Page 253 - Mathematical Techniques of Fractional Order Systems

242 Mathematical Techniques of Fractional Order Systems

As t 1 -t 2 ; the right-hand side of the above inequality tends to zero. An
application of the Arzela-Ascoli theorem yields that Φ maps B ρ into B; i.e.,
Φ:B ρ -PðBÞ is a compact operator. Thus, ΦðB ρ Þ is relatively compact.
Step 4. Φ is u.s.c. on B ρ :
Let x n -x as n-N and Ψ n -Ψ as n-N: Now derive that Ψ AΦðx Þ:
such that
Since Ψ n AΦðx n Þ means that there exists v n AN F;x n
Ψ n ðtÞ5E q2p ðAt q2p Þx 0 2At q2p E q2p;q2p11 ðAt q2p Þx 0 1tE q2p;2 ðAt q2p Þx 0
0
ð t ð s
q21 q2p
1 ðt2sÞ E q2p;q ðAðt2sÞ Þ Bu x;n ðsÞ1v n ðsÞ1 Gðθ;x n ðθÞÞdwðθÞ ds:
0 0
ð8:4Þ
One must show that there exists v AN F;x such that
Ψ ðtÞ5E q2p ðAt q2p Þx 0 2At q2p E q2p;q2p11 ðAt q2p Þx 0 1tE q2p;2 ðAt q2p Þx 0
0
ð t h
q21 q2p q2p ÞW 21
1 ðt2sÞ E q2p;q ðAðt2sÞ ÞB B E q2p;q ðA ðT2sÞ
0

3 Ex 1 2E q2p ðAT q2p Þx 0 1AT q2p E q2p;q2p11 ðAT q2p Þx 0 2TE q2p;2 ðAT q2p Þx 0
0
ð T ð s
q21 q2p
2 ðT2sÞ E q2p;q ðAðT2sÞ Þ v ðsÞ1 Gðθ;x ðθÞÞdwðθÞ ds ðsÞds
0 0
ð t ð s
q21 q2p
1 ðt2sÞ E q2p;q ðAðt2sÞ Þ v ðsÞ1 Gðθ;x ðθÞÞdwðθÞ ds:
0 0
Consider the continuous operator defined as
n
Λ:L 2 ðJ; R Þ-B; v/ΛðvÞðtÞ
such that
ð t
ΛðvÞðtÞ 5 ðt2sÞ q21 E q2p;q ðAðt2sÞ q2p Þ vðsÞ 2 BB E q2p;q ðA ðT2sÞ q2p ÞW 21

0
ð T i
3 ðT2sÞ q21 E q2p;q ðAðT2sÞ q2p ÞvðsÞds ðsÞds:
0
From Lemma 8.3, it follows that Λ3N F is a closed graph operator.
Clearly, for each tAJ; one can have

q2p q2p q2p q2p
Ψ n ðtÞ2E q2p ðAt Þx 0 1At E q2p;q2p11 ðAt Þx 0 2tE q2p;2 ðAt Þx 0
0
ð t ð s
q21 q2p
2 ðt2sÞ E q2p;q ðAðt2sÞ Þ Bu x;n ðsÞ1v n ðsÞ1 Gðθ;x n ðθÞÞdwðθÞ ds AΛðN F;x n Þ:
0 0