Page 251 - Mathematical Techniques of Fractional Order Systems

240 Mathematical Techniques of Fractional Order Systems

0 1
T 2q 2 2 2 T 2q 2 2 2 2
1 # 6n 1 1 1 6 n :B: :B : l A 1 36 n :B: :B : lE:x 1 :
@
4
4
q 2 q 2
0 1
T 2q 2 2 2
1 6n 2 1 1 6 n :B: :B : l A
@
4
q 2
0 1 0 1
T 2q 2 2 2 T 2q T 2q 2 2 2
1 6n 3 1 1 6 n :B: :B : l A 1 6 n 4 ϒ @ 1 1 6 n :B: :B : l A
@
4
4
q 2 q 2 q 2
0 1
T 2q 2 2 2 T 2q 2
1 61 1 6 n :B: :B : l A Tn 4 M 1 M G 1 1 sup E:xðtÞ:
4
b
@
q 2 q 2 tAJ
which is a contradiction to (8.3). Hence, for some ρ . 0; ΦðB ρ ÞCB ρ :
Step 3. Compactness of Φ:
To prove this, first show that the set ΦðB ρ Þ is relatively compact in B:
Subsequently, one must show that ΦðB ρ Þ is uniformly bounded. Note that by
using the same method as in the above step it can be manifested that the
operator Φ is uniformly bounded, i.e.,
0 1
T 2q 2 2 2 T 2q 2 2 2 2
6n 1 1 1 6 n :B: :B : l A 1 36 n :B: :B : lE:x 1 :
4
@
4
q 2 q 2
0 1
T 2q 2 2 2
1 6n 2 1 1 6 n :B: :B : l A
@
4
q 2
0 1 0 1
T 2q 2 2 2 T 2q T 2q 2 2 2
1 6n 3 1 1 6 n :B: :B : l A 1 6 n 4 ϒ @ 1 1 6 n :B: :B : l A
@
4
4
q 2 q 2 q 2
0 1
T 2q 2 2 T 2q 2
2
1 61 1 6 n :B: :B : l A Tn 4 M 1 M G 1 1 sup E:xðtÞ: , N
b
4
@
q 2 q 2 tAJ
and the set ΦðB ρ Þ is relatively compact. Finally, one needs to prove that
ΦðB ρ Þ is equicontinuous. For any xAB ρ and t 1 ; t 2 AJ with 0 , t 1 , t 2 # T;
then