Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems - Fluid-Structure Interaction and Biomedical Applications - page 220

Hence M is injective and has closed range, and the proof of the first statement in
(ii) is completed. In order to show the second one we introduce the quotient space

1 .0;T I X/ WD L 1 .0;T I X/= N .M/

L

where

Z T Z T

n w 2 L 1 .0;T I X/ W

U.T t/ w .t/ D 0 o ; (3.62)

U.t/ w .t/dt

N .M/ D

0 0

endowed with the usual norm

kW kWD

z 2 N .M/ k w z k ; w 2 W:

inf

Recalling that M is surjective, the operator

1 .0;T I X/ 7! M w 2 X; w 2 W;

M W W 2 L

M 1 maps dense sets of X into dense is then continuously invertible and, therefore,

1 .0;T I X/. As a result, setting

D D M 1 . R .I U// we deduce, by sets of L

1 .0;T I X/.Define

assumption, that D is dense in L

D WD fv 2 X W v 2 V for some V 2 D g ;

and notice that for any b 2 R .I U/there is v 2 D such that Mv D b. We claim

that the set

Q WD D C N .M/

is dense in L 1 .0;T I X/. In fact, pick u

2 X,and">0. By what we have just

shown, there exists u " 2 D such that

z 2 N .M/ k u u " z k <";

inf

which, in turn, by the property of the infimum, implies that we can find z 2 N .M/
such that

k u u " z k <2";

which proves the desired property. We next observe that for any b 2 R .I U/
(that, by hypothesis, is dense in X)( 3.60 ) has one and only one solution x.Now,
fromwhatwehaveprovedand( 3.62 ), we conclude that for such a b there exists
f D f 1 C f 2 2 D C

N .M/ such that

Z T Z T

b D U.T s/f 1 .s/ds D U.T s/f.s/ds;

0 0