Về sự tồn tại điểm giả bất động bộ đôi trong không gian giả meetric nón có thứ tự bộ phận

X
2
X
X ” ≤ ”
X ” ≤ ” X
x, y, z ∈ X
x ≤ x
x ≤ y y ≤ x x = y
x ≤ y y ≤ z x ≤ z
X
(X, ≤) X
X d : X × X→R
d(x, y) ≥ 0, ∀x, y ∈ X d(x, y) = 0 x = y
d(x, y) = d(y, x), ∀x, y ∈ X
d(x, z) ≤ d(x, y) + d(y, z), ∀x, y, z ∈ X
X
X d
(X, d) X
{x
n
} (X, d)
x ∈ X d(x, x
n
)→0 n→∞ x
n
→x lim
n→∞
x

n
= x
(X, d)
x
n
→x y
n
→y d(x
n
, y
n
)→d(x, y)
(X, d) {x
n
} ⊂ X
lim
n,m→∞
d(x
n
, x
m
) = 0
(X, d)
X
X K
(K = R K = C) . : X→R . X
x ≥ 0, ∀x ∈ X; x = 0 ⇔ x = 0
αx = |α| x , ∀α ∈ K, x ∈ X
x + y ≤ x + y , ∀x, y ∈ X
(X, .) X

(X, .)
d(x, y) = x − y , ∀x, y ∈ X
X
X X
E
R P E
P P = ∅, P = {0}
a b ∈ R a ≥ 0, b ≥ 0 x, y ∈ P ax + by ∈ P
x ∈ P −x ∈ P x = 0
P E E

≤

P
x ≤ y ⇔ y − x ∈ P
x < y x ≤ y x = y x  y y − x ∈ intP
P E
P k > 0
x, y ∈ E 0 ≤ x ≤ y x ≤ k y k
P
P
E
E {x
n
} E
x
1
≤ x
2
≤ ≤ x

n
≤ ≤ y y ∈ E x ∈ E x
n
− x → 0
n → ∞
k P k ≥ 1
P E a, b, c
E α
a  b b  c a  c
a ≤ b b  c a  c
a  b c  d a + c  b + d
αintP ⊂ intP αintP = {αx : x ∈ intP }
δ > 0 x ∈ intP 0 < γ < 1 γx < δ
c
1
∈ P c
2
∈ P d ∈ intP c
1
 d c
2
 d
c
1
, c
2
∈ P e ∈ intP e  c
1
, e  c
2

a ∈ P a  x x ∈ intP a = 0
E P a ≤ λa
a ∈ P, 0 < λ < 1 a = 0
0 ≤ x
n
≤ y
n
n lim
n→∞
x
n
= x, lim
n→∞
y
n
= y
0 ≤ x ≤ y
P E {x
n
}
X x
n
→0 c ∈ intP n
0
x
n
 c n ≥ n
0
X d : X
2

→E P
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, z) ≤ d(x, y) + d(y, z) x, y, z ∈ X
X d X
(X, d) X
E = R P = [0, +∞)
E = R
2
P = {(x, y) ∈ R
2
: x ≥ 0, y ≥ 0}
X = R d : X×X → E d (x, y) = (α |x − y| , β |x − y|)
∀x, y ∈ X α, β
d (X, d)
C
[a,b]
[a, b] C
[a,b]
f = sup
x∈[a,b]
|f (x)| , ∀f ∈
C
[a,b]
C
[a,b]
” ≤ ”
∀f, g ∈ C
[a,b]

, f ≤ g ⇔ f (x) ≤ g (x) , ∀x ∈ [a, b] ,
” ≤ ” R
P =

f ∈ C
[a,b]
: f ≥ 0

P
P P = ∅, P = {0}
α, β ∈ R, α, β ≥ 0 f, g ∈ P 0 ≤ αf (x) +
βf (x) , ∀x ∈ [a, b] αf + βg ∈ P
f ∈ P −f ∈ P f = 0
P E
(X, d) a ∈ X, c ∈
intP
B (a, c) := {x ∈ X : d (x, a)  c}
B(a, c)
(X, d) T
T := {G ⊂ X : ∀x ∈ G, ∃c ∈ P : B (x, c) ⊂ G}
T X
B(a, c) ∈ T x ∈ X, c ∈ intP
(X, T ) T
2
(X, T )
(X, d) {x
n
}
x y x = y
(X, d) {x

n
}
X c ∈ intP
n
0
d(x
m
, x
n
)  c m > n ≥ n
0
(X, d)
X
Y (X, d)
Y Y
(X, d)
g : X→X x ∈ X {x
n
} X x
n
→x
g(x
n
)→g(x)
(X, d)
F : X
2
→ X (x, y) ∈ X
2
F

F (x, y) = x, F (y, x) = y
F {x
n
}, {y
n
} X x
n
→
x, y
n
→ y F (x
n
, y
n
) = F (x, y)
(X, ≤)
F : X ×X → X F x, y ∈ X
x
1
, x
2
∈ X, x
1
≤ x
2
⇒ F (x
1
, y) ≤ F (x
2
, y)

y
1
, y
2
∈ Y, y
2
≤ y
1
⇒ F (x, y
1
) ≤ F (x, y
2
).
(X, ≤, d)
” ≤ ” F : X ×X →
X
A
1
) α, β, γ ≥ 0 2α+3β+3γ < 2 u ≤ x, y ≤ v
d(F (x, y), F (u, v)) ≤ α
d(x, u) + d(y, v)
2
+ β
d(x, F (x, y)) + d(u, F (u, v)) + d(y, v)
2
+ γ
d(x, F (u, v)) + d(u, F (x, y)) + d(y, v)
2
.
A

2
) x
0
, y
0
∈ X x
0
≤ F (x
0
, y
0
) F (y
0
, x
0
) ≤ y
0
A
3
) F X
{x
n
} X x
n
→ x x
n
≤ x
n = 1, 2,
{x
n

} X x
n
→ x x ≤ x
n
n = 1, 2,
F
x
n
= F (x
n−1
, y
n−1
), y
n
= F (y
n−1
, x
n−1
) n = 1, 2,
F X (A
2
)
x
0
≤ x
1
≤ ≤ x
n
≤ x
n+1

≤
≤ y
n+1
≤ y
n
≤ ≤ y
1
≤ y
0
e =
d(x
1
, x
0
) + d(y
1
, y
0
)
2
, λ =
2(α + β + γ)
2 − β − γ
.
(A
1
)
d(x
1
, x

2
) = d(F (x
1
, y
1
), F (x
0
, y
0
))
≤ α
d(x
1
, x
0
) + d(y
1
, y
0
)
2
+ β
d(x
1
, F (x
1
, y
1
)) + d(x
0

, F (x
0
, y
0
)) + d(y
1
, y
0
)
2
+ γ
d(x
1
, F (x
0
, y
0
)) + d(x
0
, F (x
1
, y
1
)) + d(y
1
, y
0
)
2
= αe + β

d(x
1
, x
2
) + d(x
0
, x
1
) + d(y
1
, y
0
)
2
≤ αe + βe +
β
2
d(x
1
, x
2
) + γ
d(x
0
, x
1
) + d(x
1
, x
2

) + d(y
1
, y
0
)
2
= (α + β + γ)e +
β + γ
2
d(x
1
, x
2
).
d(x
1
, x
2
) ≤
2(α + β + γ)
2 − β − γ
e = λe,
0 ≤ λ < 1 2α + 3β + 3γ < 2
d(y
1
, y
2
) ≤
2(α + β + γ)
2 − β − γ

e = λe.
n = 1, 2,
d(x
n+1
, x
n
) ≤
2(α + β + γ)
2 − β − γ
.
d(x
n
, x
n−1
) + d(y, y
n−1
)
2
= λ
d(x
n
, x
n−1
) + d(y, y
n−1
)
2
d(y
n
, y

n+1
) ≤
2(α + β + γ)
2 − β − γ
.
d(x
n
, x
n−1
) + d(y, y
n−1
)
2
= λ
d(x
n
, x
n−1
) + d(y, y
n−1
)
2
.
d(x
3
, x
2
) ≤ λ
d(x
2

, x
1
) + d(y
2
, y
1
)
2
≤ λ
2
e,
d(x
n+1
, x
n
) ≤ λ
d(x
n
, x
n−1
) + d(y, y
n−1
)
2
≤ λ
n
e n = 1, 2,
d(y
n
, y

n+1
) ≤ λ
n
e n = 1, 2,
{x
n
} {y
n
}
m > n
d(x
m
, x
n
) ≤ d(x
n
, x
n+1
) + + d(x
m−1
, x
n
)
≤ (λ
n
+ + λ
m−1
)e ≤
λ
n

1 − λ
e.
c ∈ intP N
d(x
n
, x
m
) ≤
λ
n
1 − λ
e  c, ∀m > n > N.
{x
n
} {y
n
}
(X, ≤, d) (x, y) ∈ X x
n
→
x, y
n
→ y
F
x = lim
n→∞
x
n
= lim
n→∞

F (x
n−1
, y
n−1
) = F (x, y)
y = lim
n→∞
y
n
= lim
n→∞
F (y
n−1
, x
n−1
) = F (y, x)
(x, y) F
F X a) b)
{x
n
} x
n
→ x {y
n
} y
n
→ y
x
n
≤ x y ≤ y

n
, n = 0, 1, 2, (A
1
)
d(F (x, y), x
n
) = d(F (x, y), F (x
n−1
, y
n−1
))
≤ α
d(x, x
n−1
) + d(y, y
n−1
)
2
+ β
d(x, F (x, y)) + d(x, x
n−1
) + d(y, y
n−1
)
2
+ γ
d(x, x
n
) + d(x
n−1

, F (x, y)) + d(y, y
n−1
)
2
≤ α
d(x, x
n−1
) + d(y, y
n−1
)
2
+ β
d(x, x
n
) + d(x, x
n−1
) + d(y, y
n−1
)
2
+ γ
d(x, x
n
) + d(x, x
n−1
) + d(y, y
n−1
)
2
+

β + γ
2
d(x
n
, F (x, y)).
2 − β − γ
2
d(F (x, y), x
n
) ≤ α
d(x, x
n−1
) + d(y, y
n−1
)
2
+ β
d(x, F (x, y)) + d(x, x
n−1
) + d(y, y
n−1
)
2
+ γ
d(x, x
n
) + d(x, x
n−1
) + d(y, y
n−1

)
2
.
c ∈ intP N n > N
d(x, x
n−1
)  c, d(x, x
n
)  c, d(y, y
n−1
)  c, d(x
n−1
, x
n
)  c
2 − β − γ
2
d(F (x, y), x
n
) 
2α + 3β + 3γ
2
c ≤ c.
x
n
→ F (x, y) x
n
→ x x = F (x, y)
F (y, x) = y (x, y)
F 

x
0
, y
0
2α + β + 3γ < 2
x = y
x
0
≤ y
0
F x
n
≤ y
n
n = 0, 1,
d(y
n
, x
n
) = d(F (y
n−1
, x
n−1
), F (x
n−1
, y
n−1
))
≤ αd(x
n

, y
n−1
) + β
d(x
n−1
, x
n
) + d(y
n−1
, y
n
) + d(x
n−1
, y
n−1
)
2
+ γ
d(x
n−1
, y
n
) + d(y
n−1
, x
n
) + d(x
n−1
, y
n−1

)
2
≤ (α +
β + γ
2
)d(x
n−1
, y
n−1
) + β
d(x
n−1
, x
n
) + d(y
n−1
, y
n
)
2
+ γ
d(x
n−1
, y
n
) + d(y
n−1
, x
n
)

2
≤ β
d(x
n−1
, x
n
) + d(y
n−1
, y
n
)
2
+ (α +
β + 3γ
2
)d(x, y)
+ (α +
β + γ
2
)[d(x
n−1
, x) + d(y, y
n−1
)]
+ γ
d(x
n−1
, x) + d(y, y
n
) + d(y

n−1
, y) + d(x, x
n
)
2
≤ β
d(x
n−1
, x
n
) + d(y
n−1
, y
n
)
2
+ (α +
β + 3γ
2
)d(x, y)
+ (α +
β
2
+ γ)[d(x
n−1
, x) + d(y, y
n−1
)] + γ
d(y, y
n

) + d(x, x
n
)
2
.
n = 0, 1, 2,
d(x, y) ≤ d(x, x
n
) + d(x
n
, y
n
) + d(y
n
, y)
d(x, y) ≤ (α +
β + 3γ
2
)d(x, y)
+ β
d(x
n−1
, x
n
) + d(y
n−1
, y
n
)
2

+ (α +
β
2
+ γ)[d(x
n−1
, x) + d(y, y
n−1
)]
+ (1 +
γ
2
)[d(y, y
n
) + d(x, x
n
)].
(1 −
2α + β + 3γ
2
)d(x, y) ≤ β
d(x
n−1
, x
n
) + d(y
n−1
, y
n
)
2

+ (α +
β
2
+ γ)[d(x
n−1
, x) + d(y, y
n−1
)]
+ (1 +
γ
2
)[d(y, y
n
) + d(x, x
n
)].
x
n
→ x, y
n
→ y c ∈ intP N
n > N
d(x
n−1
, x
n
)  c, d(y
n−1
, y
n

)  c, d(x
n−1
, x)  c, d(y
n−1
, y)  c, d(x, x
n
) 
c, d(y, y
n
)  c
(1 −
2α + β + 3γ
2
)d(x, y) ≤ (2α + 2β + 3γ + 2)c.
2α + β + 3γ < 2
d(x, y) ≤
4α + 4β + 6γ + 4
2 − 2α − β − 3γ
c, c ∈ intP.
d(x, y) = 0 x = y 
P E, ” ≤ ”
E P X
d : X × X → E
(x, y) → d(x, y).
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, z) ≤ d(x, y) + d(y, z), ∀x, y, z ∈ X.
X d
(X, d) X

L
[a,b]
[a, b] d : L
[a,b]
× L
[a,b]
→ R
d(f, g) =
b

a
|f(x) − g(x)| dx, ∀f, g ∈ L
[a,b]
d L
[a,b]
L
[a,b]
P = [0, ∞) P
R ” ≤ ” R P
R
d(f, g) ≥ 0, d(f, g) = 0 f = g d(f, g) = d(g, f), f, g ∈
L
[a,b]
f, g, h ∈ L
[a,b]
d(f, g) =
b

a
|f(x) − g(x)| dx =

b

a
|f(x) − h(x) + h(x) − g(x)| dx
≤
b

a
|f(x) − h(x)| dx +
b

a
|h(x) − g(x)| dx = d(f, h) + d(h, g)
d L
[a,b]

d X d(x, y) = 0
x = y d X
R P
P =

f ∈ C
[a,b]
: f ≥ 0

C
[a,b]
[a, b] R
” ≤ ” C
[a,b]

P ” ≤ ”
[a, b]
X =

f ∈ C
[a,b]
:

d : X×X → P d(f, g) = |f

− g

| , ∀f, g ∈ X
d(f, g)(x) = |f

(x) − g

(x)| , ∀f, g ∈ X, ∀x ∈ [a, b] d
d
X
d
f, g ∈ X f(x) = x, g(x) = x + 1, ∀x ∈ [a, b] f = g d(f, g) = 0
(X, d) d
P E ≤ 
E P
(X, d)
a ∈ X c ∈ intP
B(a, c) = {x ∈ X : d(a, x)  c}
B(a, c) a c
T := {G ⊂ X : ∀x ∈ G, ∃c ∈ intP, B(a, c) ⊂ G} .

T X
a ∈ X c ∈ intP B(a, c) ∈ T
(X, T ) T
1
∅ ∈ T X ∈ T x ∈ X c ∈ intP
B(a, c) ⊂ X {A
i
, i ∈ I} T
A
i
∈ T i ∈ I
∪ {A
i
, i ∈ I} ∈ T
x ∈ ∪ {A
i
, i ∈ I} ∃i ∈ I x ∈ A
i
A
i
∈ T
c ∈ intP B(x, c) ⊂ A
i
B(x, c) ⊂ A
i
⊂ ∪ {A
i
, i ∈ I}
∪ {A
i

, i ∈ I} ∈ T
A, B ∈ T x ∈ A ∩ B x ∈ A, x ∈ B
A, B ∈ T c
1
, c
2
∈ intP B(x, c
1
) ∈ A B(x, c
2
) ∈ B
c ∈ intP c  c
1
, c  c
2
B(x, c) ⊂ B(x, c
1
) ∩ B(x, c
2
) ⊂ A ∩ B
A ∩ B ∈ T T X
x ∈ B(a, c) 0 ≤ d(x, a)  c c

= c − d(x, a)
d(x, a)  c c

∈ intP y ∈ B(x, c

) d(y, x)  c


d(y, a) ≤ d(y, x) + d(a, x)  c

+ d(x, a) = c − d(x, a) + d(x, a) = c
y ∈ B(a, c) B(x, c

) ⊂ B(a, c)
B(a, c) ∈ T
x, y ∈ X x = y d(x, y) = 0
B(x, c) y X T
1

X d(x, y) > 0 x = y
X X T
2
X
X
X X
B(a, c)
(X, d) {x
n
} ⊂
X, a ∈ X
{x
n
} a c ∈ intP
n
c
d(x
n
, a)  c, n ≥ n

c
P x
n
→ a d(x
n
, a) → 0
x
n
→ a c ∈ intP B(a, c) ∈ T
n
c
x
n
∈ B(a, c), n ≥ n
c
d(x
n
, a)  c, ∀n ≥ n
c
c ∈ intP n
c
d(x
n
, a)  c, n ≥ n
c
U a c ∈ intP
B(a, c) ⊂ U n
c
x
n

∈ B(a, c) ⊂
U, n ≥ n
c
x
n
→ a
P 
{x
n
} X a, b ∈ X
x
n
→ a x
n
→ b d(a, b) = 0
x
n
→ a x
n
→ x x ∈ F
a
F
a
= {x ∈ X : d(a, x) = 0}
n
0 ≤ d(a, b) ≤ d(a, x
n
) + d(x
n
, b).

x
n
→ a x
n
→ b c ∈ intP
n
c
d(a, x
n
) 
c
2
, d(b, x
n
) 
c
2
n ≥ n
c
d(a, b)  c, c ∈ intP
d(a, b) = 0
x
n
→ a x
n
→ x x ∈
F
a
x
n

→ a c ∈ intP n
c
d(a, x
n
)  c, n ≥ n
c
x ∈ F
a
d(x, a) = 0
0 ≤ d(x
n
, x) ≤ d(x
n
, a) + d(a, x)  c, ∀n ≥ n
c
x
n
→ x
a ∈ F
a

(X, d) a ∈ X
c ∈ intP
V =

B(a,
c
n
) : n = 1, 2,


a X
U a
r ∈ intP B(a, r) ⊂ U
c
n
→ 0 n → ∞ r ∈ intP
n
c
n
 r
B(a,
c
n
) ⊂ B(a, r) ⊂ U.
V a
V X

(X, d) a ∈ X
F
a
= {x ∈ X : d(x, a) = 0}
b, b

∈ F
a
, x ∈ X\F
a
d(x, b) = d(x, b

)

F
a
a, b ∈ X d(x, y) = d(a, b), ∀x ∈ F
a
, ∀y ∈ F
b
b, b

∈ F
a
0 ≤ d(b, b

) ≤ d(b, a) + d(a, b

) = 0
d(b, b

) = 0, ∀x ∈ X\F
a
d(x, b) ≤ d(x, b

) + d(b, b

) = d(x, b

)
d(x, b

) ≤ d(x, b) + d(b, b


) = d(x, b).
d(x, b) = d(x, b

)
{x
n
} ⊂ F
a
x
n
→ x ∈ X X
F
a
x ∈ F
a
x
n
→ x c ∈ intP n
c
d(x
n
, x)  c
n ≥ n
c
x
n
∈ F
a
n d(x
n

, a) = 0 n = 1, 2,
d(x, a) ≤ d(x, x
n
) + d(x
n
, a) = d(x, x
n
)  c, ∀n ≥ n
c
d(x, a)  c c ∈ intP d(x, a) = 0 x ∈ F
a
F
a
a, b ∈ X
x ∈ F
a
, y ∈ F
b
d(x, y) ≤ d(x, a) + d(a, b) + d(b, y) = d(a, b)
d(a, b) ≤ d(a, x) + d(x, y) + d(y, b) = d(x, y).
d(a, b) = d(x, y) 
(X, d)
d P
E, intP = ∅, ” ≤ ” ”  ” E P
X ” ≤ ”
(X, d)
(X, d)
F : X
2
→ X (x, y) ∈ X

2
F
d(F (x, y), x) = d(F (y, x), y) = 0
F {x
n
} {y
n
} X
x
n
→ x, y
n
→ y F (x
n
, y
n
) → F (x, y)
(X, d) F : X
2
→
X X
(A
1
) α
1
, α
2
, α
3
, α

4
, α
5
≥ 0
α
2
= α
1
+ α
3
+ α
4
,
2α
1
+ 3α
3
+ 3α
4
< 1