On Semicontinuous Nonclassical Ordinary Differential Inclusions with Nonlocal Condition

In recent times, several researchers have carried out studies on differential inclusions with lots of emerging results such as semilinear evolution inclusions with nonlocal conditions and upper semicontinuous multivalued maps. See (Aitalioubrahim, 2011; Boucherif, 2009; Gatsori et al., 2004; Zhu and Li, 2008). (Cardinali et al., 2008; Cardinali and Rubbioni, 2012), established some results on local mild solutions and impulsive mild solutions for semilinear inclusions. Aitalioubrahim (2011) established some new results on mild solution of semilinear noncovex differential inclusion. The following were considered: the case when the set-valued map is a non-open multifunction with measurability and Lipschitz continuity conditions imposed on the first and second variables respectively. These recent studies have shown that differential inclusions and problems with nonlocal conditions are of more practical applications in real life when compared to problems with local conditions (Antosiewicz and Cellina, 1975; Bishop et al., 2016; Cellina, 1988). Within the setting of quantum stochastic calculus Ekhaguere (1992), not much has been done. However, Ayoola (2008), Bishop and Ayoola (2015), studied the topological properties of solution sets for Lipschitz and non Lipschitz Quantum Stochastic Differential Inclusions (QSDIs) under the local conditions with the multivalued stochastic processes been continuous. Ogundiran and Payne (2014) considered a unified treatment of existence of solution of both upper and lower semicontinuous quantum stochastic differential inclusions under the local condition. Bishop et al. (2016) established some new results on impulsive nonclassical ordinary differential equations. The initial conditions are not necessarily local and the multivalued stochastic processes are lower semicontinuous. Problems with lower semicontinuous maps have more application especially when dealing with dynamical systems. In this study, we present new results on QSDIs with nonlocal conditions where the multivalued stochastic processes are lower semicontinuous. This problem will have practical applications in the theory of quantum dynamical systems. The result of this paper generalizes some of the results of Bishop et al. (2016) and also extends the work of Aitalioubrahim (2011) to the class of noncummutative quantum setting. We consider the following lower semicontinuous quantum stochastic evolution inclusion:


Introduction
In recent times, several researchers have carried out studies on differential inclusions with lots of emerging results such as semilinear evolution inclusions with nonlocal conditions and upper semicontinuous multivalued maps.See (Aitalioubrahim, 2011;Boucherif, 2009;Gatsori et al., 2004;Zhu and Li, 2008).(Cardinali et al., 2008;Cardinali and Rubbioni, 2012), established some results on local mild solutions and impulsive mild solutions for semilinear inclusions.Aitalioubrahim (2011) established some new results on mild solution of semilinear noncovex differential inclusion.The following were considered: the case when the set-valued map is a non-open multifunction with measurability and Lipschitz continuity conditions imposed on the first and second variables respectively.These recent studies have shown that differential inclusions and problems with nonlocal conditions are of more practical applications in real life when compared to problems with local conditions (Antosiewicz and Cellina, 1975;Bishop et al., 2016;Cellina, 1988).
Within the setting of quantum stochastic calculus Ekhaguere (1992), not much has been done.However, Ayoola (2008), Bishop and Ayoola (2015), studied the topological properties of solution sets for Lipschitz and non Lipschitz Quantum Stochastic Differential Inclusions (QSDIs) under the local conditions with the multivalued stochastic processes been continuous.Ogundiran and Payne (2014) considered a unified treatment of existence of solution of both upper and lower semicontinuous quantum stochastic differential inclusions under the local condition.Bishop et al. (2016) established some new results on impulsive nonclassical ordinary differential equations.The initial conditions are not necessarily local and the multivalued stochastic processes are lower semicontinuous.Problems with lower semicontinuous maps have more application especially when dealing with dynamical systems.
In this study, we present new results on QSDIs with nonlocal conditions where the multivalued stochastic processes are lower semicontinuous.This problem will have practical applications in the theory of quantum dynamical systems.
The result of this paper generalizes some of the results of Bishop et al. (2016) and also extends the work of Aitalioubrahim (2011) to the class of noncummutative quantum setting.
We consider the following lower semicontinuous quantum stochastic evolution inclusion: The term in the bracket on the right hand of Inclusion 1 is the formulation of Hudson and Parthasarathy (1984) Boson quantum stochastic calculus.E, F, G, H are coefficients that lie in the space 2 ([0, ] ) A is a family of densely defined linear operator.However, it has been shown by Ekhaguere (2007) that the following evolution inclusion: is equivalent to Inclusion 1, where the map with its explicit form defined by Ekhaguere (2007).We organize the rest of this paper as follows: Section 2, will consist of preliminaries while in Section 3, the main result will be considered.

Notations and Preliminary Results
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 ) In what follows, as in (Bishop and Ayoola, 2015;Ekhaguere, 1992;2007;Ogundiran and Payne 2014) we employ the definitions and notations of the spaces D is some pre-Hilbert space with ℜ as its completion.Let S be a topological space, then clos(S), denotes the collection of all nonempty closed subsets of S while Comp(S) denotes the collection of all nonempty compact subsets of S. We shall employ the Hausdorff topology on clos ( ) A ɶ .By Theorem V.5 of Reed and Simon (1980), the σ-weak topology τ σw is metrizable since ∞ ⊗ D E has a countable base, hence A ɶ is metrizable.For more on Banach space, metrizable spaces, etc., see Krein (1971).

Definitions 1:
• A multivalued stochastic process Φ with values in clos ( ) A ɶ , with I + ⊆ R as its basis, is a multivalued function on the interval I • If the above holds, then a selection of Φ is a stochastic process : x I A → ɶ such that x(t)∈Φ(t) for almost all t∈I Note: All through the remaining part of this paper, Φ is multivalued stochastic process except stated otherwise.
. This will be denoted by ( ) Notations 1: is the set of maps  ( ) The following result established by Ayoola (2008) will be useful in establishing the main result.The proofs are simple adaptation of arguments employed in the reference.

Lemma 1
Let Ã be a metrizable space and let [ ] ( ) be a multivalued operator which is lower semicontinuous (l.s.c) and has nonempty closed and decomposable values.Then Φ has a continuous selection.That is there exists a continuous function P: Ã→L 1 ([0,T], Ã) such that P(y)∈Φ(y) ∀ y∈ Ã.

Theorem 1
Assume that the function g: Ã→C([0,T],Ã) is continuous and the map P:I×Ã→L Ã has non-empty compact values.Then:

•
We do this by showing that γ(B r ) is bounded ∀ r≥0, B r = {y ηξ (.) Show that γ maps bounded sets into sets that are equicontinuous • Lastly we show that for some 0<γ<1 the set ηξ,k ) k≥0 converge to y ηξ in [ ] • (S iv ) Let N > 0 and |T(t,s)|≤N for each (t,s)∈∆ • (S v ) Let m and l be non-negative constants, then Show that γ is continuous • Show that γ is bounded on bounded sets of PThe proof is presented as follows:•