Existence And Variational Stability Of Solutions Of Kurzweil Equations Associated With Quantum Stochastic Differential Equations

Abstract

The role of generalized ordinary differential equation (Kurzweil equation) in applying the technique of topological dynamics to the study of classical ordinary differential equation as outlined in [3, 4, 47, 51-58, 88-90] is a major motivation for studying this class of equations associated with the weak forms of the Lipschitzian quantum stochastic differential equations. In this work, existence and uniqueness of solution of quantum stochastic differential equations associated with the Kurzweil equations under a more general Lipschitz condition were established. The results here generalize the results in the existing literatures thereby extending the class of equations for which the theory of quantum stochastic differential equation is applicable. Existence of solution of quantum stochastic differential equation, enabled one to investigate and establish other qualitative properties of solution such as variational stability, variational attracting, variational asymptotic stability, converse variational stability and continuous dependence of solution on a parameter. The results are established within the framework of the topological linear space of processes of finite variations. The theory of Kurzweil equations associated with quantum stochastic differential equation provides a basis for future application of the technique of topological dynamics to the study of quantum stochastic differential equation.