Existence of mild solutions for impulsive fractional stochastic equations with infinite delay

Size: px

Start display at page:

Download "Existence of mild solutions for impulsive fractional stochastic equations with infinite delay"

Edmund Reynolds
6 years ago
Views:

$Malaya Journal of Maemai 4((3 3 43 Exisence of mild soluions for impulsive fracional sochasic equaions wih infinie delay Toufi Guendouzi and Khadem Mehdi Laboraory of Sochasic Models, Saisic and$

1 Malaya Journal of Maemai 4(( Exisence of mild soluions for impulsive fracional sochasic equaions wih infinie delay Toufi Guendouzi and Khadem Mehdi Laboraory of Sochasic Models, Saisic and Applicaions, Tahar Moulay Universiy POBox 38 En-Nasr, Saida, Algeria Absrac This paper is mainly concerned wih he exisence of mild soluions for a class of fracional sochasic differenial equaions wih impulses in Hilber spaces A new se of sufficien condiions are formulaed and proved for he exisence of mild soluions by means of Sadovsii s fixed poin heorem An example is given o illusrae he heory Keywords: Exisence resul, fracional sochasic differenial equaion, fixed poin echnique, infinie delay, resolven operaors MSC: 34K3, 34K5, 6A33 c MJM All righs reserved Inroducion The sochasic differenial equaions have been widely applied in science, engineering, biology, mahemaical finance and in almos all applied sciences In he presen lieraure, here are many papers on he exisence and uniqueness of soluions o sochasic differenial equaions (see,, 8 and references herein More recenly, Chang e al 4 invesigaed he exisence of square-mean almos auomorphic mild soluions o nonauonomous sochasic differenial equaions in Hilber spaces by using semigroup heory and fixed poin approach Fu and Liu 8 discussed he exisence and uniqueness of square-mean almos auomorphic soluions o some linear and nonlinear sochasic differenial equaions and in which hey sudied he asympoic sabiliy of he unique square-mean almos auomorphic soluion in he square-mean sense Recenly, fracional differenial equaions have found numerous applicaions in various fields of science and engineering The exisence of soluions for nonlinear fracional sochasic differenial equaions have been sudied by few auhors 9, 8 On he oher hand, he heory of impulsive differenial equaions is emerging as an acive area of invesigaion due o he applicaion in area such as mechanics, elecrical engineering, medicine biology, and ecology, see Benchohra and Henderson 3, Hernández e al, Lin and Hu 3, Prao and Zabczy 4 As an adequae model, impulsive differenial equaions are used o sudy he evoluion of processes ha are subjec o sudden changes in heir saes However, o he bes of our nowledge, i seems ha lile is nown abou impulsive fracional sochasic equaions wih infinie delay and he aim of his paper is o fill his gap We refer he ineresed reader, for insance, o 8 and references herein for impulsive fracional sochasic equaions Inspired by he menioned wor 8 in his paper, we are ineresed in sudying he exisence of mild soluions Corresponding auhor addresses: fguendouzi@gmailcom (T Guendouzi and mmehdi986@gmailcom (K Mehdi

2 T Guendouzi e al / Exisence of mild soluions 3 of he following impulsive fracional sochasic differenial equaions wih infinie delay in he form c D α x( g(, x = A x( g(, x f(, x, B x( σ(, x, B x( dw(, d J :=, T, T >,, x( = I (x(, =,,, m, x( = φ(, φ( B h, where c D α is he Capuo fracional derivaive of order α, < α < ; x( aes he value in he separable Hilber space H; A : D(A H H is he infiniesimal generaor of an α-resolven family S α ( The hisory x : (, H, x (θ = x( θ, θ, belongs o an absrac phase space B h, which will be described axiomaically in Secion ; g : J B h H, f : J B h H H and σ : J B h H L are appropriae funcions o be specified laer; I : B h H, =,,, m, are appropriae funcions The erms B x( and B x( are given by B x( = K(, sx(sds and B x( = P (, sx(sds respecively, where K, P C(D, IR are he se of all posiive coninuous funcions on D = {(, s IR : s T } Here = < < < m < m = T, x( = x( x(, x( = lim h x( h and x( = lim h x( h represen he righ and lef limis of x( a =, respecively The iniial daa φ = {φ(, (, } is an F -measurable, B h -valued random variable independen of w wih finie second momens The paper is organized as follows In secion, we briefly presen some basic noaions and preliminaries In secion 3, is devoed o he developmen of our main exisence resuls and our basic ool include Sadovsii s fixed poin heorem Finally, he paper is conclude wih an example o illusrae he obained resuls Preliminaries and basic properies Le H, K be wo separable Hilber spaces and L(K, H be he space of bounded linear operaors from K ino H For convenience, we will use he same noaion o denoe he norms in H, K and L(K, H, and use (, o denoe he inner produc of H and K wihou any confusion Le (Ω, F, {F }, IP be a filered complee probabiliy space saisfying he usual condiion, which means ha he filraion is a righ coninuous increasing family and F conains all IP-null ses w = (w be a Q-Wiener process defined on (Ω, F, {F }, IP wih he covariance operaor Q such ha rq < We assume ha here exiss a complee orhonormal sysem {e } in K, a bounded sequence of nonnegaive real numbers λ such ha Qe = λ e, =,, and a sequence {β } of independen Brownian moions such ha (w(, e K = λ (e, e K β (, e K,, b = Le L = L (Q / K, H be he space of all HilberSchmid operaors from Q / K ino H wih he inner produc ψ, π L = rψqπ Assume ha h : (, (, wih l = h(d < a coninuous funcion We define he absrac phase space B h by { B h = If B h is endowed wih he norm φ : (, H, for any a >, (IE φ(θ / is bounded and measurable } funcion on a, wih φ( = and φ Bh = hen (B h, Bh is a Banach space 5 We consider he space { B b = h(s sup (IE φ(θ / ds < sθ h(s sup (IE φ(θ / ds, φ B h, sθ x : (, T H such ha x J C(J, H and here exis x( and x( wih x( = x(, x = φ B h, =,,, m }, (

3 3 T Guendouzi e al / Exisence of mild soluions where x J is he resricion of x o J = (,, =,,, m he funcion Bh o be a seminorm in B b, i is defined by x Bb = φ Bh sup st (IE x(s /, x B b Lemma (6 Assume ha x B h ; hen for J, x B h Moreover, l(ie x( / l sup (IE x(s / x Bh, st where l = h(sds < Le us recall he following nown definiions For more deails see Definiion The fracional inegral of order α wih he lower limi for a funcion f is defined as I α f( = f(s ds, >, α > Γ(α ( s α provided he righ-hand side is poinwise defined on,, where Γ is he gamma funcion Definiion Riemann-Liouville derivaive of order α wih lower limi zero for a funcion f :, IR can be wrien as L D α f( = Γ(n α d n d n f(s ds, >, n < α < n ( ( s α n Definiion 3 The Capuo derivaive of order α for a funcion f :, IR can be wrien as If f( C n,, hen c D α f( = c D α f( = L D α ( Γ(n α f( n =! f (, >, n < α < n (3 ( s n α f n (sds = I n α f n (s, >, n < α < n Obviously, he Capuo derivaive of a consan is equal o zero derivaive of order α > is given as The Laplace ransform of he Capuo n L{ c D α f(; s} = s α ˆf(s s α f ( (; n α < n = Definiion 4 A wo parameer funcion of he Miag-Leffler ype is defined by he series expansion E α,β (z = = z Γ(α β = πi C µ α β e µ µ α dµ, α, β C, R(α >, z where C is a conour which sars and ends a end encircles he disc µ z / couner clocwise For shor, E α (z = E α, (z I is an enire funcion which provides a simple generalizaion of he exponen funcion: E (z = e z and he cosine funcion: E (z = cos h(z, E ( z = cos(z, and plays a vial role in he heory of fracional differenial equaions The mos ineresing properies of he Miag-Leffler funcions are associaed wih heir Laplace inegral and for more deails see e λ β E α,β (ω α d = λα β λ α ω, Reλ > ω α, ω >, Definiion 5 (3 A closed and linear operaor A is said o be secorial if here are consans ω IR, θ π, π, M >, such ha he following wo condiions are saisfied: i ρ(a Σ θ,ω = {λ C : λ ω, arg(λ ω < θ},

4 T Guendouzi e al / Exisence of mild soluions 33 ii R(λ, A = (λ A M λ ω, λ Σ θ,ω Definiion 6 Le A be a closed and linear operaor wih he domain D(A defined in a Banach space H Le ρ(a be he resolven se of A We say ha A is he generaor of an α-resolven family if here exis ω and a srongly coninuous funcion S α : IR L(H, where L(H is a Banach space of all bounded linear operaors from H ino H and he corresponding norm is denoed by, such ha {λ α : Reλ > ω} ρ(a and (λ α I A x = where S α ( is called he α-resolven family generaed by A e λ S α (xd, Reλ > ω, x H, (4 Definiion 7 Le A be a closed and linear operaor wih he domain D(A defined in a Banach space H and α > We say ha A is he generaor of a soluion operaor if here exis ω and a srongly coninuous funcion S α : IR L(H such ha {λ α : Reλ > ω} ρ(a and λ α (λ α I A x = where S α ( is called he soluion operaor generaed by A e λ S α (xd, Reλ > ω, x H, (5 The concep of he soluion operaor is closely relaed o he concep of a resolven family For more deails on α-resolven family and soluion operaors, we refer he reader o Lemma (6 If f saisfies he uniform Hölder condiion wih he exponen β (, and A is a secorial operaor, hen he unique soluion of he Cauchy problem c D α x( = Ax( f(, x, Bx(, >,, < α <, x( = φ(,, (6 is given by where x( = T α ( (x( S α ( sf(s, x s, F x(sds, (7 T α ( = E α, (A α = e λ λ α πi Br λ α dλ, (8 A S α ( = α E α,α (A α = e λ πi B r λ α dλ, (9 A here B r denoes he Bromwich pah; S α ( is called he α-resolven family and T α ( is he soluion operaor generaed by A The following resul on he operaor S α ( appeared and proved in 3 Theorem If α (, and A A α (θ, ω is a secorial operaor, hen for any x H and >, we have where C is a consan depending only on θ and ω S α ( Ce ω ( α, >, ω > ω, A he end of his secion, we recall he fixed poin heorem of Sadovsii 7 which is used o esablish he exisence of he mild soluion o he impulsive fracional sysem ( Theorem (7 Le Φ be a condensing operaor on a Banach space H, ha is, Φ is coninuous and aes bounded ses ino bounded ses, and µ(φ(b µ(b for every bounded se B of H wih µ(b > If Φ(N N for a convex, closed and bounded se N of H, hen Φ has a fixed poin in H (where µ( denoes Kuraowsi s measure of noncompacness

5 34 T Guendouzi e al / Exisence of mild soluions 3 The mild soluion and exisence In his secion, we consider he fracional impulsive sysem ( We firs presen he definiion of mild soluions for he sysem based on he paper 7 Definiion 3 An H-valued sochasic process {x(, (, T } is said o be a mild soluion of he sysem ( if x = φ B h saisfying x L (Ω, H and he following condiions hold i x( is F adaped and measurable, ; ii x is B h -valued and he resricion of x( o he inerval (,, =,,, m is coninuous; iii for each J, x( saisfies he following inegral equaion φ(, (,, x( = T α (φ( g(, φ g(, x S α ( sf(s, x s, B x(sds S α ( sσ(s, x s, B x(sdw(s,,, T α (φ( g(, φ T α ( I (x( g(, x T α ( g(, x I (x g(, x S α ( sf(s, x s, B x(sds T α (φ( g(, φ = T α ( I (x( g(, x = T α ( g(, x I (x g(, x S α ( sf(s, x s, B x(sds S α ( sσ(s, x s, B x(sdw(s, (,, S α ( sσ(s, x s, B x(sdw(s, ( m, T (3 iv x = = I (x(, =,,, m he resricion of x( o he inerval, T \{,, m } is coninuous In order o explain our heorem, we need he following assumpions (H: If α (, and A A α (θ, ω, hen for x H and > we have T α ( Me ω and S α ( Ce ω ( α, ω > ω Thus we have T α ( M T and S α ( α MS, where M T = sup T T α (, and M S = sup T Ce ω ( α (fore more deails, see 3 (H: The funcion g : J B h H is coninuous and here exiss some consan M g > such ha IE g(, ψ g(, ψ H M g ψ ψ B h, (, ψ i J B h, i =,, IE g(, ψ H M g ( ψ B h (H3: The funcion f : J B h H H saisfies he following properies: i f(,, : B h H H is coninuous for each J and for each (ψ, x B h H, f(, ψ, x : J H is srongly measurable; ii here exis wo posiive inegrable funcions µ, µ L (, T and a coninuous nondecreasing funcion Ξ f :, (, such ha for every (, ψ, x J B h H, we have ( IE f(, ψ, x H µ (Ξ f ψ B h µ (IE x Ξ f (q H, lim inf = Λ < q q

6 T Guendouzi e al / Exisence of mild soluions 35 iii here exis wo posiive inegrable funcions µ, µ L (, T such ha IE f(, ψ, x f(, ϕ, y H µ ( ψ ϕ B h µ (IE x y H, for every (, ψ, x and (, ϕ, y J B h H (H4: The funcion σ : J B h H L saisfies he following properies: i σ(,, : B h H L is coninuous for each J and for each (ψ, x B h H, σ(, ψ, x : J L is srongly measurable; ii here exis wo posiive inegrable funcions ν, ν L (, T and a coninuous nondecreasing funcion Ξ σ :, (, such ha for every (, ψ, x J B h H, we have IE σ(, ψ, x L ( ν (Ξ σ ψ B h ν (IE x Ξ σ (q H, lim inf = Υ < q q iii here exis wo posiive inegrable funcions ν, ν L (, T such ha IE σ(, ψ, x σ(, ϕ, y L ν ( ψ ϕ B h ν (IE x y H, for every (, ψ, x and (, ϕ, y J B h H (H5: The funcion I : H H is coninuous and here exiss Θ > such ha Θ = max {IE I (x H}, m, x B q where B q = {y Bb, y q, q > } Bb The se B q is clearly a bounded closed convex se in Bb for each q and for each y B q From Lemma, we have y z B h ( y ( B h z B h ( 4 l sup T 4( φ B h l q IE y( H y B h 4 l sup T IE y( H z B h (3 The main objec of his paper is o explain and prove he following heorem Theorem 3 Assume ha he assumpions (H-(H5 hold Then he impulsive sochasic fracional sysem ( has a mild soluion on (, T provided ha C 6M g l 7 M ST α η α η < (33 T (α and l M g M ST α ϑ α ϑ <, (34 T (α C is a posiive consan depending only on M T, M g and l

7 36 T Guendouzi e al / Exisence of mild soluions Proof Consider he operaor P : B b B b defined by P( = φ(, (,, T α (φ( g(, φ g(, x S α ( sf(s, x s, B x(sds S α ( sσ(s, x s, B x(sdw(s,,, T α (φ( g(, φ T α ( I (x( g(, x T α ( g(, x I (x g(, x S α ( sf(s, x s, B x(sds T α (φ( g(, φ = T α ( I (x( g(, x = T α ( g(, x I (x g(, x S α ( sf(s, x s, B x(sds S α ( sσ(s, x s, B x(sdw(s, (,, S α ( sσ(s, x s, B x(sdw(s, ( m, T (35 We shall show ha P has a fixed poin, which is hen a mild soluion for he impulsive sysem ( For φ B h, define { φ(, (, ; z( =, J Then z B b Le x( = y( z(, (, T I is easy o chec ha x saisfies ( if and only if y = and Se T α (φ( g(, φ g(, y z S α ( sf(s, y s z s, B (y(s z(sds S α ( sσ(s, y s z s, B (y(s z(sdw(s,,, T α (φ( g(, φ T α ( I (y( g(, y z T α ( g(, y z I (y z g(, y z y( = S α ( sf(s, y s z s, B (y(s z(sds S α ( sσ(s, y s z s, B (y(s z(sdw(s, (,, T α (φ( g(, φ = T α ( I (y( g(, y z = T α ( g(, y z I (y S α ( sf(s, y s z s, B (y(s z(sds z g(, y z S α ( sσ(s, y s z s, B (y(s z(sdw(s, ( m, T B b = {y B b, y = B h } Thus, for any y B b we have y b = y Bh Therefore, (B b, b is a Banach space sup st ( IE y(s ( = sup IE y(s st

8 T Guendouzi e al / Exisence of mild soluions 37 Consider he map Π on Bb defined by T α (φ( g(, φ g(, y z S α ( sf(s, y s z s, B (y(s z(sds S α ( sσ(s, y s z s, B (y(s z(sdw(s,,, T α (φ( g(, φ T α ( I (y( g(, y z T α ( g(, y z I (y z g(, y z (Πy( = S α ( sf(s, y s z s, B (y(s z(sds S α ( sσ(s, y s z s, B (y(s z(sdw(s, (,, T α (φ( g(, φ = T α ( I (y( g(, y z = T α ( g(, y z I (y S α ( sf(s, y s z s, B (y(s z(sds z g(, y z S α ( sσ(s, y s z s, B (y(s z(sdw(s, ( m, T I is clear ha he operaor P has a fixed poin if and only if Π has a fixed poin So le us prove ha Π has a fixed poin Now, we decompose Π as Π = Π Π, where,,, T α ( I (y( T α ( g(, y z I (y z g(, y z, (,, (Π y( = T α ( I (y( = = T α ( g(, y z I (y (Π y( = T α (g(, φ g(, y z z g(, y z, ( m, T, S α ( sf(s, y s z s, B (y(s z(sds S α ( sσ(s, y s z s, B (y(s z(sdw(s, J In order o use Theorem we will verify ha Π is compac and coninuous while Π is a conracion operaor For he sae of convenience, we divide he proof ino several seps Sep We show ha here exiss a posiive number q such ha Π(B q B q If his is no rue, hen for each q >, here exiss a funcion y q ( B q, bu Π(y q / B q, ha is IE (Πy q ( H > q An elemenary inequaliy can show ha, for, q IE Π(y q ( H 4IE T α (g(, φ H 4IE g(, yq z H 4IE S α ( sf(s, ys q z s, B (y q (s z(sds 4IE S α ( sσ(s, ys q z s, B (y q (s z(sdw(s H 4 = 4 I i i= H (36

9 38 T Guendouzi e al / Exisence of mild soluions Le us now esimae each erm above I i, i =,, 4 By Lemma and assumpions (H-(H, we have Togeher wih assumpion (H3 and (3, we have I M T IE g(, φ H M T M g ( φ B h, (37 ( I M g ( y q z B h M g 4 φ B h l q (38 I 3 M S M S M S S α ( s ds S α ( s IE f(s, ys q z s, B (y q (s z(s Hds ( s α ds ( s α ( µ (sξ f ys q z s B h µ (sie B (y q (s z(s H ds T α ( s α Ξ f (4( φ B α h l q µ Bµ sup IE y q (s z(s H ds st T α α Ξ f (4( φ B h l q µ Bµ q, (39 where B = sup,t K(, sds <, µ = sup µ (s, µ = sup µ (s s, s, A similar argumen involves assumpion (H4, we obain I 4 S α ( s IE σ(s, ys q z s, B (y q (s z(s Lds ( s (α Ξ σ (4( φ B h l q ν Bν ( Ξ σ 4( φ B h l q ν Bν q, M S M S T α α sup IE y q (s z(s H st ds (3 where B = sup,t P (, sds <, ν = sup ν (s, ν = sup ν (s s, s, Combining hese esimaes (36-(3 yields where q IE Π(y q ( H L 6M g l q 4 M S T α 4 M S α T α α Ξ σ ( 4( φ B h l q Ξ f (4( φ B h l q ν B ν q L = 4 M T M g ( φ B h 4M g ( 4 φ Bh µ Bµ q, (3 Dividing boh sides of (3 by q and aing q, we obain 6M g l 4 M S T α α 4Λµ Bµ 4 M S = 6M g l 4 M S T α η α η, T (α which is a conradicion o our assumpion in (33 For (,, we have T α α 4Υν Bν q IE Π(y q ( H 7 T α ( IE I (y q ( H 7 T α( IE g(, y q z I (y q z H 7 T α ( IE g(, y q z H 7IE T α(g(, φ H 7IE g(, yq z H 4IE S α ( sf(s, ys q z s, B (y q (s z(sds H 7IE S α ( sσ(s, ys q z s, B (y q (s z(sdw(s H (3

10 T Guendouzi e al / Exisence of mild soluions 39 Using assumpions (H-(H5 we obain IE Π(y q ( H L 7 M T M gl q 8M g l q 7 M S T α α Ξ f (4( φ B h l q µ Bµ q 7 M S T α ( Ξ σ 4( φ B α h l q ν Bν q, where L = 7 M T (Θ M g 6( φ B h l Θ 7 M T M g ( φ Bh 7M g ( 4 φ Bh A Similar argumen gives 7 M T M gl 8M g l 7 M S T α α 4Λµ Bµ 7 M S = 7 M T M gl 8M g l 7 M S T α η α η, T (α which is a conradicion o our assumpion in (33 Similarly for ( i, i, i =,,, m, we obain C 6M g l 7 M S T α α 4Λµ Bµ 7 M S = C 6M g l 7 M S T α η α η, T (α T α α T α α 4Υν Bν 4Υν Bν wih η = 4Λµ B µ, η = 4Υν B ν and C is a posiive consan depending only on M T, M g and l This is a conradicion o our assumpion in (33 Thus, for some posiive number q, Π(B q B q Sep The map Π is coninuous on B q Le {y n } n= be a sequence in B q wih lim y n y B q Then for ( i, i, we have IE (Π y n ( (Π y( i 3 T α ( IE I (y n ( I (y( H = IE g(, y n z I (y n z IE g(, y n z g(, y z H g(, y z I (y z H Since he funcions g, I i, i =,,, m are coninuous, hence lim n IE Π y n Π y = which implies ha he mapping Π is coninuous on B q Sep 3 Π maps bounded ses ino bounded ses in B q Le us prove ha for q > here exiss a δ > such ha for each y B q, we have IE (Π y( H δ for ( i, i, i =,,, m We have i IE (Π y( H 3 T α ( IE I (y( H IE g(, y z H = IE g(, y z I (y 3m M T z H ( Θ ( 6M g l M g M g φ B h l q which proves he desired resul := δ,

11 4 T Guendouzi e al / Exisence of mild soluions Sep 4 The se {Π y, y B q } is an equiconinuous family of funcions on J Le u, v ( i, i, i u < v i, i =,,, m, y B q We have IE (Π y(v (Π y(u H i 3 T α (v T α (u IE I (y( H IE g(, y z H = 3 IE g(, y z I (y z H ( Θ ( i 6M g l M g M g φ B h l q T α (v T α (u Since T α is srongly coninuous and i allows us o conclude ha lim u v T α (v T α (u = for all =,,, m, which implies ha he se {Π y, y B q } is equiconinuous Finally, combining Sep o Sep 4 ogeher wih Ascoli s heorem, we conclude ha he operaor Π is compac Sep 5 Π is conracive Le y, y B q and ( i, i, i =,,, m Then = IE (Π y( (Π y ( H 3 g(, y z g(, y z H 3IE S α ( s f(s, y s z s, B (y(s z(s f(s, ys z s, B (y (s z(s ds H 3IE S α ( s σ(s, y s z s, B (y(s z(s σ(s, ys z s, B (y (s z(s dw(s 3 g(, y z g(, y z H 3 S α ( s ds S α ( s IE f(s, y s z s, B (y(s z(s f(s, y s z s, B (y (s z(s H ds 3 S α ( s IE σ(s, y s z s, B (y(s z(s σ(s, y s z s, B (y (s z(s L ds 3M g y y B h 3 M S ( s α ds ( s α µ (s y s ys B h µ (sie B (y(s z(s B (y (s z(s H ds 3 M S ( s (α ν (s y s ys B h ν (sie B (y(s z(s B (y (s z(s H ds 3M g y y B h 3 M S T α ( s α α µ l sup IE y(s y(s H µ B sup IE y(s y(s H ds 3 M S ( s (α ν l sup IE y(s y(s H ν B sup IE y(s y(s H ds ( 3 l M g M S T α α (µ l µ B T (α (ν l ν B y y Bb ( = 3 l M g M S T α ϑ α ϑ T (α y y Bb H So Π is a conracion by our assumpion in (34 Hence, by Sadovsii s fixed poin heorem we can conclude ha he problem ( has a leas one soluion on (, T This complees he proof of he heorem 4 An example In his secion, we consider an example o illusrae our main heorem We examine he exisence of soluions

12 T Guendouzi e al / Exisence of mild soluions 4 for he following fracional sochasic parial differenial equaion of he form D q u(, x a(, x, s Q (u(s, xds = u(, x x H(, x, s Q (u(s, xds V (, x, s Q 3 (u(s, xds x, π,, b, u(, = = u(, π, u(, x = φ(, x, (,, x, π, u( i (x = q i ( i su(s, xds, x, π, (s, e u(s,x ds p(s, e u(s,x ds a(, x, s Q (u(s, xds dβ(, d (4 where β( is a sandard cylindrical Wiener process in H defined on a sochasic space (Ω, {F }, F, IP; D q is he Capuo fracional derivaive of order < q < ; < < < < n = T are prefixed numbers; a, Q, H, Q, V, Q 3 are coninuous; φ B h Le H = L (, π wih he norm Define A : H H by Ay = y wih he domain Then, Ay = D(A = {y H; y, y are absoluely coninuous, y H and y( = y(π = } n (y, y n y n, y D(A, where y n (x = n= π sin(nx, n =,,, is he orhogonal se of eigenvecors of A I is well nown ha A is he infiniesimal generaor of an analyic semigroup (T ( in H is given by T (y = e n (y, y n y n, for all y H, > n= I follows from he above expressions ha (T ( is a uniformly bounded compac semigroup, so ha, R(λ, A = (λ A is a compac operaor for all λ ρ(a Le h(s = e s, s <, hen l = h(sds = and define φ Bh = ( h(s sup IE φ(θ / ds sθ Hence for (, φ, T B h, where φ(θ(y = φ(θ, y, (θ, y (,, π Se u((x = u(, x, g(, φ(x = f(, φ, B u((x = σ(, φ, B u((x = I i (φ(x = a(, x, θq (φ(θ(xdθ, q i ( θφ(θ(xdθ, H(, x, θq (φ(θ(xdθ B u((x, V (, x, θq 3 (φ(θ(xdθ B u((x, where B u( = (s, e u(s,x ds and B u( = p(s, e u(s,x ds Then wih hese seings he equaions in (4 can be wrien in he absrac form of Eq ( All condiions of Theorem 3 are now fulfilled, so we deduce ha he sysem (4 has a mild soluion on (, T 5 Conclusion We have sudied he exisence of mild soluions for a class of impulsive fracional sochasic differenial equaions in Hilber spaces, which is new and allow us o develop he exisence of various fracional differenial equaions and sochasic fracional differenial equaions An example is provided o illusrae he applicabiliy of he new resul The resuls presened in his paper exend and improve he corresponding ones announced by Dabas e al 6, Dabas and Chauhan 7, Shu e al 3, Sahivel e al 8 and ohers

13 4 T Guendouzi e al / Exisence of mild soluions Acnowledgemen The wor of he corresponding auhor was suppored by The Naional Agency of Developmen of Universiy Research (ANDRU, Algeria (PNR-SMA -4 References P Balasubramaniam, JY Par, A Vincen Anony Kumar, Exisence of soluions for semilinear neural sochasic funcional differenial equaions wih nonlocal condiions, Nonlinear Anal TMA, 7 (9, J Bao, Z Hou, and C Yuan, Sabiliy in disribuion of mild soluions o sochasic parial differenial equaions, Pro American Mah Soc, 38(6(, M Benchohra, J Henderson, SK Nouyas, Exisence resuls for impulsive semilinear neural funcional differenial equaions in Banach spaces, Memoirs on Diff Equ Mah Phys, 5(, 5-4 YK Chang, ZH Zhao, G M N Guéréaa, Squaremean almos auomorphic mild soluions o nonauonomous sochasic differenial equaions in Hilber spaces, Compuers & Mahemaics wih Applicaions, 6((, J Cui, L Yan, Exisence resul for fracional neural sochasic inegro-differenial equaions wih infinie delay, J Phys A: Mah Theor 44 ( 335 (6pp 6 J Dabas, A Chauhan, M Kumar, Exisence of he mild soluions for impulsive fracional equaions wih infinie delay, In J Differ Equ, ( Aricle ID J Dabas, A Chauhan, Exisence and uniqueness of mild soluion for an impulsive neural fracional inegro-differenial equaion wih infinie delay, Mah Compuer Modelling, 57(3, M M Fu and Z X Liu, Square-mean almos auomorphic soluions for some sochasic differenial equaions, Pro American Mah Soc, 38( (, T Guendouzi, I Hamada, Exisence and conrollabiliy resul for fracional neural sochasic inegrodifferenial equaions wih infinie delay, AMO - Advanced Modeling and Opimizaion, 5((3, 8-3 E Hernández M Pierri, G Goncalves, Exisence resuls for an impulsive absrac parial differenial equaion wih sae-dependen delay, Compuers & Mahemaics wih Applicaions, 5(3-4(6, 4-4 R Hilfer, Applicaions of Fracional Calculus in Physics, World Scienific Publishing, Singapore, AA Kilbas, HM Srivasava, JJ Trujillo, Theory and Applicaions of Fracional Differenial Equaions, Elsevier Science BV, Amserdam, 6 3 A Lin, L Hu, Exisence resuls for impulsive neural sochasic funcional inegro-differenial inclusions wih nonlocal iniial condiions, Compuers & Mahemaics wih Applicaions, 59(, G Da Prao, J Zabczy, Sochasic equaions in infinie dimensions, Vol 44 Encyclopedia of Mahemaics and is Applicaions, Cambridge universiy Press, Cambridge, Mass, USA, 99 5 Y Ren, R Sahivel, Exisence, uniqueness and sabiliy of mild soluions for second-order neural sochasic evoluion equaions wih infinie delay and Poisson jumps, J Mah Phys, 53(, Y Ren, Q Zhou, L Chen, Exisence, uniqueness and sabiliy of mild soluions for ime-dependen sochasic evoluion equaions wih Poisson jumps and infinie delay, J Opim Theory Appl, 49(, BN Sadovsii, On a fixed poin principle, Func Anal Appl, (967, 7-74

14 T Guendouzi e al / Exisence of mild soluions 43 8 R Sahivel, P Revahi, Yong Ren, Exisence of soluions for nonlinear fracional sochasic differenial equaions, Nonlinear Anal TMA, 8(3, R Sahivel, P Revahi, N IMahmudov, Asympoic sabiliy of fracional sochasic neural differenial equaions wih infinie delays, Absrac and Applied Analysis V 3, Aricle ID 76957, 9 R Sahivel, P Revahi, S Marshal Anhoni, Exisence of pseudo almos auomorphic mild soluions o sochasic fracional differenial equaions, Nonlinear Anal TMA, 75(7(, R Sahivel, Yong Ren, Exponenial sabiliy of second-order sochasic evoluion equaions wih Poisson jumps, Comm Nonlinear Sci Numerical Simulaion, 7(, R Sahivel, Yong Ren, Hyunsoo Kim, Asympoic sabiliy of second-order neural sochasic differenial equaions, J Mah Phys, 5(, 57 3 XB Shu, Y Lai, Y Chen, The exisence of mild soluions for impulsive fracional parial differenial equaions, Nonlinear Anal TMA, 74 (, 3- Received: June 3, 3; Acceped: June 5, 3 UNIVERSITY PRESS

CONTRIBUTION TO IMPULSIVE EQUATIONS

European Scienific Journal Sepember 214 /SPECIAL/ ediion Vol.3 ISSN: 1857 7881 (Prin) e - ISSN 1857-7431 CONTRIBUTION TO IMPULSIVE EQUATIONS Berrabah Faima Zohra, MA Universiy of sidi bel abbes/ Algeria