Page 209 - Mathematical Techniques of Fractional Order Systems

198 Mathematical Techniques of Fractional Order Systems

r
I 3 5 4E: Ð t S α ðt2sÞf ðsÞds: 2
0
2
r
2
2
# 4M b Ð t e 2ωðt2sÞ EOf ðsÞO ds # 4M b Ð t e 2ωðt2sÞ ðμðrÞ 1 ð ^ d 1 βÞð4r 1 r ÞÞds

0 0
0 1
e 2ωb 2 1
2 A ;
2ω
# 4M bðμðrÞ 1 ð ^ d 1 βÞð4r 1 r ÞÞ @

^
I 4 5 4E: Ð t S α ðt2sÞ Ð s σðs; τ; φ τ 1y ÞdWðτÞ ds: 2
r
τ
0 0
2
2
r
# 4M b Ð t e 2ωðt2sÞ E: Ð s σðs; τ; ^ φ τ 1y ÞdWðτÞ: ds
0 0 τ
2
^
r
2
# 4M b Ð t e 2ωðt2sÞ TrðQÞ Ð s E:σðs; τ; φ τ 1y Þ: dτds
τ
0 0 Q
2 Ð t e 2ωðt2sÞ mðsÞΛ σ ðO ^ φ τ 1 y O Þds
r 2
# 4M bTrðQÞ
0 τ B
2 Ð t 2ωðt2sÞ
# 4M bTrðQÞ e mðsÞΛ σ ð4r 1 r Þds
0
0 1
e 2ωb 2 1
2
# 4M bTrðQÞ sup mðtÞΛ σ ð4r 1 r Þ @ A
tAJ 2ω
by combining the estimated I 1 -I 4 together with (7.7), one arrives at
0 1
" (
2
e 2ωb 2 1 5M OBO 4 ð b
2
2
2 A x 1 1 2EO~ b O 1 2 EO ~ φðsÞO ds
x
2ω ε 2 0
r # 4M b @
0 1
e 2ωb 2 1
2
2
2 2ωb
2

1 M e EOφð0ÞO 1 M bx 1 @ A 1 M bðμðrÞ 1 ð ^ d 1 βÞð4r 1 r ÞÞ
2ω
0 1
)
e 2ωb 2 1 ð b
2

A 1 M bTrðQÞ sup mðtÞ e 2ωðb2sÞ Λ σ ð4r 1 r Þds
3 @
2ω tAJ 0
#

1 ðμðrÞ 1 ð ^ d 1 βÞð4r 1 r ÞÞ 1 TrðQÞ sup mðtÞΛ σ ð4r 1 r Þ
tAJ
dividing both sides of the above inequality by r and taking r-N, one gets
2ωb
e 2 1
2
4M b δ 1 4ð ^ d 1 βÞ 1 ΔTrðQÞ sup mðtÞ
2ω tAJ
5OBO M bðe 2 1Þ 1 1 $ 1
4 4 2ωb
2ωε 2
which is a contradiction to our assumption (7.6). Thus, for some r . 0,
ΨðB r ÞDB r .
Step 3: ΨðB r Þ is equicontinuous. Indeed, let E . 0 be small,
0 , t 1 , t 2 # b. For each yAB r and ρ belonging to Ψy, there exists fAN F;x
such that for each tAJ,