Continuous Selections of Solution Sets of Lipschitzian Quantum Stochastic Differential Inclusions

Abstract

We prove that a multifunction associated with the set of solutions of Lipschitzian quantum stochastic differential inclusion (QSDI) admits a selection continuous from some subsets of complex numbers to the space of the matrix elements of adapted weakly absolutely continuous quantum stochastic processes. In particular, we show that the solution set map as well as the reachable set of the QSDI admit some continuous representations.

On Semicontinuous Nonclassical Ordinary Differential Inclusions with Nonlocal Condition

“…In this section we shall adopt the fundamental concepts and structures as in the references (Ayoola, 2004;Ekhaguere, 2007). We employ the space A ɶ of noncommutative stochastic processes whose topology τ w is generated by the family of seminorms…”

On Continuous Selection Sets of Non-Lipschitzian Quantum Stochastic Evolution Inclusions

“…As observed by [2], problems of continuous selection, features of reachable sets, and the solution sets of classical differential inclusions have attracted considerable attention [3][4][5][6]. The existence and nonuniqueness of solutions of such inclusions have been investigated to a large extent.…”

“…Existence of continuous selections of multifunctions associated with the sets of solutions of Lipschitzian and non-Lipschitzian quantum stochastic differential inclusions (QSDIs) has been considered in [2,8], while the existence of solution of quantum stochastic evolution arising from hypermaximal monotone coefficients was established in [9]. Also in [10,11] several results have been established concerning some properties of the solution sets of QSDIs.…”

“…Results concerning the topological properties of solution sets of Lipschitzian QSDI were also considered in [12]. In [1], results on continuous selections of solution sets of quantum stochastic evolution inclusions (QSEIs) were established under the Lipschitz condition defined in [2,13].…”

“…It was proved that certain inclusion problems do not necessarily satisfy the Lipschitz condition defined in [2,13]. In [8], the map → ( , )( , ) satisfied a general Lipschitz condition with values that are closed but not necessarily convex or bounded subsets of the field of complex numbers.…”