IMPACT: International Journal of Research in Applied, Natural and Social Sciences (IMPACT: IJRANSS) ISSN(E): 2321-8851; ISSN(P): 2347-4580 Vol. 2, Issue 1, Jan 2014, 51-60 © Impact Journals ON EXISTENCE OF SOLUTION FOR IMPULSIVE PERTURBED QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS AND THE ASSOCIATED KURZWEIL EQUATIONS S. A. BISHOP & O. O. AGBOOLA Department of Mathematics Covenant University, Ota, Ogun State, Nigeria ABSTRACT Existence of solution of impulsive Lipschitzian quantum stochastic differential equations (QSDEs) associated with the Kurzweil equations are introduced and studied. This is accomplished within the framework of the Hudson-Parthasarathy formulation of quantum stochastic calculus and the associated Kurzweil equations. Here again, the solutions of a QSDE are functions of bounded variation, that is they have the same properties as the Kurzweil equations associated with QSDEs introduced in [1, 4]. This generalizes similar results for classical initial value problems to the noncommutative quantum setting. KEYWORDS: Impulsive, Kurzweil Equations, Bounded Variation, Noncommutative Stochastic Processes INTRODUCTION Impulsive effects exist widely in many evolution processes in which states are changed abruptly at certain moments of time, involving such fields as biology, medicine, economics, mechanics, electronics, Physics, etc [2, 3, 8, 9, 12, 13, 15, 18]. Thus the qualitative properties of the mathematical theory of impulsive differential systems are very important. A lot of dynamical systems have variable structure subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of components, sudden environmental changes, etc [8, 10, 20-22]. Recently, stochastic differential equations have attracted a great attention, since they have been used extensively in many areas of application including finance and social science [8-10, 12, 13, 20-22] The existence, uniqueness and asymptotic behavior of solutions of stochastic differential equations have been considered by many authors [2, 3, 8, 9, 13]. However, within the framework of the Hudson and Parthasarathy [11] formulation of QSDE not much has been done. In [15] the existence of QSDE that exhibit impulsive effects was established using fixed point theorem. In [1], the equivalence of the non classical ODE (QSDE) and the associated Kurzweil equation was established along side with some numerical examples. It is worth mentioning that the results in [1] have proved to be very efficient when compared with results obtained from other schemes. Again using this method in [4], we studied Measure quantum stochastic differential systems (systems that exhibit discontinuous solutions) with examples. We established such results by considering the associated Kurzweil equations [4, 9, 17]. The motivation for studying the existence of solution for impulsive QSDE associated with Kurzweil equations is so that we can subsequently use the method in [1] to obtain similar approximate results for this class of QSDEs. In this paper we describe another approach to systems that exhibit impulsive behaviour. We rely on the formulations of [1] concerning the equivalent QSDE and the associated Kurzweil equation. The methods are simple extension of the methods applied in [14, 16-19] to this non commutative quantum setting involving unbounded linear