American Journal of Systems and Software, 2016, Vol. 4, No. 2, 57-68
Available online at http://pubs.sciepub.com/ajss/4/2/5
©Science and Education Publishing
DOI:10.12691/ajss-4-2-5
Existence and Stability of Mixed Stochastic Fractional
Order Differential Inclusion Equations via Cosine
Dynamical System
Salah H Abid, Sameer Q Hasan
*
, Zainab A Khudhur
Department of Mathematics, College of Education Almustansryah University
*Corresponding author: dr.sameer_kasim @yahoo.com
Abstract In this paper, we shall consider the existence and stability of stochastic fractional order differential
inclusion nonlinear equations in infinite dimensional space by mixed fractional Brownian motion in Hilbert space H.
Keywords: Neutral mixed stochastic fractional order differential inclusion equations, existence, stability, via
cosine dynamical system with fractional derivative as component in nonlinear functions 0<α, β<1
Cite This Article: Salah H Abid, Sameer Q Hasan, and Zainab A Khudhur, “Existence and Stability of Mixed
Stochastic Fractional Order Differential Inclusion Equations via Cosine Dynamical System.” American Journal of
Systems and Software, vol. 4, no. 2 (2016): 57-68. doi: 10.12691/ajss-4-2-5.
1. Introduction
In this article we study the neutral second- order
abstract differential inclusion problem
() () ( ) [ ] () () ()
( )
() ()
( )
1
2
´ , , ,
, , .
H
d x t gtxt Ax t F txt D xt dw
F txt D xt dw
α
β
− ∈ +
+
(3.1)
()
0
0 x x =
()
1
0 ,0 , 1, 1.
2
x x H αβ ′ = < ≤ < ≤
Where : → is a generator of cosine semigroup on a
Hilbert space (, ‖∙‖), {(): ≥ 0} and {
(): ≥ 0}
are −valued Brownian motion and fractional Brownian
motion respectively with afinit trace nuclear covariance
operator >0 .
1
,
2
, satisfy suitable conditions that will
be established later on. The random variable
0
∈
‖
0
‖
2
< ∞.
This problem has been studied in case 0< , ≤ 1 ,
([1,2,3,16]). Well-posedness has been established using
different fixed point theorems and the theory of strongly
continuous cosine families in Banach spaces. We refer the
reader to [27,28] for a good account on the theory of
cosine families.
The theory of integro-differential equations or inclusions
has become an active area of investigation due to their
applications in the fields such as mechanics, electrical
engineering, medicine biology, ecology and so on. On can
see ([7,8,25] and references therein). Several authors have
established the existence results of mild solutions for these
equations ([4,5,21,24]). In addition, the nonlinear integro-
differential equations with resolvent operators serve as an
abstract formulation of partial integro-differential
equations that arise in many physical phenomena. One can
see [15] and references therein. The deterministic models
often fluctuate due to noise, which is random or at least
appears to be so. Therefore, we must move from
deterministic problems to stochastic problems. As the
generalization of classic impulsive integro-differential
equations or inclusions, impulsive neutral stochastic
functional integro-differential equations or inclusions have
attracted the researchers great interest. And some works
have done on the existence results of mild solutions for
these equations (see [17,26] and references therein). To
the best of our knowledge, there is no work reported on
the existence of mild solutions for the impulsive neutral
stochastic functional integro-differential inclusions with
nonlocal initial conditions and resolvent operators, and the
aim of this paper is to close the gap. In this paper,
motivated by the previously mentioned papers, we will
study this interesting problem. Sufficient conditions for
the existence are given by means of the fixed point
theorem for multi-valued mapping due to Dhage [11] and
the fractional power of operators. Especially, the known
results appeared in [9,10] are generalized to the stochastic
settings. An example is provided to illustrate the theory.
We refer the reader to Da prato and Zabczyk [12].
throughout the paper (, ‖. ‖
) ( , ‖. ‖
)denote two
real separable Hilbert spaces. In case without confusion,
we just use ⟨.,. ⟩ for the inner product and ‖. ‖ for the
norm.
Let (, ℱ, ; ) ( ={ℱ()}
≥ 0
) be complete filtered
probability space satisfying that ℱ
0
contains all -null sets
of . An -valued random variable is an ℱ -measurable
function (): → and the collection of random
variables ={(, ): →⧵∈} is called a stochastic
process. Generally, we just write x(t) instead of (, )
and (): → in the space of S. Let {
}
=1
∞
be a
complete orthonormal basis of . Suppose that {(): ≥ 0}