# Existence Results for Stochastic Semilinear Differential Inclusions with Nonlocal Conditions

Authors: A. Vinodkumar; A. Boucherif
Journal: International Journal of Stochastic Analysis
 (2011)

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## Abstract

We discuss existence results of mild solutions for stochastic differential inclusions subject to nonlocal conditions. We provide sufficient conditions in order to obtain a priori bounds on possible solutions of a one-parameter family of problems related to the original one. We, then, rely on fixed point theorems for multivalued operators to prove our main results.


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## Body

## 1. Introduction

We investigate nonlocal stochastic differential inclusions (SDIns) of the form(1.1)dx(t)∈[Ax(t)+f(t,xt)]dt+G(t,xt)dw(t),t∈J=[0,T],x(0)=∑i=1mγix(ti),x(t)=φ(t),t∈J1=(-∞,0],
where T>0, 0<t1<t2<⋯<tm<T, γi are real numbers, f is a single-valued function, and G is multivalued map.The importance of nonlocal conditions and their applications in different field have been discussed in [1–3]. Existence results for semilinear evolution equations with nonlocal conditions were investigated in [4–7], and the case of semilinear evolution inclusions with nonlocal conditions and a nonconvex right-hand side was discussed in [8].Stochastic differential equations (SDEs) play a very important role in formulation and analysis in mechanical, electrical, control engineering and physical sciences, and economic and social sciences. See for instance [9–12] and the references therein. So far, very few articles have been devoted to the study of stochastic differential inclusions with nonlocal conditions, see [13–15] and the references therein. Our objective is to contribute to the study of SDIns with nonlocal conditions. Motivated by the above-mentioned works and using the technique developed in [11, 16, 17], we study the SDIns of the form (1.1). The paper is organized as follows: some preliminaries are presented in Section 2. In Section 3, we investigate the existence of mild solutions for SDIns by using fixed point theorems for Kakutani maps. Finally in Section 4, we give an application to our abstract result.


## 2. Preliminaries

LetX, Y be real separable Hilbert spaces and L(Y,X) be the space of bounded linear operators mapping Y into X. For convenience, we will use 〈·,·〉 to denote inner product of X and Y and ∥·∥ to denote norms in X, Y, and L(Y,X) without any confusion.Let(Ω,ℱ,P;𝔽)(𝔽={ℱt}t≥0) be a complete filtered probability space such that ℱ0 contains all P-null sets of ℱ. An X-valued random variable is an ℱ-measurable function x(t):Ω→X and the collection of random variables ℋ={x(t,ω):Ω→X:t∈J} is called a stochastic process. Generally, we just write x(t) instead of x(t,ω) and x(t):J→X is the space of ℋ. Let {ei}i≥1 be a complete orthonormal basis of Y. Suppose that {w(t):t≥0} is a cylindrical Y-valued Wiener process with finite trace nuclear covariance operator Q≥0, denote Tr(Q)=∑i=1∞λi=λ<∞, which satisfies Qei=λiei. Actually, w(t)=∑i=1∞λiwi(t)ei, where {wi(t)}i=1∞ are mutually independent one-dimensional standard Wiener processes. We assume that ℱt=σ{w(s):0≤s≤t} is the σ-algebra generated by w and ℱt=ℱ. Let μ∈L(Y,X) and define (2.1)‖μ‖Q2=Tr⁡(μQμ*)=∑n=1∞‖λnμen‖2.If∥μ∥Q<∞, then μ is called a Q-Hilbert-Schmidt operator. Let LQ(Y,X) denote the space of all Q-Hilbert-Schmidt operators μ:Y→X. The completion LQ(Y,X) of L(Y,X) with respect to the topology induced by the norm ∥·∥Q, where ∥μ∥Q2=〈μ,μ〉 is a Hilbert space with the above norm topology.We now make the system (1.1) precise. Let A:X→X be the infinitesimal generator of a compact analytic semigroup {S(t),t≥0} defined on X. Let Dτ=D((-∞,0],X) denote the family of all right continuous functions with left-hand limit φ from (-∞,0] to X and 𝒫(𝔼) is the family of all nonempty measurable subsets of 𝔼. The functions f:[0,T]×Dτ→X; G:[0,T]×Dτ→𝒫(LQ(Y,X)) are Borel measurable. The phase space D((-∞,0],X) is equipped with the norm ∥ϕ∥=sup-∞<θ≤0∥ϕ(θ)∥. We denote by Dℱ0b((-∞,0],X) the family of all almost surely bounded, ℱ0-measurable, Dτ-valued random variables. Further, let ℬ𝒯 be the Banach space of all ℱt-adapted process ϕ(t,w) which is almost surely continuous in t for fixed w∈Ω, with norm (2.2)‖ϕ‖BT=(sup⁡0≤t≤TE‖ϕ‖2)1/2,
for any ϕ∈ℬ𝒯. Here the expectation E is defined by(2.3)Eχ=∫Ωχ(w)dP.We shall assume throughout the remainder of the paper that the initial functionφ∈Dℱ0b((-∞,0],X).Some notions from set-valued analysis are in order. Denote by𝒫cl(X)={Y∈𝒫(X):Yclosed}, 𝒫bd(X)={Y∈𝒫(X):Ybounded}, 𝒫cv(X)={Y∈𝒫(X):Yconvex}, 𝒫cp(X)={Y∈𝒫(X):Ycompact}, 𝒫cp,cv(X)={Y∈𝒫(X):Ycompactandconvex}. A multivalued map F:X→𝒫(X) is convex valued if F(x)∈𝒫cv(X) for all x∈X, closed valued if F(x)∈𝒫cl(X) for all x∈X, F is compact valued if F(x)∈𝒫cp(X) for all x∈X. F is bounded on bounded sets if F(V)=∪x∈VF(x) is bounded in X, for all V∈𝒫bd(X); that is, (2.4)sup⁡x∈V{sup⁡{‖y‖:y∈F(x)}}<∞.F is called upper semicontinuous (u.s.c) on X, if for each x0∈X, the set F(x0) is non-empty, closed subset of X, and if for each open set V of X containing F(x0) there exists an open neighborhood N of x0 such that F(N)⊆V.F is said to be completely continuous if F(V) is relatively compact, for every V∈𝒫bd(X).If the multivalued mapF is completely continuous with nonempty compact values, then F is u.s.c if and only if F has a closed graph (ie., xn→x*,yn→y*,yn∈F(xn)implyy*∈F(x*)).F has a fixed point if there is x∈X such that x∈F(x). The fixed point set of the multivalued operator F will be denoted by Fix F.The Hausdorff metric on𝒫bd,cl(X) is the function H:𝒫bd,cl(X)×𝒫bd,cl(X)→ℜ+ defined by (2.5)H(A,B)=max⁡⁡{sup⁡a∈Ad(a,B),sup⁡a∈Bd(A,b)},
where d(𝔸,b)=inf{∥a-b∥2,a∈𝔸},d(a,𝔹)=inf{∥a-b∥2,b∈𝔹}.The multivalued mapM:[0,T]Pbd,cl(X) is said to be measurable if for each x∈X the function ζ:[0,T]→ℜ+ defined by(2.6)ζ(t)=d(x,M(t))=inf⁡{‖x-z‖2:z∈M(t)}ismeasurable.For more details on multivalued maps see [18–20]. Our existence results are based on the following fixed point theorem (nonlinear alternative) for Kakutani maps [21].Theorem 2.1.
LetX be a Hilbert space, C a closed convex subset of X, Y an open subset of C and 0∈Y. Suppose that F:Y¯→𝒫cl,cv(C) is an upper semicontinuous compact map. Then either (i) F has a fixed point in Y¯ or (ii) there are v∈∂Y and λ∈(0,1) with v∈λF(v).Definition 2.2.
The multivalued mapG:J×Dτ→𝒫(LQ(Y,X)) is said to be L2-Carathèodory if(i)
t↦G(t,u) is measurable for each u∈Dτ;(ii)
u↦G(t,u) is upper semicontinuous for almost all t∈J;(iii)
for eachq>0, there exists ωq∈L2(J,ℜ+) such that
(2.7)‖G(t,u)‖2:=sup⁡{‖g‖2:g∈G(t,u)}≤ωq(t),
for all ∥u∥ℬ𝒯2≤q and for a.e. t∈J.For eachx∈L2(LQ(Y,X)) define the set of selections of G by (2.8)SG,x={g∈L2=L2(LQ(Y,X)):g(t)∈G(t,xt)fora.e,t∈J}.Lemma 2.3 (see [22]).
LetI be a compact interval and X be a Hilbert space. Let G be an L2-Carathèodory multivalued map with SG,x≠ϕ and let Γ be a linear continuous mapping from L2(I,X)→C(I,X). Then the operator
(2.9)Γ∘SG:C(I,X)⟼Pbd,cl⁡,cv(C(I,X)),x⟼(Γ∘SG)(x)=Γ(SG,x),
is a closed graph operator in C(I,X)×C(I,X).Definition 2.4.
A semigroup{S(t),t≥0} is said to be uniformly bounded if there exists a constant M≥1 such that
(2.10)‖S(t)‖≤M,fort≥0.Assume that(2.11)∑i=1m|γi|<1M.
Then there exists a bounded operator B on D(B)=X given by the formula (2.12)B=(I-∑i=1mγiT(ti))-1.Definition 2.5.
A stochastic process{x(t)∈ℬ𝒯,t∈(-∞,T]} is called a mild solution of system (1.1) if(i)
x(t) is ℱt-adapted with ∫0T∥x(t)∥2dt<∞ almost surely;(ii)
x(t) satisfies the integral equation
(2.13)x(t)={φ(t),t∈J1,∑i=1mγiS(t)B[∫0tiS(ti-s)f(s,xs)ds+∫0tiS(ti-s)g(s)dw(s)]+∫0tS(t-s)f(s,xs)ds+∫0tS(t-s)g(s)dw(s),a.e.t∈J,
where g∈SG,x.


## 3. Existence Results

In this section, we discuss the existence of mild solutions of the system (1.1). We need the following hypotheses.(H1):
The functionf:J×Dτ→X is continuous and there exist two positive constants C1,C2 such that
(3.1)‖f(t,xt)‖2≤C1‖x‖2+C2,foreachx∈Dτ,t∈J.(H2):
G:J×Dτ→𝒫(LQ(Y,X)) is an L2-Carathéodory multivalued function with compact and convex values.(H3):
There exists a continuous nondecreasing functionψ:ℜ+→(0,∞) and p∈L1(J,ℜ+) such that
(3.2)‖G(t,xt)‖2=sup⁡{‖g‖2:g∈G(t,xt)}≤p(t)ψ(‖x‖2),a.et∈J,allx∈Dτ.Theorem 3.1.
Assume that(H1)–(H3) hold. Then the system (1.1) has at least one mild solution on (-∞,T], provided that
(3.3)3K1C1T<1,sup⁡ρ∈[0,∞){1-3TK1C1}ρ3TK2C2+3K1Tr(Q)‖p‖L1ψ(ρ)>1,
where
(3.4)K1=3(2mM2∑i=1m|γi|2‖B‖2+1)M2,K2=(2mM2∑i=1m|γi|2‖B‖2+T)M2.Proof.
Transform the system (1.1) into a fixed point problem. Consider the multivalued operator ℳ:ℬ𝒯→𝒫(ℬ𝒯) defined by
(3.5)M(x)=h∈BT:h(t)={φ(t),t∈J1∑i=1mγiS(t)B[∫0tiS(ti-s)f(s,xs)ds+∫0tiS(ti-s)g(s)dw(s)]+∫0tS(t-s)f(s,xs)ds+∫0tS(t-s)g(s)dw(s),g∈SG,x,a.e.t∈J.
It is clear that the fixed points of ℳ are mild solutions of system (1.1). Hence we have to find solutions of the inclusion y∈ℳ(y). We show that the multivalued operator ℳ satisfies all the conditions of Theorem 2.1. The proof will be given in several steps.Step 1.
ℳ(x) is convex for each x∈ℬ𝒯. Since G has convex values it follows that SG,x is convex; so that if g1,g2∈SG,x then αg1+(1-α)g2∈SG,x, which implies clearly that ℳ(x) is convex.Step 2.
The operatorℳ is bounded on bounded subsets of ℬ𝒯. For q>0 let Bq={x∈ℬ𝒯:∥x∥ℬ𝒯≤q} be a bounded subset of ℬ𝒯. We show that ℳ(Bq) is a bounded subset of ℬ𝒯. For each x∈Bq let h∈ℳ(x). Then there exists g∈SG,x such that for each t∈J we have
(3.6)h(t)=∑i=1mγiS(t)B[∫0tiS(ti-s)f(s,xs)ds+∫0tiS(ti-s)g(s)dw(s)]+∫0tS(t-s)f(s,xs)ds+∫0tS(t-s)g(s)dw(s),(3.7)‖h(t)‖2≤3‖∑i=1mγiS(t)B[∫0tiS(ti-s)f(s,xs)ds+∫0tiS(ti-s)g(s)dw(s)]‖2+3‖∫0tS(t-s)f(s,xs)ds‖2+3‖∫0tS(t-s)g(s)dw(s)‖2≤6m∑i=1m|γi|2M2‖B‖2M2[∫0ti‖f(s,xs)‖2ds+Tr⁡(Q)∫0ti‖g(s)‖2ds]+3M2∫0t‖f(s,xs)‖2ds+3M2Tr⁡(Q)∫0t‖g(s)‖2ds≤3(2mM2∑i=1m|γi|2‖B‖2+1)M2∫0T‖f(s,xs)‖2ds+3(2mM2∑i=1m|γi|2‖B‖2+1)Tr⁡(Q)M2∫0T‖g(s)‖2ds≤3M2(2mM2∑i=1m|γi|2‖B‖2+1)(∫0T‖f(s,xs)‖2ds+Tr⁡(Q)∫0T‖g(s)‖2ds)≤3M2(2mM2∑i=1m|γi|2‖B‖2+1)×(∫0T(C1‖x(s)‖2+C2)ds+Tr⁡(Q)∫0Tp(s)ψ(‖x(s)‖2)ds)≤K1(TC1q2+TC2+Tr⁡(Q)ψ(q2)‖p‖L1).
Hence for each h∈ℳ(Bq), we get
(3.8)‖h‖BT2=sup⁡t∈[0,T]E‖h‖2≤K1T(TC1q2+TC2+Tr⁡(Q)ψ(q2)‖p‖L1).
Then, for each h∈ℳ(x), we have ∥h∥ℬ𝒯2≤∧̂, where ∧̂:=𝒦1T(TC1q2+TC2+Tr(Q)ψ(q2)∥p∥L1).Step 3.
ℳ sends bounded sets into equicontinuous sets in ℬ𝒯. For each x∈Bq let h∈ℳ(x) be given by (3.6). Let τ1,τ2∈J with 0<τ1<τ2≤T. Then
(3.9)h(τ2)-h(τ1)=∑i=1mγi[S(τ2)-S(τ1)]B[∫0tiS(ti-s)f(s,xs)ds+∫0tiS(ti-s)g(s)dw(s)]+∫0τ2S(τ2-s)f(s,xs)ds+∫0τ2S(τ2-s)g(s)dw(s)-∫0τ1S(τ1-s)f(s,xs)ds-∫0τ1S(τ1-s)g(s)dw(s).
This implies that
(3.10)h(τ2)-h(τ1)=∑i=1mγi[S(τ2)-S(τ1)]B[∫0tiS(ti-s)f(s,xs)ds+∫0tiS(ti-s)g(s)dw(s)]+∫0τ1[S(τ2-s)-S(τ1-s)]f(s,xs)ds+∫0τ1[S(τ2-s)-S(τ1-s)]g(s)dw(s)+∫τ1τ2S(τ2-s)f(s,xs)ds+∫τ1τ2S(τ2-s)g(s)dw(s).
It follows that
(3.11)‖h(τ2)-h(τ1)‖2≤5m‖B‖2‖S(τ2)-S(τ1)‖2∑i=1m|γi|2∫0ti‖S(ti-s)‖2‖f(s,xs)‖2ds+5m‖B‖2‖S(τ2)-S(τ1)‖2∑i=1m|γi|2Tr⁡(Q)∫0ti‖S(ti-s)‖2‖g(s)‖2ds+5∫0τ1‖S(τ2-s)-S(τ1-s)‖2‖f(s,xs)‖2ds+5∫τ1τ2‖S(τ2-s)‖2‖f(s,xs)‖2ds+5Tr⁡(Q)∫0τ1‖S(τ2-s)-S(τ1-s)‖2‖g(s)‖2ds+5Tr⁡(Q)∫τ1τ2‖S(τ1-s)‖2‖g(s)‖2ds≤5mM2‖B‖2‖S(τ2)-S(τ1)‖2∑i=1m|γi|2{(C1q2+C2)+Tr⁡(Q)ψ(q2)‖p‖L1}+5(C1q2+C2)∫0τ1‖S(τ2-s)-S(τ1-s)‖2ds+5M2(C1q2+C2)(τ2-τ1)+5Tr⁡(Q)ψ(q2)∫0τ1‖S(τ2-s)-S(τ1-s)‖2p(s)ds+5M2Tr⁡(Q)ψ(q2)‖p‖L1(τ2-τ1).
Since there is δ>0 such that
(3.12)‖S(τ2)-S(τ1)‖≤δτ1τ2-τ1,
(see [23, proposition  1]) and the compactness of S(t) for t>0 implies the continuity in the uniform operator topology, we have
(3.13)‖S(τ2)-S(τ1)‖2⟶0,‖S(τ2-s)-S(τ1-s)‖2⟶0asτ2⟶τ1.
Therefore
(3.14)E‖h(τ2)-h(τ1)‖2⟶0asτ2⟶τ1.
When τ1=0 we have
(3.15)‖h(τ2)-h(0)‖2≤5mM2‖B‖2‖S(τ2)-S(0)‖2∑i=1m|γi|2{ti(C1q2+C2)+Tr⁡(Q)ψ(q2)‖p‖L1}+5M2(C1q2+C2)τ2+5M2Tr⁡(Q)ψ(q2)‖p‖L1τ2,
so that, similar to the previous situation, we have
(3.16)E‖h(τ2)-h(0)‖2⟶0asτ2⟶0.Step 4.
ℳ sends bounded sets into relatively compact sets in ℬ𝒯. Let 0<ϵ<t, for t∈J. For x∈Bq define a function hϵ by
(3.17)hϵ(t)=∑i=1mγiS(t)B[∫0tiS(ti-s)f(s,xs)ds+∫0tiS(ti-s)g(s)dw(s)]+∫0t-ϵS(t-s)f(s,xs)ds+∫0t-ϵS(t-s)g(s)dw(s),
where g∈SG,x. Since S(t) is a compact operator, the set Vϵ(t)={hϵ(t):hϵ∈ℳ(x)} is relatively compact in ℬ𝒯 for every ϵ in (0,t). Moreover, for every h∈ℳ(x) we have
(3.18)E‖h-hϵ‖2≤2ϵM2∫t-ϵt[C1E‖x(s)‖2+C2]ds+2M2Tr⁡(Q)∫t-ϵtωq(s)ds≤2ϵ2M2(C1q+C2)+2M2Tr⁡(Q)∫t-ϵtωq(s)ds.
Since ωq∈L1(J) and meas([t-ϵ,t])=ϵ it follows that
(3.19)‖h-hϵ‖BT⟶0asϵ⟶0.
As a consequence of Step 1 through Step 4, together with Ascoli-Arzela theorem, we can conclude that the multivalued operator ℳ is compact.Step 5.
ℳ has a closed graph. Let xn→x* and hn∈ℳ(xn) with hn→h*. We shall show that h*∈ℳ(x*).
There existsgn∈SG,xn such that
(3.20)hn(t)=∑i=1mγiS(t)B[∫0tiS(ti-s)f(s,xn,s)ds+∫0tiS(ti-s)gn(s)dw(s)]+∫0tS(t-s)f(s,xn,s)ds+∫0tS(t-s)gn(s)dw(s).
We must prove that there exists g*∈SG,x* such that
(3.21)h*(t)=∑i=1mγiS(t)B[∫0tiS(ti-s)f(s,xs*)ds+∫0tiS(ti-s)g*(s)dw(s)]+∫0tS(t-s)f(s,xs*)ds+∫0tS(t-s)g*(s)dw(s).
Consider the linear continuous operator Γ:L2(LQ(Y,X))→ℬ𝒯 defined by
(3.22)Γ(g)(t)=∑i=1mγiS(t)B∫0tiS(ti-s)g(s)dw(s)+∫0tS(t-s)g(s)dw(s).
Clearly, Γ is linear and continuous. Indeed, one has
(3.23)‖Γ(g)(t)‖2≤(2mM2∑i=1m|γi|2‖B‖2+1)Tr⁡(Q)M2∫0t‖g(s)‖2dsE‖Γ(g)‖2≤(2mM2∑i=1m|γi|2‖B‖2+1)Tr⁡(Q)M2‖ωq‖L1.
Let
(3.24)Θn(t)=hn(t)-∑i=1mγiS(t)B∫0tiS(ti-s)f(s,xn,s)ds-∫0tS(t-s)f(s,xn,s)ds,Θ*(t)=h*(t)-∑i=1mγiS(t)B∫0tiS(ti-s)f(s,xs*)ds-∫0tS(t-s)f(s,xs*)ds.
We have
(3.25)Θn(t)∈Γ∘SG,xn.
Since f is continuous (see (H1))
(3.26)‖Θn(t)-Θ*(t)‖2⟶0asn⟶∞.
Lemma 2.3 implies that Γ∘SG has a closed graph. Hence there exists g*∈SG,x* such that
(3.27)Θ*(t)=∑i=1mγiS(t)B∫0tiS(ti-s)g*(s)dw(s)+∫0tS(t-s)g*(s)dw(s).
Hence h*∈ℳ(x*), which shows that graph ℳ is closed.Step 6.
Letλ∈(0,1) and let x∈λℳ(x). Then there exists g∈SG,x such that
(3.28)x(t)=λ∑i=1mγiS(t)B[∫0tiS(ti-s)f(s,xs)ds+∫0tiS(ti-s)g(s)dw(s)]+λ∫0tS(t-s)f(s,xs)ds+λ∫0tS(t-s)g(s)dw(s).
Thus
(3.29)‖x(t)‖2≤3(2mM2∑i=1m|γi|2‖B‖2+T)M2∫0t‖f(s,xs)‖2ds+3(2mM2∑i=1m|γi|2‖B‖2+1)M2Tr⁡(Q)∫0t‖g(s)‖2ds.
Conditions (H1)–(H3) imply that for each t∈J(3.30)E‖x‖2≤3(2mM2∑i=1m|γi|2‖B‖2+T)M2∫0t[C1E‖x(s)‖2+C2]ds+3(2mM2∑i=1m|γi|2‖B‖2+1)M2Tr⁡(Q)∫0tp(s)ψ(E‖x(s)‖2)ds.
The function ϱ defined on [0,T] by
(3.31)ϱ(t)=sup⁡⁡{E‖x(s)‖2:0≤s≤t}
satisfies
(3.32)ϱ(t)≤3T(2mM2∑i=1m|γi|2‖B‖2+T)M2C2+3T(2mM2∑i=1m|γi|2‖B‖2+T)M2C1ϱ(t)+3(2mM2∑i=1m|γi|2‖B‖2+1)M2Tr⁡(Q)‖p‖L1ψ(ϱ(t)).
This yields
(3.33)ϱ(t)≤3TK2C2+3K1Tr⁡(Q)‖p‖L1ψ(ϱ(t))1-3TK1C1.
Since
(3.34)‖x‖BT=sup⁡0≤t≤Tϱ(t),
it follows that
(3.35)‖x‖BT≤3TK2C2+3K1Tr⁡(Q)‖p‖L1ψ(‖x‖BT)1-3TK1C1.
Therefore
(3.36)(1-3TK1C1)‖x‖BT3TK2C2+3K1Tr⁡(Q)‖p‖L1ψ(‖x‖BT)≤1.
Now, by (3.3) there exists ρ0>0 such that
(3.37){1-3TK1C1}ρ03TK2C2+3K1Tr⁡(Q)‖p‖L1ψ(ρ0)>1.
Let 𝔜={v∈ℬ𝒯:∥v∥ℬ𝒯<ρ0}. Suppose that there is v∈∂𝔜 such that v∈λℳ(v) for λ∈(0,1). Then ∥x∥ℬ𝒯=ϱ0 satisfies (3.36), which contradicts (3.37). So, alternative (ii) in Theorem 2.1. does not hold, and consequently, the multivalued operator ℳ has a fixed point, which is a solution of (1.1).We now present another existence result for system (1.1). We shall assume that the single-valued f and the multivalued G satisfy a Wintner-type growth condition with respect to their second variable.Theorem 3.2.
Assume that(H2) and the following condition hold.(HfG):
There existsℓ∈L1([0,T],ℜ+) such that(3.38)H(f(t,xt),f(t,yt))∨H(G(t,xt),G(t,yt))≤l(t)‖x-y‖2,∀t∈J,x,y∈Dτ,H(0,f(t,0))∨H(0,G(t,0))≤l(t),a.e.t∈J,
then the system (1.1) has at least one mild solution on (-∞,T].Remark 3.3.
H(f(t,xt),f(t,yt))=∥f(t,xt)-f(t,yt)∥2.Proof.
The multivalued operatorℳ defined in the proof of the previous theorem is completely continuous and upper semicontinuous. Now, we prove that
(3.39)Y={x∈BT:x∈λM(x)forsomeλ∈(0,1)}
is bounded. Let x∈𝔜. Then there exists g∈SG,x such that for each t∈J(3.40)x(t)=λ∑i=1mγiS(t)B[∫0tiS(ti-s)f(s,xs)ds+∫0tiS(ti-s)g(s)dw(s)]+λ∫0tS(t-s)f(s,xs)ds+λ∫0tS(t-s)g(s)dw(s),
for some λ∈(0,1). Then
(3.41)‖x(t)‖2≤3(2mM2∑i=1mti‖γi‖2‖B‖2+T)M2∫0t‖f(s,xs)‖2ds+3(2mM2∑i=1m‖γi‖2‖B‖2+1)M2Tr⁡(Q)∫0t‖g(s)‖2ds≤6(2mM2∑i=1mti‖γi‖2‖B‖2+T)M2∫0tl(s)(1+‖x(s)‖2)ds+6(2mM2∑i=1m‖γi‖2‖B‖2+1)M2Tr⁡(Q)∫0tl(s)(1+‖x(s)‖2)ds.
Thus
(3.42)E‖x(t)‖2≤Q1+Q2∫0tl(s)E‖x(s)‖2ds,
where
(3.43)Q1=6(2mM2∑i=1mti‖γi‖2‖B‖2+T)M2(T+Tr⁡(Q))‖l‖L1,Q2=6(2mM2∑i=1m‖γi‖2‖B‖2+1)M2(T+Tr⁡(Q)).
Using the function ϱ(t), defined by (3.31), we obtain
(3.44)ϱ(t)≤Q1+Q2∫0tl(s)ϱ(s)ds.
Gronwall’s inequality gives
(3.45)ϱ(t)≤Q1exp⁡(Q2‖l‖L1),∀t∈J.
Therefore there exists β>0 such that
(3.46)ϱ(t)≤β,∀t∈J,
which implies that
(3.47)‖x‖BT2≤β.
This shows that 𝔜 is bounded. Theorem 2.1. shows that ℳ has a fixed point, which is a solution of (1.1), and this completes the proof.


## 4. Example

Consider the following stochastic partial differential inclusion with infinite delay(4.1)∂∂tv(t,x)∈∑i,j=1n∂∂xi(aij(x)∂∂xjv(t,x))-a0v(t,x)+ϵ∑i=1n∂∂xiv(t-r,x)+∫-∞0β1(θ)v(t+θ,x)dθ+∫-∞0β2(t,x,θ)G1(v(t+θ,x))dθdβ(t)v(t,x)=0,t∈J,x∈∂Δ,v(0,x)=∑i=1nβ̂k(x)v(x,tk),x∈Δ,tk∈[0,T],v(θ,x)=φ(0,x),-∞<θ≤0,x∈Δ,
where a0,r, and ϵ  are positive constants, J=[0,T], Δ is an open bounded set in ℜn with a smooth boundary ∂Δ,β1:(-∞,0]→ℜ is a positive function, β(t) stands for a standard cylindrical Wiener process in L2(Δ) defined on a stochastic basis (Ω,ℱ,P), and φ∈Dℱ0b((-∞,0],L2(Δ)).The coefficientsaij∈L∞(Δ) are symmetric and satisfy the ellipticity condition(4.2)∑i,j=1naij(x)ξiξj≥ϰ|ξ|2,x∈Δ,ξ∈Rn,
for a positive constant ϰ.In order to rewrite (4.1) in the abstract form, we introduce X=L2(Δ) and we define the linear operator A:D(A)⊂X→X by (4.3)D(A)=H2(Δ)∩H01(Δ);A=-∑i,jn∂∂xi(aij(x)∂∂xj).HereH1(Δ) is the Sobolev space of functions u∈L2(Δ) with distributional derivative u′∈L2(Δ), H01(Δ)={u∈H1(Δ);u=0on∂Δ} and H2(Δ)={u∈L2(Δ);u′,u′′∈L2(Δ)}.ThenA generates a symmetric compact analytic semigroup e-tA in X, and there exists a constant M1>0 such that ∥e-tA∥≤M1. Also, note that there exists a complete orthonormal set {ξn}, (n=1,2,…) of eigenvectors of A with ξn(x)=(2/n)sin(nx).We assume the following conditions hold.(i)
The functionβ1(·) is continuous in J with
(4.4)∫-∞0β1(θ)2dθ<∞.(ii)
The functionβ2(·)≥0 is continuous in J×Δ×(-∞,0) with
(4.5)∫-∞0β2(t,x,θ)dθ=p1(t,x)<∞,(∫Δp12(t,x)dx)1/2<∞.(iii)
The multifunctionG1(·) is an L2-Carathèodory multivalued function with compact and convex values and
(4.6)0≤‖G1(v(θ,x))‖≤ψ0(‖v(θ,⋅)‖L2),(θ,x)∈J×Δ,
where ψ0(·):[0,∞)→(0,∞) is continuous and nondecreasing.Assuming that conditions(i)–(iii) are verified, then the problem (4.1) can be modeled as the abstract stochastic partial functional differential inclusions of the form (1.1), with (4.7)f(t,vt)=∫-∞0β1(θ)v(t+θ,x)dθG(t,vt)=∫-∞0β2(t,x,θ)G1(v(t+θ,x))dθ,γi=β̂k(x).The next result is a consequence of Theorem3.1.Proposition 4.1.
Assume that the conditions(i)–(iii) hold. Then there exists at least one mild solution v for the system (4.1) provided that
(4.8)sup⁡ρ∈[0,∞){1-3TK2C1}ρ3K1Tr(Q)‖p‖L1ψ0(ρ)>1,
where 𝕂1=(2mM12∑i=1m∥γi∥2∥B∥2+1)M12and𝕂2=(2mM12∑i=1mti∥γi∥2∥B∥2+T)M12.Proof.
Condition (i) implies that(H1) holds with C1=∫-∞0β12(θ)dθ and C2=0.  (H2) and (H3) follow from conditions (ii) and (iii) with p(t)=(∫Δp12(t,x)dx)1/2 and ψ=ψ0.



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*Source: 784638-2011-11-30.xml*