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

Abstract

We establish existence of a continuous selection of multifunctions associated with quantum stochastic evolution inclusions under a general Lipschitz condition. The coefficients here are multifunctions but not necessarily Lipschitz.

1. Introduction

We consider the following Cauchy problem defined in [1]: where , , , and are hypermaximal monotone multivalued maps and , , , and 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], 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].

2. Preliminaries

All through this work, as in [2, 8], we adopt the definitions and notations of the following spaces, subspaces, and sets: , , , , , , , and for a fixed Hilbert space . , , (resp., ), , , , , , , and so forth. For the completion of the space and the generated topology and many more we refer the reader to [2, 8].

For the definitions of lower semicontinuous multivalued map (l.s.c.) , measurability and measurability of a multifunction , decomposable subset of a space, and locally absolutely -integrable and adapted multivalued stochastic process , we refer the reader to [2, 8].

We consider the following quantum stochastic differential inclusion (QSDI) defined in [2]: where the multivalued stochastic processes and 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 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 is hypermaximal monotone and the sesquilinear form valued map satisfies a general Lipschitz condition defined in [8]. The point ranges in a subset of such that the set is compact in .

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].

Let be hypermaximal monotone, and the map appearing in (4) is assumed to satisfy the following conditions with :(1)The map is measurable.(2)There exists a map lying in and the function increasing and continuous with such that a.e., , and for each pair .(3)There exists a which lies in such that For the map , define the map by

Remark 1. Conditions (1) and (3) are similar to conditions and in [1], while condition has been modified to accommodate the general Lipschitz condition defined above. If we take , then condition (2) reduces to condition 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,(ii)the map is l.s.c.Then the map given by (7) is lower semicontinuous (l.s.c) from into if and only if there exists a continuous map such that, for every , ,

Lemma 3. Let the multivalued stochastic process be l.s.c. Assume that(i) and are continuous,(ii)for every the set defined by (10) in [1] is nonempty.Then the multivalued stochastic process is l.s.c. and therefore admits a continuous selection.

For and , we consider the Cauchy problem 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.

Let satisfy conditions (1)–(3). Consider the Cauchy problem where .

Definition 4. A function is called a solution of if there exists , a selection of such that is a weak solution of the Cauchy problem .

We denote by the set of all solutions of and prove a continuous selection theorem from the map , where Just as in [8], an important consequence of our main result is that the set map can be continuously represented in the form with the Lipschitz condition . See Corollaries and   in [2]. This generalizes all results in the literature established under the Lipschitz condition .

3. 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   in [1] and   in [8] by highlighting the major changes due to condition (iii).

Theorem 5. Assume that the maps satisfy the following conditions:(i) is hypermaximal monotone.(ii) is measurable.(iii)There exists a map lying in , such that a.e. in , where .(iv)There exists such that If , then there exists an adapted stochastic process such that(i) for every ;(ii) is continuous from to .

Proof. Let and be the unique weak solution of the Cauchy problem For and defined by (iii) and (iv), we define by By remark   in [1], the map is weakly continuous from to . Hence from (16), it follows that is continuous from to . And we have for each .
As in [8] we fix and set , . Define and by (14) and (15) in [1]. Using (16) and Lemma 2, is lower semicontinuous (l.s.c.) and for each , , and . Again by Lemma 3, there exists , a continuous selection of . Set as in [1]; then is continuous, , and If we set , where is as defined, then, for each , we can define , as follows: Thus by (19) is continuous from to since is continuous. Now if is the unique solution of the Cauchy problem then, by (11) in [1], we have for each and . Now set and assume that there exist sequences and such that, for each , (a), (b), and (d) in [1] hold in this case while (c) becomes where is due to the Lipschitz function . We now obtain the following by (22) and (11) in [1], Since is maximal monotone and hence hypermaximal monotone, we get By (24) and Lemma 2, the multivalued map defined by (19) in [1] is l.s.c. with decomposable closed nonempty values. Then by Lemma 3, the sesquilinear form valued map still admits a continuous selection of .
If we set for , , we have that satisfies the properties (a), (b) in [1] and (22); hence by (24), we obtain Again by (22) and (23), we have Since is continuous, then it is locally bounded. It follows by (26) that the sequence satisfies the Cauchy condition uniformly. If is the limit of the given sequence, then is also weakly continuous from into .
Now if we use (23) and (26), we get Hence is Cauchy in with respect to . Then the map is weakly continuous from to and so also the map uniformly and Therefore, the result (22) in [1] holds here. If we let and be the unique weak solution of the Cauchy problem (20), we obtain by (11) in [1] If , then . Therefore, is the weak solution of (20), and the result holds here under the general Lipschitz condition.

Conflict of Interests

The author declares that there is no conflict of interests.