Page 247 - Mathematical Techniques of Fractional Order Systems

236 Mathematical Techniques of Fractional Order Systems

(H2) The function G is continuous and there exists constant M 1 such that,
2 2
:Gðt; xÞ: # M 1 ð1 1 :x: Þ
n
n
(H4) The multifunction F:J 3 R -P cp ðR Þ has the property that
n
n
FðU; xÞ:J-P cp ðR Þ is measurable for each xAR :
(H5) There exists a nonnegative function mAL 2 ðJÞ such that
h 2 i
H d ðFðt; x 1 Þ; Fðt; x 2 ÞÞ # mðtÞ :x 1 2 x 2 : for every x 1 ; x 2 AR ; and
n
dð0;Fðt; 0ÞÞ # mðtÞ a:e: tAJ:
Theorem 8.2: (Convex Case) Suppose that the hypotheses (H1) (H2) are
satisfied, then the system (8.1) has at least one solution, provided that the
following holds:
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 0 1
T 2q 2 2 2 T 2q 2 2 2
1 6n 2 1 1 6 n :B: :B : l A 1 6n 3 1 1 6 n :B: :B : l A
@
4
4
@
q 2 q 2
0 1 0 1
T 2q T 2q 2 2 T 2q 2 2
2
2
1 6 n 4 ϒ @ 1 1 6 n :B: :B : l 1 61 1 6 n :B: :B : l A
A
@
q 2 q 2 4 q 2 4
T 2q 2
3 Tn 4 M 1 M G 1 1 sup E:xðtÞ: :
b
q 2 tAJ
ð8:3Þ
n
Proof: For any arbitrary function xAR ; define the following control function

q2p ÞW 21 q2p
u x ðtÞ 5 B E q2p;q ðA ðT2tÞ Ex 1 2 E q2p ðAT Þx 0
1 AT q2p E q2p;q2p11 ðAT q2p Þx 0 2 TE q2p;2 ðAT q2p Þx 0
0
ð T ð s
2 ðT2sÞ q21 E q2p;q ðAðT2sÞ q2p Þ vðsÞ 1 Gðθ; xðθÞÞdwðθÞ ds
0 0
where tAJ; vAN F;x : Using the above control, first show that the operator
Φ:B-PðBÞ; defined as

ΦðxÞ 5 ΨAB: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 ðsÞ 1 vðsÞ 1 Gðθ; xðθÞÞdwðθÞ ds
0 0
has a fixed point, which is a solution of the system (8.1).