226 - Comportement d`une suite réelle ou vectorielle définie par une

un+1 =f(un)
E X E f :X→E
f(X)⊂X
E f
(un)n un+1 =f(un)
(un)l∈X f(l) = l
(X, d)f:X→
X f l
x0∈X(un)u0=x0n∈
Nun+1 =f(un)l
d(l, un)≤kn
1−kd(u1, u0)
f:K→K K
(x, y)K d(f(x), f(y))
< d(x, y)a x0∈K
(fn(x0))na
f= [−π, π]→[−1,1] ; x7→ sin x
(X, d)L
f:L×E→E∀λ∈L, f(λ, .)
k λ λ aλ
f(λ, .)λ→aλ

IRf:I→Rf(I)⊂I
(un)∈INun+1 =f(un)
n∈N
f(un)
u0∈R∗
+un+1 =
ln(1 + un)
+∞
f(x)−x
(un)
f(u2n) (u2n+1)
f: [0,1] →[0,1], x 7→ √1−x
u0∈[0,1] (un) (−1+√5)/2u0∈ {0,1}
R
f: [0,1] →[0,1] f(un) =
un+1 (un)un+1 −un→0
(un)n∈N
u0∈Run+1 =λun+a
λ∈R\{−1}a∈R
vn=un+a
λ−1vn+1 =λvn
(un)n∈N
u0∈Run+1 =f(un)
f(x) = ax+b
cx+da, b, c, d ∈Rc6= 0 u0∈R
N
f(∞) = a/c f(−d/c) = ∞C
2h(x) = x
α β ∀n∈N
un−α
un−β=knu0−α u0−βk=a−αc a−βc

α∀n∈N
1
un−α=1 u0−α+kn k =c a−αc
f C1l
1er |f0(l)|<1
|f0(x)| ≤ k < 1J⊂I l u0∈J
(un)u0un+1 =f(un)l
f0(l)6= 0 u0∈J\ {l} |un+1 −l| ∼
+∞|f0(l)||un−l|f
C2f0(l) = 0 f00(l)6= 0
u0∈J\ {l} |un+1 −l| ∼
+∞
|f00 (l)|
2|un−l|2
2nd |f0(l)|>1
|f0(x)| ≥ k > 1J⊂I l u0∈J
fn(u0)l
f−1J
|f0(l)|= 1
n f l n ≥2
n
(un)n∈Nu0>0un+1 =th(un)
un∼q3
2n
XRdf:X→Rf(X)⊂X
(un)∈XNun+1 =f(un)n∈N
(un)∈(Rd)Np un+p=
f(un, un+1, . . . , un+p−1)f:X⊂(Rd)p→Rd
(Un)∈(Rd)pN
F: (Rd)p→(Rd)p
Un=
un
un+1
un+p−1
F
x1
x2
xp−1
xp
=
x2
x3
xp
f(x1, x2, . . . , xp)
Un+1 =F(Un)n

n
un+p=a1un+p−1+a2un+p−2+··· +apun
xp=a1xp−1+··· +apr1, . . . , rq
α1, . . . , αq(un)
un=P1(n)rn
1+··· +Pq(n)rn
qi Pi
αi
f∈C2([a, b],R)f0>0 [a, b]f(a)<0< f(b)
f(c)=0 c
F:x7→ x−f(x)
f0(x)
F
J⊂[a, b]c F
J∀x0∈J F n(x0)→c∃K > 0|Fn+1(x0)−c| ≤
K|xn−c|2
f:R+→R+, x 7→ x2−y
a=√y
x0≥a0≤xn−a≤
2ax0−a
2a2n
y0=f(t, y)
f:U→RmU= [t0, t0+T]×V
R×Rm
t0< t1<··· < tN=t0+T
y01≤n≤N
yn+1 =yn+ (tn+1 −tn)f(tn, yn)n
(tn, yn)
y
(yp)
sup{ky0
p(t)−f(t, yp(t))k | t∈[t0, t0+T]\{t1. . . tN−1}} →p→∞
0yp[t0, t0+T]y y
y0=rx r ∈R
1/n yk= (1 + r/n)ky0k=n
(1 + r/n)n→n→∞ er

(t0, y0)∈U
r0>0, T > 0C. = [t0−T, t0+T]×B(y0, r0)⊂U
Tsup(t,y)∈Ckf(t, y)k< r0
(t0, y0) [t0−T, t0+T]
f: [0,1] →[0,1]
x7→ 2x x ≤1/2
−2x+ 2 x > 1/2
(un)n∈Nu0=x∈[0,1] un+1 =
f(un)n x x =P∞
k=1 xk/2k
f
x∈[0,1]
∀ε > 0,∀x∈[0,1],∃x0∈[0,1] : |x−x0| ≤ ε:∀p∈N∃n≥
p:|fn(x)−fn(x0)| ≥ 1/2
f: [0,1] →[0,1]
µRnX
Rn
T X X T
A X µ(T−1A) = µ(A)
T A X
T−1A=A µ(A)∈ {0,1}
T:X→X
B X
x∈B k ∈N∗
Tkx∈B
x∈[0,1] x x