Page 200 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 189

Let PðHÞ denote the class of all nonempty subsets of H. Let P bd;cl ðHÞ;
P bd;cl;cv ðHÞ denote, respectively, the family of all nonempty bounded-closed,
bounded-closed-compact subsets of H (see Fitzpatrick and Petryshyn, 1974).
For xAH, and Y; ZAP bd;cl ðHÞ, we denote by dðx; YÞ 5 inffOx 2 yO: yAYg
^
and ρðY; ZÞ 5 sup aAY dða; ZÞ and the Hausdorff metric H:P bd;cl ðHÞ 3
^
1
P bd;cl ðHÞ-R by HðA; BÞ 5 maxfκðA; BÞ; κðB; AÞg.
Definition 7.1: (Balasubramaniam and Ntouyas, 2006) A multivalued map
F:H-PðHÞ is convex (closed) valued if FðxÞ is convex (closed) for all xAH.
F is bounded on bounded sets, if FðVÞ 5 , xAV FðxÞ is bounded in H, for
any bounded set V of H, i.e., sup xAV fsupfOyO: yAFðxÞgg , N.

Definition 7.2: (Balasubramaniam and Ntouyas, 2006) F is upper semicon-
tinuous (u.s.c) on H, if for each x AH the set Fðx Þ is a nonempty, closed
subset of H, and if for each open set V of H containing Fðx Þ, there exists an
open neighborhood N of x such that FðNÞDV.

Definition 7.3: (Balasubramaniam and Ntouyas, 2006) The multivalued oper-
ator F is said to be completely continuous if FðVÞ is relatively compact for
every bounded subset VDH.

Definition 7.4: (Balasubramaniam and Ntouyas, 2006) A multivalued map
F:J-P bd;cl ðHÞ is said to be measurable if for each xAH the function
t/dðx; FðtÞÞ is measurable on J.

If the multivalued map F is completely continuous with nonempty com-
pact values, then F is u.s.c if and only if F has a closed graph.
i:ex n -x ; y n -y ; y n AFx n imply y AFx .
F has fixed point if there is xAH such that xAFx.
For each ðφ; yÞAB 3 H, define the set of selections of F by

ð t
1
fAN F;φ 5 fAL ðJ; HÞ: fðtÞAFt; φ; gðt; s; φÞds for a:e tAJ :
0
Lemma 7.1: (Losta and Opial, 1965) Let J be a compact interval and H be
a Hilbert space. Let F be an L 2 -Caratheodory multivalued map with
N F;x 6¼ [ and let Θ be a linear continuous mapping from L 2 ðJ; HÞ to
CðJ; HÞ, then the operator
Θ3N F :CðJ; HÞ-P bd;cl;cv ðHÞ; x/ðΘ3N F ÞðxÞ: 5 ΘðN F;x Þ
is a closed graph operator in CðJ; HÞ 3 CðJ; HÞ.