Page 227 - Mathematical Techniques of Fractional Order Systems

216 Mathematical Techniques of Fractional Order Systems

Step 1: Π 1 is completely continuous. First, to prove that Π 1 is continu-
0
ous. Let fz n g be a xsequence such that z n -z in C as n-N. Then for tAJ,
b
EjðΠ 1 z n ÞðtÞ 2 ðΠ 1 zÞðtÞj # 2E Ð 0 t ½fðs; z n s 1y s Þ2fðs; z s 1y s ފds 2
2

2

1 Ð t α21 Ð s
1 2E ΓðαÞ 0 ðt2sÞ 2N σðτ; z τ 1y τ ÞdWðτÞ ds

Ð t 2
# 2b E fðs; z n s 1y s Þ2fðs; z s 1y s Þ ds

0
2b α ð t ð s 2
E
1 ðt2sÞ α21 ½σðτ; z n τ 1y τ Þ2σðτ; z τ 1y τ ފdWðτÞ ds

2
αΓ ðαÞ 0 2N
2
since f and σ are continuous, one has OðΠ 1 z n ÞðtÞ 2 ðΠ 1 zÞðtÞO -0 as n-N:
b
0
Now, to prove Π 1 maps bounded sets into bounded sets in C . Indeed, it is
b
enough to prove that for any q . 0, there exists a positive constant δ such
that for each zAB q , one has ðΠ 1 zÞAB δ . Let zAB q , since f and σ are continu-
ous, for each tAJ
2
2
2 2 1 Ð t α21 BðsÞuðsÞds 1 4E Ð t fðs; z s 1y s Þds

ΓðαÞ 0 ðt2sÞ 0
EjðΠ 1 zÞðtÞj # 4Ejφð0Þj 1 4E

4b α ð t ð s 2
E
1 ðt2sÞ α21 σðτ; z τ 1y τ ÞdWðτÞ ds

2
αΓ ðαÞ 0 2N
2 1 12 1 3 2
0
ðα21Þp p 1
2 ð t p21 ð t
2
p
# 4Ejφð0Þj 1 4OBO 6 B ðt2sÞ dsA EjuðsÞj ds p 7
C
6
7
2
@
Γ ðαÞ 4 0 0 5
Ð t 2
1 4b E fðs; z s 1y s Þ ds

0
α ð t " ð s 2 #
4b
1 2 ðt2sÞ α21 2M R 1 2E σðτ; z τ 1y τ ÞdWðτÞ ds
αΓ ðαÞ 0 0
2αp22 ! 2p22
OBO 2 p p
2
# 4Ejφð0Þj 1 4 b p21 OuO 2
2
Γ ðαÞ αp21 L p ðJ;UÞ
2
Ð t
1 4b nðsÞΛ f ðOz s 1 y s O Þds
0 C h
8b α ð t h i
2
1 ðt2sÞ α21 M R 1 TrðQÞmðsÞΛ σ ðOz s 1 y s O Þ ds
2 C h
αΓ ðαÞ 0
2αp22 ! 2p22
OBO 2 p p
p21
2
2

# 4Ejφð0Þj 1 4 2 b αp21 OuO 2 L p ðJ;UÞ 1 4b OnO N Λ f ðq Þ
Γ ðαÞ
8b 2α

1 2 ½ M R 1 TrðQÞOmO N Λ σ ðq ފ 5 δ , N
Γ ðα 1 1Þ
2
thus EjðΠ 1 zÞðtÞj # δ and hence Π 1 zAB δ . Now, to prove that Π 1 maps
0
bounded sets into equicontinuous sets of C . Let t 1 ; t 2 AJ; t 2 . t 1 and let B q
b
be a bounded set. Let zAB q , then if E . 0 and E # t 1 # t 2 , one has