B.2 Hypoth` ese A2”
I.2 Conditions suffisantes pour que l’hypoth` ese B3 soit satisfaite
Nous allons montrer que l’hypoth`ese B3.1 est satisfaite si (I.7), (I.8), (I.16), (I.19) et (I.22) sont v´erifi´ees. De (I.16), (I.19) et (I.22) on d´eduit l’existence d’une fonction ˜κcontinue et positive telle que :
A(x1, x2, y) ≤ |x2|(1 +|x2|+|x1|)˜κ(y)|y|2q + |x1|(1 +|x1|)˜κ(y)|y|q . (I.30)
Mais, d’apr`es le Lemme E.1 de l’annexe E, il existe une fonction ˜κ˜ telle que :
˜˜
κ(V(y)) ≥ κ(y)˜ 2+ ˜κ(y). (I.32)
D´efinissons maintenantκ par :
κ(v) = cκ(v)˜˜
√v . (I.33)
Alors :
A(x1, x2, y) ≤ c
"
1 +
Q(x1) +S(x2)#
κ(V(y))
W(y) T(x2) +c
"
1 +
Q(x1) +S(x2)
#2
κ(V(y))W(y) .
(I.34) L’in´egalit´e (2.10) est donc obtenue. De plus :
κ(V(y))∂V
∂y(y)
≤ κ(V˜˜ (y)). (I.35)
La fonction ˜κ(v) ´˜ etant continue, la propri´et´e (2.12) est v´erifi´ee.
On v´erifie de fa¸con imm´ediate que B3.2 est v´erifi´ee si et seulement si
r > 1 (I.36)
ou
q < min{n1, p1,2n2,2m2,2p2,2m1,}. (I.37) Les conditions n´ecessaires et suffisantes pour que l’hypoth`ese B3 soit v´erifi´ee avec les fonctions V,QetS sont donc obtenues.
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