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

where E1, F1, G1, andH1 are hypermaximal monotone multivalued maps and E2, F2, G2, andH2 are multifunctions but not necessarily Lipschitz. As observed by [2], problems of continuous selection, features of reachable sets, and the solution sets of classical differential inclusions have attracted considerable attention [3–6]. The existence and nonuniqueness of solutions of such inclusions have been investigated to a large extent. See [7] and the references therein. 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]. In order to generalize the results in the literature concerning QSDI, in [8] existence of continuous selections of solutions sets of non-Lipschitzian quantum stochastic differential inclusions was considered. It was proved that certain inclusion problems do not necessarily satisfy the Lipschitz condition defined in [2, 13]. In [8], themap x → P(t, x)(η, ξ) satisfied a general Lipschitz condition with values that are closed but not necessarily convex or bounded subsets of the field of complex numbers.Thiswork is concernedwith similar results established in [8] where the coefficients are not necessarily Lipschitz. The results here generalize existing results in the literature [1] concerning quantum stochastic evolution inclusions (QSEIs). The rest of this paper is organized as follows: in Section 2, we present the foundations for establishing the major results. In Section 3, we will establish the major results. Our method will be a blend of the methods applied in [1, 8].

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.See [7] and the references therein.
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].
In order to generalize the results in the literature concerning QSDI, in [8] existence of continuous selections of solutions sets of non-Lipschitzian quantum stochastic differential inclusions was considered.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.This work is concerned with similar results established in [8] where the coefficients are not necessarily Lipschitz.The results here generalize existing results in the literature [1] concerning quantum stochastic evolution inclusions (QSEIs).
The rest of this paper is organized as follows: in Section 2, we present the foundations for establishing the major results.In Section 3, we will establish the major results.Our method will be a blend of the methods applied in [1,8].
We consider the following quantum stochastic differential inclusion (QSDI) defined in [2]: where the multivalued stochastic processes , , ,  ∈  2 loc ( × Ã) mvs and (, 0) ∈  × Ã is fixed.The equivalent form of inclusion (1.2) established in [13] is given by Inclusion ( 2) is understood in the sense of Hudson and Parthasarathy [14] while inclusion (3) is a first order nonclassical ordinary differential inclusion with a sesquilinear form valued map  as the right-hand side.For existence of solution of inclusion (3) and the explicit form of the map (, ) → (, )(, ) appearing in inclusion (3) see [13] and also see [7] for nonuniqueness of solution of (3).We employ the locally convex topological ( Ã) space of noncommutative stochastic processes defined in [13].
In this work, we consider the following evolution problem given by where the sesquilinear form valued map  1 :  × Ã → 2 sesq(D⊗E) 2 is hypermaximal monotone and the sesquilinear form valued map  2 :  × Ã → 2 sesq(D⊗E) 2 satisfies a general Lipschitz condition defined in [8].The point  ranges in a subset  of Ã such that the set (, ) = ⟨, ⟩ :  ∈  is compact in C.
Motivated by the result in [8], we extend the results in [1], to a class of evolution inclusion that depends on a more general Lipschitz condition () ̸ = .Hence the results here are weaker than the results in [1].
We adopt the proof of the following results established in [1] since the proof of these results is independent of the Lipschitz function.

Lemma 2. Consider the multivalued stochastic process
, and assume that (i) the map (, ) → (, )(, ) is measurable, Then the map   given by (7) For the existence of a unique weak solution of the Cauchy problem (  ) see [15].We adopt definition 2.1 concerning the solution of (  ) and remark 2.1 all in [1].Hence condition (11) in [1] follows.

Major Results
In this section, we present our major results under the general Lipschitz condition defined above.We will establish the result by employing similar argument employed in the proof of Theorems 3.1 in [1] and 3.1 in [8] by highlighting the major changes due to condition (iii).