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EXISTENCE AND ASYMPTOTIC STABILITY OF AN IMPULSIVE STOCHASTIC DIFFERENTIAL EQUATION.

1. INTRODUCTION

Stochastic differential equations have been widely applied in science, engineering, biology, mathematical finance and in almost all sciences. In the current literature, there are many papers on the existence and uniqueness of solutions to stochastic differential equations see [9,5,6] and references therein. More recently, Fu and Liu [4] discussed the existence and uniqueness of square-mean almost automorphic solutions to some linear and nonlinear stochastic differential equations, the asymptotic stability of the unique square-mean almost automorphic solution was established in the squaremean sense.

Impulsive differential equations model problem with impulsive effects which are due to instantaneous perturbations at certain moments. The vast applications of the theory of impulsive differential equations and inclusions have attracted many authors to considering both deterministic and stochastic cases. The theory of impulsive differential equations were extensively studied in [6] and [2] for instance, while Pan [10] considered the existence of mild solution for impulsive stochastic differential equations with nonlocal conditions in PC-norm.

Correspondingly, a lot of stability results of impulsive differential equations have been obtain [1,12,13,7]. In particular, Liu [8] established comparison principles of existence and uniqueness and stability of solutions for impulsive differential systems by means of Lyapunov function method and Ito's formula. Peng and Jia [11] obtained some criteria on p-th moment stability and p-th moment asymptotical stability of impulsive stochastic functional differential equations by using Lyapunov- Razumikhin method. In [3], several criteria on the global exponential stability and instability of impulsive stochastic functional differential systems are obtained by Cheng and Deng. Inspired by [10] and [11], we extend the result to mild solutions of impulsive stochastic differential equations with local conditions and study its asymptotic stability in the p-th moment. In the sequel, preliminaries necessary for the result shall be stated in section 2 and the main result will be proved in section 3.

2. PRELIMINARIES

Let ([OMEGA], [GAMMA], P) be a complete probability space with probability measure P on Q and a normal filtration [{[[GAMMA].sub.t]}.sub.t[greater than or equal to]0]. Let X,Y be two real separable Hilbert spaces with norms [[parallel]*[parallel].sub.X], [[parallel]*[parallel].sub.Y] and Q- Wiener process on ([OMEGA], [GAMMA], P) with covariance operator Q [member of] BL(Y) such that trQ < [infinity]. Let L(X, Y) be the space of bounded linear operators mapping X into Y equipped with the usual norm [parallel] * [parallel]. We assume that there exist a complete orthonormal system [e.sub.ii[greater than or equal to]1] in Y, a bounded sequence of nonnegative real numbers [[lambda].sub.i] such that Q[e.sub.i] = [[lambda].sub.i][e.sub.i], i = 1, 2, ..., and a sequence [[beta].sub.i] i [greater than or equal to] 1 of independent Brownian motions such that <w(t),e> = [SIGMA] [absolute value of ([[lambda].sub.i])] <[e.sub.i], e> [[beta].sub.i](t), e [member of] Y, and [[GAMMA].sub.t] = [[GAMMA].sup.w.sub.t], where [[GAMMA].sup.w.sub.t] is the sigma algebra generated by w(s) : 0 [less than or equal to] s [less than or equal to] t. Let [L.sup.0.sub.2] = [L.sub.2]([Q.sup.1/2]Y; X) be the space of all Hilbert-Schmidt operators from [Q.sup.1/2] Y to X with the inner product [mathematical expression not reproducible].

We consider the existence of mild solution for the following impulsive stochastic differential equations in a Hilbert space

(2.1) [mathematical expression not reproducible].

where A : D(A) [subset or equal to] X [right arrow] X is the infinitesimal generator of strongly continuous semigroup of bounded linear operators T(t), t > 0.

X is a real Banach space. x(0) = [x.sub.0] [member of] X, G : [0,b] [right arrow] X. F : [0, b] x X [right arrow] X; let 0 < [t.sub.1] < ... < [t.sub.m] < [t.sub.m+1] = b. [I.sub.k] : X [right arrow] X, where k = 1, ..., m are impulsive functions, [DELTA]x([t.sub.k]) = x([t.sup.+.sub.k]) - x([t.sup.-.sub.k]) which is the right and left limit of x at [t.sub.k]. F and G are predictable processes with Bochner integrable trajectories on arbitrary finite interval [0,b].

Definition 2.1. The stochastic process x(t), t [member of] [0, b] [right arrow] X is called the mild solution for the impulsive SDE (2.1) if

(i) x(t) is adapted to [[GAMMA].sub.t], t [greater than or equal to] 0.

(ii) x(t) [member of] X has cadlag paths on t [member of] [0, b] a.s and for each t [member of] [0, b].

(iii) For an arbitrary t [member of] [0,b].

(2.2) [mathematical expression not reproducible].

3. MAIN RESULTS

The following important theorem and assumptions are used to obtain the existence of (2.1)

Theorem 3.1. Let F : [0,6] x X [right arrow] X be an [L.sup.1]-Caratheodory function and G : [0,6] x X [right arrow] X satisfying the following conditions:

(a) For each t [member of] [a, b], G(t, *) : X [right arrow] X is continuous for all [x.sub.0] [member of] X, G(*,x) : [0, b] [right arrow] X is measurable.

(b) The function G : [0,6] x X [right arrow] X satisfies (i) and there exist [L.sub.G] > 0 such that for [mathematical expression not reproducible].

(c) For any l > 0 there exist a function [[rho].sub.l] [member of] [L.sup.1](0,b) such that sup [mathematical expression not reproducible].

Also assume that

(i) there exist constant Ck such that [parallel][I.sub.k](x)[parallel] [less than or equal to] [C.sub.k], k = 1, 2, ..., m for each x [member of] X,

(ii) there exist a constant M such that [[parallel]T(t)[parallel].sub.B(E)] < M for each t [greater than or equal to] 0,

(iii) there exist a continuous nondecreasing function T : [0, to) [less than or equal to] [0, [infinity]) and p [member of] [L.sup.1](0,b; [R.sub.+]) such that [absolute value of (F(t,x))] [less than or equal to] p(t)[PSI]([absolute value of (x)]), for a.e t [member of] [0,b] and each x [member of] X, with

[[integral].sup.b.sub.0] m(s)ds < [[integral].sup.[infinity].sub.C] dx/x + [PSI](x),

where

m(s) = max{M[[parallel]B[parallel].sub.B(E)], [M.sub.p(s)]}, c = M [[parallel][x.sub.0][parallel] + [m.summation over (k=1)] [c.sub.k]],

(iv) For each bounded B [subset or equal to] PC(0,b; X) and t [member of] [0,b], the set

[mathematical expression not reproducible]

is relatively compact in X, then the impulsive SDE (2.1) has at least one mild solution.

Proof. Transforming the problem (2.2) into a fixed point problem. Consider the operator [PSI] : PC(0, b; X) [right arrow] PC(0, b; X) defined by

(3.1) [mathematical expression not reproducible].

Clearly, the fixed point of [PHI] are mild solutions to the SDE (2.1) Subsequently, we will prove that [PHI] has a fixed point by Schaefer's fixed point theorem.

The proof will be given in several steps.

Step 1: [PHI] is continuous

Let [{[x.sub.n]}.sup.[infinity].sub.n=1] be a sequence in PC(0, b; X) such that [x.sub.n] [right arrow] x. We will show that [PHI]([x.sub.n]) [right arrow] [PHI](x). For each t [member of] [0, b], we have

[mathematical expression not reproducible].

Then,

[mathematical expression not reproducible]

Since [I.sub.k], where k = 1, 2, ... m are continuous, and [lim.sub.n[right arrow][infinity]] [parallel][PHI][x.sub.n] - [[PHI].sub.x][parallel] [right arrow] 0 this implies that [PHI] is continuous.

Step 2 : [PHI] maps bounded sets into bounded sets in PC(0, b; X).

It is enough to show that for any q > 0, there exists a [delta] > 0 such that for each x [member of] [B.sub.q] = {y [member of] PC(0, b; X):[[parallel]x[parallel].sub.PC] [less than or equal to] q}, one has [[parallel][PHI](x)[parallel].sub.PC] [less than or equal to] [delta].

By assumptions (i)-(ii) and the fact that F is L1-Caratheodory function, we have, for each t [member of] [0,6],

[mathematical expression not reproducible]

Step 3 : [PHI] maps bounded sets into equicontinuous sets of PC(0, b; X).

Let x [member of] PC(0, b; X), [t.sub.1] [greater than or equal to] 0 and [epsilon] be sufficiently small, then

[mathematical expression not reproducible].

As a consequence of Steps 1 to 3 and asumption (iv) of Theorem 3.1 together with the Arzela-Ascoli theorem, we can deduce that [PHI] : PC(0, b; X) [right arrow] PC(0, b; X) is a completely continuous operator.

Step 4: Now we show that the set [xi]([PHI]) := {x [member of] PC(0, b; X) : x = [lambda][PHI](x), 0 < [lambda] < 1} is bounded.

Let x [member of] [xi]([PHI]), then x = [lambda][PHI](x), for some 0 < [lambda] < 1. Thus, for each t [member of] [0, 6],

[mathematical expression not reproducible].

This implies by assumption (i) to (iii) that for each t [member of] [0,6],

[mathematical expression not reproducible].

Let us denote the right hand side of the above inequality by v(t), then we have

[absolute value of (x(t))] [less than or equal to] v(t) for every t [member of] [0,6]

Using the increasing character of [psi], we get v'(t) = [psi](v(t)) + M(v(t)) for a.e t [member of] [0,6]. This shows that [epsilon]([phi])is bounded.

As a consequence of Schaefer's fixed point theorem [14], we deduce that [phi] has a fixed point which is a mild solution of eqn (2.1), hence proved.

Asymptotic Stability of the Impulsive Stochastic Differential Equations.

Lemma 3.2. For any r > 1 and for arbitrary [L.sup.0.sub.2]-valued predictable process [mathematical expression not reproducible].

Definition 3.3. Let p [greater than or equal to] 2 be an integer. Eqn (2.2) is said to be stable in pth moment if for arbitrarily given [epsilon] > 0 there exist a [delta] > 0 such that whenever [[parallel][x.sub.0][parallel].sub.X] < [delta], E([sup.sub.t[greater than or equal to]0][[parallel]x(t)[parallel].sub.X]} < [epsilon].

Definition 3.4. Let p > 2 be an integer. Eqn (2.2) is said to be asymptotically stable in p-th moment if it is stable in p-th moment and for any [x.sub.0] [member of] X, [lim.sub.T[right arrow][infinity]] E{[sup.sub.t[greater than or equal to]T] [[parallel]x(t)[parallel].sup.p.sub.X]} = 0.

Using the definitions and lemma given above, we consider the asymptotic stability in p-th moment of mild solutions of eqn (2.1) by using the contraction mapping principle. Imposing some Lipschitz and linear growth conditions on the function F and G, assume that F(t, 0) = 0, G(t, 0) = 0 and [I.sub.k](0) = 0 (k = 1, 2, ...). Then eqn (2.1) has a trival solution when [x.sub.0] = 0. Let X be the space of all [[gamma].sub.0]-adapted process [phi](t, w): [0, [infinity]) x [OMEGA] [right arrow] R which is almost certainly continuous in t for fixed w [right arrow] [OMEGA]. Moreover, [phi](0, w) = [x.sub.0] and E[[parallel][phi](t,w)[parallel].sup.p.sub.X] [right arrow] 0 as t [right arrow] [infinity]. Also X is a Banach space when it is equipped with a norm defined by

[mathematical expression not reproducible].

We impose the following conditions:

1. A is the infinitesimal generator of a semigroup of bounded linear operators S(t), t [greater than or equal to] 0 on a Banach space X with [[parallel]S(t)[parallel].sub.X] [greater than or equal to] [Me.sup.-at], t [greater than or equal to] 0 for some constants M [greater than or equal to] 1 and 0 < a [member of] [R.sub.+].

2. The functions F and G satisfy the Lipschitz conditions and there exists a constant K for every t > 0 and x,y [member of] X such that

[mathematical expression not reproducible].

3. [I.sub.k] [member of] C(X,X) and there exist a constant [q.sub.k] such that [parallel][I.sub.k](x) - [I.sub.k](y)[parallel] [less than or equal to] [q.sub.k][parallel]x - y[parallel] for each x,y [member of] X (k = 1, ..., m).

Theorem 3.5. Assume the conditions (1-3) hold. Let p > 2 be an integer. If the inequality [3.sup.p-1][M.sup.p]([K.sup.p][a.sup.-p] + [K.sup.p][c.sub.p][(2a).sup.-p/2] + L) < 1 is satisfied, then the impulsive stochastic differential equation (2.1) is asymptotically stable in p-th moment: where [c.sub.p] = [(p(p - 1)/2).sup.p/2], L = [e.sup.-apT] E([[summation].sup.m.sub.k=1][[parallel][q.sub.k][parallel].sup.p.sub.x]).

Proof. Define a nonlinear operator [psi] : X [right arrow] X by

[mathematical expression not reproducible]

To prove the asymptotic stability, it is enough to show that the operator f has a fixed point in X. To prove this result, we use the contraction mapping principle.

To apply the contraction mapping principle, we first verify the mean square continuity of [psi] on [0, [infinity]). Let x [member of] X, [t.sub.1] [greater than or equal to] 0 and [absolute value of (r)] be sufficiently small then

[mathematical expression not reproducible]

We see that E[[parallel][F.sub.i]([t.sub.1] + r) - [F.sub.i]([t.sub.1])[parallel].sup.p.sub.X] [right arrow] 0, i = 1, 2, 4 as r [right arrow] 0. Moreover, by using Holder's inequality and Lemma 3.2, we obtain

[mathematical expression not reproducible].

as r [right arrow] 0 where [c.sub.p] = [(p(p - 1)/2).sup.p/2]. Thus if is continuous in p-th moment on [0, [infinity]).

Next we show that [psi](X) [subset] X, and obtain

[mathematical expression not reproducible].

Using conditions (1) and (3) we obtain [4.sup.p-1]E [[parallel]T(t)[x.sub.0][parallel].sup.p.sub.X] [less than or equal to] [4.sup.p-1][M.sup.p][e.sup.-pat][[parallel][x.sub.0][parallel].sup.p.sub.X] [right arrow] 0 as t [right arrow] [infinity].

Now, from conditions (1) and (2) and Holder's inequality, we have

[mathematical expression not reproducible].

For any x(t) [member of] X and any [epsilon] > 0 there exist a [t.sub.1] > 0 such that E[[parallel]x(s)[parallel].sup.p.sub.X] < [epsilon] for t [greater than or equal to] [t.sub.1]. Thus we obtain

[mathematical expression not reproducible].

As [e.sup.-at] - 0 as t [right arrow] [infinity] and by assumption in Theorem 3.5, there exist [t.sub.2] [greater than or equal to] [t.sub.1] such that for any t > [t.sub.2] we have

[mathematical expression not reproducible].

we obtain for any t [greater than or equal to] [t.sub.2]

[4.sup.p-1]E[[parallel][[integral].sup.t.sub.0] T(t- s)F(s,x(s))ds[parallel].sup.p.sub.X] < [epsilon]

that is to say,

[4.sup.p-1]E [[parallel][[integral].sup.t.sub.0] T(t - s)F(s, x(s))ds[parallel].sup.p.sub.X] [right arrow] 0 as t [right arrow] [infinity].

Now for any x(t) [member of] X, t [member of] [0, [infinity]), we obtain

[mathematical expression not reproducible]

Further, we have

[4.sup.p-1]E [[parallel][[integral].sup.t.sub.0] T(t - s)G(s, x(s))dW(s)[parallel].sup.p.sub.X] [right arrow] 0 as t [right arrow] [infinity]

E[[parallel]([psi]x)(t)[parallel].sup.p.sub.X] [right arrow] 0 as t [right arrow] [infinity]. In conclusion, [psi](X) [subset] X.

Finally, we prove that [psi] is a contraction mapping. To see this, let x,y [member of] X, s [member of] [0, T]. Then,

[mathematical expression not reproducible],

where L = [e.sup.-apT] E([[summation].sup.m.sub.k=1][[parallel][q.sub.k][parallel].sup.p.sub.X]).

Therefore, [psi] is a contraction mapping and hence there exist a unique fixed point x(*) in X which is the solution of eqn (2.1) with x(0) = [x.sub.0] and E[[parallel]x(t)[parallel].sup.p.sub.x] [right arrow] 0 as t [right arrow] [t.sub.0].

REFERENCES

[1] A. Anokhin, L. Berezansky, E. Braverman, Exponential stability of linear delay impulsive differential equations, Math. Anal. Appl., 193 (1995) 923-941.

[2] M. Benchohra, J. Henderson, and S. K. Ntouyas, Impulsive differential equations and inclusions Hindawi Publishing Corporation, vol 2., New York, 2006.

[3] P. Cheng, F. Q. Deng, Global exponential stability of impulsive stochastic functional differential systems, Statist. Probab. Lett., 80 (2010) 1854-1862.

[4] M. M. Fu and Z. X. Liu, Square-mean almost automorphic solutions for some stochastic differential equations, Pro. American Math. Soc., 138(10) (2010), 3689-3701.

[5] I. Karatzas, S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, Berlin, 1991.

[6] V. Lakshmikantham, D. D. Bainov, and P. S. Simenov, Theorem of Impulsive Differential Equations, World Scientific, Singapore, 1989.

[7] B. Liu, X. Liu, K. Teo, Q. Wang, Razumikhin-type theorems on exponential stability of impulsive delay systems, IMA J. Appl. Math, 71 (2006) 47-61.

[8] ] B. Liu, Stability of solutions for stochastic impulsive systems via comparison approach, IEEE Trans. Automat. Contro, 53 (2008) 2128-2133.

[9] X. Mao, Stochastic Differential Equations and Applications, Ellis Horwood, Chichester, UK, 1997.

[10] L. Pan, Existence of Mild Solutions for Impulsive Stochastic Differential Equations with Nonlocal Conditions, Differential Equations and Application, (2012), 485-494.

[11] S. G. Peng, B. G. Jia, Some criteria on pth moment stability of impulsive stochastic functional differential equations, Statist. Probab. Lett., 80 (2010) 1085-1092.

[12] Q. Wang, X. Z. Liu, Impulsive stabilization of delay differential systems via the Lyapunov-Razumikhin method, Appl. Math. Lett., 20 (8) (2007) 839-845.

[13] Q. J. Wu, J. Zhou, L. Xiang, Global exponential stability of impulsive differential equations with any time delays, Appl. Math. Lett, 23 (2010) 143-147.

[14] E. Zeidler, Nonlinear Functional Anlysis and its Applications, Springer- Verlag New York, (1986).

F. BANDELE AND M. O. OGUNDIRAN

Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria
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Date:Mar 1, 2017
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