Existence, Uniqueness and Stability Results of impulsive Stochastic semilinear neutral functional differential equations with infinite delays”, Electronic

This article presents the results on existence, uniqueness and stability of mild solutions of impulsive stochastic semilinear neutral functional differential equations without a Lipschitz condition and with a Lipschitz condition. The results are obtained by using the method of successive approximations. 2000 Mathematical Subject Classification: 93E15,60H15,35R12.


Introduction
Neutral differential equations arise in many area of science and engineering and have received much attention in the last decades. The ordinary neutral differential equation is used extensively to study the theory of aeroelasticity [10] and lossless transmission lines (see [4] and the references therein). Partial neutral differential equations with delays are motivated from stabilization of lumped control systems and the theory of heat conduction in materials (see [7; 8] and the references therein). Hernandez and O'Regan [6] studied some partial neutral differential equations by assuming a temporal and spatial regularity type condition for the function t → g(t, x t ). In [15; 4], the authors studied several existence results of stochastic differential equations (SDEs) with unbounded delays. the existence and asymptotic stability in p-th moment of mild solutions to ISDEs with and without infinite delays through fixed point theory. Motivated by [13; 14], we generalize the existence and uniqueness of the solution to impulsive stochastic partial neutral functional differential equations (ISNFDEs) under non-Lipschitz conditions and under Lipschitz conditions. Moreover, we study the stability through the continuous dependence on the initial values by means of a corollary of Bihari's inequality. Further, we refer [3; 5; 12; 20]. This paper is organized as follows. In Section 2, we recall briefly the notation, definitions, lemmas and preliminaries which are used throughout this paper. In Section 3, we study the existence and uniqueness of ISNFDEs by relaxing the linear growth conditions. In Section 4, we study stability through the continuous dependence on the initial values. Finally in Section 5, an example is given to illustrate our results.

Preliminaries
In this article, we will examine impulsive stochastic semilinear neutral functional differential equations of the form where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear Let X, Y be real separable Hilbert spaces and L(Y, X) be the space of bounded linear operators mapping Y into X. For convenience, we shall use the same notations . to denote the norms in X, Y and L(Y, X) without any confusion. Let (Ω, B, P ) be a complete probability space with an increasing right continuous family {B t } t≥0 of complete sub σalgebra of B. Let {w(t) : t ≥ 0} denote a Y -valued Wiener process defined on the probability space (Ω, B, P ) with covariance operator Q, that is where Q is a positive, self-adjoint, trace class operator on Y . In particular, we denote by In order to define stochastic integrals with respect to the Q-Wiener process w(t), we introduce the subspace Y 0 = Q 1/2 (Y ) of Y which, endowed with the inner product where B w t is the sigma algebra generated by {w(s) : 0 ≤ s ≤ t}. Let L 0 2 = L 2 (Y 0 , X) denote the space of all Hilbert-Schmidt operators from Y 0 into X. It turns out to be a separable Hilbert space equipped with the norm µ 2 for any µ ∈ L 0 2 . Clearly for any bounded operators µ ∈ L(Y, X) this norm reduces to µ 2 L 0 2 = tr(µQµ * ).
We now make the system (2.1) precise: Let A : X → X be the infinitesimal generator We also assume that D b B 0 ((−∞, 0], X) denotes the family of all almost surely bounded, B 0 -measurable,D-valued random variables. Further, let B T be a Banach space of all B tadapted processes ϕ(t, w) which are almost surely continuous in t for fixed w ∈ Ω with norm defined for any ϕ ∈ B T by Furthermore, the fixed moments of time t k satisfy 0 < t 1 < . . . < t m < T , where x(t + k ) and x(t − k ) represent the right and left limits of x(t) at t = t k , respectively. And ∆x(t k ) = , represents the jump in the state x at time t k with I k determining the size of the jump.
Lemma 2.1. [2] Let T > 0, u 0 ≥ 0, and let u(t), v(t) be continuous functions on [0, T ]. Let K : ℜ + → ℜ + be a concave continuous and nondecreasing function such that K(r) > 0 for where G(r) = In order to obtain the stability of solutions, we use the following extended Bihari's inequality Lemma 2.2. [13] Let the assumptions of Lemma 2.1 hold. If Lemma 2.4. [3] For any r ≥ 1 and for arbitrary L 0 then the semigroup is said to be a contraction semigroup.

Existence and uniqueness
In this section, we discuss the existence and uniqueness of mild solutions of the system (2.1). We use the following hypotheses to prove our results. Hypotheses: (H 1 ) : A is the infinitesimal generator of a strongly continuous semigroup S(t), whose domain D(A) is dense in X.
(H 2 ) : For each x, y ∈D and for all t ∈ [0, T ], such that, where K(·) is a concave non-decreasing function from ℜ + to ℜ + , such that K(0) = 0, K(u) > 0, for u > 0 and 0 + du K(u) = ∞. (H 3 ) : Assuming that there exists a positive number L g such that L g < 1 10 , for any x, y ∈D and for t ∈ [0, T ], we have (H 4 ) : The function I k ∈ C(X, X) and there exists some constant h k such that for each x, y ∈D, k = 1, 2 . . . , m.
Let us next show that {x n } is Cauchy in B T . For this consider, Thus, Then, we have in the view of (3.5), Moreover, Then, we get If we take the supremum over t, and use (3.4), we get Now, for n = 1 in (3.7) we get Now, for n = 2 in (3.7), we get Thus by applying mathematical induction in (3.7) and using the above work we get Note that for any m > n ≥ 0, we have, → 0 as n → ∞. Thus, where, Q 7 = 4 L g + M 2 m m k=1 h k . Thus, Bihari's inequality yields that Thus, x 1 (t) = x 2 (t), for all 0 ≤ t ≤ T . Therefore, for all −∞ < t ≤ T , x 1 (t) = x 2 (t) a.s.
This completes the proof.

Stability
In this section, we study the stability through the continuous dependence on initial values.
wherex(t) is another mild solution of the system (2.1) with initial valueφ. Proof: By the assumptions, x(t) and y(t) are two mild solutions of equations (2.1) with initial values ϕ 1 and ϕ 2 , respectively, so that for 0 ≤ t ≤ T we have So, estimating as before, we get Thus, EJQTDE, 2009 No. 67, p. 9 where K is a concave increasing function from ℜ + to ℜ + such that K(0) = 0, K(u) > 0 for u > 0 and 0 + du K(u) = +∞. So, K 1 (u) is a concave function from ℜ + to ℜ + such that K 1 (0) = 0, K 1 (u) ≥ K(u), for 0 ≤ u ≤ 1 and Thus, there is a positive constant δ < ǫ 1 , such that Hence, for any t ∈ [0, T ], the estimate u(t) ≤ ǫ 1 holds. This completes the proof. If m = 0 in (2.1), then the system behaves as stochastic partial neutral functional differential equations with infinite delays of the form Remark 4.2.
If the system (4.2) satisfies the Remark 4.1, then by Theorem 4.1, the mild solution of the system (4.2) is stable in the mean square.
It is well known that A generates a strongly continuous semigroup S(t) which is compact, analytic and self adjoint. Moreover, the operator A can be expressed as Au = ∞ n=1 n 2 < u, u n > u n , u ∈ D(A), where u n (ζ) = ( 2 π ) 1 2 sin(nζ), n = 1, 2, . . ., is the orthonormal set of eigenvectors of A, and S(t)u = ∞ n=1 e −n 2 t < u, u n > u n , u ∈ X.
We assume that the following condition hold: (i): The function b is measurable and π 0 π 0 b 2 (y, x)dydx < ∞.
The next results are consequences of Theorem 3.1 and Theorem 4.1, respectively.