Existence and Variational Stability of Solutions of Kurzweil Equations Associated with Quantum Stochastic Differential Equations
Abstract:
The role of generalized ordinary di erential equation (Kurzweil equation) in applying the technique of topological dynamics to the study of classical ordinary di erential 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 di erential equations. In this work, existence and uniqueness of solution of quantum stochastic di erential 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 di erential equation is applicable. Existence of solution of quantum stochastic di erential 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 nite variations. The theory of Kurzweil equations associated with quantum stochastic di erential equation provides a basis for future application of the technique of topological dynamics to the study of quantum stochastic di erential equationORDER COMPLETE MATERIAL (CHAPTER 1-5)
