Page 249 - Mathematical Techniques of Fractional Order Systems

238 Mathematical Techniques of Fractional Order Systems

It is easy to see that N F;x is convex since F has convex values. So,
λv 1 1 ð1 2 λÞv 2 AN F;x : Thus, λΨ 1 1 ð1 2 λÞΨ 2 AΦðxÞ:
2
Step 2. For each positive number ρ . 0; let B ρ 5 fxAB::x: # ρg:
B
Obviously, B ρ is a bounded, closed, and convex set of B: It is to claim that
there exists a positive number ρ such that ΦðB ρ ÞCB ρ :
If this is not true, then for each positive number ρ; there exists a function
2 2
x ρ AB ρ ; but Φðx ρ Þ=2B ρ that is :Φðx ρ Þ: supf:Ψ: :Ψ ρ AðΦx ρ Þg . ρ and
B B
Ψ ρ ðtÞ 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 ρ ðsÞ 1 v ρ ðsÞ 1 Gðθ; x ρ ðθÞÞdwðθÞ ds
0 0

: Using Lemma 8.4 one can have
for some v ρ AN F;x ρ

2 2 q2p 2 21 2 2 q2p 2

E:u ρ ðtÞ: # 6:B : :E q2p;q ðA ðT2tÞ Þ: :W : fE:x 1 : 1 :E q2p ðAT Þx 0 :
2
0
1 :AT q2p E q2p;q2p11 ðAT q2p Þx 0 : 1 :TE q2p;2 ðAT q2p Þx : 2
0
ð T
1 E: ðT2sÞ q21 E q2p;q ðAðT2sÞ q2p ÞvðsÞds: 2
0
ð T ð s
1 E: ðT2sÞ q21 E q2p;q ðAðT2sÞ q2p Þ Gðθ; xðθÞÞdwðθÞ ds: 2
0 0
2 q2p 2 21 2 2 q2p 2

# 6:B : :E q2p;q ðA ðT2tÞ Þ: :W : fE:x 1 : 1 :E q2p ðAT Þx 0 :
2
1 :AT q2p E q2p;q2p11 ðAT q2p Þx 0 : 1 :TE q2p;2 ðAT q2p Þx : 2
0
0
T 2q 2 2
1 :E q2p;q ðAðT2tÞ q2p Þ: :vðtÞ:
q 2
T 2q 2 ð s
1 :E q2p;q ðAðT2tÞ q2p Þ: Gðθ; xðθÞÞdwðθÞ ds
q 2 0
T 2q
2
2
# 6:B : n 4 l E:x 1 : 1 n 1 1 n 2 1 n 3 1 n 4 ϕ ðtÞ
ρ
q 2
T 2q 2
1 Tn 4 M 1 M G 1 1 sup E:xðtÞ:
b
q 2 tAJ
also find that