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Nonlinear Analysis 81 (2013) 70–86

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Nonlinear Analysis

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Existence of solutions for nonlinear fractional stochasticdifferential equations

R. Sakthivel a, P. Revathi b, Yong Ren c,∗

a Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Koreab Department of Mathematics, Anna University, Regional Center, Coimbatore-641 047, Indiac Department of Mathematics, Anhui Normal University, Wuhu 241000, China

a r t i c l e i n f o

Article history:Received 10 April 2012Accepted 18 October 2012Communicated by S. Carl

Keywords:Existence resultFractional stochastic differential equationResolvent operators

a b s t r a c t

The fractional stochastic differential equations have wide applications in various fieldsof science and engineering. This paper addresses the issue of existence of mild solutionsfor a class of fractional stochastic differential equations with impulses in Hilbert spaces.Using fractional calculations, fixed point technique, stochastic analysis theory andmethodsadopted directly from deterministic fractional equations, new set of sufficient conditionsare formulated and proved for the existence of mild solutions for the fractional impulsivestochastic differential equation with infinite delay. Further, we study the existenceof solutions for fractional stochastic semilinear differential equations with nonlocalconditions. Examples are provided to illustrate the obtained theory.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Fractional dynamical equations have played a central role in the modeling of anomalous relaxation and diffusionprocesses. The fact that fractional derivatives introduce a convolution integral with a power-law memory kernel makesthe fractional differential equations an important one to describe memory effects in complex systems [1]. The increasinginterest of fractional equations ismotivated by their applications in various fields of science such as physics, fluidmechanics,viscoelasticity, heat conduction in materials with memory, chemistry and engineering [2–5]. Hilfer [6,7] showed thattime fractional derivatives are equivalent to infinitesimal generators of generalized time fractional evolutions that arisein the transition from microscopic to macroscopic time scales. Also, it is shown that this transition from the ordinary timederivative to the fractional time derivative arises in different physical problems [8]. Further, many different applications offractional calculus are presented in [1]. By considering some special cases of the fractional generalized Langevin equation, anew class of closed analytic expressions for the mean square displacement of single file diffusion has been obtained in [9],and also Gaussian models of retarded and accelerated anomalous diffusion are considered in [10]. Sandev et al. [11] studiedanalytically a generalized fractional Langevin equation and also general formulas for calculation of variances and the meansquare displacement are derived. Tomovski [12] proved the existence and uniqueness of solutions to Cauchy type problemsfor fractional differential equationswith composite fractional derivative operator on a finite interval of the real axis in spacesof summable functions.

The work of Yong Ren is supported by the National Natural Science Foundation of China (Nos 10901003 and 11126238), the Distinguished YoungScholars of Anhui Province (No 1108085J08), the Key Project of Chinese Ministry of Education (No 211077) and the Anhui Provincial Natural ScienceFoundation (No 10040606Q30).∗ Corresponding author.

E-mail addresses: [email protected], [email protected] (Y. Ren).

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R. Sakthivel et al. / Nonlinear Analysis 81 (2013) 70–86 71

On the other hand, the stochastic differential equations have attracted great interest due to its applications in variousfields of science and engineering. There are many interesting results on the theory and applications of stochastic differentialequations, (see [13–16] and the references therein). Chang et al. [17] investigated the existence of square-mean almostautomorphic mild solutions for a stochastic differential equation in a real separable Hilbert space by applying a compositiontheorem together with Schauder’s fixed point theorem. Ren and Sun [18] studied the existence of solutions for a class ofsecond-order neutral stochastic evolution equations with infinite delay in which the initial value belongs to the abstractspace. The existence and uniqueness of quadratic mean almost periodic mild solutions for a class of stochastic differentialequations in a real separable Hilbert space by employing the contraction mapping principle and an analytic semigroup oflinear operators has been discussed [19]. However, only few papers deal with the existence result for stochastic fractionalsystems. Cui and Yan [20] studied the existence of mild solutions for a class of fractional neutral stochastic integro-differential equations with infinite delay in Hilbert spaces by means of Sadovskii’s fixed point theorem. The existence anduniqueness for a class of fractional stochastic delay and evolution differential equations has been established in [21,22].

In particular, differential equations with impulsive conditions constitute an important field of research due to theirnumerous applications in ecology, medicine biology, electrical engineering and other areas of science. Many physicalphenomena in evolution processes are modeled as impulsive fractional differential equations and existence results for suchequations have been studied by several authors [23,24]. Feckan [5] studied the existence of a solution for a class of impulsivedifferential equations with fractional derivative. The existence of mild solutions for a class of impulsive fractional partialsemilinear differential equations has been discussed in [23]. The existence results for a class of impulsive fractional ordersemilinear evolution equations with infinite delay has been reported in [25]. Moreover, it is known that the developmentof the theory of functional differential equations with infinite delay depends on a choice of a phase space. The existenceof a mild solution for fractional semilinear differential equations with infinite delay has been discussed in [26]. Wanget al. [27] studied the solvability and optimal controls for a class of fractional integrodifferential evolution systems withinfinite delay in Banach spaces. The stochastic differential equations with infinite delay have become important in recentyears as mathematical models of phenomena in both physical and social sciences [28,18]. Wei and Wang [28] considered aclass of stochastic functional differential equations with infinite delay in which the initial value belongs to the phase space.The existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with Poissonjumps and infinite delay has been investigated in [29].

Since impulsive effects also widely exist in fractional stochastic differential systems, it is important and necessaryto discuss the qualitative properties for stochastic fractional equations with impulsive perturbations and infinite delay.However, to the authors’ knowledge, no result has been reported on the existence problem of impulsive fractional stochasticdifferential equations with infinite delay and the aim of this paper is to fill this gap.

Motivated by this consideration, in this paper, first we shall discuss the existence of solutions for a class of impulsivefractional stochastic differential equationswith infinite delay by using some appropriate fixed point theorems and evolutionsystem theory. Moreover, the nonlocal conditions give a better description in applications than standard ones, the topic ofnonlocal problems has been studied extensively for the existence of fractional differential equations [30,31]. As indicatedin [32] and references therein, the Cauchy problem with nonlocal initial condition can be applied in physics with bettereffect than the classical Cauchy problem with traditional initial conditions. Taking this into account, in this paper we alsostudy the existence of mild solutions for semilinear fractional stochastic differential equations with nonlocal conditions.

2. Preliminaries and basic properties

Let H,K be two separable Hilbert spaces and L(K,H) be the space of bounded linear operators from K into H. Forconvenience, we will use the same notation ∥ · ∥ to denote the norms in H,K and L(K,H), and use (·, ·) to denote the innerproduct of H and K without any confusion. Let (Ω,F , Ftt≥0, P) be a complete filtered probability space satisfying thatF0 contains all P-null sets of F . W = (Wt)t≥0 be a Q -Wiener process defined on (Ω,F , Ftt≥0, P) with the covarianceoperator Q such that TrQ < ∞. We assume that there exists a complete orthonormal system ekk≥1 in K, a boundedsequence of nonnegative real numbers λk such that Qek = λkek, k = 1, 2, . . . , and a sequence of independent Brownianmotions βkk≥1 such that

(w(t), e)K =

∞k=1

λk(ek, e)Kβk(t), e ∈ K, t ≥ 0.

Let L02 = L2(Q

12 K,H) be the space of all Hilbert–Schmidt operators from Q

12 K to H with the inner product ⟨ϕ,ψ⟩L0

2=

Tr[ϕQψ∗].We consider the following impulsive fractional stochastic differential equations with infinite delay in the form

Dαt x(t) = Ax(t)+ f (t, xt , B1x(t))+ σ(t, xt , B2x(t))dw(t)dt

, t ∈ J = [0, T ], T > 0, t = tk,

1x(tk) = Ik(x(t−k )), k = 1, 2, . . . ,m,x(t) = φ(t), φ(t) ∈ Bh,

(2.1)

72 R. Sakthivel et al. / Nonlinear Analysis 81 (2013) 70–86

where Dαt is the Caputo fractional derivative of order α, 0 < α < 1; x(·) takes the value in the separable Hilbert space H;A : D(A) ⊂ H → H is the infinitesimal generator of an α-resolvent family Sα(t)t≥0. The history xt : (−∞, 0] → H, xt(θ) =

x(t + θ), θ ≤ 0, belongs to an abstract phase space Bh; f : J × Bh × H → H and σ : J × Bh × L02 → H are appropriate

functions to be specified later; Ik : Bh → H, (k = 1, 2, . . . ,m), are appropriate functions. The terms B1x(t) and B2x(t)are given by B1x(t) =

t0 K(t, s)x(s)ds and B2x(t) =

t0 P(t, s)x(s)ds respectively, where K , P ∈ C(D,R+) are the set of all

positive continuous functions on D = (t, s) ∈ R2: 0 ≤ s ≤ t ≤ T . Here 0 = t0 ≤ t1 ≤ · · · ≤ tm ≤ tm+1 = T ,1x(tk) =

x(t+k ) − x(t−k ), x(t+

k ) = limh→0 x(tk + h) and x(t−k ) = limh→0 x(tk − h) represent the right and left limits of x(t) at t = tk,respectively. The initial data φ = φ(t), t ∈ (−∞, 0] is an F0-measurable, Bh-valued random variable independent of wwith finite second moments.

Now, we present the abstract space phase Bh. Assume that h : (−∞, 0] → (0,∞) with l = 0−∞

h(t)dt < ∞ acontinuous function. We define the abstract phase space Bh by

Bh =

φ : (−∞, 0] → H, for any a > 0, (E|φ(θ)|2)1/2 is bounded and measurable

function on [−a, 0] with φ(0) = 0 and 0

−∞

h(s) sups≤θ≤0

(E|φ(θ)|2)1/2ds < ∞

.

If Bh is endowed with the norm

∥φ∥Bh =

0

−∞

h(s) sups≤θ≤0

(E|φ(θ)|2)1/2ds, φ ∈ Bh,

then (Bh, ∥ · ∥Bh) is a Banach space [29,18].Now we consider the space

Bb =

x : (−∞, T ] → H such that x|Jk ∈ C(Jk,H) and there exist

x(t+k ) and x(t−k )with x(tk) = x(t−k ), x0 = φ ∈ Bh, k = 1, 2, . . . ,m,

where x|Jk is the restriction of x to Jk = (tk, tk+1], k = 0, 1, 2, . . . ,m. The function ∥ · ∥Bb to be a seminorm in Bb, it isdefined by

∥x∥Bb = ∥φ∥Bh + sups∈[0,T ]

(E∥x(s)∥2)1/2, x ∈ Bb.

We recall the following lemma [29].

Lemma 2.1. Assume that x ∈ Bb; then for t ∈ J, xt ∈ Bh. Moreover,

l(E∥x(t)∥2)1/2 ≤ l sups∈[0,t]

(E∥x(s)∥2)1/2 + ∥x0∥Bh ,

where l = 0−∞

h(s)ds < ∞.

A two parameter function of the Mittag-Leffler type is defined by the series expansion

Eα,β(z) =

∞k=0

zk

Γ (αk + β)=

12π i

C

µα−βeµ

µα − zdµ, α, β ∈ C, ℜ(α) > 0, (2.2)

where C is a contour which starts and ends at −∞ and encircles the disc |µ| ≤ |z|12 counter clockwise. For short, Eα(z) =

Eα,1(z). It is an entire function which provides a simple generalization of the exponent function: E1(z) = ez and the cosinefunction: E2(z2) = cos h(z), E2(−z2) = cos(z), and plays a vital role in the theory of fractional differential equations. Themost interesting properties of the Mittag-Leffler functions are associated with their Laplace integral

∞

0e−λt tβ−1Eα,β(ωtα)dt =

λα−β

λα − ω, Reλ > ω

1α , ω > 0, (2.3)

and for more details see [4].

Definition 2.2 ([33]). A closed and linear operator A is said to be sectorial if there are constantsω ∈ R, θ ∈ [π2 , π], M > 0,

such that the following two conditions are satisfied:

(1) ρ(A) ⊂

(θ,ω) = λ ∈ C : λ = ω, | arg(λ− ω)| < θ,(2) ∥R(λ, A)∥ ≤

M|λ−ω|

, λ ∈

(θ,ω).


Definition 2.3 ([2]). Let A be a closed and linear operator with the domain D(A) defined in a Banach space X . Let ρ(A) bethe resolvent set of A. We say that A is the generator of an α-resolvent family if there exist ω ≥ 0 and a strongly continuousfunction Sα : R+ → L(X), whereL(X) is a Banach space of all bounded linear operators fromX intoX and the correspondingnorm is denoted by ∥ · ∥, such that λα : Reλ > ω ⊂ ρ(A) and

(λα I − A)−1x =

∞

0eλtSα(t)xdt, Reλ > ω, x ∈ X, (2.4)

where Sα(t) is called the α-resolvent family generated by A.

Definition 2.4 ([25]). Let A be a closed linear operator with the domain D(A) defined in a Banach space X and α > 0. Wesay that A is the generator of a solution operator if there exist ω ≥ 0 and a strongly continuous function Sα : R+ → L(X)such that λα : Reλ > ω ⊂ ρ(A) and

λα−1(λα I − A)−1x =

∞

0eλtSα(t)xdt, Reλ > ω, x ∈ X, (2.5)

where Sα(t) is called the solution operator generated by A.

The concept of the solution operator is closely related to the concept of a resolvent family [25]. For more details onα-resolvent family and solution operators, we refer the reader to [25,34] and the references therein.

Definition 2.5 ([35]). The Caputo derivative of orderα for a function f : [0,∞) → R, which is at least n-times differentiablecan be defined as

Dαt f (t) =1

Γ (n − α)

t

0(t − s)n−α−1f (n)(s)ds = In−α f (n)(t), (2.6)

for n − 1 ≤ α < n, n ∈ N . If 0 < α ≤ 1, then

Dαt f (t) =1

Γ (1 − α)

t

0(t − s)−α f (1)(s)ds. (2.7)

Obviously, the Caputo derivative of a constant is equal to zero. The Laplace transform of the Caputo derivative of orderα > 0is given as

LDαt f (t); λ = λα f (λ)−

n−1k=0

λα−k−1f (k)(0); n − 1 ≤ α < n.

Lemma 2.6 ([25]). If f satisfies the uniform Holder condition with the exponent β ∈ (0, 1] and A is a sectorial operator, thenthe unique solution of the Cauchy problem

Dαt x(t) = Ax(t)+ f (t, xt , Bx(t)), t > t0, t0 ≥ 0, 0 < α < 1, (2.8)

x(t) = φ(t), t ≤ t0, (2.9)

is given by

x(t) = Tα(t − t0)(x(t+0 ))+

t

t0Sα(t − s)f (s, xs, Bx(s))ds, (2.10)

where

Tα(t) = Eα,1(Atα) =1

2π i

Br e

λt λα−1

λα − Adλ, (2.11)

Sα(t) = tα−1Eα,α(Atα) =1

2π i

Br e

λt 1λα − A

dλ, (2.12)

hereBr denotes the Bromwich path; Sα(t) is called the α-resolvent family and Tα(t) is the solution operator generated by A.

Next, we mention the statement of Krasnoselskii’s fixed point theorem [25].

Theorem 2.7. Let B be a nonempty closed convex of a Banach space (X, ∥ · ∥). Suppose that P and Q map B into X such that(i) Px + Qy ∈ B whenever x, y ∈ B;(ii) P is compact and continuous;(iii) Q is a contraction mapping.Then there exists z ∈ B such that z = Pz + Qz.


3. Stochastic fractional equations with infinite delay and impulses

In this section, we first establish the existence of mill solutions to nonlinear fractional stochastic equation (2.1). Moreprecisely, we will formulate and prove sufficient conditions for the existence of solutions to (2.1) with infinite delay andimpulses. In order to establish the results, we impose the following conditions.

(H1) If α ∈ (0, 1) and A ∈ Aα(θ0, ω0), then for any x ∈ H and t > 0 we have ∥Tα(t)∥ ≤ Meωt and ∥Sα(t)∥ ≤

Ceωt(1 + tα−1), ω > ω0. Thus, we have

∥Tα(t)∥ ≤ MT and ∥Sα(t)∥ ≤ tα−1MS,

where MT = sup0≤t≤T ∥Tα(t)∥, and MS = sup0≤t≤T Ceωt(1 + t1−α) (for more details, see [23]).(H2) There exist µ1, µ2 > 0 such that

E∥f (t, γ , x)− f (t, ψ, y)∥2H ≤ µ1∥γ − ψ∥

2Bh

+ µ2E∥x − y∥2H.

(H3) There exist ν1, ν2 > 0 such that

E∥σ(t, γ , x)− σ(t, ψ, y)∥2L0

2≤ ν1∥γ − ψ∥

2Bh

+ ν2E∥x − y∥2H.

(H4) For each k = 1, 2, . . . ,m, there exist nk > 0 such that

E∥Ik(x)− Ik(y)∥2H ≤ nkE∥x − y∥2

H, for all x, y ∈ H.

Now, we present the definition of mild solutions for the system (2.1) based on the paper [25].

Definition 3.1. An Ft-adapted stochastic process x : (−∞, T ] → H is called a mild solution of the system (2.1) ifx0 = φ ∈ Bh satisfying x0 ∈ L0

2(Ω,H) and the following conditions hold.

(i) x(t) is Bh-valued and the restriction of x(·) to the interval (tk, tk+1], k = 1, 2, . . . ,m is continuous.(ii) for each t ∈ J, x(t) satisfies the following integral equation

x(t) =

φ(t), t ∈ (−∞, 0], t

0Sα(t − s)f (s, xs, B1x(s))ds

+

t

0Sα(t − s)σ (s, xs, B2x(s))dw(s), t ∈ [0, t1],

Tα(t − t1)(x(t−1 )+ I1(x(t−1 )))+

t

t1Sα(t − s)f (s, xs, B1x(s))ds

+

t

t1Sα(t − s)σ (s, xs, B2x(s))dw(s), t ∈ (t1, t2],

...

Tα(t − tm)(x(t−m )+ Im(x(t−m )))+

t

tmSα(t − s)f (s, xs, B1x(s))ds

+

t

tmSα(t − s)σ (s, xs, B2x(s))dw(s), t ∈ (tm, T ].

(3.1)

(iii) 1x|t=tk = Ik(x(t−k )), k = 1, . . . ,m the restriction of x(·) to the interval [0, T ) \ t1, . . . , tm is continuous.

Now, we are in a position to state the existence theorem. Our first theorem is based on the Banach contraction principle.

Theorem 3.2. Assume that the conditions (H1)–(H4) hold. If A ∈ Aα(θ0, ω0), then the system (2.1) has a unique mild solutionprovided that

max1≤i≤m

4M2

T (1 + ni)+ 4M2S T

2α

1α2(µ1l + µ2B∗

1)+1

T (2α − 1)(ν1l + ν2B∗

2)

< 1, (3.2)

where B∗

1 = supt∈[0,t] t0 K(t, s)ds < ∞ and B∗

2 = supt∈[0,t] t0 P(t, s)ds < ∞.


Proof. Define the operatorΠ : Bb → Bb by

(Πx)(t) =

φ(t), t ∈ (−∞, 0], t

0Sα(t − s)f (s, xs, B1x(s))ds

+

t

0Sα(t − s)σ (s, xs, B2x(s))dw(s), t ∈ [0, t1],

Tα(t − t1)(x(t−1 )+ I1(x(t−1 )))+

t

t1Sα(t − s)f (s, xs, B1x(s))ds

+

t

t1Sα(t − s)σ (s, xs, B2x(s))dw(s), t ∈ (t1, t2],

...

Tα(t − tm)(x(t−m )+ Im(x(t−m )))+

t

tmSα(t − s)f (s, xs, B1x(s))ds

+

t

tmSα(t − s)σ (s, xs, B2x(s))dw(s), t ∈ (tm, T ].

(3.3)

For φ ∈ Bh, define

g(t) =

φ(t), t ∈ (−∞, 0],0, t ∈ J

then g0 = φ. Next we define the function

z(t) =

0, t ∈ (−∞, 0],z(t), t ∈ J

for each z ∈ C(J,R) with z(0) = 0. If x(·) satisfies (3.1) then x(t) = g(t) + z(t) for t ∈ J , which implies xt = gt + zt fort ∈ J and the function z(·) satisfies

z(t) =

t

0Sα(t − s)f (s, gs + zs, B1(g(s)+ z(s)))ds

+

t

0Sα(t − s)σ (s, gs + zs, B2(g(s)+ z(s)))dw(s), t ∈ [0, t1],

Tα(t − t1)((g(t−1 )+ z(t−1 ))+ I1(g(t−1 )+ z(t−1 )))+

t

t1Sα(t − s)f (s, gs + zs, B1(g(s)+ z(s)))ds

+

t

t1Sα(t − s)σ (s, gs + zs, B2(g(s)+ z(s)))dw(s), t ∈ (t1, t2],

...

Tα(t − tm)((g(t−m )+ z(t−m ))+ Im(g(t−m )+ z(t−m )))+

t

tmSα(t − s)f (s, gs + zs, B1(g(s)+ z(s)))ds

+

t

tmSα(t − s)σ (s, gs + zs, B2(g(s)+ z(s)))dw(s), t ∈ (tm, T ].

(3.4)

Set B0b = z ∈ Bb, such that z0 = 0 and for any z ∈ B0

b , we have

∥z∥B0b

= ∥z0∥Bh + supt∈J(E∥z(t)∥2)1/2 = sup

t∈J(E∥z(t)∥2)1/2,

thus (B0b , ∥ · ∥B0

b) is a Banach space.


Define the operator Ψ : B0b → B0

b by

(Ψ z)(t) =

t


+

t

0Sα(t − s)σ (s, gs + zs, B2(g(s)+ z(s)))dw(s), t ∈ [0, t1],

Tα(t − t1)(z(t−1 )+ I1(z(t−1 )))+

t


+

t


...

Tα(t − tm)(z(t−m )+ Im(z(t−m )))+

t


+

t


(3.5)

In order to prove the existence result, it is enough to show that Ψ has a unique fixed point. Let z, z∗∈ B0

b , then for allt ∈ [0, t1], we have

E(Ψ z)(t)− (Ψ z∗)(t)

2H ≤ 2E

t

0Sα(t − s)[f (s, gs + zs, B1(g(s)+ z(s)))

− f (s, gs + z∗

s , B1(g(s)+ z∗(s)))]ds

2

H

+ 2E

t

0Sα(t − s)[σ(s, gs + zs, B2(g(s)+ z(s)))

− σ(s, gs + z∗

s , B2(g(s)+ z∗(s)))]dw(s)

2

H

≤ 2 t

0∥Sα(t − s)∥ds

t

0∥Sα(t − s)∥E

f (s, gs + zs, B1(g(s)+ z(s)))

− f (s, gs + z∗

s , B1(g(s)+ z∗(s)))2

Hds

+ 2 t

0∥Sα(t − s)∥2E

σ(s, gs + zs, B2(g(s)+ z(s)))

− σ(s, gs + z∗

s , B2(g(s)+ z∗(s)))2

L02

ds

≤ 2M2S

t

0(t − s)α−1ds

t

0(t − s)α−1

[µ1∥zs − z∗

s ∥2Bh

+µ2E∥B1(g(s)+ z(s))− B1(g(s)+ z∗(s))∥2H]ds

+ 2M2S

t

0(t − s)2(α−1)

[ν1∥zs − z∗

s ∥2Bh

+ ν2E∥B2(g(s)+ z(s))

− B2(g(s)+ z∗(s))∥2H]ds

≤ 2M2STα

α

t

0(t − s)α−1

[µ1l sup E∥z(s)− z∗(s)∥2H

+µ2B∗

1 sup E∥z(s)− z∗(s)∥2H]ds

+ 2M2S

t

0(t − s)2(α−1)

[ν1l sup E∥z(s)− z∗(s)∥2H + ν2B∗


≤ 2M2ST 2α

α2[µ1l + µ2B∗

1]∥z − z∗∥2B0

b+ 2M2

ST 2α−1

2α − 1[ν1l + ν2B∗

2]∥z − z∗∥2B0

b


= 2M2S T

2α

1α2(µ1l + µ2B∗

1)+1

T (2α − 1)(ν1l + ν2B∗

2)

∥z − z∗

∥2B0

b.

For t ∈ (t1, t2], we have

E(Ψ z)(t)− (Ψ z∗)(t)

2H ≤ 4∥Tα(t − t1)∥2E∥z(t−1 )− z∗(t−1 )∥

2H + 4∥Tα(t − t1)∥2E∥I1(z(t−1 ))− I1(z∗(t−1 ))∥

2H

+ 4E

t

t1Sα(t − s)[f (s, gs + zs, B1(g(s)+ z(s)))

− f (s, gs + z∗

s , B1(g(s)+ z∗(s)))]ds

2

H

+ 4E

t

t1Sα(t − s)[σ(s, gs + zs, B2(g(s)+ z(s)))

− σ(s, gs + z∗

s , B2(g(s)+ z∗(s)))]dw(s)

2

H

≤ 4M2T [E∥z(t−1 )− z∗(t−1 )∥

2H + E∥I1(z(t−1 ))− I1(z∗(t−1 ))∥

2H]

+ 4M2S

t

t1(t − s)α−1ds

t

t1(t − s)α−1

[µ1∥zs − z∗

s ∥2Bh


+ 4M2S

t

t1(t − s)2(α−1)

[ν1∥zs − z∗

s ∥2Bh

+ ν2E∥B2(g(s)+ z(s))

− B2(g(s)+ z∗(s))∥2H]ds

≤ 4M2T (1 + n1) sup E∥z(s)− z∗(s)∥2

H

+ 4M2STα

α

t

t1(t − s)α−1


+µ2B∗


+ 2M2S

t

t1(t − s)2(α−1)

[ν1l sup E∥z(s)− z∗(s)∥2H

+ ν2B∗


≤

4M2

T (1 + n1)+ 4M2S T

2α

1α2(µ1l + µ2B∗

1)

+1

T (2α − 1)(ν1l + ν2B∗

2)

∥z − z∗

∥2B0

b.

Similarly, when t ∈ (ti, ti+1], i = 2, . . . ,m, we get

E(Ψ z)(t)− (Ψ z∗)(t)

2H ≤

4M2

T (1 + ni)+ 4M2S T

2α

1α2(µ1l + µ2B∗

1)+1

T (2α − 1)(ν1l + ν2B∗

2)

∥z − z∗

∥2B0

b.

Thus for all t ∈ [0, T ], we have

E(Ψ z)(t)− (Ψ z∗)(t)

2H ≤ max

1≤i≤m

4M2

T (1 + ni)+ 4M2S T

2α

1α2(µ1l + µ2B∗

1)

+1

T (2α − 1)(ν1l + ν2B∗

2)

∥z − z∗

∥2B0

b.

Hence, Ψ is a contraction map by the condition (3.2) and therefore it has a unique fixed point z ∈ B0b , which is a mild

solution of (2.1) on (−∞, T ]. The proof is complete.


The second result is established using Krasnoselskii’s fixed point theorem. Now, we make the following assumptions.

(H5) f : J × Bh × H → H is continuous and there exist two continuous functions µ1, µ2 : J → (0,∞) such that

E∥f (t, ψ, x)∥2H ≤ µ1(t)∥ψ∥

2Bh + µ2(t)E∥x∥2

H, (t, ψ, x) ∈ J × Bh × H,

where µ∗

1 = sups∈[0,t] µ1(s) and µ∗

2 = sups∈[0,t] µ2(s).(H6) σ : J × Bh × L0

2 → H is continuous and there exist two continuous functions ν1, ν2 : J → (0,∞) such that

E∥σ(t, ψ, x)∥2L0

2≤ ν1(t)∥ψ∥

2Bh + ν2(t)E∥x∥2

H, (t, ψ, x) ∈ J × Bh × L02,

where ν∗

1 = sups∈[0,t] ν1(s) and ν∗

2 = sups∈[0,t] ν2(s).(H7) The function Ik : H → H is continuous and there existsΛ > 0 such that

Λ = max1≤k≤m, x∈Bq

E∥Ik(x)∥2H,

where Bq = y ∈ B0b , ∥y∥2

B0b

≤ q, q > 0.

The set Bq is clearly a bounded closed convex set in B0b for each q and for each y ∈ Bq. From Lemma 2.1, we have

∥yt + zt∥2Bh

≤ 2(∥yt∥2Bh

+ ∥zt∥2Bh)

≤ 4l2 sup

t∈[0,t]E∥y(t)∥2

H + ∥y0∥2Bh

+ 4

l2 sup

t∈[0,t]E∥y(t)∥2

H + ∥z0∥2Bh

≤ 4

∥φ∥

2Bh

+ l2q. (3.6)

Theorem 3.3. Suppose that the assumptions (H1)–(H3) and (H5)–(H7) are satisfied with

q ≥ 4M2T (q +Λ)+ 4M2

S T2αλ1

α2+

λ2

T (2α − 1)

(3.7)

and 2M2S T

2α

1α2(µ1l + µ2B∗

1)+1

T (2α − 1)(ν1l + ν2B∗

2)

< 1. (3.8)

Then the impulsive stochastic fractional differential equation (2.1) has at least one mild solution on (−∞, T ].

Proof. LetΘ1 : Bq → Bq andΘ2 : Bq → Bq be defined as

(Θ1z)(t) =

0, t ∈ [0, t1],Tα(t − t1)(Z(t−1 )+ I1(Z(t−1 ))), t ∈ (t1, t2],...,

Tα(t − tm)(Z(t−m )+ Im(Z(t−m ))), t ∈ (tm, T ]

(3.9)

and

(Θ2z)(t) =

t


+

t

0Sα(t − s)σ (s, gs + zs, B2(g(s)+ z(s)))dw(s), t ∈ [0, t1], t


+

t


... t


+

t


(3.10)

In order to use Theorem 2.7 we will verify that Θ1 is compact and continuous while Θ2 is a contraction operator. For thesake of convenience, we divide the proof into several steps.


Step 1. We show thatΘ1z +Θ2z∗∈ Bq for z, z∗

∈ Bq. For t ∈ [0, t1], we have

E(Θ1z)(t)+ (Θ2z∗)(t)

2H ≤ 2E

t

0Sα(t − s)f (s, gs + z∗

s , B1(g(s)+ z∗(s)))ds2

H

+ 2E t

0Sα(t − s)σ (s, gs + z∗

s , B2(g(s)+ z∗(s)))dw(s)2

H

≤ 2 t


t

0∥Sα(t − s)∥E

f (s, gs + z∗

s , B1(g(s)+ z∗(s)))2

H ds

+ 2 t

0∥Sα(t − s)∥2E

σ(s, gs + z∗

s , B2(g(s)+ z∗(s)))2

L02ds

≤ 2M2S

t

0(t − s)α−1ds

t

0(t − s)α−1

[µ1(s)∥gs + z∗

s ∥2Bh

+µ2(s)E∥B1(g(s)+ z∗(s))∥2H]ds

+ 2M2S

t

0(t − s)2(α−1)

[ν1(s)∥gs + z∗

s ∥2Bh

+ ν2(s)E∥B2(g(s)+ z∗(s))∥2H]ds

≤ 2M2STα

α

t

0(t − s)α−1

[4µ∗

1(∥φ∥2Bh

+ l2q)+ µ∗

2B∗

1 sups∈[0,T ]

E∥z∗(s)∥2H]ds

+ 2M2S

t

0(t − s)2(α−1)

[4ν∗

1 (∥φ∥2Bh

+ l2q)+ ν∗

2B∗

2 sups∈[0,T ]

E∥z∗(s)∥2H]ds

≤ 2M2ST 2α

α2[4µ∗

1(∥φ∥2Bh

+ l2q)+ µ∗

2B∗

1q] + 2M2S

T 2α−1

2α − 1× [4ν∗

1 (∥φ∥2Bh

+ l2q)+ ν∗

2B∗

2q]

= 2M2S T

2αλ1

α2+

λ2

T (2α − 1)

.

Thus, by the condition (3.7), we obtain ∥Θ1z +Θ2z∗∥B0

b≤ q.

Similarly, for t ∈ (ti, ti+1], i = 1, 2, . . . ,m, we have the estimate

E(Θ1z)(t)+ (Θ2z∗)(t)

2H ≤ 4∥Tα(t − ti)∥2E∥z(t−i )∥

2H + 4∥Tα(t − ti)∥2E∥Ii(z(t−i ))∥

2H

+ 4E t

tiSα(t − s)f (s, gs + z∗

s , B1(g(s)+ z∗(s)))ds2

H

+ 4E t

tiSα(t − s)σ (s, gs + z∗

s , B2(g(s)+ z∗(s)))dw(s)2

H

≤ 4M2T

∥z∥2

B0b

+ E∥Ii(z(t−i ))∥2H

+ 4M2

S T2αλ1

α2+

λ2

T (2α − 1)

≤ 4M2

T (q +Λ)+ 4M2S T

2αλ1

α2+

λ2

T (2α − 1)

≤ q.

This implies that ∥Θ1z +Θ2z∗∥B0

b≤ qwith λ1 = 4µ∗

1(∥φ∥2Bh

+ l2q)+µ∗

2B∗

1q and λ2 = 4ν∗

1 (∥φ∥2Bh

+ l2q)+ ν∗

2B∗

2q. Hence,we getΘ1z +Θ2z∗

∈ Bq.

Step 2. The mapΘ1 is continuous on Bq.Let zn∞n=1 be a sequence in Bq with lim zn → z ∈ Bq. Then for t ∈ (ti, ti+1], i = 0, 1, . . . ,m, we have

E(Θ1zn)(t)− (Θ1z)(t)

2H ≤ 2∥Tα(t − ti)∥2 E∥zn(t−i )− z(t−i )∥

2H + E∥Ii(zn(t−i ))− Ii(z(t−i ))∥

2H

.

Since the functions Ii, i = 1, 2, . . . ,m are continuous, hence limn→∞ E∥Θ1zn −Θ1z∥2= 0 which implies that the mapping

Θ1 is continuous on Bq.


Step 3.Θ1 maps bounded sets into bounded sets in Bq.Let us prove that for q > 0 there exists a r > 0 such that for each z ∈ Bq, we have E ∥(Θ1z)(t)∥2

H ≤ r fort ∈ (ti, ti+1], i = 0, 1, . . . ,m. Now, we have

E ∥(Θ1z)(t)∥2H ≤ 2∥Tα(t − ti)∥2 E∥z(t−i )∥

2H + E∥Ii(z(t−i ))∥

2H

≤ 2M2

T (q +Λ) = r,

which proves the desired result.

Step 4. The mapΘ1 is equicontinuous.Let u, v ∈ (ti, ti+1], ti ≤ u < v ≤ ti+1, i = 0, 1, . . . ,m, z ∈ Bq, we obtain

E ∥(Θ1z)(v)− (Θ1z)(u)∥2H ≤ 2∥Tα(v − ti)− Tα(u − ti)∥2 E∥z(t−i )∥

2H + E∥Ii(z(t−i ))∥

2H

≤ 2(q +Λ)∥Tα(v − ti)− Tα(u − ti)∥2.

Since Tα is strongly continuous and it allows us to conclude that limu→v ∥Tα(v − ti)− Tα(u − ti)∥2= 0, which implies that

Θ1(Bq) is equicontinuous. Finally, combining Step 1 to Step 4 together with Ascoli’s theorem, we conclude that the operatorΘ1 is compact.

Now, it only remains to show that the mapΘ2 is a contraction mapping. Let z, z∗∈ Bq and t ∈ (ti, ti+1], i = 0, 1, . . . ,m,

we have

E(Θ2z)(t)− (Θ2z∗)(t)

2H ≤ 2E

t

tiSα(t − s)[f (s, gs + zs, B1(g(s)+ z(s)))

− f (s, gs + z∗

s , B1(g(s)+ z∗(s)))]ds

2

H

+ 2E

t

tiSα(t − s)[σ(s, gs + zs, B2(g(s)+ z(s)))

− σ(s, gs + z∗

s , B2(g(s)+ z∗(s)))]dw(s)

2

H

≤ 2 t

ti∥Sα(t − s)∥ds

t

ti∥Sα(t − s)∥E

f (s, gs + zs, B1(g(s)+ z(s)))

− f (s, gs + z∗

s , B1(g(s)+ z∗(s)))

2

H

ds

+ 2 t

ti∥Sα(t − s)∥E

σ(s, gs + zs, B2(g(s)+ z(s)))

− σ(s, gs + z∗

s , B2(g(s)+ z∗(s)))

2

L02

ds

≤ 2M2S

t

ti(t − s)α−1ds

t

0(t − s)α−1

[µ1∥zs − z∗

s ∥2Bh


+ 2M2S

t

ti(t − s)2(α−1)

[ν1∥zs − z∗

s ∥2Bh

+ ν2E∥B2(g(s)+ z(s))

− B2(g(s)+ z∗(s))∥2H]ds

≤ 2M2STα

α

t

ti(t − s)α−1


+µ2B∗


+ 2M2S

t

ti(t − s)2(α−1)

[ν1l sup E∥z(s)− z∗(s)∥2H


+ ν2B∗


≤ 2M2S T

2α

1α2(µ1l + µ2B∗

1)+1

T (2α − 1)(ν1l + ν2B∗

2)

∥z − z∗

∥2B0

b.

By the condition (3.8), we obtain that Θ2 is a contraction mapping. Hence, by Krasnoselskii’s fixed point theorem we canconclude that the problem (2.1) has at least one solution on (−∞, T ]. This completes the proof of the theorem.

Example 3.4. In this section, we consider an example to illustrate ourmain theorem.We examine the existence of solutionsfor the following fractional stochastic partial differential equation of the form

Dqt u(t, x) =

∂2

∂x2u(x, t)+

t

−∞

H(t, x, s − t)Q (u(s, x))ds +

t

0k(s, t)e−u(s,x)ds

+

t

−∞

V (t, x, s − t)U(u(s, x))ds +

t

0p(s, t)e−u(s,x)ds

dβ(t)dt

,

x ∈ [0, π], t ∈ [0, b], t = tku(t, 0) = 0 = u(t, π), t ≥ 0,u(t, x) = φ(t, x), t ∈ (−∞, 0], x ∈ [0, π],

1u(ti)(x) =

t

−∞

qi(ti − s)u(s, x)ds, x ∈ [0, π],

(3.11)

where β(t) is a standard cylindrical Wiener process in H defined on a stochastic space (Ω,F , P, Ft); Dqt is the Caputo

fractional derivative of order 0 < q < 1; 0 < t1 < t2 < · · · < tn < b are prefixed numbers; H,Q , V and U are continuous;φ ∈ Bh.

Let H = L2([0, π]) with the norm ∥ · ∥. Define A : H → H by Az = z ′′ with the domain D(A) = z ∈ H, z, z ′ areabsolutely continuous, z ′′

∈ H, and z(0) = z(π) = 0. Then

Az =

∞n=1

n2(z, zn)zn, z ∈ D(A),

where zn(x) =

2n sin(nx), n ∈ N is the orthogonal set of eigenvectors of A. It is well known that A is the infinitesimal

generator of an analytic semigroup (T (t))t≥0 in H and is given by

T (t)z =

∞n=1

e−n2t(z, zn)zn, for all z ∈ H, t > 0.

It follows from the above expressions that (T (t))t≥0 is a uniformly bounded compact semigroup, so that, R(λ, A) = (λ−A)−1

is a compact operator for all λ ∈ ρ(A) i.e. A ∈ Aα(θ0, ω0). Let h(s) = e2s, s < 0, then l = 0−∞

h(s)ds =12 and define

∥φ∥Bh =

0

−∞

h(s) sups≤θ≤0

(E|φ(θ)|2)1/2ds.

Hence for (t, φ) ∈ [0, b] × Bh, where φ(θ)(z) = φ(θ, z), (θ, z) ∈ (−∞, 0] × [0, π]. Put u(t) = u(t, ·), that is u(t)(x) =

u(t, x). Define f : J × Bh × L2([0, π]) → L2([0, π]) and σ : J × Bh × L02 → L2([0, π]) as follows:

f (t, φ, B1u(t))(x) =

0

−∞

H(t, x, θ)Q (φ(θ))(x)dθ + B1u(t)(x),

σ (t, φ, B2u(t))(x) =

0

−∞

V (t, x, θ)U(φ(θ))(x)dθ + B2u(t)(x),

where B1u(t)(x) = t0 k(s, t)e−u(s,x)ds and B2u(t)(x) =

t0 p(s, t)e−u(s,x)ds. Then, with the above settings the considered

equation (3.11) can be written in the abstract form of Eq. (2.1). All conditions of Theorem 3.2 are now fulfilled, so we deducethat the system (3.11) has a mild solution on (−∞, T ].

4. Stochastic fractional equations with nonlocal conditions

The initial conditions usually represent the measurements at initial time. However, in various real world problems, itis possible to require more measurements at some instances in addition to standard initial data and, therefore, the initialconditions changed to nonlocal conditions. Balasubramaniam et al. [36] studied the existence of solutions for semilinear


neutral stochastic functional differential equations with nonlocal conditions. In the last few years, there has been anincreasing interest in study of fractional differential equations involving nonlocal conditions [34]. In this section, we areconcerned with the existence and uniqueness of mild solutions for semilinear stochastic fractional differential equationswith nonlocal conditions in the form

Dαt x(t)+ Ax(t) = f (t, x(t))+ σ(t, x(t))dw(t)dt

, t ∈ J = [0, T ], 0 < α < 1,

x(0)+ g(x) = x0, (4.1)

where A;D(A) ⊂ H → H;Dαt and w are defined as in Eq. (2.1). Further, the functions f , σ and g are given functions to bedefined later. The collection of all strongly-measurable, square-integrableH-valued random variables, denoted by L2(Ω,H),is a Banach space equipped with norm ∥x(·)∥L2 = (E ∥ x(·;w ∥

2H))

12 , where E is defined by E(h) =

Ωh(w)dP, w ∈ Ω .

Let C(J, L2(Ω,H)) be the Banach space of all continuous maps from J into L2(Ω,H) satisfying the condition supt∈J E∥x(t)∥2

< ∞.Let H2 be the closed subspace of all continuous processes x that belong to the space C(J, L2(Ω,H)) consisting of

Ft-adapted measurable processes such that the F0-adapted processes x(0)with a seminorm ∥ · ∥H2 in H2 be defined by

∥x∥H2 =

supt∈J

∥x(t)∥2L2

1/2

.

It is easy to verify that H2 furnished with the norm topology as defined above, is a Banach space.Now, we define the definition of the mild solution of (4.1).

Definition 4.1. A continuous stochastic process x : J → H is called a mild solution of (4.1) if the following conditions hold.

(i) x(t) is measurable and Ft-adapted.(ii) x(0)+ g(x) = x0.(iii) x satisfies the following equation

x(t) = Tα(t)(x0 − g(x))+

t

0Sα(t − s)f (s, x(s))ds +

t

0Sα(t − s)σ (s, x(s))dw(s),

where Tα(t) = Eα,1(Atα) =1

2π i

Br eλt λα−1

λα−Adλ, Sα(t) = tα−1Eα,α(Atα =1

2π i

Br eλt 1λα−Adλ)here Br denotes the

Bromwich path, Sα(t) is called the α-resolvent family and Tα(t) is the solution operator generated by −A.

Further, we assume the following conditions.

(H8) There exists a constant Lg > 0 such that E∥g(x)− g(y)∥2H ≤ LgE∥x − y∥2

H.(H9) The nonlinear maps f : J × H → H and σ : J × H → L0

2 are continuous and there exist constants Lf and Lσ such that

E∥f (t, x)− f (t, y)∥2H ≤ Lf E∥x − y∥2

H, E∥σ(t, x)− σ(t, y)∥2L0

2≤ Lσ E∥x − y∥2

H

for all x, y ∈ H and t ∈ [0, T ].(H10) f ∈ C(J × H,H), g ∈ C(H,H), and σ ∈ C(J × H,L0

2). Moreover, there exists a constant C1 > 0 such that for x ∈ H,

E∥g(x)∥2H ≤ C1

and for s ∈ J, x ∈ Br there exist two continuous functionsLf , Lσ : J → (0,∞) such that

E∥f (t, x)∥2H ≤Lf (t)φ(E∥x∥2

H), E∥σ(t, x)∥2L0

2≤Lσ (t)ψ(E∥x∥2

H).

(H11) T

0ξ(s)ds ≤

∞

c

dsφ(s)+ ψ(s)

,

where

ξ(t) = max

4M2

s Tα

αtα−1Lf (t), 4M2

s t2(α−1)Lσ (t) , c = 4M2

T

E∥x0∥2

H + C1.

Theorem 4.2. Under the assumptions (H1), (H8) and (H9), the fractional stochastic equation (4.1) has a unique mild solutionon J provided that

3M2T Lg + 3M2

S T2α

Lfα2

+Lσ

T (2α − 1)

≤ 1. (4.2)


Proof. Let λ : H2 → H2 be the operator defined by

(λx)(t) = Tα(t)(x0 − g(x))+

t


t

0Sα(t − s)σ (s, x(s))dw(s).

It can be seen that λ maps H2 into itself. Let us show that λ is a contraction on H2. For t ∈ J , it follows from the assump-tions (H1), (H8) and (H9) that

E∥λx(t)− λy(t)∥2H ≤ 3∥Tα(t)∥2E∥g(x)− g(y)∥2

H + 3 t


×

t

0∥Sα(t − s)∥E∥f (s, x(s))− f (s, y(s))∥2

Hds

+ 3 t

0∥Sα(t − s)∥2E∥σ(s, x(s))− σ(s, y(s))∥2

L02ds

≤ 3M2T LgE∥x − y∥2

H + 3M2STα

α

t

0(t − s)α−1Lf E∥x − y∥2

Hds

+ 3M2S

t

0(t − s)2(α−1)Lσ E∥x − y∥2

Hds

≤

3M2

T Lg + 3M2S Lf

T 2α

α2+ 3M2

S LσT 2α−1

2α − 1

E∥x − y∥2

H

=

3M2

T Lg + 3M2S T

2α

Lfα2

+Lσ

T (2α − 1)

E∥x − y∥2

H.

Hence by the condition (4.2), λ is a contractionmapping. Therefore, by the Banach contraction principle λ has a unique fixedpoint. The proof is complete.

Next our second result is based on the following Schaefer’s fixed point theorem.

Theorem 4.3. Let K be a closed convex subset of a Banach space X such that 0 ∈ K . Let P : K → K be a completely continuousmap. Then the set x ∈ K ; x = νPx; 0 ≤ ν ≤ 1 is unbounded or P has a fixed point.

Theorem 4.4. Assume that (H1), (H8), (H10) and (H11) hold. Then the fractional stochastic equation (4.1) has at least one mildsolution on [0, T ].

Proof. Define the operator λ : H2 → H2 as in Theorem 4.2. Now, we have to prove that λ is a completely continuousoperator. Note that λ is well defined in H2. For the sake of convenience, we divide the proof into several steps.Step 1. We prove that λ is continuous.

Let xn∞n=0 be a sequence in H2 such that xn → x in H2. Since the functions f , g and σ are continuous,

limn→∞

E∥λxn(t)− λx(t)∥2H = 0

in H2 for every t ∈ J . This implies that the mapping λ is continuous on H2.Step 2. Next we prove that λmaps bounded sets into bounded sets in H2.

To prove that for any r > 0, there exists a γ > 0 such that for x ∈ Br = x ∈ H2 : E∥x∥2H ≤ r, we have E∥λx∥2

H ≤ γ . Forany x ∈ Br , t ∈ J , we have

E∥λx(t)∥2H ≤ 4∥Tα(t)∥2E∥x0∥2

H + 4∥Tα(t)∥2E∥g(x)∥2H + 4

t


×

t

0∥Sα(t − s)∥E∥f (s, x(s))∥2

Hds

+ 4 t

0∥Sα(t − s)∥2E∥σ(s, x(s))∥2

L02ds

≤ 4M2T r + 4M2

T C1 + 4M2STα

αφ(r)

t

0(t − s)α−1Lf (s)ds

+ 4M2Sψ(r)

t

0(t − s)2(α−1)Lσ (s)ds

= γ , t ∈ J.


Step 3. We show that λmaps bounded sets into equicontinuous sets of Br .Let 0 < u < v ≤ T , for each x ∈ Br , we have

E∥λx(v)− λx(u)∥2H ≤ 6∥Tα(v)− Tα(u)∥2E∥x0∥2

H + 6∥Tα(v)− Tα(u)∥2E∥g(x)∥2H

+ 6E u

0[Sα(v − s)− Sα(u − s)]f (s, x(s))ds

2H

+ 6E v

uSα(v − s)f (s, x(s))ds

2H

+ 6E u

0[Sα(v − s)− Sα(u − s)]σ(s, x(s))dw(s)

2H

+ 6E v

uSα(v − s)σ (s, x(s))dw(s)

2H.

Therefore we obtain

E∥λx(v)− λx(u)∥2H ≤ 6(r + C1)∥Tα(v)− Tα(u)∥2

+ 6 u

0∥Sα(v − s)− Sα(u − s)∥ds

×

u

0∥Sα(v − s)− Sα(u − s)∥E∥f (s, x(s))∥2

Hds

+ 6 v

u∥Sα(v − s)∥ds

v

u∥Sα(v − s)∥E∥f (s, x(s))∥2

Hds

+ 6 u

0∥Sα(v − s)− Sα(u − s)∥2E∥σ(s, x(s))∥2

L02ds

+

v

u∥Sα(v − s)∥2E∥σ(s, x(s))∥2

L02ds

≤ 6(r + C1)∥Tα(v)− Tα(u)∥2

+ 6φ(r) u

0∥Sα(v − s)− Sα(u − s)∥ds

×

u

0∥Sα(v − s)− Sα(u − s)∥Lf (s)ds

+ 6M2S(v − u)α

αφ(r)

v

u(v − s)α−1Lf (s)ds

+ 6ψ(r) u

0∥Sα(v − s)− Sα(u − s)∥2Lσ (s)ds

+ 6M2Sψ(r)

v

u(v − s)2(α−1)Lσ (s)ds.

Since Tα(t) and Sα(t) are strongly continuous, ∥Tα(v) − Tα(u)∥ → 0 and ∥Sα(v − s) − Sα(u − s)∥ → 0 as u → v. Thus,from the above inequality we have limu→v E∥λx(v) − λx(u)∥2

H = 0. Thus, the set λx, x ∈ Br is equicontinuous. Finally,combining Step 1 to Step 3 with Ascoli’s theorem, we conclude that the operator λ is compact.Step 4. Next, we show that the set

N = x ∈ H2 such that x = qλx(t) for some 0 < q < 1

is bounded. Let x ∈ N then x(t) = qλx(t) for some 0 < q < 1. Then for each t ∈ J , we have

x(t) = qTα(t)(x0 − g(x))+

t


t

0Sα(t − s)σ (s, x(s))dw(s)

which implies that

E∥x(t)∥2H ≤ 4∥Tα(t)∥2E∥x0∥2

H + 4∥Tα(t)∥2E∥g(x)∥2H

+ 4 t


t

0∥Sα(t − s)∥E∥f (s, x(s))∥2

Hds


+ 4 t

0∥Sα(t − s)∥2E∥σ(s, x(s))∥2

L02ds

≤ 4M2T E∥x0∥2

H + 4M2T C1

+ 4M2STα

α

t

0(t − s)α−1Lf (s)φ(E∥x(s)∥2

H)ds

+ 4M2S

t

0(t − s)2(α−1)Lσ (s)ψ(E∥x(s)∥2

H)ds.

Consider the function µ(t) defined by

µ(t) = supE∥x(s)∥2H, 0 ≤ s ≤ t, 0 ≤ t ≤ T .

µ(t) ≤ 4M2T [E∥x0∥2

H + C1] + 4M2STα

α

t

0(t − s)α−1Lf (s)φ(µ(s))ds + 4M2

S

t

0(t − s)2(α−1)Lσ (s)ψ(µ(s))ds.

Denoting by v(t) the right hand side of the last inequality, we have v(0) = c = 4M2T [E∥x0∥2

H + C1], µ(t) ≤ v(t), t ∈ J .Moreover,

v′(t) = 4M2STα

αtα−1Lf (t)φ(µ(t))+ 4M2

S t2(α−1)Lσ (t)ψ(µ(t))

≤ 4M2STα

αtα−1Lf (t)φ(v(t))+ 4M2

S t2(α−1)Lσ (t)ψ(v(t)),

or equivalently by (H11), we have v(t)

v(0)

dsφ(s)+ ψ(s)

≤

T

0ξ(s)ds <

∞

c

dsφ(s)+ ψ(s)

, 0 ≤ t ≤ T .

This inequality implies that there is a constant k such that v(t) ≤ k, t ∈ J , and hence,µ(t) ≤ k, t ∈ J . Furthermore, we get∥x(t)∥2

≤ µ(t) ≤ v(t) ≤ k, t ∈ J . By the Schaefer’s fixed point theorem, we deduce that λ has a fixed point on J which is asolution to (4.1). This completes the proof of the theorem.

Example 4.5. Now, we present an example to illustrate Theorem 4.4. Consider the fractional partial stochastic differentialequation in the following form

Dαt [z(t, x)] =∂2

∂x2[z(t, x)] + h(t, z(t, x))+ ψ(t, z(t, x))

dβ(t)dt

, 0 ≤ t ≤ b, 0 ≤ x ≤ π,

z(t, 0) = z(t, π) = 0,

z(0, x)+

pi=0

π

0K(x, y)z(t, y)dy = z0(x), 0 ≤ x ≤ π, (4.3)

where p is a positive integer, b ≤ π, 0 < t0 < t1, . . . < tp < b, z0(x) ∈ H = L2([0, π]), K(x, y) ∈ L2([0, π] × [0, π]) andDαt is the Caputo fractional derivative of order 0 < α < 1.

To study this system, we take H = L2([0, π]) and let A be the operator defined by Ay = y′′ with domain D(A) = y ∈

H : y, y′ are absolutely continuous y′′∈ H, y(0) = y(π) = 0. It is well known that A is the infinitesimal generator of an

analytic semigroup (T (t))t≥0 in H. Furthermore, A has a discrete spectrum with eigenvalues of the form −n2, n ∈ N and

corresponding normalized eigenfunctions are given by xn(z) =

2πsin(nz). In addition xn : n ∈ N is an orthonormal basis

for H,

T (t)y =

∞n=1

e−n2t(y, xn)xn, for all y ∈ H, and every t > 0.

From these expressions it follows that (T (t))t≥0 is a uniformly bounded compact semigroup, so that R(λ, A) = (λ− A)−1 isa compact operator for all λ ∈ ρ(A) i.e. A ∈ Aα(θ0, ω0).

To represent the above fractional system (4.3) into the abstract form of (4.1), we introduce the functions f : J × H →

H, σ : J × H → L02 and g : H → H by f (t, z)(x) = h(t, z(x)), σ (t, z)(x) = ψ(t, z(x)) and g(w) =

pi=0 Kw(ti), where

K(z)(x) = π0 K(x, y)z(y)dy. Thus, f , σ and g satisfy the assumption of Theorem 4.4. Hence, by Theorem 4.4 the system

(4.3) has a mild solution on [0, T ].

Note 4.6. The differential inclusion system is considered as a generalization of the system described by differentialequations. Ning and Liu [37] discussed the existence of mild solutions of a class of impulsive neutral stochastic evolution


inclusions inHilbert space in the casewhere the right hand side is convex or nonconvex-valuedbyusing fixedpoint theoremsformultivaluedmappings and evolution system theory. The existence of solutions of nonlinear neutral stochastic differentialinclusions with infinite delay in a Hilbert space has been reported in [38]. Upon making some appropriate assumption onfunctions f and σ , one can establish the approximate controllability of fractional stochastic differential inclusions withnonlocal conditions by adapting the techniques and ideas established in this paper and suitably introducing the techniqueof single valued maps defined in [39].

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