International Journal of Stochastic Analysis
Volume 2015, Issue 1 834194
Research Article
Open Access

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

Sheila Bishop, 

Corresponding Author

Sheila Bishop

Department of Mathematics, Covenant University, KM 10 Idiroko Road, PMB 1023, Ota, Ogun State, Nigeria covenantuniversity.edu.ng

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First published: 28 July 2015
https://doi.org/10.1155/2015/834194
Citations: 1
Academic Editor: Alexander M. Krasnosel’skii

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 E1, F1, G1, and H1 are hypermaximal monotone multivalued maps and E2, F2, G2, and H2 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 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. 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 r. , clos(S), (resp., Comp(S)), , ρ(A, B), , ak, aηξ,k, Φηξ,k(·), 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 (t, x) → Φ(t, x)(η, ξ), decomposable subset of a space, and locally absolutely P-integrable and adapted multivalued stochastic process Φ, 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 P as the right-hand side. For existence of solution of inclusion (3) and the explicit form of the map (t, X) → P(t, X)(η, ξ) 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 a ranges in a subset A of such that the set A(η, ξ) = 〈η, aξ〉 : a ∈ A 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 W(t) ≠ t. Hence the results here are weaker than the results in [1].

Let P1 be hypermaximal monotone, and the map (t, x) → P2(t, x)(η, ξ) appearing in (4) is assumed to satisfy the following conditions with W(t) ≠ t:
  • (1)

    The map (t, x)∈→P2(t, x)(η, ξ) is measurable.

  • (2)

    There exists a map lying in and the function increasing and continuous with W(0) = 0 such that

    ()
    a.e., t ∈ [0, T], and for each pair .

  • (3)

    There exists a β which lies in such that

    ()

For the map , define the map 〈η, aξ〉 → ϕP(aηξ) by
()

Remark 1. Conditions (1) and (3) are similar to conditions S(i) and S(iii) in [1], while condition S(ii) has been modified to accommodate the general Lipschitz condition defined above. If we take W(t) = t, then condition (2) reduces to condition S(ii) 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 (t, x) → P(t, x)(η, ξ) is measurable,

  • (ii)

    the map (t, ·) → P(t, ·)(η, ξ) is l.s.c.

Then the map ϕP given by (7) is lower semicontinuous (l.s.c) from A(η, ξ) into if and only if there exists a continuous map such that, for every a ∈ A, aηξ ≡ 〈η, aξ〉 ∈ A(η, ξ),
()

Lemma 3. Let the multivalued stochastic process be l.s.c. Assume that

  • (i)

    and are continuous,

  • (ii)

    for every aηξ ∈ A(η, ξ) the set Φ(aηξ) 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 (Hp) see [15]. We adopt definition 2.1 concerning the solution of (Hp) and remark 2.1 all in [1]. Hence condition (11) in [1] follows.
Let satisfy conditions (1)–(3). Consider the Cauchy problem
()
where a ∈ A0.

Definition 4. A function is called a solution of (Ha) if there exists , a selection of P2(·, x(·, a))(η, ξ) such that x(·, a) is a weak solution of the Cauchy problem Hp(·,a).

We denote by ST(a) the set of all solutions of (Ha) and prove a continuous selection theorem from the map ST(a)(η, ξ), where
()
Just as in [8], an important consequence of our main result is that the set map 〈η, aξ〉 → ST(a)(η, ξ) can be continuously represented in the form
()
with the Lipschitz condition W(t) ≠ t. See Corollaries 3.2 and  3.3 in [2]. This generalizes all results in the literature established under the Lipschitz condition W(t) = t.

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  3.1 in [1] and  3.1 in [8] by highlighting the major changes due to condition (iii).

Theorem 5. Assume that the maps satisfy the following conditions:

  • (i)

    P1 is hypermaximal monotone.

  • (ii)

    (t, x) → P2(t, x)(η, ξ) is measurable.

  • (iii)

    There exists a map lying in , such that

    ()
    a.e. in I, where W(t) ≠ t.

  • (iv)

    There exists such that

    ()

If , then there exists an adapted stochastic process such that
  • (i)

    〈η, x(·, a)ξ〉 ∈ ST(a)(η, ξ) for every a ∈ A0;

  • (ii)

    〈η, aξ〉 → 〈η, x(·, a)ξ〉 is continuous from A0(η, ξ) to .

Proof. Let a ∈ A0 and be the unique weak solution of the Cauchy problem

()
For Kηξ and β defined by (iii) and (iv), we define by
()
By remark  2.1 in [1], the map 〈η, aξ〉 → 〈η, x0(·, a)ξ〉 is weakly continuous from A0(η, ξ) to . Hence from (16), it follows that b(·) is continuous from A0(η, ξ) to . And we have
()
for each aηξ ∈ A0(η, ξ).

As in [8] we fix ϵ > 0 and set ϵ = ϵ/2n+1, . Define and by (14) and (15) in [1]. Using (16) and Lemma 2, ϕ0(·) is lower semicontinuous (l.s.c.) and for each aη,ξ ∈ A0(η, ξ), ϕ0(aηξ) ≠ 0, and W(t) ≠ t. Again by Lemma 3, there exists , a continuous selection of ϕ0(·). Set p0(t, a)(η, ξ) = φ0(aηξ)(t) as in [1]; then p0(·, a)(η, ξ) is continuous, p0(t, a)(η, ξ) ∈ P2(t, a)(η, ξ), and

()
If we set , where aηξ is as defined, then, for each a ∈ A0, we can define βn(aηξ)(t), n ≥ 1 as follows:
()
Thus by (19) βn(·) is continuous from A0(η, ξ) to since b(·) is continuous. Now if is the unique solution of the Cauchy problem
()
then, by (11) in [1], we have
()
for each aηξ ∈ A0(η, ξ) and t ∈ I∖{0}. Now set 〈η, pn(s, a)ξ〉 ≡ pn(s, a)(η, ξ) and assume that there exist sequences and such that, for each n ≥ 1, (a), (b), and (d) in [1] hold in this case while (c) becomes
()
where W is due to the Lipschitz function K. We now obtain the following by (22) and (11) in [1], t ∈ I∖{0}
()
Since P 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 φn+1(aηξ)(t) still admits a continuous selection of Φn+1(·).

If we set 〈η, pn+1(t, a)ξ〉 = φn+1(aηξ)(t) for aηξ ∈ A0(η, ξ), t ∈ I, we have that pn+1 satisfies the properties (a), (b) in [1] and (22); hence by (24), we obtain

()
Again by (22) and (23), we have
()
Since aηξ → ‖b(aηξ)(t)‖ is continuous, then it is locally bounded. It follows by (26) that the sequence satisfies the Cauchy condition uniformly. If p(·, a) is the limit of the given sequence, then aηξ → 〈η, p(·, a)ξ〉 is also weakly continuous from A0(η, ξ) into .

Now if we use (23) and (26), we get

()
Hence {〈η, xn(·, a)ξ〉} is Cauchy in with respect to a. Then the map aηξ → 〈η, xn(·, a)ξ〉 is weakly continuous from A0(η, ) to and so also the map 〈η, xn(·, a)ξ〉 → 〈η, x(·, a)ξ〉 uniformly and
()
Therefore, the result (22) in [1] holds here. If we let p0 = p and x1(·, a) be the unique weak solution of the Cauchy problem (20), we obtain by (11) in [1]
()
If n → ∞, then 〈η, x1(·, a)ξ〉 ≡ 〈η, x(·, a)ξ〉. Therefore, x(·, a) 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.

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