Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems - Fluid-Structure Interaction and Biomedical Applications - page 219

one. We next prove property (ii). If R .I U/ ¤ X, there exists b 2 X such that
the equation

.I U/xD b (3.60)

does not have a solution. Therefore, since the existence of a T -periodic solution is
equivalent to the resolution of ( 3.55 ), the first stated property in part (ii) follows
provided we show that for any b 2 X there is f 2 L 1 .0;T I X/such that

Z T

b D U.T s/f.s/ds: (3.61)

0

Once such an f is found, we may extend it periodically of period T to the whole real
line. 13 For >0we set U.t/ WD e t U.t/, and observe that since k U.T/k <1,
I U.T/is continuously invertible. Equation ( 3.61 ) becomes

Z T

U.T s/ w .s/ds;

b D

0

where w .s/ WD e .T s/ f.s/. Replacing w .s/ by w .T s/ and keeping the same
notation, the question then reduces to show that the bounded linear operator

M W L 1 .0;T I X/ ! X

defined by

Z T

U.s/ w .s/ds

M w WD

0

is surjective. It is clear that R .M/ is dense in X. To see this we observe that w .t/ D
. U.T/I/ 1 .A/b satisfies M w D b, which in turn implies D .A/ R .M/, and,
therefore, the density property. We shall next prove that the range of M is closed. To
this end, by the Banach closed range theorem, this is equivalent to show that R .M /
is closed, where M is the conjugate of M. From the identity

Z T
0 h U.s/ w .s/;x ids D

Z T
0 h w .s/; U .s/x ids

hM w ;x iD

it follows that M W X ! L 1 .0;T I X/ L 1 .0;T I X / is given by M x .s/ D
U .s/x , s 2 Œ0;T. This implies

kx k

0 s T k U .s/x k DkM x k L 1 .0;T I X / :

sup

13 The argument that follows is due to Professor Jan Prüss, to whom we are indebted.