Page 221 - Mathematical Techniques of Fractional Order Systems

210 Mathematical Techniques of Fractional Order Systems

this integral equation can be written in the following form
1 ð t α21
xðtÞ 5 hðtÞ 1 ðt2sÞ AxðsÞds; t $ 0 ð7:17Þ
ΓðαÞ 0
where
1 ð t α21 ð t
hðtÞ 5 φð0Þ 1 ðt2sÞ BðsÞuðsÞds 1 fðs; x s Þds
ΓðαÞ 0 0
1 ð t α21 ð s
1 ðt2sÞ σðτ; x τ ÞdWðτÞ ds:
ΓðαÞ 0 2N

Let us assume that the integral Eq. (7.17) has an associated resolvent
operator fSðtÞg t $ 0 on H.

Definition 7.11: (Pru ¨ss, 2013) A one parameter family of bounded linear
operators fSðtÞg on H is called a resolvent operator for (7.17) if the fol-
t $ 0
lowing conditions hold
(i) SðUÞxACð½0;NÞ; HÞ and Sð0Þx 5 x for all xAH,
(ii) SðtÞDðAÞCDðAÞ and ASðtÞx 5 SðtÞAx, ’xADðAÞ and every t $ 0,
(iii) for every xADðAÞ and t $ 0, SðtÞx 5 x 1 1 Ð t ðt2sÞ α21 ASðsÞxds:
ΓðαÞ 0

Definition 7.12: (Pru ¨ss, 2013) A resolvent operator fSðtÞg t $ 0 for (7.17) is
1
called differentiable if SðUÞxAW 1;1 ðR ; HÞ for all xADðAÞ and there exists
loc
1
1
ϕ AL ðR Þ such that OS ðtÞxO # ϕ ðtÞOxO ½DðAފ for all xADðAÞ, where the
0
A loc A
notation ½DðAފ stands the domain of the operator A provided with the graph
norm OxO ½DðAފ 5 OxO 1 OAxO.
Definition 7.13: (Pru ¨ss, 2013) A resolvent operator fSðtÞg t $ 0 for (7.17) is
called analytic if the operator function SðUÞ : ð0;NÞ-LðHÞ admits an ana-
P π
lytic extension to a sector 5 fλAC:jargðλÞj , θ 0 g for some 0 , θ 0 # .
0;θ 0 2
Definition 7.14: (Pru ¨ss, 2013) A function xACðJ; HÞ is called a mild solution
Ð t α21
of the integral Eq. (7.17) on J,if ðt2sÞ xðsÞdsADðAÞ for all
0
tAJ; hðtÞACðJ; HÞ and
1 ð t
xðtÞ 5 ðt2sÞ α21 AxðsÞds 1 hðtÞ; ’tAJ
ΓðαÞ 0
Lemma 7.5: (Herna ´ndez et al., 2013) Under the above conditions the follow-
ing properties are valid

(i) if xðUÞ is a mild solution of (7.17) on J, then the function
t- Ð t Sðt 2 sÞhðsÞds is differentiable on J and
0