On the Existence, Uniqueness and Stability Behavior of a Random Solution to a Non local Perturbed Stochastic Fractional Integro-Differential Equation

Transcription

1 On he Exisence, Uniqueness and Sabiliy ehavior of a Random Soluion o a Non local Perurbed Sochasic Fracional Inegro-Differenial Equaion Mahmoud M. El-orai,*, M.A.Abdou, Mohamed Ibrahim M. Youssef Dearmen of Mahemaics, Faculy of Science, Alexandria Universiy, Alexandria, Egy Dearmen of Mahemaics, Faculy of Educaion, Alexandria Universiy, Alexandria, Egy m_m_elborai@yahoo.com; abdella_777@yahoo.com; m_ibrahim83@yahoo.com Absrac: In his aer, we rove he exisence and uniqueness of a nonlinear erurbed sochasic fracional inegro-differenial equaion of Volerra-Iô ye involving nonlocal iniial condiion by using he heory of admissibiliy of inegral oeraor and anach fixed-oin rincile. Also he sabiliy and boundedness of he second momens of he sochasic soluion are sudied. In addiion, an alicaion o fracional sochasic feedback sysem is resened. [Mahmoud M. El-orai, M.A.Abdou, Mohamed Ibrahim M. Youssef. On he Exisence, Uniqueness and Sabiliy ehavior of a Random Soluion o a Non local Perurbed Sochasic Fracional Inegro-Differenial Equaion. Life Sci J 3; (4): ]. (ISSN:97-835). 448 Keywords: Fracional inegral, random inegro-differenial equaions, admissibiliy heory, anach s fixed-rincile, Iô-Doob inegral. Inroducion Inegro-differenial equaions arise quie naurally in he sudy of many hysical henomena in life science and engineering, for examle, equaions of his form occur in he formulaion of roblems in reacor dynamics, in he sudy of he growh of biological oulaion models and in he heory of auomaic sysems resuling in he delaydifferenial equaions, see for more deails []. Many invesigaions have been carried ou concerning he exisence and uniqueness of soluion of deerminisic and sochasinegro-differenial equaions of Volerra ye, see [-5]. However, due o he comlex naure of he roblems being characerized by such equaions, many auhors, in he las few decades, oined ou ha fracional sochasic models are very suiable for he descriion of roeries of various real maerials, e.g. olymers. I has been shown ha new models are more adequae han reviously ineger-order models [6-8]. In many cases, i is beer o have more iniial informaion o obain a good descriion of he evoluion of a hysical sysem. The local iniial condiion is relaced hen by a nonlocal condiion, which gives beer effec han he iniial condiion, since he measuremen given by a nonlocal condiion is usually more recise han he only one measuremen given by a local condiion, see [9]. Therefore, in his aer we shall be concerned wih exending he resuls in El-orai e al. [], William J. adge and Chris P. Tsokos [5]. Tha is, we shall consider a nonlinear sochasic erurbed facional inegro-differenial equaion of Volerra-Iôye of he form: α x ;ω α =, x ; ω + k, ; ω f, x ; ω d + k, ; ω f, x ; ω dβ (.) wih he nonlocal condiion x ; ω + i= x i, ω = x ω. where < α, R + =,, < < < <. The fracional derivaive is rovided by he Cauo derivaive and (i) ω Ω, he suoring se of a robabiliy measure sace Ω, A, P ; (ii) x ; ω is he unknown sochasic rocess for R + ; (iii), x is called he sochasic erurbing erm and i is a scalar funcion of R + and x R ; (iv) k, ; ω, k, ; ω are scalar sochasic kernels defined for and saisfying < ; (v) f, x, f, x are scalar funcions of R +, x R o be secified laer; (vi) β is a sochasic rocess o be defined laer. The urose of his aer is o obain he condiions which guaranee he exisence and uniqueness of random soluion x ; ω of he roblem.,. and o invesigae he asymoic momen behavior of such a random soluion. In addiion, he usefulness of he resuls will be illusraed wih an alicaion o fracional sochasic feedback sysem. Equaions (.), (.) generalize he resuls of El-orai e al. [], and he 3368 lifesciencej@gmail.com

2 resuls of Padge and Tsokos [5]. The considered nonlocal Cauchy roblem (.), (.) consiss of wo ars, he firs inegral being a lebesgue inegral and he second a sochasinegral of he Iô-Doob ye. In our work we shall uilize he saces of funcions and admissibiliy heory which were inroduced ino he sudy of sochasinegral equaions by Tsokos []. The nonlocal Cauchy roblem (.), (.) has alicaions in many fields such as elecromagneic heory, viscoelasiciy, and fluid mechanics [-3].. Preliminaries Le Ω, A, P denoes a robabiliy measure sace, ha is Ω is a nonemy se known as he samle sace, A is a sigma-algebra of subses of Ω, and P is a comlee robabiliy measure on A. Le L Ω, A, P be he sace of all random variables x ; ω, R +, which have a second momen wih resec o P-measure for each R +. Tha is: E x ; ω = x ; ω d ω <. Ω The norm of x ; ω in L Ω, A, P is defined for each R + by: x ; ω = E x ; ω. Le L Ω, A, P be he sace of all measurable and P -essenially bounded random variables of ω Ω. The norm of k, ; ω in L Ω, A, P will be defined by: k, ; ω = P ess su ω Ω k, ; ω. Wih resec o he random rocess β, we shall assume ha β is adaed o he filraion A which is an increasing family of sub sigma-algebras A A. furhermore, we shall assume ha: (i) The rocess β, A, < is a real maringale. (ii) There is a coninuous monoone nondecreasing funcion F on R +, such ha, if s <, hen E β ; ω β s; ω = E β ; ω β s; ω \ A s = F F s P a. e. Noe ha: If F = c, s a consan, wih almos all is samle funcions are coninuous, hen β is a rownian moion rocess, (see [4], ), and his is he mos imoran secial case. Definiion. We define he sace C c = C c R +, L Ω, A, P o be he sace of all coninuous funcions x ; ω from R + ino L Ω, A, P, such ha for each R +, x ; ω is A -measurable. We define a oology in he sace C c by means of he following family of seminorms: x ; ω n = su n x ; ω, n =,,3, I is known ha his oology is merizable and he sace C s Freche sace. Definiion. We define he sace C g = C g R +, L Ω, A, P o be he sace of all coninuous funcions from R + ino L Ω, A, P, such ha here exis a consan a > and a osiive coninuous funcion g on R + saisfying x ; ω a g. The norm in C g R +, L Ω, A, P will be defined by: x ; ω Cg = su R+ x ; ω g. Definiion.3 We define he sace C = C R +, L Ω, A, P o be he sace of all coninuous and bounded funcions on R + wih values in L Ω, A, P, ha is, C is he sace of all second order sochasic rocesses on R + which are bounded and coninuous in mean square. The norm in C is defined by: x ; ω C = su R+ x ; ω < I is clear ha C, C g are anach saces and he following inclusion hold: C C g C c. Definiion.4 The air of anach saces, D wih, D C s said o be admissible wih resec o he oeraor T: C c C f and only if T D. Definiion.5 The anach sace is said o be sronger han C c, if every convergen sequence in, wih resec o is norm, will also converge in C c. (bu he converse is no rue in general). Definiion.6 We call x ; ω a random soluion of he equaion (.) if x ; ω C c for each R +, saisfies he equaion (.) for every > and saisfies he nonlocal iniial condiion almos surely. We now sae he following lemma which is given by Tsokos [4]. Lemma. Le T be a coninuous linear oeraor from C c R +, L Ω, A, P ino iself, if and D are anach saces sronger han C c and if, D is admissible wih resec o T, hen T is a coninuous linear oeraor from ino D lifesciencej@gmail.com

3 3. Main resuls Using he definiions of he fracional derivaives and inegrals, i is suiable o rewrie he considered roblem in he form: x ; ω = x ; ω s s α, x ; ω d s α k s, ; ω s α k s, ; ω f, x ; ω dds f, x ; ω dβ ds Changing he order of inegraion (noe ha: he assumions on he funcions k and f ermi his oeraion on he las inegral and he roof is essenial he same as he one given in ([4],.43-43) x ; ω = x ; ω Where α, x ; ω d K, ; ω f, x ; ω d K, ; ω f, x ; ω dβ 3. K, ; ω = s α k s, ; ω ds 3. K, ; ω = s α k s, ; ω ds 3.3 Now define he inegral oeraors T, T and T 3 on C c R +, L Ω, A, P as follows: T x ; ω = T x ; ω = T 3 x ; ω = α x ; ω d 3.4 K, ; ω x ; ω d 3.5 K, ; ω x ; ω dβ 3.6 In lemma (3.) in [], El-orai e al. roved ha T is coninuous oeraor from C no C c. Now we shall rove a lemma concerning he coninuiy of T and T 3 as maings from C c R +, L Ω, A, P ino iself. Lemma 3. Suose ha (i) The funcions k, ; ω and k, ; ω are A measurable and P -ess bounded for each, saisfying < ; (ii) k, ; ω and k, ; ω are coninuous as mas from =, : < ino L Ω, A, P. Then he oeraors T and T 3 defined by he equaion 3.5 and 3.6 are coninuous maings from he sace C c R +, L Ω, A, P ino iself. Proof: The asserion abou T follows from lemma (3.) in El-orai e al. []. Hence, we shall only rove he asserion regarding he oeraor T 3. Se, we shall show ha T 3 : C c C c. We need o rove ha T 3 x ; ω L Ω, A, P and is coninuous funcion in mean square sense for each R +. y he same way which was used in lemma (3.) in El-orai e al. [], i is easy o rove ha he assumions (i),(ii) on k, ; ω imly ha K, ; ω L Ω, A, P, also for each,, K, ; ω is A measurable and is a coninuous ma from ino L Ω, A, P, hence for each x ; ω C c and for each, he funcion K, ; ω x ; ω is ddp measurable and T 3 x ; ω = K, ; ω x ; ω dβ K, ; ω x ; ω df < Thus, he sochasic kernel in 3.6 is well defined, and T 3 x ; ω L Ω, A, P. Now i remains only o rove ha T 3 is coninuous in he mean square sense for each R + as follow: Le x ; ω C c, <,,, n R +, hen = T 3 x ; ω T 3 x ; ω K, ; ω x ; ω dβ K, ; ω x ; ω dβ K, ; ω K, ; ω x ; ω dβ + K, ; ω x ; ω dβ Using he inequaliy A + A +, yields 337 lifesciencej@gmail.com

4 T 3 x ; ω T 3 x ; ω K, ; ω K, ; ω x ; ω dβ + K, ; ω x ; ω dβ Aling Iô-Doob Isomery, yields K, ; ω K, ; ω. x ; ω df + K, ; ω. x ; ω df x ; ω n K, ; ω K, ; ω df + x ; ω n K, ; ω df, as Since K is coninuous from =, : < ino L Ω, A, P, and F is coninuous, hen T 3 is coninuous in he mean square sense for each R +, and hence, T 3 : C c C c. Se, we shall show ha T 3 : C c C s a coninuous oeraor as follow: Le x ; ω C c, hen T 3 x ; ω = Ω K, ; ω x ; ω dβ Aling Iô-Doob Isomery, yields Ω Ω K, ; ω x ; ω df dp dp K, ; ω x ; ω dp df K, ; ω. x ; ω df x ; ω n K, ; ω df Now, T 3 x ; ω x ; ω n K, ; ω df Thus, T 3 x ; ω n = su n T 3 x ; ω x ; ω n su n K, ; ω df N x ; ω n where N is a consan deends uon n and α. Since K, ; ω is coninuous, i follows ha T 3 is coninuous oeraor from C no C c, (see [5]. 4). Hence he required resuls. Now le he oeraors T, T and T 3 be defined by equaions 3.4, 3.5 and 3.6 resecively, and le, D C c R +, L Ω, A, P be anach saces sronger han C c, such ha, D is admissible wih resec o each of he oeraors T, T and T 3. Then, I follows from lemma (.), ha T, T and T 3 are coninuous from ino D, hence here exis consans M, M, and M 3 such ha T i x ; ω D M i x ; ω i =,,3 The infimum of such consans M, M, and M 3 is called he norm of he oeraors T, T and T 3 resecively. In wha follow we shall assume ha f and f are mas from C no C c and ha k, ; ω and k, ; ω saisfy he condiions of lemma (3.). Lemma 3. Assume ha i=, hen he nonlocal Cauchy roblem.,. is equivalen o he following inegral equaion x ; ω = Ax ω A i= + T i f x i, ω + T 3i f x i, ω T i x i, ω + T x ; ω + T f x ; ω + T 3 f x ; ω 3.7 where A = + i= 337 lifesciencej@gmail.com

5 T, T and T 3 are defined by 3.4, 3.5 and 3.6 resecively and T i x i ; ω = T i x i ; ω = T 3i x ; ω = i i i i α x ; ω d K i, ; ω x ; ω d K i, ; ω x ; ω dβ i =,,. The roof is analogous o ha of lemma (3.3) in El-orai e al. [] and hence omied. We now go o he following exisence heorem. Theorem 3. Suose he inegral equaion. saisfies he following condiions: (i) and D are anach saces sronger han C c and he air, D is admissible wih resec o each of he oeraors, T, T and T 3 defined by 3.4, 3.5 and 3.6 resecivly; (ii) x ; ω, x ; ω is an oeraor on S = x ; ω D: x ; ω D ρ, wih values in saisfying:, x ; ω, y ; ω λ x ; ω y ; ω D for x ; ω, y ; ω S and ρ >, λ > are consans; (iii) x ; ω f, x ; ω is an oeraor on S wih values in saisfying: f, x ; ω f, y ; ω λ x ; ω y ; ω D for x ; ω, y ; ω S and λ > consan; (iv) x ; ω f, x ; ω is an oeraor on S wih values in saisfying: f, x ; ω f, y ; ω λ 3 x ; ω y ; ω D for x ; ω, y ; ω S and λ 3 > consan; (v) k, ; ω and k, ; ω saisfy he condiions of lemma (3.) (vi) x ω D. Then here exiss a unique random soluion x ; ω S of equaion., rovided ha: M λ + M λ + M 3 λ 3 + A A x ω D + M, + A i= i= M f, + M 3 f, + A i= < + ρ M λ + M λ + M 3 λ 3 + A i= where M and M and M 3 are he norms of T, T and T 3, resecively. Proof: y condiion (i) and lemmas (3.) in [], (.), and (3.), T, T and T 3 are coninuous from ino D. Hence, heir norms M and M and M 3 exis. Define he oeraor U: S D by Ux ; ω = Ax ω + T x ; ω + T f x ; ω + T 3 f x ; ω A i= T i x i, ω + T i f x i, ω + T 3i f x i, ω 3.8 We mus show ha U S S and ha he oeraor U is a conracion oeraor on S. Then, we may aly anach s fixed-oin heorem o obain he exisence of a unique random soluion. Le x ; ω S, hen ake he norm of 3.8, we ge Ux ; ω D Ax ω D + T x ; ω D + T f x ; ω D + T 3 f x ; ω D + A T i x i ; ω + T i fx i ; ω i= + T 3i f x i, ω A x ω D + T x ; ω D + T f x ; ω D + T 3 f x ; ω D + A T i x i ; ω D i= + A T i f x i ; ω D i= + A T 3i f x i, ω D i= A x ω D + M, x ; ω + A M i, x i ; ω i= + A M f i, x i ; ω i= + A M 3 f i, x i ; ω i= +M f, x ; ω + M 3 f, x ; ω A x ω D + M λ x ; ω D +, + A M λ x i ; ω D + i, i= D 337 lifesciencej@gmail.com

6 + A i= + A M 3 i= M λ x i ; ω D + f i, λ 3 x i ; ω D + f i, +M λ x ; ω D + f, +M 3 λ 3 x ; ω D + f, A x ω D + M, + A +ρ M λ + M λ + M 3 λ 3 + A i= i= + M f, + M 3 f, + A ρ i= +M 3 f, x ; ω f, y ; ω A i= M λ x i ; ω y i ; ω D + A M λ x i ; ω y i ; ω D i= + A M 3 λ 3 x i ; ω y i ; ω D i= +M λ x ; ω y ; ω D +M λ x ; ω y ; ω D + M 3 λ 3 x ; ω y ; ω D M λ + M λ + M 3 λ 3 + A y ; ω D i= x ; ω Thus, U S S, by he las condiion of he heorem. Le y ; ω be anoher elemen of S, from he assumions, i is clear ha: Ux ; ω Uy ; ω D, since he difference of wo elemens of a anach sace is in he anach sace. Ux ; ω Uy ; ω D A T i x i ; ω T i y i ; ω i= + A T i f x i ; ω T i f y i ; ω i= + A T 3i f x i ; ω T 3i f y i ; ω i= + T x ; ω T y ; ω D + T f x ; ω T f y ; ω D + T 3 f x ; ω T 3 f y ; ω D A i= M i, x i ; ω i, y i ; ω + A M f i, x i ; ω i= f i, y i ; ω + A M 3 f i, x i ; ω i= f i, y i ; ω +M, x ; ω, y ; ω +M f, x ; ω f, y ; ω D D D Since by hyohesis: M λ + M λ + M 3 λ 3 + A i= <, hen U is a conracion oeraor on S. Alying anach s fixed-oin heorem, here exiss a unique elemen of S so ha Ux ; ω = x ; ω. Tha is, here is a unique random soluion of he random equaion., comleing he roof. We now sae he following corollary when he sochasic erurbing erm, x ; ω is zero which is a generalizaion of he inegro-differenial equaion sudied by Tsokos [5] and El-orai e al. []. Corollary 3. If he sochasic fracional inegro-differenial equaion α x ; ω α = k, ; ω f, x ; ω d + k, ; ω wih he nonlocal condiion x ; ω + i= f, x ; ω dβ 3.9 x i, ω = x ω 3. saisfies he following condiions: (i) and D are anach saces sronger han C c and he air, D is admissible wih resec he oeraor T and T 3 defined by 3.5, 3.6 ; (ii) x ; ω f, x ; ω is an oeraor on S = x ; ω D: x ; ω D ρ, wih values in saisfying: f, x ; ω f, y ; ω λ x ; ω y ; ω D for x ; ω, y ; ω S and ρ >, λ > are 3373 lifesciencej@gmail.com

7 consans; (iii) x ; ω f, x ; ω wih values in saisfying: is an oeraor on S f, x ; ω f, y ; ω λ 3 x ; ω y ; ω D for x ; ω, y ; ω S and λ 3 > consan; (vii) k, ; ω and k, ; ω saisfy he condiions of lemma (3.), and (iv) x ω D. Then here exiss a unique random soluion x ; ω S of equaion 3.9, rovided ha: A x ω D + λ M + λ 3 M 3 + A i= < M f, + M 3 f, + A i= ρ λ M + λ 3 M 3 + A i= Where M, and M 3 are he norm of T and T 3 resecively. Since 3.9 is he equivalen of 3.7 wih, x equal o zero, he roof follows from ha heorem (3.) wih T being he null oeraor. 4. oundedness and Asymoic ehavior of Random Soluion. Using he saces C g and C, we now give some resuls concerning he asymoic behavior of he random soluion of.. We firs consider he unerurbed case 3.9. Theorem 4. Suose ha equaion 3.9 saisfies he following condiions: (i) k s, ; ω Λ e γ for some consans Λ > and γ >, s ; (ii) k s, ; ω ds Λ for some consan Λ > and s ; (iii) α e β df Λ 3 consan Λ 3 > ; (iv) x ; ω f, x ; ω for some saisfies f, x ; ω Λ 4 e β,, for some Λ 4 >, γ > β >, and f, x ; ω f, y ; ω λ e β x ; ω y ; ω for x ; ω and y ; ω ρ a each and λ > consan; (v) x ; ω f, x ; ω saisfies f, x ; ω Λ 5 e β,, for some Λ 5 >, γ > β >, and f, x ; ω f, y ; ω λ 3 e β x ; ω y ; ω for x ; ω and y ; ω ρ a each and λ 3 > consan; and (vi) x ω = P a. e. Then here exiss a unique random soluion o equaion 3.9 such ha su x ; ω = su E x ; ω ρ, where E is he mahemaical execaion, rovided ha: λ, λ 3, f, Cg and f, Cg are small enough. Proof: I is sufficien o show ha condiions (i), (ii) and (iii) imlies he admissibiliy of he air of saces C g, C wih resec o he oeraors T and T 3 defined by 3.5, 3.6, and ha condiions (iv) and (v) are equivalen o condiion (ii) and (iii) of corollary (3.) wih = C g, D = C, g = e β, β >. In [], El-orai e al. roved ha C g, C is admissible wih resec o he oeraors T. Now le us consider T 3, le x ; ω C g R +, L Ω, A, P, aking he norm in L Ω, A, P of 3.6, we obain x ; ω T 3 K, ; ω x ; ω dβ K, ; ω x ; ω df K, ; ω. x ; ω df Λ Λ 3 α x ; ω C g This imlies ha su T 3 x ; ω is bounded, which imlies T 3 x ; ω C, and hus, C g, C is admissible wih resec o he oeraors T 3. Now we will show ha condiions (iv) and (v) are equivalen o condiion (ii) and (iii) of corollary (3.), le f, x ; ω, f, y ; ω C g, hen f, x ; ω f, y ; ω C g = su f, x ; ω f, y ; ω e β 3374 lifesciencej@gmail.com

8 su λ e β x ; ω y ; ω e β = λ x ; ω y ; ω C and similarly for he oher condiion, alying corollary (3.), we ge on he required resul. Now we sae he resuls concerning he erurbed equaion. Theorem 4. Assume ha equaion. saisfies he following condiions: (i) k s, ; ω Λ for some consan Λ >, s ; (ii) k s, ; ω ds Λ for some Λ >, s ; (iii) α df Λ 3, for some Λ 3 > ; (iv) x ; ω, x ; ω saisfies, for some Λ 4 > and β >,, x ; ω Λ 4,, and, x ; ω, y ; ω λ x ; ω y ; ω for x ; ω and y ; ω ρ, and λ > consan; (v) x ; ω f, x ; ω saisfies, for some consan Λ 5 >, f, x ; ω Λ 5,, and f, x ; ω f, y ; ω λ x ; ω y ; ω for x ; ω and y ; ω ρ a each and λ > consan; (vi) x ; ω f, x ; ω saisfies, for some consan Λ 6 >, f, x ; ω Λ 6,, and f, x ; ω f, y ; ω λ 3 x ; ω y ; ω for x ; ω and y ; ω ρ a each and λ 3 > consan; (vii) x ω C Then, here exiss a unique random of soluion of. saisfying su x ; ω = su E x ; ω ρ,, Provided ha: λ, λ, λ 3, x ω C,, C f, C, and f, C are sufficienly small. Proof: I will suffice o show ha he air of saces C, C is admissible wih resec o he inegral oeraors defined by 3.4, 3.5, 3.6 under condiions (i), (ii) and (iii). In [], El-orai e al. roved ha C, C is admissible wih resec o T, T, so we need o rove ha C, C is admissible wih resec o T 3 as follow: Le x ; ω C. hen from 3.6 we have ha T 3 x ; ω K, ; ω x ; ω dβ K, ; ω x ; ω df K, ; ω. x ; ω df Λ Λ 3 α x ; ω C <. This imlies ha su T 3 x ; ω is bounded, which imlies T 3 x ; ω C and hus, C, C is admissible wih resec o he oeraors T 3. Therefore, he condiions of heorem (3.) hold wih = C, g =, and D = C, and hen, here exiss a unique random soluion x ; ω of.,., which is bounded in he mean square by ρ for all R +. and hence, su x ; ω ρ. 5. Alicaion o a Sochasic Fracional Feedback Sysem. Consider he following nonlinear sochasic fracional differenial sysem: dy ; ω = Π ω y ; ω + b ω Φ, σ ; ω d + b ω Φ, σ ; ω dβ 5. α σ ; ω α = C T ; ω y ; ω, wih he following iniial condiions 5. σ ; ω + i= σ i, ω = σ ω, y ; ω = y ω where < α, R + =,, < < < <. The fracional derivaive is rovided by he Cauo derivaive. Π ω is an n n marix of measurable funcions, x ; ω and C ; ω are n vecors of random variables for each R +, b i ω, i =,, is an n vecor of random variables, σ ; ω is a scalar random variable for each R +, Φ i, σ, i =,, is a scalar funcion for each R +, and T denoes he ransose of a marix. β is a sandard rownian moion. Now inegraing 5., we have y ; ω = e Π ω y ; ω + e Π ω b ω Φ, σ ; ω d + e Π ω b ω Φ, σ ; ω dβ 5.3 Subsiuing from 5.3 ino 5., we obain α σ ; ω α = c T ; ω e Π ω y ω 3375 lifesciencej@gmail.com

9 + e Π ω c T ; ω b ω Φ, σ ; ω d + e Π ω c T ; ω b ω Φ, σ ; ω dβ Now assume ha c T ; ω K for all and K > a consan. Also, le y ω C, and b i ω L Ω, A, P, i =,, if we assume ha he marix Π ω is sochasically sable, ha is, here exis an > such ha P ω: Re ψ k ω <, k =,,, n =, where ψ k ω, k =,,, n, are he characerisic roos of he marix Π ω, hen i has been shown by Morozan [6] ha e Π ω K e < K for some consan K >. We also le Φ i, σ ; ω C c R +, L Ω, A, P, i =, for each R +, and Φ i, σ ; ω Φ i, σ ; ω λ i σ ; ω σ ; ω. where λ i >, i =,, is a consan, also le h, σ ; ω = c T ; ω e Π ω y ω hen h, σ ; ω c T ; ω. e Π ω. y ω ZK K e ZK K where Z > is a consan, since y ω C. Thus, by definiion h, σ ; ω C, also, since does no deend on σ, hen h, σ ; ω h, σ ; ω = ha is, i saisfies a Lischiz condiion. Now, by he assumions on c T ; ω, b ω, and Π ω, we have k s, ; ω = e Π ω s c T s; ω b ω Saisfying k s, ; ω e Π ω s c T s; ω b ω s K K e b ω K K b ω and k s, ; ω = e Π ω s c T s; ω b ω saisfying Moreover, k s, ; ω ds K K b ω ds α df = a = K K b ω < α d 5.4 a = α α <. Therefore, all condiions of heorem (4.) are saisfied, and hence, here exiss a unique random soluion of he sysem 5., 5. which is bounded in he mean square on R +. Acknowledgemen The auhors are very graeful o he anonymous referees for heir careful reading of he aer and heir valuable commens. REFERENCES [] J. M. Gushing, Volerra inegro-differenal equaions in oulaion dynamics, Mahemaics of iology, C.I.M.E. Summer Schools, Vol. 8,. 8-48,. [] David N. Keck and Mark A. McKibben. Funcional inegro-differenial sochasic evoluion equaions in Hilber sace, J. Al. Mah. Sochasic Anal., 6():4 6, 3. [3] D. ahuguna,, Inegro-differenial equaions wih analyic semigrous, J. Al. Mah. Sochasic Anal. 6(), 77 89, 3. [4] W. J. Padge and C. P. Tsokos, Random Inegral Equaions wih Alicaions o Life Sciences and Engineering, vol. of Mahemaics in Science and Engineering, Academic Press, London, UK, 974. [5] A. N. V. Rao and C. P. Tsokos On he exisence, uniqueness and sabiliy behaviour of a random soluion o a nonlinear erurbed sochasinegro-differenial equaion Informaion and Conrol 7,. 6-74, 975. [6] Yu. A. Rossikhin and M.V. Shiikova, Alicaions of fracional calculus o dynamic roblems of linear and nonlinear herediary mechanics of solids, Al. Mech. Rev., Vol. 5, no., January 997, [7] Ronald L. agley and Peer J. Torvik, Fracional calculus in he ransien analysis of viscoelasically damed srucures. AIAA J., 3:98 95, 985. [8] M. M. El-orai. On some fracional differenial equaions in he Hilber sace. Discree Conin. Dyn. Sys., (sul.):33 4, 5. [9] Mahmoud M. El-orai On Some Sochasic Fracional Inegro-Differenial equaions, Advances in Dynamical Sys. and Al.vol, , 6. [] Mahmoud M. El-orai, M.A. Abdou, Mohamed Ibrahim M. Youssef On some nonlocal erurbed random fracional inegro-differenial equaions acceed in Life Science Journal 3 (o aear). [] C.P. Tsokos, On a sochasinegral equaion of he Volerra ye, Mah. Sysems Theory, 3,. -3, 969. [] C. Friedrich, Linear viscoelasic behavior of branched olybuadiens: a fracional calculus aroach, Aca Polym., 46:385 39, 995. [3] T. E. Govindan and M. C. Joshi. Sabiliy and oimal conrol of sochasic funcional-differenial equaions wih memory, Numer. Func. Anal. Oim.,3(3-4):49 65, 99. [4] J.L. Doob Sochasic Processes Wiley, New York, 953. [5] V. Skorohod Sudies in he heory of random rocesses, Addison Wesley, Reading, MA.965. [6] T. Morozan, Sabiliy of linear sysems wih random arameers J. Diff. Equaions, Vol.3,. 7-78, // lifesciencej@gmail.com