Page 201 - Mathematical Techniques of Fractional Order Systems

190 Mathematical Techniques of Fractional Order Systems

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Lemma 7.2: (Wang and Zhou, 2011b) Let H be a Hilbert space, S be a non-
empty subset of H, which is bounded, closed and convex. Suppose
~
~
F:S-P bd;cl;cv ðHÞ is u.s.c with closed, convex values and such that FðSÞCS ~
~
and FðSÞ is compact, then F has a fixed point.
Definition 7.5: (Podlubny, 1998) The Riemann Liouville fractional integral
of order α . 0 for the function f: ð0;bŠ-H is defined by
α 1 ð t α21
J fðtÞ 5 ðt2sÞ fðsÞds
t
ΓðαÞ 0
where Γ is the Euler’s Gamma function.

Definition 7.6: (Podlubny, 1998) The Caputo fractional derivative of order
α . 0 for the function f:J-H is defined by
n
α
c D fðtÞ 5 1 ð f ðsÞ ds:
a t α2n11
Γðn2αÞ a ðt2sÞ
Definition 7.7: (Lizama and N’Gue ´re ´kata, 2013) Let H be a Hilbert space,
1
1
1
kACðR Þ; k 6¼ 0 and let aAL ðR Þ; a 6¼ 0. Assume that A is a linear
loc
operator with domain DðAÞ. A strongly continuous family fS α ðtÞg t $ 0 of
bounded linear operators from H into H is called an ða; kÞ-regularized resol-
vent family on H having A as a generator if the following hold
(i) S α ð0Þ 5 kð0ÞI
(ii) S α ðtÞxADðAÞ and S α ðtÞAx 5 AS α ðtÞx for all xADðAÞ and t $ 0
(iii) S α ðtÞx 5 kðtÞx 1 Ð t aðt 2 sÞAS α ðsÞxds; t $ 0; xADðAÞ.
0
Assume that a and k are both positive and one of them is nondecreasing.
Let fS α ðtÞg t $ 0 be an ða; kÞ-regularized family with generator A such that
OS α ðtÞO # MkðtÞ; t $ 0 for some constant M . 0. Then, we have

S α ðtÞx 2 kðtÞx
Ax 5 lim ; xADðAÞ:

t-0 1 ða kÞðtÞ
We say that fS α ðtÞg t $ 0 is of type ðM; ωÞ, if there exist constants M $ 0
and ωAR such that
ωt
OS α ðtÞO # Me ; ’t $ 0:

Theorem 7.1: (Lizama, 2000) Let A be a closed linear densely defined oper-
ator in a Hilbert space H. Then fS α ðtÞg is an ða; kÞ-regularized family of
t $ 0
type ðM; ωÞ if and only if the following conditions hold