Existence results fortime-dependent neutral functional integrodifferential equations driven by a fractional Brownian motion

Journal of Numerical Mathematics and Stochastics,6(1) : 84-99, 2014 © JNM@S http://www.jnmas.org/jnmas6-7.pdf Euclidean Press, LLC Online: ISSN 2151-2302 Existence Results for Time-Dependent Stochastic Neutral Functional Integrodifferential Equations Driven by a Fractional Brownian Motion∗ T.CARABALLO1,M.A.DIOP2,and A.A.NDIAYE2 1Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apartado 1160, 41080-Sevilla, Spain; 2Université Gaston Berger de Saint-Louis, UFR SAT, Département de Mathématiques, 234, Saint-Louis, Sénégal, E-mail: [email protected] Abstract.This article presents some results on existence and uniqueness of mild solutions to neutral stochastic functional evolution integrodifferential equations driven by a fractional Brownian motion. The existence of mild solutions for the equations are discussed by means of theory of resolvent operators. Under some sufficient conditions, results are obtained by using a Banach contraction principle. Key words : Resolvent Operators, Evolution Operators, Existence, Uniqueness, C0-semigroup, Wiener Process, Mild Solutions, Fractional Brownian Motion. AMS Subject Classifications : 60H15, 93E15, 35R12 1.Introduction In this paper, we study the existence of mild solutions for a class of abstract stochastic partial neutral functional integro-differential equations modeled in the form dutGt,ut−rtAtutGt,ut−rtdt 0 tBt,susGs,us−rsds Ft,ut−t dt tdBHtfor t ∈0,T, u0.,−≤t≤0, 1 where Atis a linear operator which generates a linear evolution system Rt,s,t≥0on a —————————— ∗This research has been partially supported by FEDER and Ministerio de Economía y Competitividad (Spain) under grant MTM2011-22411, and Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under Proyecto de Excelencia P12-FQM-1492. 84 85 T.CARABALLO,M.A.DIOP,andA.A.NDIAYE Hilbert space X,Btis a closed linear operator on Xwith domain DB⊃DAwhich is independent of t,BHis a fractional Brownian motion on a real and separable Hilbert space Y. The functions r,defined from0,into 0,0are measurable, and G,F:0,XX,0,L2 0Y,Xare appropriate functions. Here L2 0Y,X denotes the space of all Q-Hilbert-Schmidt operators from Yinto X(see Section 2). Neutral integro-differential equations arise in many areas of applied mathematics. For instance, the system of rigid heat condition with finite wave speeds, studied in [6], can be modeled in the form of integrodifferential equations of neutral type with delay, and for this reason these equations (with an initial condition or a nonlocal condition) have received much attention in the last few decades. One of the important techniques to discuss these topics is the semigroup approach; see , for example [13, 8, 9]. In the paper [5], Caraballo and Diop investigated the existence of solutions for the following stochastic functional differential equation: dutGt,ut−rtAutGt,ut−rtdt 0 tBt,susGs,us−rsds Ft,ut−t dt tdBHtfor t ∈0,T, u0.,−≤t≤0, 2 which differs from (1) only in its At, by using a semigroup approach and classical fixed point arguments. In this work, the linear part in our equation is an operator independent of time tand generates a strongly continuous semigroup, so that the semigroup approach can be employed. Our purpose in the present paper is to establish some results concerning existence and uniqueness of the solutions for the non-autonomous stochastic integrodifferential equations (1). A motivating example for this type of equations is the following non-autonomous boundary problem: ∂ ∂txt,gt,xt−1,∂2 ∂2t,xt,gt,xt−1, 0 tbt−s∂2 ∂2xs,gs,xs−1,ds ft,xt−2, tdBH dt t, xt,0gt,xt−rt,0 0for t ≥0, xt,gt,xt−rt, 0for t ≥0, x,x0,,s,.∈L20,T,−≤≤0, 0 ≤≤. 3 Such problems arise in the study of stochastic systems in the presence of hereditary influences on the state variable. For example, stochastic integrodifferential systems which cover a large area of system dynamics including reactor dynamics [4, 14, 16], heat transfer by conduction and radiation [17, 15], mathematical modeling of system hysteresis [11, 14], models of transmission of infection of diseases [3]. Therefore, it is meaningful to deal with 1 to acquire some results applicable to problem 3. Existence Results for Stochastic Neutral Functional Integrodifferential Equations 86 As we know, non-autonomous evolution equations are much more complicated, to be dealt with, than autonomous ones. Our approach here is to assume that At:t≥0is a family of linear operators on Xwith dense domain such that it generates a linear evolution system. The results in this paper are natural continuation and generalization of the results reported by Caraballo and Diop [5]. Let us now describe the remaining contents of the paper. In Section 2, we introduce some notations, concepts of resolvent operators, basic results about fractional Brownian motion and Wiener integral over Hilbert space. In Section 3, we prove the existence and uniqueness of mild solutions for the system (1). An example to illustrate our previous abstract results is analyzed in Section 4. 2.Wiener Process and Deterministic Integrodifferential Equations 2.1. Wiener process In this section we introduce the fractional Brownian motion as well as the Wiener integral with respect to it. We also need to establish some important results which will be needed throughout the paper. So, first, let ,F,Pbe a complete probability space. Definition 2.1.GivenH∈0,1, a continuous centered Gaussian process Ht,t∈R, with covariance function RHs,tEHtHs1 2t2Hs2H−|t−s|2H,t,s∈R, is called a two-sided one-dimensional fractional Brownian motion (fBm), and His the Hurst parameter. Now we aim at introducing the Wiener integral with respect to the one-dimensional fBm H.LetT0 and denote by the linear space of R-valuedstepfunctionon0,T, that is ∈if t∑ i1 n−1 xi1ti,ti1t, where t∈0,T,xi∈Rand 0 t1t2tnT.For∈we define its Wiener integral with respect to Has 0 TsdHs∑ i1 n−1 xiHti1−Hti. Let Hbe the Hilbert space defined as the closure of with respect to the scalar product 〈10;t,10;sHRHt,s.Then the mapping t∑ i1 n−1 xi1ti,ti1t→0 TsdHs 87 T.CARABALLO,M.A.DIOP,andA.A.NDIAYE is an isometry between and the linear space span H ,t∈0,T, which can be extended to an isometry between Hand the first Wiener chaos of the fBm spanL2H,t∈0,T (see [21]). The image of an element ∈Hby this isometry is called the Wiener integral of with respect to H. Our next goal is to give an explicit expression for this integral. To this end,consider the Kernel KHt,scHs1 2−Hs tu−sH−3 2uH−1 2du , where cHH2H−1 B 2−2H,H−1 2, with B denoting the Beta function and t≤s. It is not difficult to see that ∂KH ∂tt,scHt s 1 2−Ht−sH−3 2. Consider the linear operator KH ∗:L20,T given by KH ∗ss tt∂K ∂tt,sdt. Then KH ∗10;tsKHt,s10;ts, and KH ∗is an isometry between and L20,T that can be extended to (see [1]). Considering WWt,t∈0,T defined by WtHKH ∗−110;t, it turns out that Wis a Wiener process and Hhas the following Wiener integral representation: Ht0 tKHt,sdWs. In addition, for any ∈, 0 TsdHs0 TKH ∗tdWt, if and only if KH ∗∈L20,T. Also denoting LH 20,T ∈,KH ∗∈L20,T,sinceH1 2, allows for L1 H0,T ⊂LH 20,T,4 see [18]. Moreover, the following useful result holds. Lemma 2.1. [19] For ∈L1 H0,T, H2H−10 T0 T|r||u||r−u|2H−2drdu ≤cH‖‖L1 H0,T 2. Next we are interested in considering a fBm with values in a Hilbert space and giving the definition of the corresponding stochastic integral. Let X,‖.‖X,.,.Xand Y,‖.‖Y,.,.Ybe separable Hilbert spaces. Let LX,Ydenote the space of all bounded linear operator from Xto Y.LetQ∈LX,Ybe a non-negative self-adjoint operator. Denote by L2 0Y,Xthe space of ∈LY,Xsuch that Q1 2is a Hilbert-Schmidt operator.The norm is given by ||L2 0Y,X 2Q1 2HS trQ∗. Existence Results for Stochastic Neutral Functional Integrodifferential Equations 88 Then is called a Q-Hilbert-Schmidt operator from Yto X. Let n Htn∈Nbe a sequence of two-sided one-dimensional standard fractional Brownian motions mutually independent on ,F,P. When one considers the following series ∑ n1  n Hten,t≥0, where enn∈Nis a complete orthonormal basis in X, this series does not necessarily converge in the space Y. Thus we consider a Y-valued stochastic process BQ Ht∑ n1  n HtQ1 2en,t≥0. If Qis a non-negative self-adjoint trace class operator, this series converges in the space Y, that is, it holds that BQ Ht∈L2,Y. Then, we say that the above BQ Htis a Y-valued Q-cylindrical fractional Brownian motion with covariance operator Q. For example, if nn∈N is a bounded sequence of non-negative real numbers such that Qennen, assuming that Qis a nuclear operator in Y(that is, ∑n1 n), then the stochastic process BQ Ht∑ n1  n HtQ1 2en∑ n1  nn Hten,t≥0, is well-defined as a Y-valued Q-cylindrical fractional Brownian motion. Then let :0,T→LQ 0Y,Xsuch that ∑ n1  KH ∗Q1 2enL20,T;X.5 Definition 2.2.GivenH∈0,1, and let :0,T→LH 0Y,Xsatisfy (5). Then, its stochastic integral with respect to the fBm BQ His defined, for t≥0, as follows 0 tsdBQ Hs:∑ n1 0 tsQ1 2enn Hs∑ n1 0 tKH ∗Q1 2ensdWs.6 Notice that if ∑ n1  ‖Q1 2en‖L1 H0,T;X, then in particular (5) holds, which follows immediately from (4). Now we end this subsection by stating the following result which is crucial for proving our main result. It can be proved by similar arguments to those used in Lemma 2 in Caraballo et al. [6]. 89 T.CARABALLO,M.A.DIOP,andA.A.NDIAYE Lemma 2.2.If :0,T→L2 0X,Ysatisfies 0 T‖‖L2 0 2ds ,then the above sum in (6) is well defined as a X-valued random variable and E0 tsdBHs2≤2Ht 2H−10 t‖s‖L2 0 2ds. Proof. See [2].  2.2. The stochastic convolution integral Here we present some properties of the stochastic convolution integral of the form R t0 tRt,ssdBHs,t∈0,T, where s∈L2 0X,Yand Rt,s,t≥0is an evolution system of operators. The following result on the stochastic convolution integral R should always hold. Lemma 2.3.Suppose that :0,T→L2 0X,Ysatisfies supt∈0,T‖t‖L2 0X,Y 2, and suppose that Rt,s,t≥0is an evolution system of operators satisfying ‖Rt,s‖≤Me−t−s, for some constants 0and M ≥1, for all t s. Then E0 tRt,ssdBHs2≤CM2t2H t∈0,T sup ‖t‖L2 0X,Y 2. Proof.Letenn∈Nbe a complete orthonormal basis of Yand n Hn∈Nis a sequence of independent, real-valued standard fractional Brownian motion each with the same Hurst parameter H∈1 2,1. Thus, using the fractional Itô isometry one can write E0 tRt,ssdBHs2 ∑ n1  E0 tRt,ssendn Hs2 ∑ n1 0 t0 t〈Rt,ssen,Rt,rrenH2H−1|s−r|2H−2dsdr ≤H2H−10 t‖Rt,ss‖0 t‖Rt,rr‖|s−r|2H−2dr ds ≤H2H−1M20 te−t−s‖s‖L2 0X,Y0 te−t−r|s−r|2H−2‖r‖L2 0X,Ydr ds. Since is bounded, one can then conclude that E0 tRt,ssdBHs2≤H2H−1M2 t∈0,T sup ‖t‖L2 0X,Y 22 0 te−t−s0 te−t−r|s−r|2H−2dr ds. Performing the change of variables vt−sfor the first integral, and ut−rfor the second Existence Results for Stochastic Neutral Functional Integrodifferential Equations 90 one, we obtain E0 tRt,ssdBHs2≤H2H−1M2 t∈0,T sup ‖t‖L2 0X,Y 22 0 te−v0 te−u|u−v|2H−2du dv. From [29] it follows that E0 tRt,ssdBHs2≤CM2t2H t∈0,T sup ‖t‖L2 0X,Y 22, and the proof is then complete.  2.3. Partial integro-differential equations in Banach spaces Let us recall some fundamental results needed to establish our results. The resolvent operators play an important role to study the existence of solutions and to give a variation of constants formula for nonlinear systems. We need to know when the linear system (7) has a resolvent operator. For more details on resolvent operators, we refer the reader to [12]. The following assumptions are: ∙(i) Atgenerates a strongly continuous semigroup of evolution operators. ∙(ii) Suppose Yrepresents the Banach space DAequipped with the graph norm defined by |y|Y:|Ay||y|for y ∈Y. Atand Bt,sare in the set of bounded linear operators from Yto X,LY,Xfor 0 ≤t≤T and 0 ≤s≤Trespectively. Atand Bt,sare continuous on 0 ≤t≤Tand 0 ≤s≤t≤T, respectively, into LY,X. To obtain the results, we consider the following abstract integrodifferential Cauchy problem dvtAtvt0 tBt,svsds dt,for 0≤s≤t≤T, v0v0∈X. 7 Definition 2.3. [12] A resolvent operator for Eq(7) is a bounded linear operator valued function Rt,s∈LXfor 0 ≤s≤t≤T, satisfying the following properties: ∙(i) Rt,tIand |Rt,s|≤Net−s,t,s∈0,Tfor some constants Nand . ∙(ii) Rt,sis strongly continuous in sand t. ∙(iii) For y∈Y,Rt,syis continuously differentiable in sand t,andfor0≤s≤t≤T, ∂ ∂tRt,syAtRt,sys tBt−rRr,sydr, ∂ ∂sRt,sy−Rt,sAsy−s tRt,rBr−sydr, with ∂ ∂tRt,syand ∂ ∂sRt,sare strongly continuous on 0 ≤s≤t≤T.HereRt,scan be 91 T.CARABALLO,M.A.DIOP,andA.A.NDIAYE extracted from the evolution operator of the generator At. For the family of linear operators At:0≤t≤T, the following assumptions need to be imposed: ∙(H1) The domain DAof At:0≤t≤Tis dense in Xand independent of t;Atis a closed linear operator. ∙(H2) For each t∈0,T, the resolvent operator R ,At exists for all with Re≤0and there exists K0 such that ‖R ,At‖≤K ||1. ∙(H3) There exists 0 ≤1andK0 such that ‖At−AsA−1r‖≤K|t−s|for all t,s,r∈0,T. ∙(H4) For each t∈0,Tand some ∈At, the resolvent set of At, the resolvent R ,At, is a compact operator. Under these assumptions, the family At:0≤t≤Tgenerates a unique linear evolution system, also called linear evolution operator. Definition 2.4. [20] A two parameter family of bounded linear operators Ut,s,0≤s≤t≤T,onXis called an evolution system if the following two conditions holds ∙(i) Us,sI,Ut,rUr,sUt,s, for 0 ≤s≤r≤t≤T. ∙(ii) t,sUt,sis strongly continuous for 0 ≤s≤t≤T. Lemma 2.4. [20] Assume that H1−H3hold. Then, there exist a unique evolution system Ut,s,0≤s≤t≤T and a constant K 0such that ∙(i) Ut,s≤Kfor0≤s≤t≤T, ∙(ii) for 0≤s≤t≤T,Ut,s:X→Y and t →Ut,sis strongly differentiable in X. The derivative ∂ ∂tUt,sbelongs to LXand it is strongly continuous on 0≤s≤t≤T. Moreover, for all 0≤s≤t≤T, it holds ∂ ∂tUt,sAtUt,s0, ∂ ∂tUt,s‖AtUt,s‖≤K t−s, ‖AtUt,sAs−1‖≤K, ∙(iii) for each y ∈Y and t ∈0,T,Ut,sy is differentiable with respect to s on 0≤s≤t≤T and ∂ ∂tUt,syUt,sAsy. Lemma 2.5. [10] Let At,t∈0,T be a family of linear operators satisfying (H1)-(H4). If Ut,s,0≤s≤t≤Tis the linear evolution system generated by At,t∈0,T, then Ut,s,0 ≤s≤t≤Tis a compact operator whenever t −s0. 3.Existence of Mild Solutions for Eq (1) In this section, we establish the existence and uniqueness of mild solutions of Eq (1) using a contraction mapping principle. For this reason we introduce the following technical assumptions. ∙(H5) There exists a resolvent operator Rt,swhich is compact and continuous in the uniform Existence Results for Stochastic Neutral Functional Integrodifferential Equations 92 operator topology for ts. ∙(H6) The function F:0,X→Xsatisfies the following conditions: there exist positive constants C1,C2such that, for all t∈0,Tand x,y∈X ‖Ft,x−Ft,y‖≤C1‖x−y‖, ‖Ft,x‖2≤C21‖x‖2. ∙(H7) The function G:0,X→Xsatisfies the following conditions: there exist positive constants C3,C4,0 C31suchthat,forallt∈0,Tand x,y∈X ‖Gt,x−Gt,y‖≤C3‖x−y‖, ‖Gt,x‖2≤C41‖x‖2. ∙(H8) The function Gis continuous in the mean square sense. For all x∈u∈C0,T,L2,X, it holds that limt→sE‖Gt,xt −Gs,xs‖20. ∙(H9) The function :0,→L2 0Y,Xsatisfies 0 T‖s‖L2 0 2ds ,∀T0. Moreover, we assume that ∈C−,0,L2,X. Next, we introduce the concept of mild solution for Eq (1). Definition 3.1.AnX-valued process ut,t∈−,T, is called a mild solution of Eq (1) if u∈C−,T,L2,X,uttfor t∈−,0, and, for t∈0,T, satisfies utGt,ut−rs Rt,00−G0,−r00 tRt,sFs,us−sds 0 tRt,ssdBHsP−a.s. To prove our main results we first recall the next lemma, which is Lemma 1 of [7] by Caraballo et al. Lemma 3.1.[7]For x,y∈X and 0c1, ‖x‖X 2≤1 1−c‖x−y‖X 21 c‖y‖X 2. Theorem 3.1.Under the assumptions H1−H9, for every ∈C−,0,L2,X there exists a unique mild solution u to Eq (1). Proof. Assume that T0 is a fixed time and let CT:C−,T,L2,X be the Banach space of all continuous functions from −,Tinto L2,Xequipped with the supremum norm ‖‖CT z∈−,T sup E‖z‖21/2, and let us consider the set ST:u∈C−,T,L2,X :uss,for s ∈−,0. 99 T.CARABALLO,M.A.DIOP,andA.A.NDIAYE [16] J. J. Levin, and J. A. Nohel, The integro-differential equations of a class of nuclear reactors with delayed neutrons, Archive for Rational Mechanics and Analysis 31, (1968), 151-172. [17] R. K. Miller, An integro-differential equation for heat conductors with memory, Journal of Mathematical Analysis and Applications 66, (1978), 313-333. [18] Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Topics,in Lecture Notes in Mathematics 1929, Springer-Verlag, Berlin, 2008. [19] D. Nualart, The Malliavin Calculus and Related Topics, 2nd Ed., Springer-Verlag, Berlin, 2006. [20] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. [21] S. Tindel, C. Tudor, F. Viens, Stochastic evolution equations with fractional Brownian motion, Probability Theory and Related Fields 127(2), (2003), 186-204. ______ Article history: Submitted May, 05, 2014; Revised March, 27, 2015 ; Accepted June, 06, 2015.