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

Authors: Sheila Bishop
Journal: International Journal of Stochastic Analysis
 (2015)

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## 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.


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## Body

## 1. Introduction

We consider the following Cauchy problem defined in [1]: (1)dXt∈E2t,Xtd∧πt+F2t,XtdAgt+G2t,XtdAf+t+H2t,Xtdt-E1t,Xtd∧πt+F1t,XtdAgt+G1t,XtdAf+t+H1t,XtdtX0=a,t∈0,T,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 Section2, 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: R⊗Γ(Lγ2R+), Ad(A~), Ad(A~)wac, Lloc⁡p(A~), (Ad(A~)wac,τw), L1(I,D⊗_E), C(I,D⊗_E), and Lr,Loc∞(R+) for a fixed Hilbert space r. τ1,τc,A~, clos(S), (resp., Comp(S)), clos(A~), ρ(A,B), clos(C), ak, aηξ,k, Φηξ,k(·), and so forth. For the completion of the space (Ad(A~)wac,τwac) 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.)Φ:S→2A~, 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]:(2)dXt∈Et,Xtd∧πt+Ft,XtdAgt+Gt,XtdAf+t+Ht,Xtdt,X0=a,almost  allt∈I=0,T,where the multivalued stochastic processes E,F,G,H∈Lloc⁡2(I×A~)mvs and (a,0)∈I×A~ is fixed. The equivalent form of inclusion (1.2) established in [13] is given by(3)ddtη,Xtξ∈Pt,Xη,ξXt0=X0,t∈t0,T.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 (A~) space of noncommutative stochastic processes defined in [13].In this work, we consider the following evolution problem given by(4)ddtη,Xtξ∈P2t,Xη,ξ-P1t,Xη,ξX0=a,t∈0,T,where the sesquilinear form valued map P1:I×A~→2sesq(D⊗_E)2 is hypermaximal monotone and the sesquilinear form valued map P2:I×A~→2sesq(D⊗_E)2 satisfies a general Lipschitz condition defined in [8]. The point a ranges in a subset A of A~ such that the set A(η,ξ)=η,aξ:a∈A 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 W(t)≠t. Hence the results here are weaker than the results in [1].LetP1 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 mapKηξp:[0,T]→R+ lying in Lloc⁡I([0,T]) and the function W:[0,∞)→R increasing and continuous with W(0)=0 such that(5)ρP2t,xη,ξ-P2t,yη,ξ≤KηξptWx-yηξ,a.e., t∈[0,T], and for each pair x,y∈A~.(3)
There exists aβ which lies in Lloc⁡1([0,T]) such that(6)d0,P2t,0η,ξ≤βηξ,a.e.,t∈I.For the map P:I×A~→2sesq(D⊗_E)2, define the map η,aξ→ϕP(aηξ) by(7)ϕPaηξ=vηξ∈L1I,D⊗_E:vηξ∈Pt,aη,ξ,a.e.,t∈I.Remark 1.
Conditions (1) and (3) are similar to conditionsS(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 processP:I×A~→2sesq(D⊗_E)2, 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 D if and only if there exists a continuous map β:A(η,ξ)→L1(I,R) such that, for every a∈A, aηξ≡η,aξ∈A(η,ξ),(8)d0,Pt,aη,ξ≤βaηξt,a.e.,t∈I.Lemma 3.
Let the multivalued stochastic processϕ:A(η,ξ)→D be l.s.c. Assume that(i)
φ:A(η,ξ)→L1(I,D⊗_E) and ϕ:A(η,ξ)→L1(I,R) are continuous,(ii)
for everyaηξ∈A(η,ξ) the set Φ(aηξ) defined by (10) in [1] is nonempty.Then the multivalued stochastic process ϕ:A(η,ξ)→D is l.s.c. and therefore admits a continuous selection.Fora∈D(P1(t,·)(η,ξ))¯ and p∈L1(I,D⊗_E), we consider the Cauchy problem(9)Hpddtη,Xtξ∈η,ptξ-P1t,Xtη,ξ,X0=a.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.LetP2:I×A~→2sesq(D⊗_E)2 satisfy conditions (1)–(3). Consider the Cauchy problem(10)Haddtη,Xtξ∈P2t,Xtη,ξ-P1t,Xtη,ξ,X0=a,where a∈A0.Definition 4.
A functionx(·,a):I→A~ is called a solution of (Ha) if there exists η,p(t)ξ∈L1(I,sesq(D⊗_E)), a selection of P2(·,x(·,a))(η,ξ) such that x(·,a) is a weak solution of the Cauchy problem Hp(·,a).We denote byST(a) the set of all solutions of (Ha) and prove a continuous selection theorem from the map ST(a)(η,ξ), where(11)STaη,ξ=η,Φξ∖Φ∈STa.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(12)gη,aξ,u=STaη,ξwith 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 Theorems3.1 in [1] and  3.1 in [8] by highlighting the major changes due to condition (iii).Theorem 5.
Assume that the mapsP1,P2:I×A~→2sesq(D⊗_E)2 satisfy the following conditions:(i)
P1 is hypermaximal monotone.(ii)
(t,x)→P2(t,x)(η,ξ) is measurable.(iii)
There exists a mapKηξ:[0,T]→R+ lying in Lloc1([0,T]), such that(13)ρP2t,xη,ξ,P2t,yη,ξ≤KηξtWx-yηξa.e. in I, where W(t)≠t.(iv)
There existsβ∈Lloc1(I) such that(14)d0,P2t,0η,ξ≤βηξ,a.e.,t∈I.If A0⊂A~, then there exists an adapted stochastic process x:I×A0→sesq(D⊗_E) 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 C(I,D⊗_E).Proof.
Leta∈A0 and x0(·,a):I→A~ be the unique weak solution of the Cauchy problem(15)ddtη,Xtξ∈-P1t,Xtη,ξ,X0=a.For Kηξ and β defined by (iii) and (iv), we define b:A0(η,ξ)→Lloc⁡1(I) by(16)baηξt=βηξt+KηξtWx0t,aηξ.By remark  2.1 in [1], the map η,aξ→η,x0(·,a)ξ is weakly continuous from A0(η,ξ) to C(I,D⊗_E). Hence from (16), it follows that b(·) is continuous from A0(η,ξ) to Lloc⁡1(I). And we have(17)d0,P2t,x0t,aη,ξ≤baηξt,a.e.  inIfor each aηξ∈A0(η,ξ).
As in [8] we fix ϵ>0 and set ϵ=ϵ/2n+1, n∈N. Define Φ0:A0(η,ξ)→2L(I,D⊗_E) and ϕ0:A0(η,ξ)→2L(I,D⊗_E) 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 φ0(η,ξ)→L(I,D⊗_E), 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(18)p0t,aη,ξ≤baηξt+ϵ0,a.e.t∈I.If we set Mηξ=∫0tKηξ(s)ds, where aηξ is as defined, then, for each a∈A0, we can define βn(aηξ)(t), n≥1 as follows:(19)βnaηξt=∫0taηξsMηξt-Mηξsn-1dsn-1!+WT∑i=0nϵiMηξtn-1n-1!,t∈I.Thus by (19) βn(·) is continuous from A0(η,ξ) to Lloc⁡1(I,R) since b(·) is continuous. Now if x1(·,a):I→A~ is the unique solution of the Cauchy problem(20)ddtη,Xtξ∈η,p0tξ-P1t,Xtη,ξ,X0=athen, by (11) in [1], we have(21)η,x1tξ-η,x0tξ≤∫0tp0s,aη,ξds≤∫0tbaηξsds+ϵ0T<β1aηξt,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 {pn(·,a)}n∈N and {xn(·,a)}n∈N such that, for each n≥1, (a), (b), and (d) in [1] hold in this case while (c) becomes(22)η,p1t,aξ-η,pn-1t,aξ≤WKηξtβnt,a.e.t∈I,where W is due to the Lipschitz function K. We now obtain the following by (22) and (11) in [1], t∈I∖{0}(23)η,xn+1t,aξ-η,xnt,aξ≤∫0tpns,aη,ξ-pn-1s,aη,ξds≤W∫0tKηξsβnaηξsds=W∫0tbaηξsMηξt-Mηξsnn!ds+WT∑i=0nϵiMηξtnn!<Wβ1aηξt.Since P is maximal monotone and hence hypermaximal monotone, we get(24)dη,pt,aξ,P2t,xn+1t,aη,ξ≤KηξtWxn+1t,a-xnt,aηξ<KηξtWβn+1aηξt.By (24) and Lemma 2, the multivalued map ϕn+1:A0(η,ξ)→2L1(I,D⊗_E) 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 setη,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(25)Φn+1aηξ=cl⁡vηξ∈ϕn+1aηξ:vηξ-η,pnkt,aξ<KηξtWβn+1aηξt,t∈I.Again by (22) and (23), we have(26)η,pn·,aξ-η,pn-1·,aξ=∫0Tpns,aη,ξ-pn-1s,aη,ξds≤W∫0TbaηξsMηξT-Mηξsnn!ds+WT∑i=0nϵiMηξTnn!≤WKηξt1nn!baηξ+Tϵ.Since aηξ→b(aηξ)(t) is continuous, then it is locally bounded. It follows by (26) that the sequence {pn(·,a)}n∈N 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 L1(I,D⊗_E).
Now if we use (23) and (26), we get(27)η,xn+1·,aξ-η,xn·,aξ≤η,pn·,aξ-η,pn-1·,aξ≤WKηξt1nn!baηξ1+Tϵ.Hence {η,xn(·,a)ξ} is Cauchy in C(I,D⊗_E) with respect to a. Then the map aηξ→η,xn(·,a)ξ is weakly continuous from A0(η,) to C(I,D⊗_E) and so also the map η,xn(·,a)ξ→η,x(·,a)ξ uniformly and(28)dη,pnt,aξ,Pt,xt,aη,ξ≤KηξtWxn·,a-x·,aηξ.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](29)η,xnt,aξ-η,x1t,aξ≤∫0tη,pns,aξ-η,ps,aξds.If n→∞, then η,x1(·,a)ξ≡η,x(·,a)ξ. Therefore, x(·,a) is the weak solution of (20), and the result(30)η,x·,aξ∈STaη,ξholds here under the general Lipschitz condition.



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*Source: 834194-2015-07-28.xml*