Page 222 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 211

d ð t
xðtÞ 5 Sðt 2 sÞhðsÞds; ’tAJ;
dt 0
γ
(ii) if fSðtÞg t $ 0 is analytic and hAC ðJ; HÞ for some γAð0; 1Þ, then the
function defined by
ð t
xðtÞ 5 SðtÞðhðtÞ 2 hð0ÞÞ 1 S ðt 2 sÞ½hðsÞ 2 hðtފds 1 SðtÞhð0Þ; tAJ
0
0
is a mild solution of (7.17) on J;
(iii) if fSðtÞg t $ 0 is differentiable and hACðJ; ½DðAÞŠÞ then the function
x:J-H defined by

ð t
xðtÞ 5 S ðt 2 sÞhðsÞds 1 hðtÞ; tAJ
0
0
is a mild solution of (7.17).

Lemma 7.6: Let H be a Banach space, G is a closed, convex subset of H, U ^
^
^
an open subset of G and 0AU. Suppose that F:U-G is a continuous, com-
^
pact (i.e., FðUÞ is a relatively compact subset of G) map, then either
^
(i) F has a fixed point in U; or
^
(ii) there is a yA@U and λAð0; 1Þ with y 5 λFðyÞ.
The following hypotheses are considered to prove the main results

(H 14 ) SðtÞ is compact for all t . 0.
(H 15 ) f:J 3 C h -H satisfies the following
(i) fðt; UÞ:C h -H is continuous for each tAJ and for each
xAC h ; fðU; xÞ:J-H is strongly measurable.
1
(ii) There is a positive integrable function nAL ð½0;bŠÞ and a continuous
nondecreasing function Λ f : ½0;NÞ-ð0;NÞ such that for every
ðt; xÞAJ 3 C h ,

2
2
Ejfðt; xÞj # nðtÞΛ f ðOxO Þ; lim inf Λ f ðrÞ 5 ϒ 1 , N:
H C h r-N r
2
2
(iii) For arbitrary ξ ; ξ AC h ; satisfying Oξ O ; Oξ O # ρ, there exists a
1 2 1 C h 2 C h
M f ðρÞ . 0 such that
2 2
E fðt; ξ Þ2fðt; ξ Þ # M f ðρÞ:ξ 2ξ : :

1 2 1 2 C h
(H 16 ) The function σ:J 3 C h -LðK; HÞ satisfies the following
(i) For each tAJ; σðt; UÞ:C h -LðK; HÞ is continuous and for each
xAC h ; σðU; xÞ:J-LðK; HÞ is strongly measurable.