Giới hạn của dãy điểm bất động của dãy các ánh xạ trong không gian mêtric

BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC VINH
- - - - - -  - - - - - -
VÕ VĂN CẨM
GIỚI HẠN CỦA DÃY ĐIỂM BẤT ĐỘNG CỦA DÃY
CÁC ÁNH XẠ TRONG KHÔNG GIAN MÊTRIC
LUẬN VĂN THẠC SĨ TOÁN HỌC
Nghệ An - 2014
BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC VINH
- - - - - -  - - - - - -
VÕ VĂN CẨM
GIỚI HẠN CỦA DÃY ĐIỂM BẤT ĐỘNG CỦA DÃY
CÁC ÁNH XẠ TRONG KHÔNG GIAN MÊTRIC
Chuyên ngành: TOÁN GIẢI TÍCH
Mã số: 60. 46. 01. 02
LUẬN VĂN THẠC SĨ TOÁN HỌC
Người hướng dẫn khoa học: TS. KIỀU PHƯƠNG CHI
Nghệ An- 2014


X d : X × X → R
X
d(x, y)  0 x, y ∈ X d(x, y) = 0 x = y.
d(x, y) = d(y, x) x, y ∈ X
d(x, y)  d(x, z) + d(z, y) x, y, z ∈ X
(X, d)
(X, d) {x
n

} ⊂ X
x ∈ X x
n
→ x x
{x
n
} lim
n→∞
d(x
n
, x) = 0.
X
X X
X
a ∈ X r > 0 B(a, r) = {x ∈ X : d(x, a) < r}
(X, d)
{x
n
} ⊂ X lim
m,n→∞
d(x
m
, x
n
) = 0
(X, d) X
X
B ⊂ X
δ[B] = sup{d(x, y) : x, y ∈ B}
B B

{B
n
} ⊂ X
B
n+1
⊂ B
n
lim
n→∞
δ[B
n
] = 0
(X, d) (Y, ρ)
f : X → X
f {x
n
} ⊂ X x
n
→ x
f(x
n
) → f(x)
f ε > 0 δ = δ(ε)
ρ(fx, fy) < ε, ∀x, y ∈ X, d(x, y) < δ.
(X, d) (Y, ρ)
X × Y
D((x
1
, y
1

), (x
2
, y
2
)) = d(x
1
, x
2
) + ρ(y
1
, y
2
)
(x
1
, y
1
), (x
2
, y
2
) ∈ X × Y
d : X × X → R
(x
n
, y
n
) (x, y) ∈ X × X d(x
n
, y

n
)
d(x, y) R
T
α
: (X, d) → (Y, ρ) (α ∈ I)
a ∈ X ε > 0 δ > 0
α ∈ I ρ(T
α
x, T
α
a) < ε x d(x, a) < δ
(T
α
) X ε > 0
δ > 0 α ∈ I ρ(T
α
x, T
α
y) < ε x, y ∈ X
d(x, y) < δ
d, ρ X
d ρ i
d
: (X, d) →
(X, ρ)
d ρ i
d
:
(X, d) → (X, ρ)

d ρ
a, b > 0
ad(x, y)  ρ(x, y)  bd(x, y)
x, y ∈ X
(T
n
)
(X, d) (Y, ρ)
(T
n
) X T : X → Y
x ∈ X T
n
x T x
(T
n
) X T : X → Y
lim
n→∞
sup
x∈X
ρ(T x
n
, T x) = 0.
(T
n
) X T :
X → Y K X
lim
n→∞

sup
x∈K
ρ(T x
n
, T x) = 0.
(T
n
)
(T
n
) K
T
n
T K T
n
T K
ε > 0 T K
T K δ
1
= δ
1
(ε)
d(T x, Ty) <
ε
3
x, y ∈ K d(x, y) < δ
1
(T
n
) ε > 0 δ

2
> 0
d(T
n
x, T
n
y) <
ε
3
x, y ∈ K d(x, y) < δ
2
(T
n
) T K a ∈ K
n
0
(a) = n
0
(a, ε)
d(T
n
, T a) 
ε
3
n  n
0
(a) δ = min{δ
1
, δ
2

} {B(a, δ) :
a ∈ K} K K a
1
, , a
k
K ⊂
k

i=1
B(a
i
, δ).
N = max{n
0
a(a
1
) : i = 1, 2 k}. x ∈ K a
i
x ∈ B(a
i
, ε) n  N
d(T
n
x, T x)  d(T
n
x, T
n
a
i
) + d(T

n
a
i
, T a
i
) + d(T a
i
, T x) 
ε
3
+
ε
3
+
ε
3
= ε.
sup
x∈K
d(T
n
x, T x)  ε
n  N lim
n→∞
sup
x∈K
d(T
n
x, T x) = 0 T
n

K T
(M, d) T : X →
X T k  0
(T x, Ty)  kd(x, y)
x, y ∈ M k
T k(T )
k(T ) T T
q k(T ) = 1 T
T, S : M → M k(T ◦
S)  k(T )k(S)
d(T x, T y)  k(T )d(x, y)
x, y ∈ M
d(Sx, Sy)  k(S)d(x, y)
x, y ∈ M
d

T (Sx), T (Sy)

 k(T )d(Sx, Sy)  k(T )k(S)d(x, y)
x, y ∈ X T ◦ S
k(T ◦ S)  k(T )k(S)
T : M → M
T
n
k(T
n
)  [k(T )]
n
n = 1, 2,
T : M → M

k
∞
(T ) := lim
n→∞

k(T
n
)

1
n
= inf{

k(T
n
)

1
n
: n = 1, 2, }.
n k(T
n
)  [k(T )]
n
inf
n1

k(T
n
)


1
n
 sup
n1

k(T
n
)

1
n
 k(T ).
n

k(T
n
)

1
n


k(T
n−1
)k(T )

1
n




k(T
n−1
)

1
n−1

n−1
n

k(T )]
1
n
.
sup
n1

k(T
n
)

1
n
 inf
n1


k(T

n−1
)

1
n−1

n−1
n

k(T )]
1
n
. = inf
n1

k(T
n−1
)

1
n−1
.
lim
n→∞

k(T
n
)

1

n
= inf
n1

k(T
n
)

1
n
= sup
n1

k(T
n
)

1
n
.
k
∞
(T )
T : M → M
ρ d M
lim
n→∞

k
d

(T
n
)

1
n
= lim
n→∞

k
ρ
(T
n
)

1
n
.
ρ d M
a, b
aρ(x, y)  d(x, y)  bρ(x, y)
x, y ∈ M T d ρ
ρ(T x, T y) 
1
a
d(T x, T y) 
1
a
k
d

(T )d(x, y) 
b
a
k
d
(T )ρ(x, y).
k
ρ
(T ) 
b
a
k
d
(T ).
k
d
(T ) 
b
a
k
ρ
(T ).
T
n
a
b
k
d
(T
n

)  k
ρ
(T
n
) 
b
a
k
d
(T
n
)
n
lim
n→∞

a
b
k
d
(T
n
)

1
n
 lim
n→∞

k

ρ
(T
n
)

1
n
 lim
n→∞

b
a
k
d
(T
n
)

1
n
k
∞
(T ) = lim
n→∞

k
d
(T
n
)


1
n
= lim
n→∞

k
ρ
(T
n
)

1
n
.
T : M → M
k
∞
(T ) = inf k
ρ
(T ),
ρ d
k
∞
(T ) = lim
n→∞

k
d
(T

n
)

1
n
= lim
n→∞

k
ρ
(T
n
)

1
n
.

k
ρ
(T
n
)

1
n


(k
ρ

(T ))
n

1
n
= k
ρ
(T )
k
∞
(T )  k
ρ
(T )
ρ d
λ ∈ [0,
1
k
∞
(T )
)
ρ
λ
(x, y) =
∞

n=0
λ
n
d(T
n

x, T
n
y).
λ
n+1
d(T
n+1
x, T
n+1
y)
λ
n
d(T
n
x, T
n
y)
 λ
k
∞
(T )d(T
n
x, T
n
y)
λ
n
d(T
n
x, T

n
y)
= λk
∞
(T ) < 1
ρ
λ
(x, y) M
d(x, y)  ρ
λ
(x, y) 

∞

n=0
[k
∞
(T )λ]
n

d(x, y)
x, y ∈ M ρ
λ
d
ρ
λ
(T x, T y) =
∞

n=0

λ
n
d(T
n+1
x, T
n+1
y) =
1
λ
[ρ
λ
(x, y)−d(x, y)] 
1
λ
ρ
λ
(x, y)
x, y ∈ M
k
ρ
λ
(T ) 
1
λ
.
λ =
1
k
∞
(T ) + ε

ε > 0
k
ρ
λ
(T )  k
∞
(T ) + ε.
k
∞
(T ) = inf k
ρ
(T ),
ρ d
T
M x
0
∈ M
{T
n
(x
0
)} T
T q = k(T ) x
0
∈ M
{x
n
}
x
n+1

= fx
n
, n = 0, 1, 2, . . .
d(x
1
, x
2
) = d(T x
0
, T x
1
)  qd(x
0
, x
1
)
d(x
2
, x
3
) = d(T x
1
, T x
2
)  qd(x
1
, x
2
)  q
2

d(x
0
, x
1
).
d(x
n
, x
n+1
)  q
n
d(x
0
, x
1
), ∀n = 1, 2, . . .
d(x
n
, x
n+p
)  d(x
n
, x
n+1
) + d(x
n+1
, x
n+2
) + · · · + d(x
n+p−1

, x
n+p
)
 (q
n
+ q
n+1
+ · · · + q
n+p−1
)d(x
0
, x
1
)

q
n
1 − q
d(x
0
, x
1
)
n, p ∈ N
∗
0  q < 1 lim
n→∞
d(x
n
, x

n+p
) = 0 {x
n
}
M
a = lim
n→∞
x
n
.
T
a = lim
n→∞
x
n+1
= lim
n→∞
fx
n
= T ( lim
n→∞
x
n
) = T a.
a T
b f
d(a, b) = d(T a, Tb)  qd(a, b)
q ∈ [0, 1) d(a, b) = 0 a = b T
a = inf{d(x, T x) : x ∈ M} k = k(T )
a = 0 ε > 0 x ∈ M d(x, T x)  a + ε

a  d(T x, T
2
x)  kd(x, T x)  k(a + ε).
k < 1 ε > 0 a = 0 n = 1, 2,
M
n
= {x ∈ M : d(x, T x) 
1
n
.
M
ε
x, y ∈ M
n
d(x, y)  d(x, T x) + d(Tx, T y) + d(T y, y) 
2
n
+ kd(x, y).
d(x, y) 
2
n(1 − k)
lim
n→∞
d(M
n
) = 0.
M
n
0
M


M
n
= {u}.
d(u, T u) 
1
n
n ∈ N d(u, T u) = 0
k = k(T ) ϕ(x) =
d(x,T x)
1−k
x ∈ M
d(T x, T
2
x)  kd(x, T x)
d(x, T x) − d(Tx, T x)  d(x, T x) − kd(x, T
2
x)
x ∈ M
d(x, T x)  ϕ(x) − ϕ(T x), x ∈ M.
x
0
∈ M m, n ∈ N n < m
d(T
n
x
0
, T
m+1
x

0
) 
m

i=n
d(T
i
x
0
, T
i+1
x
0
)  ϕ(T
n
x
0
) − ϕ(T
m+1
x
0
).
S
n

∞
i=1
d(T
i
x

0
, T
i+1
x
0
)
ϕ(x
0
) {d(T
n
x
0
, T
n+1
x
0
)}
M T
n
x
0
x ∈ X T
x = lim
n→∞
T
n
x
0
= lim
n→∞

T
n+1
x
0
= T ( lim
n→∞
T
n
x
0
) = T x.
d(T
n
x
0
, x) 
k
n
1 − k
d(x
0
, T x
0
).
N
d(m, n) =

0 m = n
1 +
1

m + n
m = n.
(N, d)
f : N → N fn = n + 1 f
d(fm, fn) < d(m, n), ∀m = n.
m = 0
d(fm, f0) = d(m + 1, 1) = 1 +
1
m + 2
< 1 +
1
m
= d(m, 0).
m = n mn = 0
d(fm, fn) = d(m + 1, n + 1) = 1 +
1
m + n + 2
< 1 +
1
m + n
= d(m, n).
X
X
f : X → X
d(fx, fy) < d(x, y), ∀x, y ∈ X x = y
f
f
ϕ(x) = d(fx, x), x ∈ X.
f d ϕ X
ϕ a ∈ X fa = a

d(f
2
a, fa) < d(fa, a).
ϕ(fa) < ϕ(a) ϕ a
fa = a a f
b = a f
d(a, b) = d(fa, fb) < d(a, b).
f
X
C
[0,1]
[0, 1]
x = sup{|x(t)| : t ∈ [0, 1]}
x ∈ C
[0,1]
M = {x ∈ C
[0,1]
: 0 = x(0)  x(t)  x(1) = 1}.
M C
[0,1]
T : M → M
T x(t) = tx(t), t ∈ [0, 1]
x ∈ M
d(T x, T y) = T x − T y = sup{|tx(t) − ty(t)| : t ∈ [0, 1]}
= sup{|t||x(t) − y(t)| : t ∈ [0, 1]}
 sup{|x(t) − y(t)| : t ∈ [0, 1]} = x − y = d(x, y)
x, y ∈ M T T
T x = x x ∈ M
tx(t) = x(t)
t ∈ [0, 1] x(t) = 0 0  t  1

x(1) = 1 x [0, 1]
(X, d)
f : X → X f h ∈ [0, 1)
d(fx, fy)  hM
f
(x, y)
x, y ∈ X
M
f
(x, y) = max{d(x, y), d(x, fx), d(y, fy), d(x, fy), d(y, fx)}.
f : X → X x ∈ X
O(x) := {x, fx, f
2
x, , f
n
x, }
O(x, n) = {x, fx, , f
n
x}, n = 1, 2,
f n = 1, 2,
i, j ∈ {1, , n}
d(fx
i
, fx
j
)  hδ[O(x, n)].
k  n d(x, f
k
x) = δ[O(x, n)]
x X n = 1, 2,

i, j ∈ {1, 2, , n}
f
i−1
x, f
i
x, f
j−1
x, f
j
x ∈ O[x, n].
d(f
i
x, f
j
x) = d(ff
i−1
x, ff
j−1
x)
 h max{d(f
i−1
x, f
j−1
x), d(f
i
x, f
i−1
x),
d(f
j−1

x, f
j
x), d(f
i−1
x, f
j
x), d(f
i
x, f
j−1
x)}
 hδ[O(x, n)].
δ[O(x, n)] = 0
δ[O(x, n)] > 0 0 < i < j  n
δ[O(x, n)] = d(f
i
x, f
j
x).
δ[O(x, n)] = d(f
i
x, f
j
x) = d(ff
i−1
x, ff
j−1
x)
 h max{d(f
i−1

x, f
j−1
x), d(f
i
x, f
i−1
x),
d(f
j−1
x, f
j
x), d(f
i−1
x, f
j
x), d(f
i
x, f
j−1
x)}
 hδ[O(x, n)].
h < 1
δ[O(x, n)] = d(f
i
x, f
j
x)
i = 0
f
δ[O(x)] 

1
1 − h
(x, fx).
x ∈ X O(x, 1) ⊂ O(x, 2) ⊂ ⊂ O(x)
δ[O(x, 1)]  δ[O(x, 2)]   δ[O(x, n)]  δ[O(x)]
n = 1, 2, O(x, n) O(x)
δ[O(x)] = sup
n∈N
δ[O(x, n)].
n = 1, 2, k ∈ {1, 2, , n}
δ[O(x, n)] = d(x, f
k
x)
d(x, f
k
x)  d(x, fx) + d(fx, f
k
x)
 d(x, fx) + hδ[O(x, n)]
d(x, fx) + hd(x, f
k
x).
δ[O(x, n)] = d(x, f
k
x) 
1
1 − h
d(x, fx)
n n
δ[O(x)] 

1
1 − h
(x, fx).
(X, d)
f : X → X f α ∈ [0, 1)
d(fx, fy)  αM
f
(x, y)
x, y ∈ X
M
f
(x, y) = max{d(x, y), d(x, fx), d(y, fy), d(x, fy), d(y, fx)}.
(X, d) f :
X → X f
x ∈ X (x
n
) ⊂
x
n
= fx
n−1
, n = 1, 2
d(x
n
, x
n+1
) = d(fx
n−1
, fx
n

)  αM(x
n−1
, x
n
)
= αM(x
n−1
, x
n
)
= αM(x
n−1
, x
n
).
d(f
n
x, f
n+1
x) = d(ff
n−1
x, f
2
f
n−1
x)  αδ[O(f
n−1
x, 2)]
1  k
1

 2
δ[O(f
n−1
x, 2)] = d(f
n−1
x, f
k
1
f
n−1
x)
d(x
n
, x
n+1
) = d(f
n
x, f
n+1
x)  αd(f
n−1
x, f
k
1
f
n−1
x).
n  2
d(f
n−1

x, f
k
1
f
n−1
x) = d(ff
n−2
x, f
k
1
+1
f
n−2
x)
 αδ[O(f
n−2
x, k
1
+ 1)]  αδ[O(f
n−2
x, 3)]
.
d(f
n
x, f
n+1
x)  αδ[O(f
n−1
x, 2)]  α
2

δ[O(f
n−2
x, 3)]
d(f
n
x, f
n+1
x)  αδ[O(f
n−1
x, 2)]   α
n
δ[O(x, n + 1)].
δ[O(x, n + 1)]  δ[O(x, ∞)] 
1
1 − α
d(x, fx).
d(f
n
x, f
n+1
x) 
α
n
1 − α
d(x, fx)
n = 1, 2, p ∈ N
d(f
n
x, f
n+p

x)  d(f
n
x, f
n+1
x) + d(f
n+1
x, f
n+2
x) + + d(f
n+p−1
x, f
n+p
x)

α
n
1 − α
(1 + α + + α
p−1
)d(x, fx)
=
α
n
1 − α
1 − α
p
1 − α
d(x, fx)

α

n
(1 − α)
2
d(x, fx)
0  α < 1 lim
n→∞
d(x
n
, x
n+p
) = 0 p ∈ N
(x
n
) X u = lim
n→∞
x
n
d(u, fu)  d(u, x
n+1
) + d(x
n+1
, fu) = d(f
n+1
x, u) + d(f
n+1
x, fu)
 d(f
n+1
x, u) + α max{d(f
n

x, u), d(f
n
x, f
n+1
x), d(u, fu),
d(f
n+1
x, u), d(f
n
x, fu)}
 d(f
n
x, u) + α

d(f
n
x, u) + d(f
n
x, f
n+1
x)
+ d(u, fu) + d(f
n+1
x, u) + d(f
n
x, fu)

d(u, fu) 
1
1 − α


(1 + α)d(x
n+1
, u) + αd(u, x
n
) + αd(x
n
, x
n+1
)

.
n → ∞ d(u, fu) = 0 u
f
(X, d) (T
n
)
X X (a
n
) X
(T
n
) T
n
(a
n
) = a
n
n = 1, 2,
(X, d) (T

n
)
X X (a
n
)
(T
n
) T
n
X T lim
n→∞
a
n
= a
a T
T (T
n
)
T lim
n→∞
a
n
= a lim
n→∞
T a
n
= T a