BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC VINH

ĐẶNG THÀNH TRUNG
SỰ TỒN TẠI ĐIỂM BẤT ĐỘNG CHUNG
CỦA CÁC ÁNH XẠ T – CO
TRONG KHÔNG GIAN MÊTRIC NÓN
LUẬN VĂN THẠC SĨ TOÁN HỌC

NGHỆ AN – 2014
BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC VINH


.
C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
class="bi x18 y33 w4 hd"
class="bi x18 y33 w4 he"
X τ X
X
T
1
∅ X ∈ τ
T
2
G
i
∈ τ, i ∈ I

i∈I
G
i
∈ τ
T
3
G
1
, G
2
∈ τ G
1

G
2

X
x ∈ X B(x)
X T
2
x, y ∈ X, x = y U
x
, V
y
x y U
x

V
y
= ∅
X X
X, Y f : X → Y
f x ∈ X V f(x)
U x f(U) ⊂ V f
x ∈ X
X d : X X → R d
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
(X, d) X
X (x
n
) X
ε > 0 n

X ≤
X ≤ X
x ≤ x, ∀x ∈ X
x ≤ y y ≤ x x = y, ∀x, y ∈ X
x ≤ y; y ≤ z x ≤ z, ∀x, y, z ∈ X
X
(X, ≤) X
≤ X A ⊂ X
x ∈ X A
a ≤ x x ≤ a a ∈ A
x ∈ X
A x A y
A x ≤ y y ≤ x
x = sup A x = inf A
E
R P E E
P P = ∅, P =
a, b ∈ R, a, b ≥ 0 x, y ∈ P ax + by ∈ P
x ∈ P −x ∈ P x = 0
R
P = x ∈ R : x ≥ 0
E = R
2
, P = {(x, y) ∈ E : x, y ≥ 0} ⊂ R
2
P
P P = ∅, P =
(x, y), (u, v) ∈ P a, b ∈ R; a, b ≥ 0 a(x, y)+b(u, v) ∈
P
(x, y) ∈ P (−x, −y) ∈ P (x, y) = (0, 0)

P E
P K > 0
x, y ∈ E 0 ≤ x ≤ y x ≤ Ky K
P
P E a, b, c ∈ E
{x
n
}, {y
n
} E α
a  b b  c a  c
a ≤ b b  c a  c
a  b, c  d a + c  b + d
αintP ⊂ intP
δ > 0 x ∈ intP 0 < γ < 1 γx < δ
c
1
∈ intP c
2
∈ P d ∈ intP c
1
 d
c
2
 d
c
1
, c
2
∈ intP e ∈ intP e  c

αintP ⊂ intP
δ > 0 x ∈ intP n > 1
δ
nx
< 1
γ =
δ
nx
0 < γ < 1
γx ≤ γx ≤
δ
nx
x ≤
δ
n
< δ
δ > 0 c
1
+ B(0, δ) ⊂ intP B(0, δ) = {x ∈ E :
x < δ} B(0, δ) m > 1 c
2
∈ mB(0, δ)
−c
2
∈ mB(0, δ) mc
1
− c
2
∈ intP d = mc
1

) c
2
∈ mB(0, δ

) −c
1
∈ mB(0, δ

) −c
2
∈ mB(0, δ

)
mc
1
− c
1
∈ intP mc
2
− c
2
∈ intP e = mc
1
− c
1
+ mc
2
− c
2
e  c

a ∈ P −a ∈ P a −a ∈ P
P a = 0
x
n
≤ y
n
y
n
− x
n
∈ P P lim
n→∞
(y
n
− x
n
) ∈ P
lim
n→∞
x
n
= x lim
n→∞
y
n
= y lim
n→∞
(y
n
− x

∈ N
x
n
 < δ, ∀n > n
0
c − x
n
∈ intP ∀n > n
0
x
n
 c, ∀n > n
0
P E
intP = 0 ≤ E P
X d : X × X → E
d 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
(X, d) X
E = R P = {x ∈ R : x ≥ 0}
E = R
2
P = {(x, y) ∈ R
2
: x, y ≥ 0}
X = R d : X × X → E
d(x, y) = (α|x − y|, β|x − y|), ∀x, y ∈ X

i∈I
U
i
∈ F
x ∈

i∈I
U
i
i = i
0
∈ I x ∈ U
i
0
⊂ F c ∈ intP
B(x, c) ⊂ U
i
0
B(x, c) ⊂

i∈I
U
i

i∈I
U
i
∈ F
U, V ∈ F U ∩ V ∈ F x ∈ U ∩ V
x ∈ U x ∈ V c

d(z, y)  c − d(y, x) d(z, y) + d(y, x)  c
d(z, x)  c z ∈ B(x, c) B(y, c

) ⊂ B(x, c) B(x, c) ∈ F
x, y ∈ X x = y
(X, F) T
2
c
1
, c
2
∈ intP
B(x, c
1
)

B(y, c
2
) = ∅ c
1
, c
2
∈
intP B(x, c
1
)

B(y, c
2
) = ∅ c ∈ intP

c
n
≤ c n = 1, 2, d(x, y)  c c
intP d(x, y) = 0 x = y
x = y (X, F) T
2
x ∈ X x
c ∈ intP
U = {B(x,
c
n
) : n = 1, 2, }
U ⊂ F V x y ∈ intP
B(x, y) ⊂ V y ∈ intP  > 0 B
E
(y, ) ⊂ P
B
E
(y, ) y  n ∈ N n >
c

y − (y −
c
n
) =
c
n
< 
y −
c

lim
n→∞
x
n
= x
x
n
→ x n → ∞
(X, d) {x
n
} ⊂
X x y x = y
(X, d) {x
n
} ⊂ X
x
n
→ x ∈ X c ∈ intP n
c
d(x
n
, x)  c n ≥ n
c
{x
n
} ⊂ X c ∈ intP B(x, c)
x n
c
x
n

n
→ x ∈ X
(X, d) {x
n
} ⊂
X c ∈ intP N
d(x
m
, x
n
)  c m, n > N
(X, d)
{x
n
} (X, d)
{x
n
} X x
n
→ x ∈ X
0  c ∈ E N d(x
n
, x) 
c
2
n > N
m, n > N
d(x
m
, x

} x
N
d(x
n
, x
m
) 
c
2
, ∀m, n ≥ N
d(x
n
k
, x) 
c
2
, ∀n
k
≥ N
d(x
n
, x) ≤ d(x
n
, x
n
k
) + d(x
n
k
, x)  c, ∀n, n

0
f(x
n
) ∈ f(U) ⊂ V, ∀n ≥ n
0
f(x
n
) → f(a)
{x
n
} X x
n
→ a f(x
n
) → f(a)
f a
f a
y
0
∈ intP c ∈ intP
f(B(a, c)) ⊂ B(f(a), y
0
)
n = 1, 2, x
n
∈ B(a,
c
n
) f(x
n

(y
n
)
T (y
n
) T (y
n
)
(y
n
)
T (x
n
) lim
n→∞
x
n
= x
lim
n→∞
T x
n
= T x
x ∈ M S Sx = x
x ∈ M S T Sx =
T x = x
(M, d) T, S :
M → M
S T T K
1

= |
x
2
4
−
y
2
4
|e
t
≤
1
3
[|T x − T Sx| + |T y − T Sy|]e
t
=
1
3
[d(T x, T Sx) + d(T y, TSy)]
S T K
1
T : M → M
S : M → M T K
1
x
0
∈ M
lim
n→∞
d

x
0
)
x
0
M (x
n
)
x
n.+1
= Sx
n
= S
n+1
x
0
S T K
1
d(T x
n
, T x
n+1
) = d(T Sx
n−1
, T Sx
n
)
≤ b[d(T x
n−1
, T Sx


b
1 − b

n
d (T x
0
, T Sx
0
) ∀n = 1, 2,


d

T S
n
x
0
, T S
n+1
x
0



≤

b
1 − b


x
0
, T S
n+1
x
0

= 0.
m, n ∈ N m > n
d(T x
n
, T x
m
) ≤ d(T x
n
, T x
n+1
) + + d(T x
m−1
, T x
m
)
≤


b
1 − b

n
+ +

x
0
, T S
m
x
0
) ≤

b
1 − b

n
1
1 −

b
1 − b

d (T x
0
, T Sx
0
) .
d (T S
n
x
0
, T S
m
x

m,n→∞
d (T S
n
x
0
, T S
m
x
0
) = 0
(T S
n
x
0
) M M
v ∈ M
lim
n→∞
T S
n
x
0
= v.
T (S
n
x
0
)
u ∈ M (x
n

x
0
, T S
n
i
+1
x
0

+ d

T S
n
i
+1
x
0
, T u

≤ b

d (T u, T Su) + d

T S
n
i
−1
x
0
, T S


T S
n
i
−1
x
0
, T S
n
i
x
0

+
1
1 − b

b
1 − b

n
i
d (T x
0
, T Sx
0
)
+
1
1 − b


+
K
1 − b

b
1−b

n
i
d (T x
0
, T Sx
0
)
+
K
1 − b


d

T S
n
i
+1
x
0
, T u


x
0
= u
(S
n
x
0
) S
T : M → M
S : M → M T K
2
x
0
∈ M
lim
n→∞
d

T S
n
x
0
, T S
n+1
x
0

= 0;
v ∈ M
lim

n
x
0
S T K
2
d(T Sx
n
, T Sx
n+1
) ≤ c[d(T x
n
, T Sx
n+1
) + d(T x
n+1
, T Sx
n
)]
≤ c[d(T Sx
n−1
, T Sx
n
) + d(T Sx
n
, T Sx
n+1
)]
d(T Sx
n
, T Sx

, T Sx
n+1
) ≤ h
n
Kd(T Sx
0
, T Sx
1
)
h ∈ [0, 1)
lim
n→∞
d(T Sx
n
, T Sx
n+1
) = 0
lim
n→∞


d(T S
n
x
0
, T S
n+1
x
0
)

n
1 − h
d(T Sx
0
, T Sx
1
)
d(T Sx
n
, T Sx
m
) ≤
h
n
1 − h
Kd(T Sx
0
, T Sx
1
)
lim
m,n→∞
d(T Sx
n
, T Sx
m
) = 0.
(T S
n
x

n→∞
T fx
2n
= lim
n→∞
T gx
2n+1
= z
x
.
{fx
2n
} {gx
2n+1
}
w
x
∈ X fw
x
= gw
x
= w
x
{fx
2n
} {gx
2n+1
} w
x
x

, T gx
2n+1
)
≤ q(x
2n
, x
2n+1
)d(T x
2n
, T x
2n+1
) + r(x
2n
, x
2n+1
)d(T x
2n
, T fx
2n
)
+s(x
2n
, x
2n+1
)d(T x
2n+1
, T gx
2n+1
)
+t(x

2n
, x
2n+1
)d(T x
2n+1
, T x
2n+2
)
+t(x
2n
, x
2n+1
)[d(T x
2n
, T x
2n+2
) + d(T x
2n+1
, T x
2n+1
)]
≤ (q + r + t)(x
2n
, x
2n+1
)d(T x
2n
, T x
2n+1
)

2n+1
) − t(x
2n
, x
2n+1
)
d(T x
2n
, T x
2n+1
).
q(x, y) + r(x, y) + t(x, y)
1 − s(x, y) − t(x, y)
≤ λ
x, y ∈ X
d(T x
2n+1
, T x
2n+2
) ≤ λd(T x
2n
, T x
2n+1
).
d(T x
2n+2
, T x
2n+3
) ≤ λd(T x
2n+1