STOCHASTIC DIFFERENTIAL EQUATIONS WITH NON-INSTANTANEOUS IMPULSES DRIVEN BY A FRACTIONAL BROWNIAN MOTION

This paper is concerned with the existence and continuous dependence of mild solutions to stochastic differential equations with non–instantaneous impulses driven by fractional Brownian motions. Our approach is based on a Banach fixed point theorem and Krasnoselski-Schaefer type fixed point theorem.

1. Introduction.Stochastic differential equations have many applications in science and engineering, and for this reason, these equations have been receiving much attention over the last decades (see, e.g., [4,8,10,34,15,19,30,29,1] and references therein).Also, in recent years, stochastic differential equations driven by fractional Brownian motions (fBm) have attracted much attention and there are only a few papers published in this field (see, e.g.[9,11,12]).Boudaoui et al. [6] discussed the existence of mild solutions to stochastic impulsive evolution equations with time delays, driven by fBm with Hurst index H > 1  2 .On the other hand, impulsive effects exist in several evolution processes in which states are changed abruptly at certain moments of time, related to fields such as economics, bioengineering, chemical technology, medicine and biology etc (see [20,33,13,27,28]).Boudaoui et al. [8] obtained several new sufficient conditions to ensure the local and global existence and attractivity of mild solutions for stochastic neutral functional differential equations with instantaneous impulses, driven by fractional Brownian motions.Recently, Hernández and O'Regan [18] initially study on Cauchy problems for a new type of first order evolution equations with non-instantaneous impulses.For example, impulses start abruptly at the instant t k and their action continue on a finite time interval (t k , s k ].This type of problem motivates to study certain dynamical changes of evolution processes in pharmacotheraphy [26,14,32].As a motivation, we can mention a simple situation concerning the hemodynamical behavior of a person who has a decompensation of the glucose level (either high or low level).Then, this person can be prescribed some intravenous insulin to compensate the level.Since the introduction of the drug in the bloodstream and its absorption are gradually continuous processes, we can interpret the above situation as an impulsive action which remains active for a period of time, so a non-instantaneous impulse is taking place.
Hence, it is important to take into account the effect of impulses in the investigation of stochastic delay differential equations driven by f Bm.
In this paper, our main objective is to establish sufficient conditions ensuring existence and continuous dependence of mild solutions to the following first order stochastic impulsive differential equation: ) = φ(t) ∈ D F0 , for a.e.t ∈ J 0 = (−∞, 0], (3) in a real separable Hilbert space H with inner product (•, •) and norm • , where A : D(A) ⊂ H −→ H is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators {S(t), t ≥ 0} satisfying S(t) 2 ≤ M , B H Q is a fractional Brownian motion on a real and separable Hilbert space K, with Hurst parameter H ∈ (1/2, 1), and with respect to a complete probability space (Ω, F, F t , P ) furnished with a family of right continuous and increasing σ- As for y t we mean the segment solution which is defined in the usual way, that is, if y(•, •) : (−∞, T ] × Ω → H, then for any t ≥ 0, y t (•, •) : (−∞, 0] × Ω → H is given by: y t (θ, ω) = y(t + θ, ω), for θ ∈ (−∞, 0], ω ∈ Ω, Before describing the properties fulfilled by the operators f, g, σ and h k , we need to introduce some notation and describe some spaces.
In this work, we will use an axiomatic definition of the phase space D F0 introduced by Hale and Kato [17].
Definition 1.1.D F0 is a linear space of family of F 0 -measurable functions from (−∞, 0] into H endowed with a norm • D F 0 , which satisfies the following axioms: (A-1): If y : (−∞, T ] −→ H, T > 0 is such that y 0 ∈ D F0 , then for every t ∈ [0, T ) the following conditions hold (i): Now, for a given T > 0, we define ), for all ω ∈ Ω, for k = 0, . . .m, y 0 ∈ D F0 , and there exist y(t − k ) and y(t endowed with the norm where Then we will consider our initial data φ ∈ D F0 .Assume that B H Q is a K-valued fractional Brownian motion with increment covariance given by a non-negative trace class operator Q (see next section for more details), and let us denote by L(K, H) the space of all bounded, continuous and linear operators from K into H.
Then we assume that ) denotes the space of all Q-Hilbert-Schmidt operators from K into H, which will be also defined in the next section.
The plan of this paper is as follows.In Section 2 we introduce notations, definitions, and preliminary facts which are useful throughout the paper.In Section 3 we state and prove our main results by using the Banach fixed point theorem and Krasnoselskii-Schaefer type fixed point theorem [3].The continuous dependence of mild solutions to problem (1)-( 3) is investigated in Section 4. Finally, in Section 5, an example is exhibited to illustrate the applicability of our results.

2.
Preliminaries.In this section we introduce notations, definitions, and preliminary facts which will be used throughout this paper.In particular, we consider a fractional Brownian motion as well as the Wiener integral with respect to it.We also establish some important and helpful results for our analysis.Needless to say that we could omit this whole section and refer to other works already published for these preliminaries (see, e.g.[6] and the references therein), but we aim at making this paper as much self contained as possible and this is why we decided to include this material in this section.
Recall that (Ω, F, F t , P ) is a complete probability space furnished with a family of right continuous increasing σ-algebras {F t , t ∈ J} satisfying F t ⊂ F. Definition 2.1.Given H ∈ (0, 1), a continuous centered Gaussian process β H = {β H (t), t ∈ R}, with the covariance function is called a two−sided one−dimensional fractional Brownian motion, and H is the Hurst parameter.
Now we aim at introducing the Wiener integral with respect to the one-dimensional β H .
Let T > 0 and denote by Λ the linear space of R−valued step functions on [0, T ], that is, ψ ∈ Λ if For ψ ∈ Λ we define its Wiener integral with respect to β H as Let H be the Hilbert space defined as the closure of Λ with respect to the scalar product 1 [0,t] , 1 [0,s] H = R H (t, s).Then, the mapping is an isometry between Λ and the linear space span {β H (t), t ∈ [0, T ]}, which can be extended to an isometry between H and the first Wiener chaos of the fractional Brownian motion span L 2 (Ω) {β H (t), t ∈ [0, T ]} (see [31]).The image of an element ψ ∈ H by this isometry is called the Wiener integral of ψ with respect to β H .Our next goal is to give an explicit expression of this integral.To this end, consider the kernel , with B(•, •) denoting the Beta function, and t ≤ s.
Next we are interested in considering a fractional Brownian motion with values in a Hilbert space and giving the definition of the corresponding stochastic integral.
Let {β H n (t)} n∈N be a sequence of two-sided one-dimensional standard fractional Brownian motions mutually independent on (Ω, F, P ).When one considers the following series where {e n } n∈N is a complete orthonormal basis in K, this series does not necessarily converge in the space K. Thus we consider a K−valued stochastic process B H Q (t) given formally by the following series: 5).Then, its stochastic integral with respect to the fractional Brownian motion B H Q is defined, for t ≥ 0, as follows then in particular (5) holds, which follows immediately from (4).
then the series in ( 6) is well defined as a H-valued random variable and we have (iii): for each q > 0, there exists , for all v 2 D F 0 ≤ q and for a.e.t ∈ J.
Definition 2.6.The map g : The following result is known as Gronwal-Bihari Theorem.
[5] Let u, g : J → R be positive real continuous functions.Assume there exist c > 0 and a continuous nondecreasing function h : R → (0, +∞) such that Then Here H −1 refers to the inverse of the function H(u) = u c dy h(y) for u ≥ c.One of the key tools in our approach is the following form of Burton-Kirk's fixed point theorem Theorem 2.8 (Burton-Kirk's fixed point theorem [3]).Let E be a Banach space, and G 1 , G 2 : E → E be two operators satisfying: 1. G 1 is a contraction, and 2. G 2 is completely continuous Then, either the operator equation y = G 1 (y) + G 2 (y) possesses a solution, or the set Ξ = y ∈ E : λG 1 ( y λ ) + λG 2 (y) = y, for some λ ∈ (0, 1) is unbounded.

Existence of mild solution.
In this section, we first establish the existence of mill solutions to stochastic differential equations with non-instantaneous impulses driven by fractional Brownian motions (1)-( 3).More precisely, we will formulate and prove sufficient conditions for the existence of solutions to (1)-( 3) with infinite delay.In order to establish the results, we will need to impose some of the following conditions.
• (H1) There exist constants • (H4) f and g are a L 2 -Caratheodory map and for every t ∈ [0, T ] the function t → f (t, y t ) and t → g(t, y t ), y t ∈ D F0 are mesurable • (H5) For the initial value φ ∈ D F0 , there exists a continuous nondecreasing functions ψ, where and • (H6) Operator A : D(A) ⊂ H → H is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators {S(t)}, t ∈ J which is compact for t > 0 in H. Now, we present the definition of mild solutions to our problem.Definition 3.1.Given φ ∈ D F0 , a H−valued stochastic process {y(t), t ∈ (−∞, T ]} is called a mild solution of the problem (1)-(3) if y(t) is measurable and F t -adapted, for each t > 0, y(t) = φ(t) on (−∞, 0], for each t > 0, y(t) = h k (t, y t ) for all t ∈ (t k , s k ] and each k = 1, • • • , m, and and For our main consideration of Problem ( 1)-( 3), a Banach fixed point is used to investigate the existence and uniqueness of solutions for impulsive stochastic differential equations.Theorem 3.2.Assume conditions (H1)-(H3) are satisfied and Then, for every initial function φ ∈ D F0 there exists a unique associated mild solution y ∈ D F T of the problem (1)-(3).
Remark 1.It is worth mentioning that the assumption L 0 < 1 in Theorem 3.2 implies some kind of smallness of the functions f, g and h k in comparison with the periods of time when the impulses are active or viceversa.
Proof.We first transform problem (1)-(3) into a fixed point one.Consider operator Φ : For φ ∈ D F0 , we define φ by Set D F T = {z ∈ D F T , such that z 0 = 0 ∈ D F0 } and for any z ∈ D F0 we have Let the operator Φ : From the assumption it is easy to see that Φ is well defined.Now we only need to show that Φ is a contraction mapping.Case 1: For u, v ∈ D F T and for t ∈ [0, t 1 ], we have Taking the supremum over t, we obtain From above, we obtain which implies that Φ is a contraction and therefore has a unique fixed point z ∈ D F T .Since y(t) = z(t) + φ(t), t ∈ (−∞, T ], then y is a fixed point of the operator Φ which is a mild solution of the problem (1)-( 3).This completes the proof.
The second result is established using a Krasnoselskii-Schaefer type fixed point theorem.As we can easily see, we will weaken the assumption L 0 < 1 in Theorem 3.2, but at the same time we need to impose some Carathéodory and Nagumo type of assumptions as well as an additional smallness hypothesis.The counterpart is that we can prove existence of at least one bounded solution for any initial value satisfying appropriate assumptions.Theorem 3.3.Assume that h k (t, 0) = 0, t ≥ 0, k ∈ N * , and hypotheses (H3)−(H6) Now we split our operator Φ into two parts in the following way: To use Theorem 2.8 we will verify that Φ 1 is a contraction while Φ 2 is a completely continuous operator.For the sake of convenience, we split the proof into several steps.
Taking the supremum over t, we obtain Thus Φ 1 is a contraction.
Next, we aim to prove that the operator Φ 2 is completely continuous.
Let z n be a sequence such that z n → z in D F T .Then, for t ∈ (s k , t k+1 ], k = 0, 1, • • • , m, and thanks to (H4) , we have by the dominated convergence theorem as n → +∞.Thus Φ 2 is continuous on D F T .
Step 3. Φ 2 maps bounded sets into bounded sets in D F T .

Let us first define
≤ q, where q > 0 is a given number.
The set B q is clearly a bounded closed convex set in D F T .We then have Now, to prove that Φ 2 maps bounded sets into bounded sets in D F T , it is enough to show that for any q > 0, there exists a positive constant l such that for each and therefore Step 4 : The map Φ 2 is equicontinuous .
Step 5. ( Φ 2 B q )(t) is precompact in H As a consequence of Steps 3 to 4, together with the Arzelá-Ascoli theorem, it suffices to show that Φ 2 maps B q into a precompact set in H.
Let s k < t < t k+1 be fixed and let be a real number satisfying s k < < t.For z ∈ B q we define Since S(t) is a compact operator, the set is precompact in H for every , s k < < t.Moreover, for every z ∈ B q we have Thus, there are precompact sets arbitrarily close to the set precompact in H, and therefore, the operator Φ 2 is completely continuous .
Step 6 : A priori bounds.Now it remains to show that the set This implies, for each t ∈ [0, But If we set w(t) the right hand side of the above inequality we have that and therefore (7) becomes Using (8) in the definition of w, we have that Thus, we obtain where Let us denote the right-hand side of the inequality (10) This implies, for each t ∈ J, we have .
Hence, there exists M t0 > 0 such that This implies, for each t ∈ (t k , s k ], If and therefore (12) becomes Using (13) in the definition of w(t), we have that Thus, we obtain This implies there is a constant M t k > 0 such that From equation ( 13) we obtain that Cas 3: This implies, for each t ∈ J, But If we set w(t) the right hand side of the above inequality we have that , and therefore (17) becomes Using (18) in the definition of w, we have that Thus, we obtain where Let us denote by v(t) the right-hand side of inequality (20).Then we have Using the increasing character of ψ and ψ 1 we obtain This implies, for each t ∈ J, we have .
By Lemma 2.7, we have where .
Hence, there exists M t k+1 > 0 such that Thus, there exists L t k+1 > 0 such that This implies that the set Ξ is bounded.
As a consequence of Theorem 2.8 we deduce that Φ has a fixed point, since y(t) = z(t) + φ(t), t ∈ (−∞, T ].Then y is a fixed point of the operator Ψ which is a mild solution of the problem (1)-(3).

2 A
. BOUDAOUI, T. CARABALLO AND A. OUAHAB (1)-(3) possesses at least one mild solution on (−∞, T ].Proof.As in the proof of Theorem 3.2, we first transform our problem (1)-(3) into the same fixed point formulation, and keep the same notation for the operators Φ, Φ and the spaces D F T and D F T .