Page 223 - Mathematical Techniques of Fractional Order Systems

212 Mathematical Techniques of Fractional Order Systems

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(ii) There is a positive integrable function mAL ð½0;bŠÞ and a continuous
nondecreasing function Λ σ : ½0;NÞ-ð0;NÞ such that for every
ðt; xÞAJ 3 C h ,

ð t Λ σ ðrÞ
2
2
Ejσðt; xÞj 0ds # mðtÞΛ σ ðOxO Þ; lim inf 5 ϒ 2 , N:
L C h r-N
0 2 r
2
2
(iii) For arbitrary ξ ; ξ AC h ; satisfying Oξ O ; Oξ O # ρ, there exists a
1
2
2 C h
1 C h
M σ ðρÞ . 0 such that
ð t
2 2
E σðs; ξ Þ2σðs; ξ Þ ds # M σ ðρÞ:ξ 2ξ : :
1

1
2
2 C h
0
Ð 0
(H 17 ) For each φAC h ; RðtÞ 5 lim a-N 2a σðt; φÞdWðsÞ exists and is con-
tinuous. Further there exists a positive constant M R such that
2
EjRðtÞj # M R .
H
(H 18 ) Let U be a separable reflexive Hilbert space from which the control
u takes the values. Operator BAL N ðJ; LðU; HÞÞ; OBO stands for the norm
of operator B on Hilbert space L N ðJ; LðU; HÞÞ.
U
(H 19 ) The multivalued map A:J-2 f[g has closed, convex, and
bounded values. AðUÞ is graph measurable and AðUÞDΣ, where Σ is a
bounded set of U.
Let the admissible set be,

ð b
2
A ad 5 u : J 3 Ω-H= u is F t 2 adapted stochastic process and E OuðtÞO dt , N :
0
7.3.1.1 Existence of Mild Solution
The subsection deals with the existence result for (7.15) (7.16), this prob-
lem is equivalent to the following integral equation
8
φðtÞ; tAð2N; 0Š
>
> ð t ð t
> 1 1
> α21 α21
φð0Þ 1 AxðsÞds 1 BðsÞuðsÞds
>
> ðt2sÞ ðt2sÞ
<
xðtÞ 5 ΓðαÞ 0 ΓðαÞ 0
> ð t ð s
> Ð t 1 α21
>
> 1 fðs; x s Þds 1 ðt2sÞ σðτ; x τ ÞdWðτÞ ds; tAJ
> 0
>
: ΓðαÞ 0 2N
By Lemma 7.5, and the above representation, the mild solution of
(7.15) (7.16) can be defined as follows