Existence et comportement asymptotique des solutions d'une équation de viscoélasticité non linéaire de type hyperbolique

1.1.2 Hilbert spaces

The proper setting for the rigorous theory of partial differential equation turns out to be the most important function space in modern physics and modern analysis, known as Hilbert spaces. Then, we must give some important results on these spaces here.

Definition 1.1.8 A Hilbert space H is a vectorial space supplied with inner product (u,v) such that Jkuk = (u, u) is the norm which let H complete.

Theorem 1.1.2 ([42], Theorem 1.1.1)

Let _(un)mEN is a bounded sequence in the Hilbert space H, then it possess a subsequence which converges in the weak topology of H.

Theorem 1.1.3 ([42], Theorem 1.1.2)

In the Hilbert space, all sequence which converges in the weak topology is bounded.

Theorem 1.1.4 ([42], Corollary 1.1.1)

Let _(un)mEN be a sequence which converges to u, in the weak topology and _(vfl)mEN is an other sequence which converge weakly to v, then

lim Th-400

(vn,un) = (v,u). (1.5)

Theorem 1.1.5 ([42], Theorem 1.1.3)

Let X be a normed space, then the unit ball

{ }

B' = i 2 X^' : klk < 1 , (1.6)

of X^' is compact in a (X^', X).

Proposition 1.1.4 ([42], Proposition 1.1.1)

Let X and Y be two Hilbert spaces, let (un)nEN _E X be a sequence which converges weakly to u E X, let A E £(X, Y ). Then, the sequence (A (u_n))_nEN converges to A(u) in the weak topology of Y.

Proof. For all u E X, the function

u i-- (A(u), v)

is linear and continuous, because

1(A(u), v)1 < IIAII,c(x, _IT) IIuII_X IIvII_I , Vu E X, v E Y. So, according to Riesz theorem, there exists w E X such that

(A(u), v) = (u, w), Vu E X.

Then,

lim n-->o

(A (un) , v) = lim n-->o

(um, w)

= (u, w) = (A(u), v). (1.7)

This completes the proof.