μt≤2p-1MbEφBp+2p-1KbpExtHp+2p-1Ex′tHp≤M~+12p-1Lgμt+N1Mpbp-1Lge-pδt∫0tepδsμsds+N1Mpe-pδtN2∫0te2pδsds+N1Mpe-pδtN3∫0te2pδsμsds+N1CpMpbp/2-1e-pδt∫0tepδsmfs·Θfμs+bp-1∫0smhs,τΘhμτdτds+10p-1Kbp+12p-1·mp-1M2pe-pδtN∗p∑k=1mck+ϵkμt+10p-1Kbp+12p-1mp-1M2pe-pδtN∗p∑k=1mdk+ϵkμt,where (72)M~=2p-1MbEφBp+12p-1KbpLg+N1mpM∗pN∗pMpC1,N1=10p-1Kbp+12p-1,δ=min⁡α,β,γ,N2=2p-11apMpe-pδbM12pbp-14p-1Ex~bHp+Cpbp/2-1∫0bEϕ~ηHpdη+Mpe-pδbEφ0Hp+2p-1ϕHp+g0,φ,ϕHp+Mpbp-1Lg∫0be-pδb-ημη+1dη+CpMpbp/2-1∫0be-pδb-ηmfηΘfμη+bp-1∫0ηmhη,νΘhμνdνdη+mpMpe-δpbN∗pC1,N3=2p-11apMpe-pδbM12pbp-1M∗pN∗pMpmp-1∑k=1mck+ϵk+dk+ϵk.Since L~=[12p-1Lg+(10p-1Kbp+12p-1)mp-1M2p∑k=1m(ck+dk+2ϵk)]N∗p<1, we obtain (73)eδptμt≤11-L~M~N∗p+N1Mpbp-1Lg∫0tepδsμsds+N1MpN2∫0te2pδsds+N1MpN3∫0te2δpsμsds+N1CpMpbp/2-1∫0tepδsmfsΘfμs+bp-1∫0smhs,τΘhμτdτds.Denoting by ζ(t) the right-hand side of the above inequality, we have (74)epδtμt≤ζt∀t∈0,b,ζ(0)=1/1-L~M~N∗p, and(75)ζ′t=11-L~N1Mpbp-1Lgepδtμt+N1MpN2e2pδt+N1MpN3e2pδtμt+N1CpMpbp/2-1epδtmftΘfμt+bp-1∫0tmht,sΘhμsds≤11-L~N1Mpbp-1Lgepδte-pδtζt+N1MpN2e2pδt+N1MpN3e2pδte-pδtζt+N1CpMpbp/2-1epδtmftΘfe-pδtζt+bp-1∫0tmht,sΘhe-pδsζsds.Let ξ(t)=e-pδtζ(t)+bp-1∫0tmh(t,s)Θh(e-pδsζ(s))ds; then ξ(0)=ζ(0),e-pδtζ(t)≤ξ(t), and for each t∈J we have (76)ξt′=-pδe-pδtζt+ζ′te-pδt+bp-1mht,t·Θhe-pδtζt≤-pδξt+11-L~N1Mpbp-1Lgξt+N1MpN2epδt+N1MpN3epδtξt+11-L~·N1CpMpbp/2-1mftΘfξt+bp-1mht,t·Θhξt≤m~tξt+1+Θfξt+Θhξt,t∈J,where (77)m~t=max⁡-pδ+11-L~N1Mpbp-1Lg+N1M2N3epδt,11-L~·N1M2N2epδt,11-L~·KbN1CpMpbp/2-1mft,bp-1mht,t.This implies that (78)∫ξ0ξtdϑ1+ϑ+Θfϑ+Θhϑ≤∫0bm~tdt<∞.This inequality shows that there is a constant K~ such that ξt≤K~, t∈J, and hence xPC1≤μ(t)≤e-pδtζ(t)≤ξ(t)≤K~, where K~ depends only on M, δ, p, Cp, Kb, and b and on the functions mf(·), mh(·,·), Θf(·) and Θh(·). This indicates that G is bounded on J. Consequently, by Lemma 13, we deduce that Λn+Υn has a fixed point xn∈Bb0, which is a mild solution of problem (27). Then, we have(79)xnt=∫0tCt-sgs,x¯n,s,x¯n′sds+∫0tSt-sBux¯nasds+∫0tSt-sfsdws+∑0<tk<tCt-tkCσnIkx¯n,tk,x¯n′tk+∑0<tk<tSt-tkSσnJkx¯n,tk,x¯n′tkfor t∈[0,b], and some fn∈SF,x¯n.
Next we will show that the set{xn:n∈N} is relatively compact in Bb0. We consider the decomposition xn=xn1+xn2, where(80)xn1t=∫0tCt-sgs,x¯n,s,x¯n′sds+∫0tSt-sBux¯nasds+∫0tSt-sfsdwsfor some fn∈SF,x¯n, and(81)xn2t=∑0<tk<tCt-tkCσnIkx¯n,tk,x¯n′tk+∑0<tk<tSt-tkSσnJkx¯n,tk,x¯n′tk.Step 4.{xn1(t):n∈N} is relatively compact in Bb0.
1
{
x
n
1
:
n
∈
N
} is equicontinuous on J.
Forε>0, xn∈Br(0,Bb0), there exists a constant 0<η~<ε such that, for all t∈(0,b] and ξ∈(0,η~) with t+ξ≤b, we have (82)Exn1t+ξ-xn1tHp≤6p-1E∫tt+ξCt+ξ-s·gs,x¯n,s,x¯n′sdsHp+6p-1E∫0tCt+ξ-s-Ct-s·gs,x¯n,s,x¯n′sdsHp+6p-1∫tt+ξSt+ξ-s·Bux¯nasdsHp+6p-1∫0tSt+ξ-s-St-sBux¯nasdsHp+6p-1E∫tt+ξSt+ξ-sfnsdwsHp+6p-1E∫0tSt+ξ-s-St-s·fnsdwsHp≤6p-1ξp-1Mp∫tt+ξe-pαt+ξ-sLGr∗+r+1ds+6p-1bp-1∫0tCt+ξ-s-Ct-sHpLGr∗+r+1ds+6p-1MpM1pξp-1∫tt+ξe-βt+ξ-sM2ds+6p-1M1pbp-1∫0tSt+ξ-s-St-sHp·M2ds+6p-1CpMpΘfr∗∗2δp-1p-21-p/2·∫tt+ξe-βt+ξ-smfsds+6p-1CpΘfr∗∗·∫0tSt+ξ-s-St-sHpmfs2/pdsp/2.Using the compact operator property, we can choose ξ∈(0,t) such that(83)6p-1ξp-1Mp∫tt+ξe-pαt+ξ-sLGr∗+r+1ds<ε6,6p-1bp-1∫0tCt+ξ-s-Ct-sHp·LGr∗+r+1ds<ε6,6p-1MpM1pξp-1∫tt+ξe-βt+ξ-sM2ds<ε6,6p-1M1pbp-1∫0tSt+ξ-s-St-sHpM2ds<ε6,6p-1CpMpΘfr∗∗2δp-1p-21-p/2·∫tt+ξe-βt+ξ-smfsds<ε6,6p-1CpΘfr∗∗·∫0tSt+ξ-s-St-sHpmfs2/pdsp/2<ε6.By (83) one has (84)Exn1t+ξ-xn1tHp<ε.Therefore, {xn1(t):n∈N} is equicontinuous for t∈(0,b]. Clearly {xn1(0):n∈N} is equicontinuous.
(
2
)
{
x
n
1
(
t
)
:
n
∈
N
} is relatively compact in H.
Lett∈(0,b], ε>0, xn∈Br(0,BPC); there exists ξ∈(0,t) such that (85)Exn1t-xnξtHp≤3p-1∫t-ξtCt-sgs,x¯n,s,x¯n′sdsHp+3p-1∫t-ξtSt-sBux¯nasdsHp+3p-1∫t-ξtSt-sfnsdwsHp≤3p-1ξp-1MpLGr∗+r+1∫t-ξte-pαt-sds+3p-1MpM1pβ1-p∫t-ξte-βt-sM2ds+3p-1CpMpΘfr∗∗2βp-1p-21-p/2·∫tt-ξe-βt-smfsds<ε,where (86)xnξt=∫0t-ξCt-sgs,x¯n,s,x¯n′sds+∫0t-ξSt-sBux¯nasds+∫0t-ξSt-sfnsdwsfor some fn∈SF,x¯n. By the compactness of C(t), S(t) for t>0, we see that the set {xnξ(t):n∈N} is relatively compact in H. Combining the above inequality, one has {xn1(t):n∈N} which is relatively compact in H.
Step 5.
{
x
n
2
(
t
)
:
n
∈
N
} is relatively compact in Bb0.
1
{
x
n
2
:
n
∈
N
} is equicontinuous on J.
For anyε>0, 0<t<b. Since C(σn), S(σn) are compact operators, we find that the sets W1={C(σn)Ik(x¯n,tk,x¯n′(tk)):xn∈Br(0,Bb0)} and W2={S(σn)Jk(x¯n,tk,x¯n′(tk)):xn∈Br(0,Bb0)} are relatively compact in H. From the strong continuity of (C(t))t≥0, for ε>0, we can choose 0<η~<b-t such that (87)ECt+ξ-Ctv1Hp<ε22mp-1,v1∈W1,ESt+ξ-Stv2Hp<ε22mp-1,v2∈W2when |ξ|<η~. For each xn∈Br(0,Bb0), t∈(0,b) is fixed and t∈[tk,tk+1], such that (88)Exn2^kt+ξ-xn2^ktHp≤2p-1E∑0<tk<tCt+ξ-tk-Ct-tkCσn·Ikx¯n,tk,x¯n′tkHp+2p-1E∑0<tk<tSt+ξ-tk-St-tkSσn·Jkx¯n,tk,x¯n′tkHp≤2mp-1·∑k=1mECt+ξ-tk-Ct-tkCσn·Ikx¯n,tk,x¯n′tkHp+2mp-1·∑k=1mESt+ξ-tk-St-tkSσn·Jkx¯n,tk,x¯n′tkHp<ε.As ξ→0 and ε is sufficiently small, the right-hand side of the above inequality tends to zero independently of xn, so [xn2^]k, k=1,2,…,m, are equicontinuous.
(
2
)
{
x
n
2
(
t
)
:
n
∈
N
} is relatively compact in H.
Fort∈tk,tk+1,k=1,…,m, and xn∈Br(0,Bb0), we have that there exists r′>0 such that (89)xn2^kt∈∑i=1kCt-tiCσnIix¯n,ti,x¯n′ti+∑i=1kSt-tiSσnJix¯n,ti,x¯n′ti,t∈tk,tk+1,xn∈Br′0,Bb0,∑i=1kCtk+1-tiCσnIix¯n,ti,x¯n′ti+∑i=1kStk+1-tiSσnJix¯n,ti,x¯n′ti,t=tk+1,xn∈Br′0,Bb0,∑i=1k-1Ctk-tiCσnIix¯n,ti,x¯n′ti+CσnIix¯n,ti,x¯n′ti+∑i=1k-1Stk-tiSσnJix¯n,ti,x¯n′ti+SσnJix¯n,ti,x¯n′tit=tk,xn∈Br′0,Bb0,where Br′(0,Bb0) is a closed ball of radius r′. One has [xn2^]k(t),k=1,2,…,m, which is relatively compact for every t∈[tk,tk+1], and {xn2(t):n∈N} is relatively compact in H.
These facts imply the relatively compact of{xn:n∈N} in Bb0. Therefore, without loss of generality, we may suppose that (90)xn⟶x∗∈Bb0asn⟶∞.Obviously, x∗∈Bb0; taking limits in (79) one has(91)x∗t=∫0tCt-sgs,x¯∗,s,x¯∗′sds+∫0tSt-sBux¯∗asds+∫0tSt-sf∗sdws+∑0<tk<tCt-tkIkx¯∗,tk,x¯∗′tk+∑0<tk<tSt-tkJkx¯∗,tk,x¯∗′tkfor t∈[0,b], and some f∗∈SF,x¯∗, which implies that x∗ is a mild solution of the problem (1) and the proof of Theorem 16 is complete.


## 4. Approximate Controllability of Impulsive Stochastic Control System

In this section, we present our main result on approximate controllability of system (1). To do this, we also need the following assumptions:(B1)
The functiong:J×B×H→H is continuous and there exists a constant C~1>0 such that (92)Egt,ψ,yHp≤C~1for0≤t≤b,(ψ,y)∈B×H.(B2)
There exists a constantC~2>0 such that (93)Ft,ψ,y,zHp≤C~2for0≤t≤b,(ψ,y,z)∈B×H×H, where (94)Ft,ψ,y,zHp=sup⁡EfHp:f∈Ft,ψ,y,z.Theorem 17.
Assume that assumptions of Theorem16 hold and, in addition, hypotheses (S1), (B1), and (B2) are satisfied. Then system (1) is approximately controllable on J.Proof.
Letxa(·) be a fixed point of Φ in Bb0. By Theorem 16, any fixed point of Φ is a mild solution of system (1). This means that there is xa∈Φ(xa); that is, there is f∈SF,x¯a such that (95)xat=∫0tCt-sgs,x¯sa,x¯a′sds+∫0tSt-sBux¯asds+∫0tSt-sfsdws+∑0<tk<tCt-tkIkx¯tka,x¯a′tk+∑0<tk<tSt-tkJkx¯tka,x¯a′tk,t∈J,where (96)uxat=B∗S∗b-taI+Γ0b-1Ex~b+∫0bϕ~sdws-Cbφ0-Sbϕ-g0,φ,ϕ-B∗S∗b-t·∫0baI+Γsb-1Cb-sgs,x¯sa,x¯a′sds-B∗S∗b-t∫0baI+Γsb-1Sb-sfsdws-B∗S∗b-taI+Γsb-1·∑k=1mCb-tkIkx¯tka,x¯a′tk-B∗Sα∗b-t·aI+Γsb-1·∑k=1mSb-tkJkx¯tka,x¯a′tk,t∈J,and by using the stochastic Fubini theorem, it is easy to see that (97)xab=∫0bCb-sgs,x¯sa,x¯a′sds+∫0bSb-sBux¯asds+∫0bSb-s·fsdws+∑0<tk<bCb-tk·Ikx¯tka,x¯a′tk+∑0<tk<bSb-tk·Jkx¯tka,x¯a′tk=xb-aaI+Γ0b-1Ex~b+∫0bϕ~sdws-Cbφ0-Sb·ϕ-g0,φ,ϕ-a∫0baI+Γsb-1Cb-s·gs,x¯sa,x¯a′sds-a∫0baI+Γsb-1·Sb-sfsdws-aaI+Γsb-1·∑k=1mCb-tkIkx¯tka,x¯a′tk-aaI+Γsb-1·∑k=1mSb-tkJkx¯tka,x¯a′tk.By conditions (B1) and (B2), we get that the sequences {g(s,x¯sa,(x¯a)′(s))} and {f(s)} are uniformly bounded on J. Thus there are subsequences, still denoted by {g(s,x¯sa,(x¯a)′(s))} and {f(s)} that converge weakly to, say, g(s) in H and f∗∗(s) in L(K,H), respectively. The compactness of St, t>0, implies that S(b-s)[g(s,x¯sa,(x¯a)′(s))-g(s)]→0, S(b-s)[f(s)-f∗∗(s)]→0. On the other hand, by Lemma 9, for all t∈J, a(aI+Γtb)-1→0 strongly as a→0+ and a(aI+Γtb)-1≤1. Therefore, by the Lebesque dominated convergence theorem it follows that(98)Exab-xbHp≤8p-1EaaI+Γ0b-1Ex~b-Cb·φ0-Sbϕ-g0,φ,ϕHp+8p-1E∫0baaI+Γ0b-1ϕ~sH2dsp/2+8p-1∫0baaI+Γsb-1Cb-s·gs,x¯sa,x¯a′s-gsHdηp+8p-1E∫0baaI+Γsb-1Cb-sgsHdsp+8p-1E∫0baaI+Γsb-1Sb-s·fs-f∗∗sH2dsp/2+8p-1E∫0baaI+Γsb-1Sb-s·f∗∗sH2dsp/2+8p-1EaaI+Γsb-1·∑k=1mCb-tkIkx¯tka,x¯a′tkHp+8p-1EaaI+Γsb-1∑k=1mSb-tk·Jkx¯tka,x¯a′tkHp⟶0asa⟶0+.So xa(b)→xb holds, which shows that system (1) is approximately controllable and the proof is complete.


## 5. Example

Consider the following impulsive partial stochastic neutral differential inclusions of the form(99)d∂∂tzt,x-∫-∞tb1s-tzs,xds-b2sz′t,x∈∂2∂2tzt,xdt+u~t,xdt+∫-∞tϖ1t,t-s,x,zs,xds+ς1tz′t,x+∫0t∫-∞sς2tϖ2s,τ-s,x,zτ,xdτds+∫0tς3sz′s,xdsdwt,0≤t≤b,0≤x≤π,t≠tk,zt,0=zt,π=0,0≤t≤b,zt,x=φt,x,∂∂tz0,x=ϕx,t≤0,0≤x≤π,Δztk,x=∫-∞tkηks-tkzs,xds+k1tk·z′tk,x,k=1,2,…,m,Δz′tk,x=∫-∞tkη~ks-tkzs,xds+k2tk·z′tk,x,k=1,2,…,m,where (tk)k∈N is a strictly increasing sequence of positive numbers and u~(·) is a real function of bounded variation on [0,b]. w(t) denotes a standard cylindrical Wiener process in H defined on a stochastic space (Ω,F,P).LetH=L2([0,π]) with the norm ·H and define the operator A by Aω=ω′′ with the domain (100)DA≔ω·∈H:ω,ω′are absolutely continuous,ω′′∈H,ω0=ωπ=0.It is well known that A is the infinitesimal generator of a strongly continuous cosine family {C(t):t∈R} in H and is given by (101)Ctω=∑n=1∞cos⁡ntω,enen,ω∈H,where enθ=2/πsin⁡nθ,i=1,2,…, is the orthogonal set of eigenvalues of A. The associated sine family S(t), t>0, is compact and is given by (102)Stω=∑n=1∞1nsin⁡ntω,enen,ω∈H.Additionally, we will assume the following:(i)
The functionsbi:R→R, i=1,2, are continuous, and L~1=(∫-∞0(b1(s))2/h~(s)ds)1/2<∞.(ii)
The functionsϖi:R4→R, i=1,2, are continuous and there exist continuous functions aj:R→R, j=1,2,3,4, such that (103)ϖ1t,s,x,y≤a1ta2sy,t,s,x,y∈R4,ϖ2t,s,x,y≤a3ta4sy,t,s,x,y∈R4withL^1=(∫-∞0(a2(s))2/h~(s)ds)1/2<∞, L^2=(∫-∞0(a4(s))2/h~(s)ds)1/2<∞.(iii)
The functionsςi,ki:R→R, i=1,2, are continuous.(iv)
The functionsηk,η~k:R→R, k=1,2,…,m, are continuous, ϑk=(∫-∞0(ηk(s))2/h~(s)ds)1/2<∞, and ϑ~k=(∫-∞0(η~k(s))2/h~(s)ds)1/2<∞ for every k=1,2,…,m.TakeB=PC0×L2(h,H) which is the space introduced in Example 2. Set φ(θ)(x)=φ(θ,x)∈B, defining the maps g:[0,b]×B×H→H, F:[0,b]×B×H×H→P(H) by (104)gt,ψ,ψ′x=∫-∞tb1sψs,xds+b2tψ′t,x,ft,ψ,ψ′,B1ψx=∫-∞0ϖ1t,s,x,ψs,xds+ς1tψ′t,x+B1ψx,B1ψx=∫0t∫-∞0ς2tϖ2s,τ,x,ψτ,xdτds+∫0tς3sψ′s,xds,Ikψ,ψ′x=∫-∞0ηksψs,xds+k1tkψ′tk,x,Jkψ,ψ′x=∫-∞0ηksψs,xds+k2tkψ′tk,x.Using these definitions, we can represent system (99) in the abstract form (1). Moreover, it is easy to see that G, Ik, and Jk are continuous, and G, Ik, Jk, and F are bounded linear operators with EGL(B,H)p≤LG, EIkL(B,H)p≤ck, EJkL(B,H)p≤dk, k=1,2,…,m, and EFL(B,H)p≤LF, where LG=[L~1+b2∞]p, ck=[ϑk+k1∞]p, dk=[ϑ~k+k2∞]p, and LF=[a1∞L^1+ς1∞+ς2∞a3L1L^2+ς3L1]p. Further, we can impose some suitable conditions on the above-defined functions to verify the assumptions on Theorem 16. Therefore, assumptions (H1)–(H6), (B1), and (B2) all hold, and the associated linear system of (99) is not exactly controllable but it is approximately controllable. Hence by Theorems 16 and 17, system (99) is approximately controllable on [0,b].



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*Source: 925209-2015-08-30.xml*