Corrigé - Le parc

Q
Q={(x, y)∈R2|x≥0y≥0}
S(un)
u0≥0
u1≥0
∀n≥0, un+2 =1
2(u2
n+u2 n+1)
(x, y)Qu(x, y)Su0=x u1=y
n u(x, y)un(x, y)
un
λ∈R∪ {−∞,+∞} Eλ(x, y)∈ Q
u(x, y)λ
S
Sα
Sα=1
2(2α2)α=α2α= 0 α= 1
u0=u1= 0 (un)
u0=u1= 1 (un) 1
S1
S
un+2 =1
2(u2
n+u2 n+1)
(un) 0,1 +∞
S
k uk=uk+1 =uk+2
uk+3 =1
2(u2
k+1 +u2 k+2)
=1
2(u2
k+1 +u2 k)uk+2 =uk
=uk+2
uk+1 =uk+2 =uk+3 ∀n≥k, un=
uk
(un)
uk

uk−1ukuk+1 (un)
uk−1=q2uk+1 −u2
k
=q2uk+2 −u2
k+1 uk=uk+1 =uk+2
=uk
n≤k un=uk
(un)
S1
k uk=uk+1 = 1 uk+2 =
1
2(12+ 12) = 1
(un)
S
k≥2uk= 0 u2
k−2+
u2
k−1= 0 uk−2=uk−1= 0
(un)
x y
0≤x≤y≤1
2(x2+y2)y≥1x=y= 0
y < 1x=y= 0
y < 1y2≤y y ≤1
2(x2+y2)y2≤1 2(x2+y2)
y2≤x2
0≤x≤y x =y t 7→ t2R+
y≤1
2(x2+y2)y≤y2
y2≤y y2=y y = 0
1y < 1
x=y= 0
0≤x≤y≤1
2(x2+y2)y≥1x=y= 0
1
2(x2+y2)≤x≤y y ≤1
y > 1
x xy ≥x x
1
2(x2+y2)≤xy (x−y)2= 0 x=y
1
2(x2+y2)≤y y2≤y y > 1
1
2(x2+y2)≤x≤y y ≤1
(un)Sn≥2un+1 −unun−un−2

S
un+1 −un=1
2(u2
n+u2 n−1−2un)
u2
n−u2 n−2un−un−2(un)
t7→ t2R+
u2
n−u2 n−2=u2 n−(2un−u2 n−1) = u2 n+u2 n−1−2un
un+1 −un
n≥2un+1 −unun−un−2
(un)SN≥0uN+2
uNuN+1
uN+1 ≤uN+2 ≤uN+3
(un)u2
N+2 ≥u2 N
uN+3 =1
2(u2
N+1 +u2 N+2)≥1
2(u2
N+1 +u2 N) = uN+2
uN+2 ≥uN+1 uN+1 ≤uN+2 ≤uN+3
(un)
k≥N+ 1 uk≤uk+1 ≤uk+2
N+ 1
(un)
n≥N+ 3 un+1 =unun+1 −un=
0un+1 −un
un−un−2= 0 un−2=un
n−2≥N+ 1 N+ 1
un−2≤un−1≤unun−2=un−1=un
(un)N+ 3
uN+2 ≥1
uN+2 uNuN+1 uN+2 =
1
2(u2
N+u2 N+1)uN+2 ≥1x=uNy=uN+1
uNuN+1
+∞
+∞(un) 0,1 +∞
N+ 3
0 1 uN+2 ≥1

1N+2
N+ 3
+∞
(un)SN≥0uN+2
uNuN+1
0
•0≤uN+2 ≤uN
uN+3 =1
2(u2
N+2 +u2 N+1)≤1
2(u2
N+u2 N+1) = uN+2 ≤uN+1
•
∀k≥N+ 1, uk+2 ≤uk+1 ≤uk
N+ 1
•un+1 =unn≥N+ 3 un=un−2
N+ 1 n−2≥N+ 1
un=un+1 =un+2 (un)
(un)N+ 3
•uN+2 ≤uN, uN+1 uN+2 ≤1
•0 (un)
0 1 N+ 1 uN+2 ≤1
N+ 3
1 (un) 0
u(√2,0) u(2,0)
+∞
0
− ∗∗ − ∗∗
10
u5u3u4u(√2,0) 0

u4u2u3u(2,0) +∞
(un)S+∞0
u06=u1
u0=u1u2u0u1
u2u0u1(un) 0
+∞u06=u1
n≥0un+2 unun+1
n un+2
un+1 unun+1 un
0 +∞
n≥0un+2 unun+1
u0< u1
(u2n) (u2n+1)
u2u0u1u0<
u2< u1
u3u1u2u0< u2< u3< u1
n n ∈Nu2n< u2n+2 < u2n+3 < u2n+1
•n= 0
•n∈N
u2n< u2n+2 < u2n+3 < u2n+1
u2n+4 u2n+2 u2n+3 u2n+5 u2n+3 u2n+4
u2n< u2n+2 < u2n+4 < u2n+5 < u2n+3 < u2n+1
u2n+2 < u2n+4 < u2n+5 < u2n+3
n∈Nu2n< u2n+2 < u2n+3 <
u2n+1
n∈Nu2n< u2n+2
(u2n)
n∈Nu2n+3 < u2n+1
(u2n+1)
(un) 1
(u2n+1)n∈Nu2n+1 ≤u1
n∈Nu2n< u2n+1 (u2n)
u1
a