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

1.3 Existence Methods

1.3.1 The Contraction Mapping Theorem

Here we prove a very useful fixed point theorem called the contraction mapping theorem. We will apply this theorem to prove the existence and uniqueness of solutions of our nonlinear problem.

Definition 1.3.1 Let f : X - X be a map of a metric space to itself. A point x 2 X is called a fixed point off if f(x) = x.

Definition 1.3.2 Let (X, dx) and (Y, dY ) be metric spaces. A map çü : X -p Y is called a contraction if there exists a positive number C < 1 such that

dy (ço(x),ço(y)) Cdx(x,y), (1.35)

for all x,y 2 X.

Theorem 1.3.1 (Contraction mapping theorem [45] )

Let (X, d) be a complete metric space. If çü : X -p X is a contraction, then çü has a unique fixed point.

1.3.2 Gronwell's lemma

Theorem 1.3.2 ( In integral form)

Let T > 0, and let çü be a function such that, çü 2 L¹(0, T), çü ~ 0, almost everywhere and q be a function such that, q 2 L¹(0,T), q ~ 0, almost everywhere and qço 2 L¹ [0, T], Ci, C2 ~ 0. Suppose that

q(t) ~ Ci + C2 _Zt ço(s)q(s)ds, for a.e t 2 ]0,T[, (1.36)

0

then,

t

_{0 1}

_f

0

q(t) Ci exp _@C2 '(s)ds ^A , for a.e t 2 ]0,T[. (1.37)

Proof. Let

F(t) = C1 + C2 _Zt ço(s)q(s)ds, for t 2 [0, T], (1.38)

0

we have,

q(t) F(t),

From (1.38) we have

F^'(t) = C2ço(t)q(t)

~ C2ço(t)A(t), for a.e t 2 ]0,T[. (1.39)

d

₈

<

:

dt

Consequently,

_{0 1 9}

f t =

F (t) exp @_ C2'(s)ds _A 0, (1.40)

0

;

then,

t

_{0 1}

_f

F (t) ~ Ci exp _@C2 (s)ds _A , for a.e t 2 ]0, T[. (1.41)

0

Since q F, then our result holds.

In particle, if C1 = 0, we have q = 0 for almost everywhere t 2 ]0, T[.

1.3.3 The mean value theorem

Theorem 1.3.3 Let G : [a, b] -p be a continues function and çü : [a, b] -p is an integral positive function, then there exists a number x in (a, b) such that

_Zb G(t)cp(t)dt = G(x) _Zb ço(t)dt. (1.42)

a a

m _Zb ço(t)dt ~ _Zb G(t)ço(t)dt M _Zb ço(t)dt. (1.46)

a a a

In particular for ço(t) = 1, there exists x 2 (a, b) such that

_Zb G(t)dt = G(x) (b - a). (1.43)

a

Proof. Let

m = inf {G(x), x 2 [a, b]} (1.44)

and

M = sup {G(x), x 2 [a, b]} (1.45)

of course m and M exist since [a, b] is compact. Then, it follows that

By monotonicity of the integral. dividing through by f a b ço(t)dt, we have that

f a b G(t)cp(t)dt

m < f b < M. (1.47)

_a co(t)dt

Since G(t) is continues, the intermediate value theorem implies that there exists x 2 [a, b] such that

: (1.48)

_a co(t)dt

f a b C(t)'(t)dt G(x) = f b

Which completes the proof.