Mild solutions for semi-linear fractional order functional stochastic differential equations with impulse effect

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1 Malaya J. Ma. 3(3)(25) Mild soluions for semi-linear fracional order funcional socasic differenial equaions wi impulse effec Mod Nadeem a, and Jaydev Dabas b a,b Deparmen of Applied Science and Engineering, II Roorkee, Saaranpur Campus, Saaranpur-247, India. Absrac is paper is concerned wi e exisence resuls of mild soluion for an impulsive fracional order socasic differenial equaion wi infinie delay subjec o nonlocal condiions. e resuls are obained by using e fixed poin ecniques and soluion operaor generaed by secorial operaor on a Hilber space. Keywords: Fracional order differenial equaion, nonlocal condiions, exisence and uniqueness, impulsive condiions, socasic differenial equaions. 2 MSC: 26A33,34B,34A2,34A37,34K5. c 22 MJM. All rigs reserved. Inroducion Recenly, fracional differenial equaions ave been proved o be valuable ools in e modeling of many penomena in various fields of engineering, pysics, economics and science. Fracional models ave various applicaions suc as nonlinear oscillaions of earquakes, viscoelasiciy, elecrocemisry, seepage flow in porous media, and elecromagneic, ec. ere as been a significan developmen in fracional differenial equaions since las few years for more deails one can see e papers ([7],[8],[9],[],[4],[5],[9]) and references cied erein. e deerminisic sysems ofen flucuae due o environmenal noise due o is reason i is imporan and necessary for researcer o sudy ese sysems. ese sysems are modeled as socasic differenial sysems. In many evoluion processes impulsive effecs exis in wic saes are canged abruply a cerain momens of ime. erefore e socasic differenial equaions wi impulsive effecs exis in real sysems and provide a more accurae maemaical model. For more deails one can see e papers ([6],[7],[8]) and references erein. Furer, if we combine e socasic differenial equaion wi a nonlocal iniial condiion srengens e model even furer. ese fac moivae us o sudy suc model in is paper. e basic ools are used in is paper including fixed-poin ecniques, e eory of linear semi-groups, resuls for probabiliy measures, and resuls for infinie dimensional socasic differenial equaions. e resuls are imporan from e viewpoin of applicaions since ey cover nonlocal generalizaions of inegro-differenial socasic differenial equaion arising in various fields suc as elecromagneic eory, populaion dynamics, and ea conducion in maerials wi memory, for more deail one can see e papers ([6],[3],[6],[23],[24],[25]) and references erein. In [4] Bauguna, considered e following problem { u () + Au() = f (, u(), u(b ()), u(b 2 ()),..., u(b m ())), (, ], (u) = φ on [ τ, ], Corresponding auor. address: modnadeem.jmi@gmail.com (Mod Nadeem), jay.dabas@gmail.com(jaydev Dabas).

2 278 Mod Nadeem e al. / Mild soluions for... and found e exisence, uniqueness and coninuaion of a mild soluion on e maximal inerval of exisence. e auor also proved some regulariy resuls under various condiions. Cauan e al. [5] considered e following semi-linear fracional order differenial equaions wi nonlocal condiion d α d α x() + Ax() = f (, x(), x(a ()),..., x(a m ())), [, ], = i, x() + g(x) = x, x( i ) = I i (x( i )), and discussed e exisence and uniqueness resuls of soluions using e applicaions of classical fixed poin eorems. Balasubramaniam e al. [2] sudied e exisence of soluions for e e following semi-linear neural socasic funcional differenial equaions d[x() + F(, x(), x(b ()),..., x(b m ()))] = Ax()d + G(, x(), x(a ()),..., x(a n ()))dw(), J = [, b], x() = x + g(x), were A is a infiniesimal generaor of an analyic semigroup of bounded linear operaors (),, on a separable Hilber space. By using fracional power of operaors and Sadovskii fixed poin eorem, e auors esablised e exisence of mild and srong soluions. Sakivel e al. [22] considered e following impulsive fracional socasic differenial equaions wi infinie delay in e form { D α x() = Ax() + f (, x, B x()) + σ(, x, B 2 x()) dw() d, [, ], = k, x( k ) = I k (x( k )), k =, 2,..., m x() = φ(), φ() B, and sudied e exisence resuls of mild soluions and esablised e sufficien condiions for e exisence of mild soluions by using fixed poin ecniques. Moivaed by e works of ese auor s ([2],[4],[5],[22]), we sudy e exisence of mild soluions of e following semi-linear socasic fracional funcional differenial equaion of e form: c D α x() = Ax() + f (, x, x(a ()),..., x(a m ())) +σ(, x, x(a ()),..., x(a m ())) dw(), J, = d k, (.) x( k ) = I k (x( k )), k =, 2,..., p, (.2) x() + g(x) = φ(), (, ], (.3) were J = [, ] and c D α denoes e Capuo s fracional derivaive of order α (, ). A : D(A) H H is a closed linear secorial operaor defined on a Hilber space (H, ). e funcions f, σ are given and saisfy some assumpions o be defined laer. We assume a x : (, ] H, x (s) = x( + s), s, belong o an absracase space B. Here < < < < +, I k C(H, H), (k =, 2,..., p), are bounded funcions, x( k ) = x( + k ) x( k ), x(+ k ) = lim x( k + ) and x( k ) = lim x( k ) represen e rig and lef-and limis of x() a = k, respecively, also we ake x( i ) = x( i ). e nonlocal condiion g : H H is defined as g(x) = p k= c kx( k ) were c k, k =,..., p, are given consans and < < 2 < < <. Suc nonlocal condiions were firs inroduced by Deng []. e iniial daa φ = {φ(), (, ]} is an F -measurable, B -valued random variable independen of w() wi finie second momens. o e bes of our knowledge, e exisence and uniqueness of mild soluion for e sysem (.) (.3) wi non local condiion is an unreaed opic ye in e lieraure and is fac is e moivaion of e presen work. Our work is divided in four secions, Second secion provides e basic definiions and preliminaries resuls wic are used in proving our main resuls. In e ird secion, we sae and prove e exisence resuls of e considered problem in is e paper. e four secion includes examples. 2 Preliminaries Le H, K be wo separable Hilber spaces and L(K, H) be e space of bounded linear operaors from K ino H. For convenience, we will use e same noaion o denoe e norms in H, K and L(K, H), and

3 Mod Nadeem e al. / Mild soluions for use (, ) o denoe e inner produc of H and K wiou any confusion. Le (Ω, F, {F }, P) be a complee filered probabiliy space saisfying a F conains all P-null ses of F. W = (W ) be a Q-Wiener process defined on (Ω, F, {F }, P) wi e covariance operaor Q suc a rq <. We assume a ere exiss a complee oronormal sysem {e k } k in K, a bounded sequence of nonnegaive real numbers λ k suc a Qe k = λ k e k, k =, 2,..., and a sequence of independen Brownian moions {β k } k suc a (w(), e) K = λk (e k, e) K β k (), e K,. k= Le L 2 = L 2(Q 2 K, H) be e space of all Hilber Scmid operaors from Q 2 K o H wi e inner produc < ϕ, ψ > L 2 = r[ϕqψ ]. Now, we inroduce absrac space pase B. Assume a : (, ] (, ) wi l = a coninuous funcion. An absracase B defined by ()d <, B = {φ : (, ] H, for any a >, (E φ(θ) 2 ) /2 is bounded and measurable funcion on[ a, ] wi φ() = and If B is endowed wi e norm (s) sup (E φ(θ) 2 ) /2 ds < }. s θ φ B = en (B, B ) is a Banac space ([2],[2]). Now we consider e space (s) sup (E φ(θ) 2 ) /2 ds, φ B, s θ B = {x : (, ] H suc a x Jk C(J k, H) and ere exis x( + k ) and x( k ) wi x( k) = x( k ), x = φ B, k =, 2,..., p}, were x Jk is e resricion of x o J k = ( k, k+ ], k =,, 2,..., p. e funcion B o be a semi-norm in B, i is defined by x B = φ B + sup (E x(s) 2 ) /2, x B. s [,] Lemma 2.. ([2]) Assume a x B, en for J, x B. Moreover, l(e x() 2 ) /2 l sup (E x(s) 2 ) /2 + x B, were l = (s)ds <. s [,] Definiion 2.. e Reimann-Liouville fracional inegral operaor for order α >, of a funcion f : R + R and f L (R +, X) is defined by were Γ( ) is e Gamma funcion. J f () = f (), Jα f () = ( s) α f (s)ds, α >, >, Γ(α) Definiion 2.2. Capuo s derivaive of order α > for a funcion f : [, ) R is defined as D α f () = for n < α < n, n N. If < α <, en ( s) n α f (n) (s)ds = J n α f (n) (), Γ(n α) D α f () = ( s) α f () (s)ds. Γ( α) Obviously, Capuo s derivaive of a consan is equal o zero.

4 28 Mod Nadeem e al. / Mild soluions for... Definiion 2.3. A wo parameer funcion of e Miag Lefller ype is defined by e series expansion E α,β (z) = k= z k Γ(αk + β) = 2πι c µ α β e µ µ α dµ, α, β >, z C, z were c is a conour wic sars and ends a and encircles e disc µ z α couner clockwise. e mos ineresing properies of e Miag Lefller funcions are associaed wi eir Laplace inegral e λ β E α,β (ω α )d = λα β λ α ω, Reλ > ω α, ω >. Definiion 2.4. [2] A closed and linear operaor A is said o be secorial if ere are consans ω R, θ [ π 2, π], M >, suc a e following wo condiions are saisfied: () (θ,ω) = {λ C : λ = ω, arg(λ ω) < θ} ρ(a), (2) R(λ, A) L(X) M λ ω, λ (θ,ω). Definiion 2.5. [] Le A be a closed and linear operaor wi e domain D(A) defined in a Banac space X. Le ρ(a) be e resolven se of A. We say a A is e generaor of an α-resolven family if ere exis ω and a srongly coninuous funcion α : R + L(X), were L(X) is a Banac space of all bounded linear operaors from X ino X and e corresponding norm is denoed by., suc a {λ α : Reλ > ω} ρ(a) and (λ α I A) = e λ α ()xd, Reλ > ω, x X, were α () is called e α-resolven family generaed by A. Definiion 2.6. [] Le A be a closed and linear operaor wi e domain D(A) defined in a Banac space X and α >. We say a A is e generaor of a soluion operaor if ere exis ω and a srongly coninuous funcion S α : R + L(X), suc a {λ α : Reλ > ω} ρ(a) and λ α (λ α I A) = were S α () is called e soluion operaor generaed by A. e λ S α ()xd, Reλ > ω, x X, eorem 2.. [26](Scauder fixed poin eorem) If U is a closed, bounded, convex subse of a Banac space X and e mapping : U U is compleely coninuous, en as a fixed poin in U. Definiion 2.7. A measurable F adaped socasic process x : (, ] H is called a mild soluion of e sysem (.)-(.3) if x() = φ() g(x) B on (, ], x =k = I k (x( k )), k =, 2,..., p, e resricion of x( ) o e inerval [, )\,...,, is coninuous and x() saisfies e following fracional inegral equaion S α ()(φ() g(x)) + α( s) f (s, x s, x(a (s)),..., x(a m (s)))ds + α( s)σ(s, x s, x(a (s)),..., x(a m (s)))dw(s), [, ], S α ( )[x( ) + I (x( ))] + α ( s) f (s, x s, x(a (s)),..., x(a m (s)))ds x() = + α ( s)σ(s, x s, x(a (s)),..., x(a m (s)))dw(s), (, 2 ], (2.4)... S α ( )[x( p ) + I p (x( p ))] + α ( s) f (s, x s, x(a (s)),..., x(a m (s)))ds + α ( s)σ(s, x s, x(a (s)),..., x(a m (s)))dw(s), (, ], were S α () = e λ λ α (λ α I A) dλ, α () = e λ (λ α I A) dλ, 2πi Γ 2πi Γ are called analyic soluions operaor and α resolven family and Γ is a suiable pa lying on θ,ω for more deails one can see []. Furer we inroduce e following assumpions o esablis our resuls:

5 Mod Nadeem e al. / Mild soluions for (H) If α (, ) and A A α (θ, ω ) en for any x H and > we ave α () Me ω and S α () Ce ω ( + α ), ω > ω. us we ave α () M and S α () α M S, were M = sup α () and M S = sup Ce ω ( + α )(for more deails, see [2]). (H) ere exis a consans L g >, suc a E g(x) g(y) 2 H L g x y 2 H. (H2) e nonlinear maps f : J B H m H and σ : J B H m L(K, H) are coninuous and ere exis consans L f, L σ, suc a E f (, ϕ, x, x 2,..., x m ) f (, ψ, y, y 2,..., y m ) 2 H L f [ ϕ ψ 2 B + E σ(, ϕ, x, x 2,..., x m ) σ(, ψ, y, y 2,..., y m ) 2 L(K,H) L σ [ ϕ ψ 2 B + for all (x, x 2,..., x m ) and (y, y 2,..., y m ) H m, J and ϕ, ψ B. m i= m i= E x i y i 2 H ], E x i y i 2 H ], (H3) e funcions I k : H H are coninuous and ere exiss L k >, suc a x, y H, k =, 2,..., p, L = max{l k } > L g. E I k (x) I k (y) 2 H L ke x y 2 H, 3 Exisence and uniqueness of soluions eorem 3.2. Le e assumpions (H)-(H3) are saisfied and [ Θ = 3 M S 2( + L) + 3 M 2 2α α 2 L f (l + m) + 3 M 2 2α ] 2α L σ(l + m) <, en e problem (.)-(.3) as a unique mild soluion x H on J. Proof. Firs we conver e problem (.)-(.3) ino a fixed poinroblem. Consider e operaor P : B B defined by S α ()(φ() g(x)) + α( s) f (s, x s, x(a (s)),..., x(a m (s)))ds + α( s)σ(s, x s, x(a (s)),..., x(a m (s)))dw(s), [, ], S α ( )[x( ) + I (x( ))] + α ( s) f (s, x s, x(a (s)),..., x(a m (s)))ds (Px)() = + α ( s)σ(s, x s, x(a (s)),..., x(a m (s)))dw(s), (, 2 ],... S α ( )[x( p ) + I p (x( p ))] + α ( s) f (s, x s, x(a (s)),..., x(a m (s)))ds + α ( s)σ(s, x s, x(a (s)),..., x(a m (s)))dw(s), (, ]. Le y(.) : (, ] H be e funcion defined by y() = { φ(), (, ], J, en y = φ. For eac z : J H wi z k C(J k, H), k =,..., p and z() =, we denoe by z e funcion defined by {, (, ] z = z(), J.

6 282 Mod Nadeem e al. / Mild soluions for... If x( ) saisfies e sysem (2.4), en we can decompose x( ) as x() = y() + z(), wic implies x = y + z for J and e funcion z( ) saisfies S α ()(φ() g(y + z)) + α( s) f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))ds + α( s)σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))dw(s), [, ], S α ( )[y( ) + z( ) + I (y( ) + z( ))] + z() = α ( s) f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))ds + α ( s)σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))dw(s), (, 2 ],... S α ( )[y( p ) + z( p ) + I p (y( p ) + z( p ))] + α ( s) f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))ds + α ( s)σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))dw(s), (, ]. Se B, suc a z = and for any z B, we ave z B = z B + sup(e z() 2 ) 2 J = sup (E z() 2 ) 2. J us (B, B ) is a Banac space. Define an operaor N : B B by S α ()(φ() g(y + z)) + α( s) f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))ds + α( s)σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))dw(s), [, ], S α ( )[y( ) + z( ) + I (y( ) + z( ))] + (Nz)() = α ( s) f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))ds + α ( s)σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))dw(s), (, 2 ],... S α ( )[y( p ) + z( p ) + I p (y( p ) + z( p ))] + α ( s) f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))ds + α ( s)σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))dw(s), (, ]. In order o prove exisence resuls, i is enoug o sow a N as a unique fixed poin. Le z, z B en for [, ], we ave E (Nz)() (Nz )() 2 H 3E S α()[g(y + z) g(y + z )] 2 H α ( s)[ f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s))) f (s, y s + z s, y(a (s)) + z (a (s)),..., y(a m (s)) + z (a m (s)))]ds 2 H α ( s)[σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s))) σ(s, y s + z s, y(a (s)) + z (a (s)),..., y(a m (s)) + z (a m (s)))]dw(s) 2 H, by applying assumpions, we ave E (Nz)() (Nz )() 2 H (3 M S 2 L g + 3 M 2 2α α 2 L f (l + m) + 3 M 2 2α 2α L σ(l + m)) z z 2 B.

7 Mod Nadeem e al. / Mild soluions for For (, 2 ], we ave E (Nz)() (Nz )() 2 H 3E S α( )[z( ) z ( ) + I (y( ) + z( )) I (y( ) + z ( ))] 2 H α ( s)[ f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s))) f (s, y s + z s, y(a (s)) + z (a (s)),..., y(a m (s)) + z (a m (s)))]ds 2 H by applying assumpions, we obain E (Nz)() (Nz )() 2 H (3 M S 2( + L ) + 3 M 2 2α Similarly, for (, ], we ave α ( s)[σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s))) σ(s, y s + z s, y(a (s)) + z (a (s)),..., y(a m (s)) + z (a m (s)))]dw(s) 2 H. α 2 L f (l + m) + 3 M 2 2α 2α L σ(l + m)) z z 2 B. E (Nz)() (Nz )() 2 H 3E S α( )[z( p ) z ( p ) + I p (y( p ) + z( p )) I p (y( p ) + z ( p ))] 2 H by applying assumpions, we ave α ( s)[ f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a n (s)) + z(a n (s))) f (s, y s + z s, y(a (s)) + z (a (s)),..., y(a m (s)) + z (a m (s)))]ds 2 H E (Nz)() (Nz )() 2 H (3 M S 2( + L p) + 3 M 2 2α us for all [, ], we esimae α ( s)[σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s))) σ(s, y s + z s, y(a (s)) + z (a (s)),..., y(a m (s)) + z (a m (s)))]dw(s) 2 H, α 2 L f (l + m) + 3 M 2 2α 2α L σ(l + m)) z z 2 B. E (Nz)() (Nz )() 2 H {3 M S 2( + L) + 3 M 2 2α α 2 L f (l + m) +3 M 2 2α } 2α L σ(l + m) z z 2 B, Θ z z 2 B. Since Θ < as in e eorem 3.2, erefore N is a conracion. Hence N as a unique fixed poin by Banac conracion principle. is complees e proof of e eorem. e second resul is proved by using e Scauder fixed poin eorem. For is we ake e following assumpions (H4) ere exis a consans M >, suc a E g(x) 2 H M. (H5) e funcions I k : H H are coninuous and ere exiss M 2 >, suc a E I k (x) 2 H M 2. (H6) f, σ : J B H m H are coninuous and ere exis consans M 3, M 4, suc a E f (, ϕ, x, x 2,..., x m ) 2 H M 3, E σ(, ϕ, x, x 2,..., x m ) 2 H M 4. eorem 3.3. Le e assumpions (H3)-(H6) are saisfied en e impulsive socasic differenial equaion (.)-(.3) as a leas one mild soluion.

8 284 Mod Nadeem e al. / Mild soluions for... Proof. le us consider e space B r = {y B : y r}. I is obvious a B r is closed convex and bounded subse of B. Consider e operaor N : B r B r defined by S α ()(φ() g(y + z)) + α( s) f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))ds + α( s)σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))dw(s), [, ], S α ( )[y( ) + z( ) + I (y( ) + z( ))] + (Nz)() = α ( s) f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))ds + α ( s)σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))dw(s), (, 2 ],... S α ( )[y( p ) + z( p ) + I p (y( p ) + z( p ))] + α ( s) f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))ds + α ( s)σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))dw(s), (, ]. Firs we sall sow a N is coninuous, for is le {z n } n= be a sequence in B r suc a lim z n z B r. Wen [, ], we ave E (Nz n )() (Nz)() 2 H 3E S α()[g(y + z n ) g(y + z) 2 H α ( s)[ f (s, y s + z n s, y(a (s)) + z n (a (s)),..., y(a m (s)) + z n (a m (s))) f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))]ds 2 H α ( s)[σ(s, y s + z n s, y(a (s)) + z n (a (s)),..., y(a m (s)) + z n (a m (s))) σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))]dw(s) 2 H. en for ( i, i+ ], were i =, 2,..., p, en we ave E (Nz n )() (Nz)() 2 H 3E S α( i )[z n ( i ) z( i ) + I i (y( i ) + z n ( i )) I i (y( i ) + z( i ))] 2 H i α ( s)[ f (s, y s + z n s, y(a (s)) + z n (a (s)),..., y(a m (s)) + z n (a m (s))) f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))]ds 2 H i α ( s)[σ(s, y s + z n s, y(a (s)) + z n (a (s)),..., y(a m (s)) + z n (a m (s))) σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))]dw(s) 2 H. Since e funcions f, σ, g and I i, i =, 2,..., p, are coninuous, ence lim n E (Nz n )() (Nz)() 2 H. is implies a e mapping N is coninuous on B r. Now we sow a N maps bounded se ino bounded ses in B r. Le z B r en we ave E (Nz)() 2 H ˆM, for ( i, i+ ], i =,, 2,..., p. en for [, ], we ave E (Nz)() 2 H 3E S α()[φ() + g(y + z)] 2 H α ( s)[ f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))]ds 2 H α ( s)[σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))]dw(s) 2 H, 3 M 2 S [r + M ] + 3 M 2 2α 2α α 2 M M 2 2α M 4.

9 Mod Nadeem e al. / Mild soluions for For ( i, i+ ], i =, 2,..., p, en we ave E (Nz)() 2 H 3E S α( i )[y( i ) + z( i ) + I i (y( i ) + z( i ))] 2 H i i α ( s)[ f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))ds 2 H α ( s)[σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))]dw(s) 2 H 3 M S 2[r + M 2] + 3 M 2 2α α 2 M M 2 2α 2α M 4 = ˆM. Iroves a N maps bounded se ino bounded ses in B r for all sub inerval ( i, i+ ], i =, 2,..., p. Finally, we sow a N maps bounded se ino equi-coninuous ses in B r. le l, l 2 ( i, i+ ], i l < l 2 i+, i =,, 2,..., p, z B r, we obain for [, ] E (Nz)(l 2 ) (Nz)(l ) 2 H 3E [S α(l 2 ) S α (l )][φ + g(y + z)] 2 H [ α (l 2 s) α (l s)][ f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))]ds 2 H + 3E [ α (l 2 s) α (l s)] [σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))]dw(s) 2 H, 3[M + r]e [S α (l 2 ) S α (l )] 2 H + 3M 3E +3M 4 E [ α (l 2 s) α (l s)] 2 H. [ α (l 2 s) α (l s)] 2 H For ( i, i+ ], i =, 2,..., p, we ave E (Nz)(l 2 ) (Nz)(l ) 2 H 3E [S α(l 2 i ) S α (l i )][y( i ) + z( i ) + I i (y( i ) + z( i ))] 2 H i [ α (l 2 s) α (l s)][ f (s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))]ds 2 H + 3E i [ α (l 2 s) α (l s)] [σ(s, y s + z s, y(a (s)) + z(a (s)),..., y(a m (s)) + z(a m (s)))]dw(s) 2 H, 3[M 2 + r]e [S α (l 2 i ) S α (l i )] 2 H + 3M 3E +3M 4 E [ α (l 2 s) α (l s)] 2 H. i i [ α (l 2 s) α (l s)] 2 H Since α () and S α () are srongly coninuous is implies a lim l2 l [S α (l 2 i ) S α (l i )] 2 H = and lim l2 l [ α (l 2 i ) α (l i )] 2 H = is implies a N is equi-coninuous on all subinervals ( i, i+ ], i =, 2,..., p. us by Arzela -Ascoli eorem, i follows a N is a compac operaor. Hence N is compleely coninuous operaor. erefore, by Scauder fixed poin eorem, e operaor N as a fixed poin, wic in urns implies a (.)-(.3) as a leas one soluion on [, ]. is complees e proof of e eorem. 4 Example

10 286 Mod Nadeem e al. / Mild soluions for... Example 4.. Consider e following nonlocal impulsive fracional parial differenial equaion of e form u(, x) + m k= q 2 q u(, x) = u(, x) + H(, x, s )Q y2 25 (u(s, x), u(a (s),..., u(a m (s)))ds [ ] dw() + V(, x, s )Q 2 (u(s, x), u(a 25 (s),..., u(a m (s)))ds, (4.5) d u(, ) = u(, π) =,, (4.6) c k u(x, k ) = φ(, x), (, ], x [, π], (4.7) u( i )(x) = 9 i q i ( i s)u(s, x)ds, x [, π], (4.8) were q q is Capuo s fracional derivaive of order < q <, < < 2 < < n are prefixed numbers, φ B. Le H = L 2 [, π] and define e operaor A : D(A) H H by Aω = ω wi e domain D(A) := {ω X : ω, ω are absoluely coninuous, ω H, ω() = = ω(π)}. en Aω = n= n2 2 (ω, ω n )ω n, ω D(A), were ω n (x) = π sin(nx), n N is e orogonal se of eigenvecors of A. I is well known a A is e infiniesimal generaor of an analyic semigroup (()) in H and is given by ()ω = e n2 (ω, ω n )ω n, for all ω H, and every >. n= e subordinaion principle of soluion operaor (eorem 3. in [3]) implies a A is e infiniesimal generaor of a soluion operaor {S α ()}. Since S α () is srongly coninuous on [, ), by uniformly bounded eorem, ere exiss a consan M >, suc a S α () L(H) M, for [, ]. Le (s) = e 2s, s < en l = (s)ds = 2 and define φ B = (s) sup φ(θ) L 2ds. θ [s,] Hence for (, φ) [, ] B, were φ(θ)(x) = φ(θ, x), (θ, x) (, ] [, π]. Se u()(x) = u(, x), f (, φ, u(a ()),..., u(a m ())))(x) = 25 σ(, φ, u(a ()),..., u(a m ())))(x) = 25 I i (φ)(x) = q 9 i ( θ)φ(θ)(x)dθ, g(x) = m c k u(x, k ). k= H(, x, θ)q (φ(θ, u(a ()),..., u(a m ()))(x))dθ, V(, x, θ)q 2 (φ(θ, u(a ()),..., u(a m ()))(x))dθ, en wi ese seings e equaions (4.5)-(4.8) can be wrien in e absrac form of equaions (.)-(.3). Furer we ave ere L f = 25, L σ = 25, L = 9, =, l = 2, M =, M S = 5 and m = 2. In is formulaion of e problem we can verify e assumpions of eorem (3.2). We ge e value of condiion in eorem (3.2) as Θ =.73 <. is implies a ere exiss a unique mild soluion u on [, ]. Example 4.2. Here we consider e following non-rivial problem u(, x) + m k= q 2 q u(, x) = y 2 u(, x) + e + e 25 + e H(, x, s )[Q (u(s, x), u(a (s),..., u(a m (s))) + 7 ]ds 25 + e V(, x, s )[Q 2 (u(s, x), u(a (s),..., u(a m (s))) + ]dw(s) 7 (4.9) u(, x) = u(, π) =,, (4.) c k u(x, k ) = φ(, x), (, ], x [, π], (4.) u = 2 = sin( 9 u( 2, x) ),, x π, (4.2)

11 Mod Nadeem e al. / Mild soluions for were q (, ). In e perspecive of Example we se f (, φ, u(a ()),..., u(a m ())))(x) = σ(, φ, u(a ()),..., u(a m ())))(x) = e 25 + e e 25 + e H(, x, θ)[q (φ(θ, u(a ()),..., u(a m ()))(x)) + 7 ]dθ, V(, x, θ)[q 2 (φ(θ, u(a ()),..., u(a m ()))(x)) + 7 ]dθ. en wi ese seings e equaions (4.9)-(4.2) can be wrien in e absrac form of equaions (.)-(.3). Hence e our problem (4.9)-(4.2) ave a unique mild soluion on [, ]. References [] D. Araya and C. Lizama, Almos auomorpic mild soluions o fracional differenial equaions, Nonlinear Anal. MA, 69()(29), [2] P. Balasubramaniam, J. Y. Park and A. V. A. Kumar, Exisence of soluions for semilinear neural socasic funcional differenial equaions wi nonlocal condiions, Nonlinear Anal., 7(29), [3] E. Bazlekova, Fracional Evoluion Equaions in Banac Spaces, in: Universiy Press Faciliies, Eindoven Universiy of ecnology, 2. [4] D. Buguna, Exisence, uniqueness and regulariy of soluion o semilinear nonlocal funcional differenial problems, Nonlinear Anal., 57 (24), [5] A. Cauan and J. Dabas, Exisence of mild soluions for impulsive fracional-order semilinear evoluion equaions wi nonlocal condiions, Elec. J. of Diff. Equ., 2(7)(2), -. [6] A. Cauan and J. Dabas, Local and global exisence of mild soluion o an impulsive fracional funcional inegro-differenial equaion wi nonlocal condiion, Commun Nonli. Sci Numer Simu., 9(24), [7] A. Cauan, J. Dabas and M. Kumar, Inegral boundary-value problem for impulsive fracional funcional differenial equaions wi infinie delay, Elec. J. of Diff. Equ., 22(229)(22), -3. [8] J. Dabas and A. Cauan, Exisence and uniqueness of mild soluion for an impulsive neural fracional inegro-differenial equaion wi infinie delay, Ma. and Comp. Model., 57(23), [9] J. Dabas and G. R. Gauam, Impulsive neural fracional inegro-differenial equaions wi sae dependen delays and inegral condiions, Elec. J. of Diff. Equ., 23(273)(23), -3. [] K. Deng, Exponenial decay of soluions of semilinear parabolic equaions wi nonlocal iniial condiions, J. of Ma. anal. and appl., 79(993), [] G. R. Gauam and J. Dabas, Mild soluion for fracional funcional inegro-differenial equaion wi no insananeous impulse, Malaya J. of Maemaik, 2(3)(24), [2] M. Haase, e funcional calculus for secorial operaors, Operaor eory, Advances and Applicaions, vol. 69, Birkauser-Verlag, Basel, 26. [3] D. N. Keck and M. A. McKibben, Funcional inegro-differenial socasic evoluion equaions in ilber spaces, J. of Appl. Ma. and So. Anal., 6(2)(23), 4-6. [4] A. A. Kilbas, H. M. Srivasava and J. J. rujillo, eory and Applicaions of Fracional Differenial Equaions, Elsevier Science B.V, Amserdam, 26. [5] V. Laksmikanam, eory of fracional differenial equaions, Nonlinear Anal., 69(28), [6] C. Li, J. Sun and R. Sun, Sabiliy analysis of a class of socasic differenial delay equaions wi nonlinear impulsive effecs, J. of e Franklin Insiue, 347(2), [7] A. Lin, Y. Ren and N. Xia, On neural impulsive socasic inegro-differenial equaions wi infinie delays via fracional operaors, Ma. and Comp. Model., 5(2),

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