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

1.2.4 Some Algebraic inequalities

Since our study based on some known algebraic inequalities, we want to recall few of them here. Lemma 1.2.10 ([48],The Cauchy-Schwarz inequality)

Every inner product satisfies the Cauchy-Schwarz inequality

(x₁, X2) kxiM X2M (1.30)

The equality sign holds if and only if X1 and X2 are dependent.

Young's inequalities :

Lemma 1.2.11 For all a, b 2 R⁺, we have

ab < 8a² + b2

48, (1.31)

where 8 is any positive constant.

Proof. Taking the well-known result

(28a -- b)² ~ 0 Va, b 2 R

for all 8 > 0, we have

48²a² + b² - 48ab ~ 0.

This implies

48ab < 48²a² + b2

consequently,

ab < 8a² + 48 1 b2.

This completes the proof.

Lemma 1.2.12 ([43])

For all a, b ~ 0, the following inequality holds

_bq

+ ;

q

_a°

ab < p

1
q

= 1.

+

1

where, p

Proof. Let G = (0,1), and f : G --> IR is integrable, such that

p log a, 0 < x < 1

q log b, 1

P

P

< x < 1

)

₈

<>

>:

f(x) =

1
q

= 1.

+

for all a, b > 0 and 1

P

Since cp(t) = e^t is convex, and using Jensen's inequality

(,⁰(¹

(1.32)

,u (G) If (x)dx) <

-- ,u (G)

f cp (f (x)) dx.

G G

Consequently, we have

_Z

1

~ (G)

G

1

I

0 _of (x) dx =

1 /p 1

I

0 _ep log adx + I

1/p

cp (f(x)) dx =

_eq log b_dx

1 / p

I

0

1

I

1/p

=

ap dx +

bq dx

(a^p) + (1 -- ) b^q .

=

P

(1.33)

ap	bq
=		+ P	, q

G

where, ,u (G) = 1 and

40 ( ¹ I

f (x)dx = e(f0 f (x)dx) = e (V/p p log _{adx#177;fi1/p} q log bdx)

= _e(log a#177;log b) = _clog ab

= ab. (1.34)

Using (1.32), (1.33) and (1.34) to conclude the result.