Page 207 - Mathematical Techniques of Fractional Order Systems

196 Mathematical Techniques of Fractional Order Systems

Let 0 # λ # 1, then for each tAJ, one can get

Ð t Ð t b 21
ðλρ 1 ð1 2 λÞρ ÞðtÞ 5 0 S α ðsÞx 1 ds 1 0 S α ðt 2 sÞBB S ðb 2 sÞðEI1Π Þ

α
2
1
0
~
n Ð b Ð b
E~ x b 1 φðsÞdWðsÞ 2 S α ðbÞφð0Þ 2 S α ðsÞx 1 ds
0 0
2 Ð b S α ðb 2 sÞ½λf 1 ðsÞ 1 ð1 2 λÞf 2 ðsފds
0
o
Ð b Ð s ^
2 S α ðb 2 sÞ σðs; τ; φ 1 y τ ÞdWðτÞ ds ds
0 0 τ
Ð t
1 S α ðt 2 sÞ½λf 1 ðsÞ 1 ð1 2 λÞf 2 ðsފds
0

Ð t Ð s ^
1 S α ðt 2 sÞ σðs; τ; φ 1 y τ ÞdWðτÞ ds
0 0 τ
It is easy to see that N F;x is convex, since F has convex values, so
λf 1 1 ð1 2 λÞf 2 AN F;x . Thus λρ 1 ð1 2 λÞρ AΨy:
1
2
Step 2: Next, one can show that there exists a positive number r such
that ΨðB r ÞDB r . If it is not true, then there exists E . 0 such that for every
r
positive number r and tAJ, there exists a function y ðUÞAB r but Ψðy r Þ=2B r ,
r 2
r
r
2
r
i.e., EOðΨy ÞðtÞO fOρ O : ρ AðΨy Þg . r. For such E . 0,
b
r # EOðΨy ÞðtÞO 2
r
2
ε
^
# 4E: Ð t S α ðsÞx 1 ds: 1 4E: Ð t S α ðt2sÞBu ðs; φ1y Þds: 2
r
0 0
Ð s 2
^
t
2
t
1 4E: S α ðt2sÞf ðsÞds: 1 4E: S α ðt2sÞ σðs; τ; φ 1y ÞdWðτÞ ds: ð7:7Þ
Ð
r
r
Ð
τ
0 0 0 τ
4
X
I i :
i51
By using ðH 1 Þ 2 ðH 5 Þ, one can obtain
0 2ωb 1
2
I 1 5 4E: Ð t S α ðsÞx 1 ds: # 4M bx 1 @ e 2 1 A
2
0 2ω
ð7:8Þ
^
I 2 5 4E: Ð t S α ðt2sÞBu ðs; φ1y Þds: 2
E
r
0
^
E
2
r
# 4M bOBO 2 Ð t e 2ωðt2sÞ EOu ðs; φ 1 y ÞO ds:
2
0
Now, in view of ðH 4 Þ, there exist positive constants, β; γ, such that, for
^
^
2
2
2
2
all OφO . γ, EOFðt; φ; LφÞO # ðd 1 βÞOφO . Let F 1 5 fφ: OφO # γg;
B B B
2
F 2 5 fφ: OφO . γg thus
B
^
2
2
EOFðt; φ; ^ LφÞO # μðγÞ 1 ðd 1 βÞOφO :
B
and