Mediterr. J. Math. DOI 10.1007/s00009-016-0737-1 c Springer International Publishing 2016 

Existence and Dependence Results for Semilinear Functional Stochastic Differential Equations with Infinite Delay in a Hilbert Space Lahcene Guedda

and Paul Raynaud de Fitte

Abstract. Using techniques of measures of noncompactness, we prove existence, uniqueness, and dependence results for semilinear stochastic differential equations with infinite delay on an abstract phase space of Hilbert space valued functions defined axiomatically, where the unbounded linear part generates a noncompact semigroup and the nonlinear parts satisfies some growth condition and, with respect to the second variable, a condition weaker than the Lipschitz one. These results are applied to a stochastic parabolic partial differential equation with infinite delay. Mathematics Subject Classification. Primary 34K50; Secondary 47H08, 35K58, 60H15. Keywords. Stochastic functional differential equation, infinite delay, measure of noncompactness, condensing map.

1. Introduction Stochastic differential equations with delay are equations where the history of the solution process is considered, which makes them more suitable and effective in modeling many economical, biological, and engineering problems than equations without delay, see [7,17,19]. This explains why such equations have received an extensive attention by many researchers in recent years. In the case of stochastic equations with finite delay, the literature is rich, see the survey of Ivanov–Kazmerchuk–Swishchuk [13]. Stochastic equations with infinite delay are more complicated (see [4,15,20,21,23]), the most general approach, initiated by Hale and Kato [10,16], consists of taking the initial condition in an abstract seminormed space phase space. This kind of space is defined axiomatically, of course, a concrete choice of this space depends on the particular problem under investigation.

L. Guedda, P. Raynaud de Fitte

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In this paper, we aim to establish existence, uniqueness, and dependence results for stochastic differential equations with infinite delay in a Hilbert space, where the paths of the initial condition lie in an abstract phase space defined axiomatically, the unbounded linear part generates a noncompact semigroup, and the nonlinear parts satisfy some growth condition and, with respect to the second variable, a condition weaker than the Lipschitz one. As a first step, using techniques of measures of noncompactness, as in [3,5,14,21], we first define a measure of noncompactness adapted to our formulation of the problem; then, we prove that the sequence of approximating solutions constructed via Tonelli’s scheme has a subsequence which converges to the unique desired solution (Theorem 3.2). As a second step, we prove that this solution depends continuously on the initial condition (Theorem 4.1). We illustrate our results in the last section by showing how they can be applied to a parabolic stochastic partial differential equation with infinite delay.

2. Main Setting In all this paper, p > 2 is a fixed number, T > 0 is a fixed time horizon, F = (Ω, F, (Ft )t∈[0,T ] , P) is a stochastic basis satisfying the usual conditions, H and U are two separable Hilbert spaces, and W is a (possibly cylindrical) (Ft )t∈[0,T ] –Brownian motion on U. We denote by L, the space of Hilbert– Schmidt operators from U to H. The Phase Space B For any σ ∈ R, for any function x : (−∞, σ + T ] → H, and for every t ∈ (σ, σ + T ], xt denotes the function from (−∞, 0] to H defined by xt (θ) = x(t + θ), −∞ < θ ≤ 0. Throughout this paper, we employ the axiomatic definition of the phase space B introduced by Hale and Kato [16]. We assume that the phase space B is a separable Banach space of functions mapping (−∞, 0] into H endowed with a norm .B satisfying the following axioms. If x : (−∞, σ + T ] → E is continuous on [σ, σ + T ] and xσ ∈ B; then, for every t ∈ [σ, σ + T ], we have (B1) xt ∈ B. (B2) there exists l > 0, such that x(0)H ≤ l x0 B . (B3) xt  ≤ K(t − σ) supσ≤s≤t x(t) + N (t − σ) xσ B , where K, N : [0, +∞) −→ [0, +∞) are independent of x, K is positive and continuous, and N is locally bounded. (B4) the function t → xt is continuous. Let us give two examples of phase spaces satisfying the axioms cited above, see [11] (see also [12, page 20]). 1. The Space Cg Let g(θ), θ ≤ 0, be a positive, continuous function, such that g(θ) → +∞ when θ → −∞. The space Cg is the set of continuous functions ψ : (−∞, 0] →

Semilinear Functional SDEs with Infinite Delay H, such that ψ(θ)/g(θ) has a limit in H as θ → −∞. Set ψB = sup {ψ(θ) /g(θ) : θ ∈ (−∞, 0]} . Thus, Cg is a separable Banach space satisfying Axioms (B1)–(B4). A simple example of such space is given by g(θ) = eγ θ , where γ is a real negative constant. In this situation, l = 1, K(t) = 1, and M (t) = e−γt . 2. The Space Lγ For a real constant γ > 0, the space Lγ is the set of measurable functions ψ, such that eγθ ψ(θ) is integrable over (−∞, 0]. Set  0 ψLγ = ψ(0) + eγθ ψ(θ) dθ. −∞

Thus, Lγ is a separable Banach space satisfying axioms (B1)–(B4). Spaces of Processes For any t ∈ [0, T ], and for any separable Banach space E, we denote by Ncp (F, [0, t]; E), the separable Banach space of continuous adapted E-valued processes X defined on the time interval [0, t], such that   p p XNcp (F,[0,t];E) := E sup X(s)E < +∞, 0≤s≤t

p

and we denote by N (F, (−∞, t], H), the space of H-valued processes X : is in (−∞, t) → H, such that X0 ∈ Lp (Ω, F0 , P; B), and the restriction X Ncp (F, [0, t]; H). We endow N p (F, (−∞, t], H) with   p p XN p (F,(−∞,t],H) := E X0 B + X

[0,t]

the norm    p

[0,t] Nc (F,[0,t];H)

.

Remark 2.1. No measurability assumption is made for the variable X(t) for any particular t < 0; thus, X is not necessarily a stochastic process on (−∞, T ] in the “classical” sense. This is because we are not interested in the values taken by X(t) for particular t < 0, only by the global history function Xt for t ∈ [0, T ]. On the other hand, the process (Xt )0≤t≤T is a B-valued (Ft )-adapted B-valued stochastic process, as shows the next lemma. Lemma 2.2. For any t ∈ [0, T ], the space N p (F, (−∞, t], H) is a separable Banach space. Furthermore, for any X ∈ N p (F, (−∞, t], H), the process (Xs )0≤s≤t is in Ncp (F, [0, t]; B), and the linear mapping  p N (F, (−∞, t], H) → Ncp (F, [0, t]; B) Π: X → X. is continuous. Proof. The separability of N p (F, (−∞, t], H) is obvious. For the completeness, let (X n ) be a Cauchy sequence in N p (F, (−∞, t], H). Then, (X0n ) converges ) converges in Ncp (F, [0, t]; H) to a in Lp (Ω, F0 , P; B) to a limit Y , and (X n [0,t]

limit Z. By (B2), the sequence (X n (0)) converges in Lp (Ω, F0 , P; H) to Y (0), thus Y (0) = Z(0) a.s., which means that (X n ) converges for .N p (F,(−∞,t],H)

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to the element X of N p (F, (−∞, t], H) which coincides with Y on (−∞, 0] and with Z on [0, t]. Let X ∈ N p (F, (−∞, t], H). Since X0 takes its values in B, we deduce by (B1) that Xs is B-valued for every s ∈ [0, t]. Furthermore, By (B4), the trajectories s → Xs are continuous on [0, t]. Let s ∈ [0, t] be fixed. Let E = {x : (−∞, s] → H; x0 ∈ B and x is continuous on [0, s]} . Let us endow the vector space E with the norm xE = x0 B + supr∈[0,s] x(r)H . Then, the space E is separable (and complete by (B2), but we do not need this fact). By (B3), we have xs B ≤ K(s) sup x(r) + N (s) x0 B , r∈[0,s]

which shows that the linear mapping x → xs from E to B is continuous, thus measurable. Now, by definition of N p (F, (−∞, t], H), the variable X0 : Ω → B : Ω → C([0, s], H) is Fs -measurable. is F0 -measurable, and the variable X [0,s]

Thus, X

(−∞,s]

: Ω → E is Fs -measurable. This proves that Xs : Ω → B is

Fs -measurable. Thus (Xs )0≤s≤t is adapted. Finally, the functions K and N of Axiom (B3) are bounded on [0, t]. Using Axiom (B3), we have, for every X ∈ N p (F, (−∞, t], H)   p

Π(X)Ncp (F,[0,t];B) = E

p

sup Xs B

s∈[0,t]

⎧   ⎪ ⎪ p ≤ 2p−1 ⎪ ⎪ max K p (s) E sup X(s)H ⎩ s∈[0,t] s∈[0,t] ⎫ ⎪ p ⎪ p + sup N (s) E (X0 B ) ⎪ ⎪ ⎭ s∈[0,t] p

≤ α XN p (F,(−∞,t],H) ,

  where α = 2p−1 maxs∈[0,t] K p (s) + sups∈[0,t] N p (s) < +∞. This simultaneously proves that (Xs )0≤s≤t is in Ncp (F, [0, t]; B) and that Π is continuous.  Let us now consider the problem  dX(t) ∈ AX(t) + f (t, Xt ) dt + g(t, Xt ) dW (t), X0 = ϕ,

t ∈ [0, T ],

(1)

where the initial condition ϕ is an H-valued stochastic process {ϕ(θ), θ ≤ 0} is an F0 -measurable B-valued random variable, A is a linear operator on H generating a C0 semigroup, and f and g are mappings defined on [σ, σ +T ]×B with values in H and L, respectively, which are continuous in the second variable, and satisfy an assumption which is more general than the usual Lipschitz one. More precisely, we assume the following hypotheses: (HS) A is the generator of a C0 semigroup (S(t))t≥0 on H. (HFG) f : [0, T ] × B → H and g : [0, T ] × B → L are measurable mappings which satisfy the following conditions:

Semilinear Functional SDEs with Infinite Delay (i) There exists a constant Cgrowth > 0, such that, for all (t, u) ∈ [0, T ] × B, f (t, u) ≤ Cgrowth (1 + uB ) g(t, u) ≤ Cgrowth (1 + uB ). (ii) For all (t, u, v) ∈ [0, T ] × B × B, p

p

p

p

f (t, u) − f (t, v)) ≤ L(t, u − vB ) g(t, x) − g(t, y)) ≤ L(t, u − vB ), where L : [0, T ] × [0, +∞] → [0, +∞] is a given measurable mapping, such that (a) for every t ∈ [0, T ], the mapping L(t, .) is continuous, nondecreasing, and concave, (b) for every measurable mapping, z : [0, T ] → [0, +∞[ and for every constant, K > 0, the following implication holds true:    t L(s, z(s)) ds ⇒ z = 0. (2) (∀t ∈ [0, T ]) z(t) ≤ K 0

In particular, we have L(t, 0) = 0 for all t ∈ [0, T ], thus Hypothesis (HFG)-(ii) entails that, for each t ∈ [0, T ], the mappings f (t, ·) and g(t, ·) are continuous. Such a function L is considered in e.g., [3,5,6,18,21,22,25]. Concrete examples can be found in [24, pages 35–37]. (HI) The initial process ϕ is an F0 -measurable B-valued random variable, with ϕ ∈ Lp (Ω, F0 , P; B). Let us recall a well-known result. Lemma 2.3. ([8, Theorem 1.1] or [9, Proposition 7.3]) Under Hypothesis (HS), there exists a constant CConv , such that, for any predictable process Z ∈ Lp (Ω × [0, T ]; L), we have  s p    t   p  ≤ CConv t(p/2)−1 E S(s − r)Z(r) dW (r) Z(s)L ds. (3) E sup    s≤t

0

Furthermore, the process

. 0

0

S(.−s)Z(s) dW (s) has a continuous modification.

3. Existence Result Definition 3.1. We say that X ∈ N p (F, (−∞, T ], H) is a mild solution to (1) if a.s., for all t ∈ [0, T ] ⎧  t  t ⎨ X(t) = S(t)ϕ(0) + S(t − s)f (s, Xs ) ds + S(t − s)g(s, Xs ) dW (s). 0 0 ⎩ X0 = ϕ. (4) We can now state the main result of this paper.

L. Guedda, P. Raynaud de Fitte

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Theorem 3.2. Assume that the phase space B satisfies the axioms (B1)–(B4). Under Hypothesis (HS), (HFG), and (HI), Eq. (1) has a unique mild solution. Before we give the proof of the main result, we introduce some auxiliary notions and we prove some auxiliary results. Remark that Axiom (B2) means that the linear mapping x → x(0), B → H, is continuous, and, therefore, measurable. Thus, by (HI), ϕ(0) is p F0 -measurable. We also deduce from (B2) that E ϕ(0)H ≤ l E ϕB . Hence, for t ∈ [0, T ], the set D(ϕ, t) = {X ∈ Ncp (F, [0, t]; H), X(0) = ϕ(0) P -a.s.} is a well-defined closed convex subset of Ncp (F, [0, t]; H). For X ∈ D(ϕ, T ), we define the process X[ϕ] = (X[ϕ](t))t≤T by  ϕ(t) if −∞ < t < 0, X[ϕ](t) = X(t) if 0 ≤ t ≤ T. Then, for every t ≤ T , we get a new process X[ϕ]t = (X[ϕ]t (θ))θ≤0 defined by  ϕ(t + θ) if −∞ < θ < −t, X[ϕ]t (θ) = X(t + θ) if −t ≤ θ ≤ 0. It is clear that the process X[ϕ] is in N p (F, (−∞, T ], H). Furthermore, by Lemma 2.2, for any t ∈ [0, T ] and for any X ∈ D(ϕ, t), the process (X[ϕ]s )0≤s≤t is in Ncp (F, [0, t]; B). Let us denote by Φ, the mapping which, with every element X ∈ D(ϕ, T ), associates the process  t S(t)ϕ(0) + S(t − s)f (s, X[ϕ]s ) ds 0  t S(t − s)g(s, X[ϕ]s ) dW (s), 0 ≤ t ≤ T. + 0

Considering Lemma 2.2, it is clear that under conditions (HS), (HFG) and (HI), the operator Φ maps D(ϕ, T ) into itself. Remark 3.3. Each mild solution to (1) has the form X[ϕ](.), for a fixed point X of Φ. We need to study some further properties of Φ. Recall that, if A is a subset of a metric space E, an -net of A in E is a family (xi ) of points of E, such that A ⊂ ∪i Bi , where Bi denotes the open ball of radius  centered on xi . Let Λ be a subset of D(ϕ, T ). For any t ∈ [0, T ], we denote by Λ , the t subset of D(ϕ, s) of restrictions to [0, t] of elements of Λ, and   ΨD (Λ)(t) :=, inf  > 0; Λ t has a finite -net in D(ϕ, t) ,   p ΨNcp (F,[0,t];H) (Λ)(t) := inf  > 0; Λ has a finite -net in Nc (F, [0, t]; H) . t

We have ΨNcp (F,[0,t];H) (Λ) (t) ≤ ΨD (Λ)(t) ≤ 2 ΨNcp (F,[0,t];H) (Λ) (t).

Semilinear Functional SDEs with Infinite Delay For every t ∈ [0, T ], since D(ϕ, t) is a convex closed subset of Ncp (F, [0, t]; B), the mapping ΨD (.)(t) : 2D(ϕ,T ) → R+ is a measure of noncompactness in the sense of the general definition of [3], i.e., ΨD (.)(t) is invariant under passage to the closed convex hull. The nondecreasing mapping ΨD (Λ) : t → ΨD (Λ)(t) is called the measure of noncompactness of Λ in D(ϕ, T ). A subset Λ of D(ϕ, T ) is relatively compact in D(ϕ, T ) if and only if ΨD (Λ)(T ) = 0. Lemma 3.4. Let Λ be a bounded subset of D(ϕ, T ). Assume Hypotheses (HS), (HFG), (HI) and (B1)–(B4). We then have  t p L (s, K p (s) ΨpD (Λ)(s)) ds ΨD (Φ ◦ Λ)(t) ≤ α 0

for some constant α which depends only on T , p, and the semigroup S. Proof. The main arguments are inspired from [3, Lemma 4.2.6]. Since (S(t))t is a C0 semigroup, there exists, for each t ∈ [0, T ], a constant Mt > 0, such that sup S(s) ≤ Mt .

(5)

0≤s≤t

For more details, see e.g. [2, Theorem 1.3.1]). Let Λ be a bounded subset of D(ϕ, T ). Let  > 0. As the function t → Ψ(Λ)(t) is nondecreasing and bounded, there exist at most a finite number of points 0 ≤ t1 ≤ · · · ≤ tn ≤ T , for which Ψ(Λ) makes a jump greater than . Remove these points with their disjoint δ1 -neighborhoods from the segment [0, T ]. Using points βj , j = 1, . . . , m, divide the remaining part into intervals, on which the increment of the function Ψ(Λ) is smaller than , i.e., sup s,t∈[βj−1 ,βj ]

|Ψ(Λ)(s) − Ψ(Λ)(t)| < .

(6)

Now, to get a net of processes with continuous trajectories, surround the points βj , j = 1, . . . , m, by δ2 -neighborhoods, and consider the family Z in D(ϕ, T ) obtained by taking all continuous processes which coincide on each ]βj−1 + δ2 , βj − δ2 [ (2 ≤ j ≤ m) with some element of a finite (Ψ(Λ)(βj ) + )– net of Λ in D(ϕ, βj ) and which have affine trajectories on the complementary segments. Note that, if t1 > 0, we have β1 = 0, and if t1 = 0, we have β1 = δ1 . Thus, in every case, the elements of Z are affine on [0, β1 + δ2 ]. We need them to stay in D(ϕ, T ); therefore, we choose them equal to ϕ(0) at t = 0. The set Z in D(ϕ, T ) is finite. It remains to prove that for every t ∈   1/p t [0, T ], the finite set Φ(Z) is a α 0 L (s, K p (s) ΨpD (Λ)) (s) ds -net of (Φ ◦ Λ)

[0,t]

[0,t]

for some constant α which depends only on T , p MT , and CConv .

Consider a fixed X ∈ Λ. We can find an element Z of Z, such that, for each j = 1, . . . , m dD(ϕ, βj ) (X − Z) := X − ZNcp (βj ) ≤ Ψ(Λ)(βj ) + . For t ∈ ]βj−1 + δ2 , βj − δ2 [, we have, using (6), p

E X(t) − Z(t)H ≤ E

sup

βj−1 +δ2 ≤s≤βj −δ2

p

X(s) − Z(s)H

L. Guedda, P. Raynaud de Fitte

MJOM

p

≤ X − ZNcp (βj )

= dpD(ϕ, βj ) (X − Z) p

≤ (Ψ(Λ)(βj ) + ) p ≤ (Ψ(Λ)(t) + 2) .

(7)

We have, by (B3), for any s ∈ [0, t], p

f (s, X[ϕ]s ) − f (s, Z[ϕ]s ) ≤ L (s, X[ϕ]s − Z[ϕ]s B )   p  p ≤ L s, K (s) sup X(τ ) − Z(τ ) . 0≤τ ≤s

(8)

and p

g(s, X[ϕ]s ) − g(t, Z[ϕ]s ) ≤ L (s, X[ϕ]s − Z[ϕ]s B )   p  p ≤ L s, K (s) sup X(τ ) − Z(τ ) . 0≤τ ≤s

(9)

p

Using (HFG) and the convexity of x → |x| , we obtain p

E sup Φ(X)(τ ) − Φ(Z)(τ ) 0≤s≤t  s  ≤ E sup  S(s − τ )(f (τ, X[ϕ]τ ) − f (τ, Z[ϕ]τ )) dτ  0≤s≤t  s

0

p  + S(s − τ )(g(τ, X[ϕ]τ ) − g(τ, Z[ϕ]τ )) dW (τ )  0  s p   p−1  ≤2 E sup  S(s − τ )(f (τ, X[ϕ]τ ) − f (τ, Z[ϕ]τ )) ds  0≤s≤t

0

 s p    + E sup  S(s − τ )(g(τ, X[ϕ] − g(τ, Z[ϕ] )) dW (τ ) τ τ   0≤s≤t 0   t p ≤ 2p−1 Mtp E f (τ, X[ϕ]τ ) − f (τ, Z[ϕ]τ ) ds 0   t p p/2−1 + CConv T E g(τ, X[ϕ]τ ) − g(τ, Z[ϕ]τ ) ds , 0

where Mt < ∞ satisfies (5). Let us denote   α = 2p−1 MTp + CConv T p/2−1 , (10) J(t) = [0, t] ∩ ((∪1≤i≤n ]ti − δ1 , ti + δ1 [) ∪ (∪1≤j≤m ]βj − δ2 , βj + δ2 [)) , I(t) = [0, t] \ J(t). Using (8) and (9), Axiom (HFG), and the boundedness of Λ and Z in D(ϕ, T ), and taking δ1 and δ2 sufficiently small, we get    p p p dNcp (t) (Φ(X), Φ(Z)) ≤ α E L(s, K (s) sup X(τ )−Z(τ ) ) ds+ . I(t)

0≤τ ≤s

Semilinear Functional SDEs with Infinite Delay We deduce, using the concavity of L(t, .), and (7),  t  p p p dNcp (t) (Φ(X), Φ(Z)) ≤ α L (s, K (s) (Ψ(Λ)(s) + 2) ) ds +  . 0



As  and X are arbitrary, the result follows.

Proof of Theorem 3.2. Let us begin by with the uniqueness. According to Remark 3.3, is it enough to prove that Φ has at most one fixed point. Assume the contrary, i.e., Φ has two fixed points, say X and Y , then following the same line of calculations as above, we get   p E sup Φ(X)(τ ) − Φ(Y )(τ ) 0≤τ ≤t



=E

p



sup X(τ ) − Y (τ )

0≤τ ≤t

p−1



≤2



Mt

t

p

E f (t, X[ϕ]t ) − f (t, Y [ϕ]t ) ds

0

  t p +CConv T p/2−1 E g(t, X[ϕ]t ) − g(t, Y [ϕ]t ) ds 0    t  p ≤α L s, K p (s) E sup X(τ ) − Y (τ ) ds, 0≤τ ≤s

0

where α is set as in (10). Let β = max0≤t≤T K p (t). As L(s, .) is nondecreasing, we have   p β E sup Φ(X)(τ ) − Φ(Y )(τ ) 0≤τ ≤t



≤ αβ

0

t

   p L s, β E sup X(τ ) − Y (τ ) ds. 0≤τ ≤s

p

It results that E(sup0≤τ ≤T X(τ ) − Y (τ ) ) = 0. Hence, for every t ∈ [0, T ], X(t) = Y (t) a.s. Hence, the problem (1) has at most one solution. Let us now prove the existence. We construct a sequence (X n )n≥1 in D(ϕ, T ) of approximating solutions to the equation X = Φ(X) using the Tonelli scheme: for each integer n ≥ 1, and for each t ∈ [0, T ], we define X n (t) by X n (t) = ϕ(0) X n (t) = S(t −  + 0

if 1 )ϕ(0) + n

1 t− n

 0

1 t− n

0≤t≤

1 , n

if t ≥

1 . n

S(t − s)f (s, X n [ϕ]s ) ds

S(t − s)g(s, X n [ϕ]s ) dW (s)

Let us prove that (X n )n≥1 has a subsequence which converges to a solution to (1). Claim 1. The sequence (X n ) is bounded in Ncp (F, [0, T ]; H).

L. Guedda, P. Raynaud de Fitte For 1 ≤ s ≤ T and n ≥ 1, set  υn (s) = E

MJOM

 p

n

sup X (τ )

,

τ ∈[0,s]

λK = 2p−1 sup K p (s),

(11)

λN = 2p−1 sup N p (s),

(12)

s∈[0,T ]

s∈[0,T ]

where the functions K and N are defined in (B3). From (B3), we immediately get   E (X n [ϕ]s ) ≤ K(s) E sup X n (τ ) + N (s) E (ϕB ) . 0≤τ ≤s

Consequently p

p

E (X n [ϕ]s  ) ≤ λK υn (s) + λN E ϕB for all s ∈ [0, T ].

(13)

Let n1 ≤ t ≤ T . Considering Inequality (13), we get, for every n ≥ 1   E

p

sup X n (s)

1 s∈[ n ,t]



p−1

≤3



E sup

1 s∈[ n ,t]



p

S(s − 1/n)ϕ(0) +

s−1/n 0

p

S(s−τ )f (τ, X n [ϕ]τ ) dτ

p     s−1/n   n S(s − τ )g(τ, X [ϕ]τ ) dW (τ ) +   0   t

p

p E MTp E ϕ(0)H + MTp Cgrowth

≤ 3p−1

p

sup 0 τ ∈[1/n,t]

(1 + X n [ϕ]τ ) ds

 p p + T 2 −1 CConv E g(s, X n [ϕ]s ) ds 0  p p ≤ 3p−1 MT E ϕ(0)H 

+

p Cgrowth (MTp

t

+T

p 2 −1

 CConv ) E

 p ≤ 3p−1 MTp E ϕ(0)H +

p Cgrowth (MTp

p−1

≤3



+T

p 2 −1

p

n

sup (1 + X [ϕ]τ ) ds .

0 τ ∈[0,t]

 CConv ) E



t



t

p−1

sup 2 0 τ ∈[0,t]

n

p

(1 + X [ϕ]τ  ) ds .

p

MTp E ϕ(0)H p

p + 2p−1 Cgrowth (MTp + T 2 −1 CConv )



t 0

 p (1 + λK υn (s) + λN E ϕB ) ds

Semilinear Functional SDEs with Infinite Delay   p p p p ≤ 3p−1 MTp E ϕ(0)H +2p−1 Cgrowth (MTp +T 2−1 CConv )T (1+λN E ϕB )    t p p p p−1 −1 2 +6 CConv )λK υn (s) ds Cgrowth (MT + T 0

 ≤a+a

t 0

υn (s) ds,

with a = 3p−1 max {a1 , a2 } , p

p

p

p (MTp + T 2 −1 CConv )T (1 + λN E ϕB ), a1 = MTp E ϕ(0)H + 2p−1 Cgrowth p

p (MTp + T 2 −1 CConv )λK . a2 = 2p−1 Cgrowth

Since X n (t) = ϕ(0) for t ∈ [0, n1 ], we can extend the preceding inequality to p all t ∈ [0, T ], by replacing a by max {a, E ϕ(0)H }. We then have, for any n ≥ 1 and any t ∈ [0, T ]    p

E

sup X n (t)

s∈[0,t]

t

= υn (t) ≤ a + a

0

υn (s) ds.

Thus, by Gronwall’s lemma, we have, for all n ≥ 1   p

sup X n (t)

E

t∈[0,T ]

≤ a ea T .

(14) 

Claim 2. The sequence (X n ) is relatively compact in Ncp (F, [0, T ]; H). For each n ≥ 1, let Φn be the mapping which, with every process X ∈ D(ϕ, T ), associates the process Φn (X), such that Φn (X)(t) = ϕ(0) for t ∈ [0, n1 ], and, for t ∈ [ n1 , T ]  t−1/n Φn (X)(t) = S(t − 1/n)ϕ(0) + S(t − s)f (s, X[ϕ]s ) ds 0



t−1/n

S(t − s)g(s, X[ϕ]s ) dW (s).

+ 0

Let us show that n n ΨD ({Φn (X ); n ≥ 1}) ≤ MT ΨD ({Φ(X ); n ≥ 1}) ,

(15)

where MT has been defined in (HS). Since the values of ϕ(0) do not enter in the calculation of ({Φn (X n ); n ≥ 1}) and ({Φn (X n ); n ≥ 1}), we can assume without loss of generality that ϕ(0) = 0. We then have  t−1/n n Φn (X )(t) = S(1/n) S((t − 1/n) − s)f (s, X n [ϕ]s ) ds 0



t−1/n

+ 0

 n

S((t − 1/n) − s)g(s, X [ϕ]s ) dW (s)

L. Guedda, P. Raynaud de Fitte

MJOM

= S(1/n)Φ(X n )(t − 1/n) = τn (Φ(X n ))(t), where τn : D(ϕ, T ) → D(ϕ, T ) is defined by  Y (t − 1/n) if τn (Y )(t) = ϕ(0) if

t ∈ [1/n, T ] t ∈ [0, 1/n].

By Claim 1 and (HFG), the set ({Φ(X n ); n ≥ 1}) is bounded in Ncp (F, [0, t]; H). Furthermore, if Θ is a finite -net of a bounded subset Λ of Ncp (F, [0, t]; H), then τn (Θ) is a finite -net of τn (Λ). Hence, we get n n ΨD ({τn (Φ(X )); n ≥ 1}) ≤ ΨD ({Φ(X ); n ≥ 1}).

It results that n n ΨD ({Φn (X ); n ≥ 1}) = ΨD (S(1/n)τn ({Φ(X ); n ≥ 1})) ≤ MT ΨD ({Φ(X n ); n ≥ 1}) .

Now, considering that X n = Φn (X n ) for each n, using Lemma 3.4 and the inequality (15), we obtain, for every t ∈ [0, T ] p p n n ΨD ({X ; n ≥ 1}) (t) ≤ ΨD ({Φn (X ); n ≥ 1})(t)

≤ MTp ΨpD ({Φ(X n ); n ≥ 1}) (t)  t p ≤ MT α L (s, K p (s) ΨpD ({X n ; n ≥ 1}) (s)) ds, 0

where α is the constant we obtained in Lemma 3.4, see (10). Since the function K is positive, we get K p (s) ΨpD ({X n ; n ≥ 1})(t)    sup K p (t) MTp α

≤

t∈[0,T ]

t

p

L (s, K p (s) ΨD ({X n ; n ≥ 1})(s)) ds.

0

By (2) in Hypothesis (HFG)-(b), we get that K(t) ΨD ({X n ; n ≥ 1})(t) = 0, for each t, which implies that ΨD ({X n ; n ≥ 1})(t) = 0; hence (X n ) is relatively compact. We can extract a subsequence of (X n ) which converges to some X ∞ in D(ϕ, T )) (recall that D(ϕ, T )) is closed). For simplicity, we denote this extracted sequence by (X n ). It remains to prove that X is a mild solution to (1). Claim 3. We have Φ(X n ) → Φ(X ∞ ) in Ncp (F, [0, T ]; H).  E

For every t ∈ [0, T ], we have n

∞

 p

sup Φ(X )(t) − Φ(X )(t)

t∈[0,T ]

p   t  t   n ∞ E sup  S(t−s)f (s, X [ϕ]s ) ds− S(t−s)f (s, X [ϕ]s ) ds ≤2   t∈[0,T ] 0 0   t  p−1 E sup  S(t − s)g(s, X n [ϕ]s ) dW (s) +2 

p−1

t∈[0,T ]

0

Semilinear Functional SDEs with Infinite Delay  −

t

0

p−1

≤2

p   S(t − s)g(s, X [ϕ]s ) dW (s) .  ∞

MTp

p



n

T E

p

∞

sup f (t, X [ϕ]t ) − f (t, X [ϕ]t )

t∈[0,T ]

+ 2p−1 CConv T p/2−1 E



T

0

g(s, X n [ϕ]s ) − g(s, X ∞ [ϕ]s )p ds.

By Claim 2, the sequence (X n ) converges in Ncp (F, [0, T ]; H) to X ∞ , i.e., p

sup X n (t) − X ∞ (t) → 0 in L1 .

t∈[0,T ]

In particular, the sequence (supt∈[0,T ] X n ) is uniformly integrable. From Lemma 2.2 and the growth condition (HFG)-(i), this entails that the sequences   p

sup f (t, X n [ϕ]t ) − f (t, X ∞ [ϕ]t )

t∈[0,T ]

and

n≥1



 n

∞

p

sup g(t, X [ϕ]t ) − g(t, X [ϕ]t )

t∈[0,T ]

n≥1

are uniformly integrable too. Since f and g are of Carath´eodory type, we deduce by Vitali’s theorem that these sequences converge to 0 in L1 , which proves the claim. Claim 4. We have Φ(X ∞ ) = X ∞ , i.e., X ∞ is a solution to (1). Let q, such that 2 < q < p, and let us prove that (Φ(X n ) − X n ) → 0 in Ncq (F, [0, T ]; H). As X n → X ∞ and Φ(X n ) → Φ(X ∞ ) in Ncq (F, [0, T ]; H), this will prove the claim and Theorem 3.2. For every t ∈ [0, T ] and every n ≥ 1, we have X n (t) = Φn (X n )(t)     (t− n1 )+ 1 S(t − s)f (s, X n [ϕ]s ) ds ϕ(0) + t− =S n + 0  (t− n1 )+ + S(t − s)g(s, X n [ϕ]s ) dW (s), 

where t −



1 n +

0

  = max 0, t − n1 . Thus

Φ(X n )(t) − X n (t)   S(t) − S

=

 +

t 1 )+ (t− n

1 t− n

 

 ϕ(0) +

+

t

1 (t− n )+

S(t − s)g(s, X n [ϕ]s ) dW (s).

S(t − s)f (s, X n [ϕ]s ) ds (16)

L. Guedda, P. Raynaud de Fitte

MJOM

Since (S(t))t is a C0 -semigroup, we have, for any x ∈ H        1   t− x → 0, sup  S(t) − S   n t∈[0,T ] + and thus        1   t− sup  S(t) − S ϕ(0) → 0 a.s. and in Lp when n → ∞.  n + t∈[0,T ]  Therefore, we only need to prove that both integrals in (16) converge to 0 in Ncq (F, [0, T ]; H). Using the growth condition (HFG)-(i) and Axiom (B3), we obtain, for every t ∈ [0, T ]  q    t    S(t − s)f (s, X n [ϕ]s ) ds E sup  1   t∈[0,T ] (t− n )+    t q    q n ≤ MT E sup f (s, X [ϕ]s ) 1l[(t− n1 )+ ,t] (s) ds t∈[0,T ]

1 (t− n )+

q/p   (p−q)/p 1 ≤ f (s, X [ϕ]s ) ds E sup 1 n t∈[0,T ] (t− n )+   q/p  (p−q)/p  t 1 q ≤ MTq Cgrowth E sup (1 + X n [ϕ]s )p ds 1 n t∈[0,T ] (t− n )+ 

MTq



t

p

n

→ 0 as n → ∞. Similarly 

 q   t    n E sup  S(t − s)g(s, X [ϕ]s ) dW (s) 1   t∈[0,T ] (t− n )+  t q     ≤ CConv T q/2−1 E g(s, X n [ϕ]s ) 1l[(t− n1 )+ ,t] (s) ds 1 (t− n )+

q/p   (p−q)/p 1 g(s, X [ϕ]s ) ds 1 n (t− n )+ q/p     (p−q)/p t 1  q/2−1 q n n p ≤ CConv T Cgrowth E (1 + X [ϕ]s ) ds 1 n (t− n )+    q/2−1 E ≤ CConv T

t

n

p

→ 0 as n → ∞. 

4. Dependence on the Initial Process For every t ∈ [0, T ], let us denote Bt = Lp (Ω, Ft , P; B).

Semilinear Functional SDEs with Infinite Delay Under the conditions of Theorem 3.2, let us consider the map  B0 → Ncp (F, [0, T ]; H) Σ: ψ → X ψ , where X ψ is the mild solution to the equation obtained by replacing in (1), the initial condition ϕ by ψ, and let us also consider the map  B0 → BT π: ψ → X ψ [ψ]T . Theorem 4.1. Assume Hypotheses (HS), (HFG), and (B1)–(B4). Then, the operators Σ and π are continuous. Proof. Let us prove that Σ is continuous. Let (ψ n )n be a sequence in B0 n converging to some ψ, and let us prove that Σ(ψ n ) = X ψ converges to Σ(ψ) = X ψ . We have   p E sup Σ(ψ n )(τ ) − Σ(ψ)(τ ) 0≤τ ≤t

p

≤ 3p−1 MTp E ψ n (0) − ψ(0)   p−1

+3

τ

Mt E

+ 3p−1 CConv T p/2−1 Set

f (s, X [ψ]s ) − f (s, X[ψ]s )

sup τ ∈[0,t]

0



 p

n

t

0

ds p

E g(s, X n [ψ]s ) − g(s, X[ψ]s ) ds.

  λ = 3p−1 MTp + CConv T p/2−1 .

Since, for every t ∈ [0, T ], the function L(t, .) is nondecreasing and concave, the preceding inequality gives 

E

p  sup Σ(ψ n )(τ ) − Σ(ψ 0 )(τ )



0≤τ ≤t



p    n E L s, X ψ [ψ]s − X ψ [ψ]s  ds 0    t  ψ p p p−1 n p ψn   ds ≤3 MT E ψ(0) − ψ (0) + λ E L s, sup X [ψ]τ − X [ψ]τ

≤ 3p−1 MTp E ψ(0) − ψ n (0)p + λ

t

0

≤3

p−1

+λ

MTp



t

0

n

0≤τ ≤s

p

E ψ(0) − ψ (0)    p n L s, λK E( sup X ψ (τ ) − X ψ (τ ) ) + λN E ψ − ψ n p ds, 0≤τ ≤s

where λK has been defined in (11) and λN in (12). By (B2), we have immediately that p

p

E ψ − ψ n B ) → 0 ⇒ E ψ(0) − ψ n (0)H ) → 0. Set, for every t ∈ [0, T ]

 p n   Vn (t) = λK E sup X ψ (s) − X ψ (s) . 0≤s≤t

L. Guedda, P. Raynaud de Fitte

MJOM

Then, using Fatou’s Lemma and the fact that the function L is nondecreasing in the second argument, we get   t   p   ψ  ψn lim Vn (t) ≤ lim λλK L s, λK E sup X (τ ) − X (τ ) n→∞

n→∞

0≤τ ≤s

0

p

+ λN E ψ − ψ n  ) ds    t  p  n   ≤ λλK lim L s, λK E sup X ψ (τ ) − X ψ (τ ) 0 n→∞

0≤τ ≤s

n p

+ λN E ψ − ψ  ) ds ⎧    t  p   ψ  ⎪ ψn ⎪ ≤ λλK lim L s, lim ⎩λK E sup X (τ ) − X (τ ) n→∞ n→∞ 0≤τ ≤s 0 ⎫ n p⎭ ds + λN E ψ − ψ   t   ≤ λλK L s, lim Vn (s) ds, 0

n→∞

which implies  p n   lim λK E sup X ψ (s) − X ψ (s) = 0.

n→∞

0≤s≤t

Thus, the map Σ is continuous. Now, for the continuity of π, let D be the subset of B0 × Ncp (F, [0, T ]; H) defined by D = {(ψ, X) : ψ(0) = X(0) a.s.} . It is easy to see, by Axiom (B2), that D is a closed subset of Lp (Ω, F0 ; B) × Ncp (F, [0, T ]; H). Remark that the operator π can be represented as π = π3 ◦ π2 ◦ π1 , where the maps πi , i ∈ {1, 2, 3}, are given by  B0 → D π1 : ψ  ψ → (ψ, X p), D → N (F, (−∞, T ], H) π2 : (ψ, X) → X[ψ],  p N (F, (−∞, T ], H) → BT π3 : Y → YT . Since Σ is continuous, the map π1 is continuous too. The continuity of π2 is  obvious. Finally, the continuity of π3 results from Axiom (B3).

5. Application to a Stochastic Parabolic Equation with Delay Example 5.1. Let us consider the following stochastic partial differential equation:

Semilinear Functional SDEs with Infinite Delay ⎧  2 ∂ u ⎪ d u(t, x) = (t, x) ⎪ ∂x2  ⎪ ⎪  ⎪ x t x ⎪ ⎪ + F t, 0 a0 (σ) u(t, σ)dσ, −∞ 0 a(θ − t)u(θ, σ) dσ dθ dt ⎨    t 1 1 + G t, b (σ) u(t, σ)dσ, b(θ − t)u(θ, σ) dσ dθ dW (t) ⎪ 0 ⎪ 0 −∞ 0 ⎪ ⎪ ⎪ u(t, 0) = u(t, 1) = 0, t ∈ [0, 1], ⎪ ⎪ ⎩ u(θ, x) = β(θ, x), θ ≤ 0, x ∈ [0, 1], (17) 2

where F : [0, 1] × R × R → R, and G : [0, 1] × R × R → LHS (L ([0, 1])), LHS (L2 ([0, 1])) denotes the space of Hilbert-Schmidt operators from L2 ([0, 1]) to L2 ([0, 1]), β(ω, ·, ·) ∈ Lγ (H) P-a.e., (for the definition of Lγ (H), see Sect. 2), and W is an L2 ([0, 1])-valued Wiener process. Fix p > 2. Let us consider the following hypotheses. (HI) The initial process β is an F0 -measurable Lγ (H)-valued random variable, p with E β(·)(·)Lγ (H) < ∞. (A) F : [0, 1] × R × R → R and G : [0, 1] × R × R → LHS (L2 ([0, 1])) are measurable. (B) There exists a constant C > 0, such that  p p p |F (t, x, y) − F (t, x , y  )| ≤ C α |x − x | + |y − y  | ,  p p p G(t, x, y) − G(t, x , y  )LHS (L2 ([0,1])) ≤ C α |x − x | + |y − y  | , where α(·) : [0, +∞[→ [0, +∞[ is a continuous, monotone nondecreasing  +∞ 1 and concave function with α(0) = 0 and 0+ α(s) ds = ∞. (C) There exists a constant c > 0, such that |F (t, x, y)| ≤ c (1 + |x| + |y|), G(t, x, y)LHS (L2 ([0,1])) ≤ c (1 + |x| + |y|). (D) The functions a0 (·) and b0 (·) belong to L2 ([0, 1]), and the functions a(·) and b(·) are continuous, with   −γθ −γθ max sup e |a(θ)| , sup e |b(θ)| < ∞. θ≤0

θ≤0

Let U = H = L2 ([0, 1]) and B = Lγ (H). Set X(t) = u(t, ·), t ∈ [0, 1], ϕ(ω)(θ)(x) ˆ = β(θ, x), θ ≤ 0, x ∈ [0, 1]. It is clear that ϕ(ω)(·)(·) ˆ ∈ Lγ (H) P-a.e., define the operator A by A(h)(t) = h (t), where D(A) = {h ∈ H; h , h ∈ H

and

h(0) = h(1) = 0} .

(18)

It is well-known that A generates a C0 semigroup. With these data, the problem (17) can be written in abstract form, as follows:  dX(t) ∈ AX(t) + f (t, Xt ) dt + g(t, Xt ) dW (t), t ∈ [0, 1], ˆ X0 (ω) = ϕ(ω)(·)(·),

L. Guedda, P. Raynaud de Fitte where, for a.e., x ∈ [0, 1] and ψ ∈ Lγ (H)   x  a0 (σ) ψ(0)(σ) dσ, f (t, ψ)(x) = F t, 0

  g(t, ψ) = G t,



1

b0 (σ) ψ(0)(σ) dσ,

0



0

MJOM

x

a(θ) −∞

0

0





1

b(θ) −∞

 ψ(θ)(σ) dσ dθ ;

ψ(θ)(σ) dσ dθ . 0

Note that  ϕLγ (H) = ϕ(0)(·)L2 ([0,1]) + and





t

−∞

a(θ − t)

0

γθ



1

e −∞

0



x 0



0

u(θ, σ)dσdθ =

 12 |ψ(θ)(x)| dx <∞ 2

x

a(θ) −∞  0

u(t + θ, σ)dσdθ 

0 x

a(θ)

= −∞

0

ut (θ, σ)dσdθ.

Theorem 5.2. Assume that the hypotheses (HI) and (A)–(D) are satisfied. Then, the problem (17) has a unique mild solution in N p (F, ] − ∞, T ], L2 ([0, 1])). Moreover, the stochastic translation operator which assigns to each ψˆ ∈ B0 = Lp (Ω, F0 , P, B, ), the element uT in BT = Lp (Ω, F0 , P, B, ), ˆ where u is the mild solution of the problem (17) with β(ω) = ψ(ω) P-a.e., in 2 Lγ (L ([0, 1])), is continuous. Proof. According to Theorems 3.2 and 4.1, we have only to prove that f and g satisfy the hypotheses (HFG)-(i)-(ii). Define in the space B = Lγ (H), an ◦ equivalent norm ·B by ◦

ϕB = v ψLγ (H) , where v is a positive constant chosen, so that   −γθ −γθ v ≥ max a0 L2 + sup e |a(θ)| , b0 L2 + sup e |b(θ)| . θ≤0

θ≤0

(19)

Using Hypothesis (B) and the fact that the function α(·) is nondecreasing, for ψ1 , ψ2 ∈ Lγ (H), and for a.e., x ∈ [0, 1], we have |f (t, ψ1 )(x)) − f (t, ψ2 )(x))|p  x p ≤ Cα |a0 (σ)| |ψ1 (0)(σ) − ψ2 (0)(σ)| dσ 0    +

−γθ

sup e θ≤0

|a(θ)|

0

x

γθ

e

−∞

0

|ψ1 (θ)(σ) − ψ2 (θ)(σ)| dσ dθ

 x ≤ Cα |a0 (σ)| |ψ1 (0)(σ) − ψ2 (0)(σ)| dσ 0    +

−γθ

sup e θ≤0

|a(θ)|

0

−∞

x

γθ

e

0

p 

|ψ1 (θ)(σ) − ψ2 (θ)(σ)| dσ dθ

p 

Semilinear Functional SDEs with Infinite Delay  ≤ Cα

a0 L2 ψ1 (0)(·) − ψ2 (0)(·)L2

 −γθ

sup e

+

θ≤0

 ≤ Cα 

 |a(θ)|

0

γθ



1

e

−∞

0

2

|ψ1 (θ)(σ) − ψ2 (θ)(σ)| dσ

  v ψ1 (0)(·) − ψ2 (0)(·)L2

0

+ −∞

γθ



1

e

0

|ψ1 (θ)(σ) − ψ2 (θ)(σ)| dσ

dθ

p 

1/2

2

p 

1/2

,

dθ

where v is as in (19). Hence, for a.e., x ∈ [0, 1], we have  p ◦ p  |f (t, ψ1 (·)(x)) − f (t, ψ2 (·)(x))| ≤ Cα ψ1 − ψ2 B . Therefore,  1  2p ◦ p  . |f (t, ψ1 (·)(x)) − f (t, ψ2 (·)(x))| dx ≤ C 2 α2 ψ1 − ψ2 B 0

Since the function x → xp is convex (recall that p > 2), using Jensen’s inequality, we get  1 p  1  2 2p |f (t, ψ1 )(x) − f (t, ψ2 )(x)| dx ≤ |f (t, ψ1 )(x) − f (t, ψ2 )(x)| dx 0 0  p  . ≤ C 2 α2 ψ1 − ψ2 ◦B Thus p

f (t, ψ1 )(·) − f (t, ψ2 )(·)L2 ([0,1]) ≤ Cα

 ◦ p  . ψ1 − ψ2 B

Now, let us show that f satisfies hypothesis (HFG)-(i). Using Hypothesis (C), for a.e., x ∈ [0, 1], we have    −γθ |a(θ)| |f (t, ψ)(x))| ≤ 1 + a0 L2 + sup e 

θ≤0

× ψ(0)(·)L2 +



0

γθ



1

e −∞

0

2

|ψ(σ)(x)| dσ



1/2 dθ

 ◦ ≤ c (1 + v ψB ) ≤ c 1 + ψB , where v satisfies (19). Hence

 ◦ f (t, ψ)(·))L2 (0,1) ≤ c 1 + ψB .

Bearing in mind (19), by a similar reasoning as above, one can show that  p ◦ p  g(t, ψ1 ) − g(t, ψ2 )LHS (L2 ([0,1])) ≤ Cα ψ1 − ψ2 B and

 ◦ g(t, ψ)LHS (L2 ([0,1])) ≤ c 1 + ψB . 

L. Guedda, P. Raynaud de Fitte

MJOM

Remark 5.3. As an example of such function α : [0, +∞[→ [0, +∞[, one can write ⎧ ⎪ z = 0, ⎨0, α(z) = −z ln z, 0 < z < 1/e, ⎪ ⎩ 1/e, z > 1/e. The function α is continuous nondecreasing, concave, and  dz lim+ =∞ ε→0 ε α(z) (see [1, Example 1.4.2, page 14]). Remark 5.4. Note that, in Example 5.1, the function L of Hypothesis (HFG)(ii) has the form L(t, x) = C α(x), we are in the case of Osgood’s type conditions. Remark 5.5. Let us give a simple example of a function G satisfying the assumptions (B) and (C). Let G : [0, 1] × R × R → LHS (L2 ([0, 1])) be defined by G(t, x, y)h(·) = g(x, y)Sh(·), where S is a fixed Hilbert-Schmidt operator, and g : R × R → R is a given function, such that p

p

p

|g(x, y) − g(x , y  )| ≤ C α(|x − x | + |y − y  | ), |g(x, y)| ≤ c (1 + |x| + |y|) , where C and c are positive constants, p > 2, and α(·) satisfies Hypothesis (B). Then, it is clear that G(t, x, y)LHS (L2 ([0,1])) ≤ c (1 + |x| + |y|) and

 p p p G(t, x, y) − G(t, x , y  )LHS (L2 ([0,1])) ≤ C α |x − x | + |y − y  | .

Hence, Hypotheses (B) and (C) are satisfied by G. Acknowledgments We thank an anonymous referee for his remarks which led to the addition of Sect. 5.

References [1] Agarwal, R.P., Lakshmikantham, V.: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations, Series in Real Analysis, vol. 6. World Scientific Publishing Co., Inc., River Edge (1993) [2] Ahmed, N.U.: Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series, vol. 246. Longman Scientific and Technical, Harlow; copublished in the United States with Wiley, New York (1991)

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L. Guedda, P. Raynaud de Fitte

MJOM

[22] Taniguchi, T.: Successive approximations to solutions of stochastic differential equations. J. Differ. Equ. 96(1), 152–169 (1992) [23] Tudor, C.: On stochastic functional-differential equations with unbounded delay. SIAM J. Math. Anal. 18(6), 1716–1725 (1987) [24] Walter, W.: Differential and integral inequalities. Translated from the German by Lisa Rosenblatt and Lawrence Shampine. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 55. Springer, New York (1970) [25] Yamada, T.: On the successive approximation of solutions of stochastic differential equations. J. Math. Kyoto Univ. 21(3), 501–515 (1981) Lahcene Guedda Faculty of Mathematics and informatics University of Tiaret BP78 Tiaret 14000 Algeria e-mail: lahcene [email protected] Paul Raynaud de Fitte Normandie Univ., Laboratoire Rapha¨el Salem UMR CNRS 6085 Rouen, France e-mail: [email protected] Received: April 15, 2015. Revised: March 14, 2016. Accepted: May 18, 2016.