Page 255 - Mathematical Techniques of Fractional Order Systems

244 Mathematical Techniques of Fractional Order Systems

Hence vAN F;x : Then, for each tAJ; Ψ n ðtÞ-ΨðtÞ; where
q2p q2p q2p q2p
ΨðtÞ 5 E q2p ðAt Þx 0 2 At E q2p;q2p11 ðAt Þx 0 1 tE q2p;2 ðAt Þx 0
0
ð t ð s
1 ðt2sÞ q21 E q2p;q ðAðt2sÞ q2p Þ Bu x;1 ðsÞ 1 vðsÞ 1 Gðθ; x 1 ðθÞÞdwðθÞ ds:
0 0
ð8:6Þ
So, ΨAΦðxÞ:
Step 2. There exists γ , 1 such that H d ðΦðx 1 Þ; Φðx 2 ÞÞ # γ:x 1 2 x 2 : for
B
such
each x 1 ; x 2 AB: Let x 1 ; x 2 AB and ΨAΦðxÞ: Then, there exists vAN F;x 1
that ΨðtÞ is defined in (8.6). From (H5), it follows that
2
h i
H d ðFðx 1 ÞðtÞ; Fðx 2 ÞðtÞÞ # mðtÞ :x 1 ðtÞ 2 x 2 ðtÞ: :
such that
Hence, there exists ωAN F;x 2
2 h 2 i
:vðtÞ 2 ωðtÞ: # mðtÞ :x 1 ðtÞ 2 x 2 ðtÞ: ; tAJ:
n
Consider the map S:J-PðR Þ defined by

n
SðtÞ 5 ωðtÞjω:J-R is Lebesgue integrable and:vðtÞ

h 2 i
2
2 ωðtÞ: # mðtÞ :x 1 ðtÞ 2 x 2 ðtÞ: :
Since the multivalued operator SðtÞ - Fðt; x 2 ðtÞÞ is measurable
(Proposition 8.1), there exists a function vðtÞ which is a measurable selection
; and for each tAJ;
for S: So, vðtÞAN F;x 2
2 h 2 i
:vðtÞ 2 vðtÞ: # mðtÞ :x 1 ðtÞ 2 x 2 ðtÞ: :
Define the following

ΨðtÞ 5 E q2p ðAt q2p Þx 0 2 At q2p E q2p;q2p11 ðAt q2p Þx 0 1 tE q2p;2 ðAt q2p Þx 0
0
ð t ð s
1 ðt2sÞ q21 E q2p;q ðAðt2sÞ q2p Þ Bu x;2 ðsÞ 1 vðsÞ 1 Gðθ; x 2 ðθÞÞdwðθÞ ds:
0 0