Page 229 - Mathematical Techniques of Fractional Order Systems

218 Mathematical Techniques of Fractional Order Systems

is bounded, since

1
ð t2E ð t2E
E φð0Þ 1 ð t2E2sÞ α21 BðsÞuðsÞds 1 fðs; z s 1 y s Þds

ΓðαÞ 0
0

t2E ð s
ð
1
1 ð t2E2sÞ α21 σðτ; z τ 1 y τ ÞdWðτÞ ds
ΓðαÞ 0 2N
2

2 1 Ð t2E α21

ΓðαÞ 0
# 4E φð0Þ 1 4E ð t2E2sÞ BðsÞuðsÞds

1 4E Ð t2E fðs; z s 1y s Þds 2

0
2

1 Ð t2E α21 Ð s
ΓðαÞ 0 ð t2E2sÞ 2N σðτ; z τ 1y τ ÞdWðτÞ ds
1 4E

2 1 3 2
0 ðα21Þp 1 12 1
2 ð t2E p21 p ð t2E
2
p

# 4E φð0Þ 1 4OBO 6 @ ð t2E2sÞ ds A E uðsÞ ds p7

7
6

2
Γ ðαÞ 4 0 0 5
t2E 2
Ð
ð
1 4ðt 2 EÞ E fs; z s 1y s Þ ds

α 0 ð t2E ð s
ð
4 t2EÞ α21 2
1 2 ð t2E2sÞ 2M R 1 2Tr QðÞ E:σðτ; z τ 1y τ Þ: 0 dτ ds
L
αΓ ðαÞ 0 0 2
2αp22 ! 2p22
2
:B: p p
2 p21 2
# 4E φð0Þ 1 4 2 ð t2εÞ αp21 :u:

Γ ðαÞ L p ðJ;UÞ
2α
ð
2
1 4 t2EÞ :n: Λ f ðq Þ 1 8 t2EÞ M R 1 TrðQÞ:m: Λ σ qðÞ

ð
N 2 N
Γ ðα 1 1Þ

then for t . 0, the set Z E ðtÞ 5 ð Π 1E zÞðtÞ; zAB q is precompact in H for
every 0 , E , t. Furthermore
2 5OBO ð t2E 2
2
E ðΠ 1 zÞðtÞ2ðΠ 1ε zÞðtÞ # 2 E ð t2sÞ α21 2 t2E2sÞ α21 uðsÞds
ð

Γ ðαÞ 0
5OBO 2 ð t 2 ð t
2
1 E ð t2sÞ α21 uðsÞds 15E Ejfðs;z s 1y s Þj ds

2
Γ ðαÞ t2E t2E
ð t2E ð s
5
1 2 E ð t2sÞ α21 2 t2E2sÞ α21 σðτ;z τ 1y τ ÞdWðτÞ ds
ð
Γ ðαÞ 0 2N
5 ð t ð s
1 2 E ð t2sÞ α21 σðτ;z τ 1y τ ÞdWðτÞ ds
Γ ðαÞ t2E 2N
therefore, the set ZðtÞ 5 fðΠ 1 zÞðtÞ: zAB q g is precompact in H. Hence the
operator is completely continuous.
Step 2: Π 2 is completely continuous.
By Step 1, Π 1 is continuous. Now, to prove the operator Π 2 is continu-
ous, let B q be a bounded set as in Step 1. For zAB q , one has
2
EjðΠ 2 zÞðtÞj 5 E Ð t S ðt2sÞðΠ 1 zÞðsÞds # b Ð t ðϕ ðt2sÞÞ EjðΠ 1 zÞðsÞj ds
2
2
2
0

0 0 A
2
2 Ð t EjðΠ 1 zÞðsÞj ds # b Oϕ O 1 δ 5 δ 0
2
2
A L A L
# bOϕ O 1 0