Sự tồn tại điểm bất động của ánh xạ cyclic trong không gian meetric nón



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

∈ τ
X τ
X τ X
X
τ
A ⊂ X A X \ A
X A X
x ∈ X V ⊂ X x ∈ V ⊂ A
X x ∈ X U(x)
B(x) ⊂ U(x) U ∈ U(x)
V ∈ B(x) V ⊂ U
{x
n
} X
x ∈ X n
0
∈ N
x
n
∈ U n ≥ n
0
.
x
n
→ x lim
n→∞
x
n
= x.
X

x ∈ X B
X T
2
x, y ∈ X, x = y U
x
, U
y
U
x
∩ U
y
= ∅.
X X
X, Y f : X → Y
x ∈ X f(x)
U x f(U) ⊂ V 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
0
∈ N

n, m ≥ n
0
d(x
n
, x
m
) < ε.
X X
A ⊂ X
A A
X A ⊂ X
{x
n
} A {x
n
k
}
A
f : (X, d) → (Y, ρ) (X, d)
(Y, ρ) α ∈ [0, 1]
ρ(f(x), f(y)) ≤ αd(x, y) x, y ∈ X
(X, d) f : X → X
X x
∗
∈ X
f(x
∗
) = x
∗
.

x
∗
f(x
∗
) = x
∗
E K = R
K = C p : E −→ R E
p(x) ≥ 0, ∀x ∈ E p(x) = 0 ⇔ x = 0
p(λx) = |λ|p(x), ∀x ∈ E, ∀λ ∈ K
p(x + y) ≤ p(x) + p(y), ∀x, y ∈ E
p(x) x ∈ E
x E
E
d(x, y) = x − y, ∀x, y ∈ E
E
x → x, ∀x ∈ E;
(x, y) → x + y, ∀(x, y) ∈ E × E;
(λ, x) → λx, ∀(λ, x) ∈ K × E
E a ∈ E
λ ∈ K, λ = 0
x → x + a, x → λx, ∀x ∈ E
E E
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 A ⊂ X

x ∈ X A a ≤ x
x ≤ a a ∈ A.
x ∈ X
A A
x ≤ y y ≤ x)
x = sup A x = inf A
E R
P E E
P P = ∅, P = {0}
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 = {0}
(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
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].
P = {f ∈ C
[a,b]
: 0 ≤ f}.
P P = ∅, P = {0}
a, b ∈ R, a, b ≥ 0 f, g ∈ P
0 ≤ af(x) + bg(x) ∀x ∈ [a, b].
af + bg ∈ P
f ∈ P −f ∈ P f = 0
P E
P E E

≤

P x ≤ y y − x ∈ P
x < y x ≤ y x = y x  y y − x ∈ intP intP
P
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
1
e  c
2
a ∈ P a ≤ x x ∈ intP a = 0
a ≤ λa a ∈ P, 0 < λ < 1 a = 0

0 ≤ x
n
≤ y
n
n ∈ N lim
n→∞
x
n
= x, lim
n→∞
y
n
= y
0 ≤ x ≤ y
intP + intP ⊂ intP a  b
b  c b−a ∈ intP c−b ∈ intP c−a = c−b+b−a ∈ intP +intP ⊂
intP a  c
αintP ⊂ intP
intP + P =

x∈P
(x + intP ) P
x + intP ⊂ P P + intP ⊂ intP a ≤ b b  c b − a ∈ P
c − b ∈ intP c − a = c − b + b − a ∈ intP + P ⊂ intP c − a ∈ intP
a  c
a  b c  d b − a ∈ intP d − c ∈ intP
b − a + d − c ∈ intP (b + d) − (a + c) ∈ intP a + c  b + d
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
δ


> 0 c
1
+ B(0, δ

) ⊂ intP c
2
+ B(0, δ

) ⊂ intP,
B(0, δ

) = {x ∈ E : x < δ

} B(0, δ

) m > 0
c
1
∈ mB(0, δ

), 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
x ∈ intP a ≤
x
n
n = 1, 2,
x
n
− a ∈ P n = 1, 2, 
x
n
 =
x

n
→ 0
x
n
→ 0
x
n
− a → −a

x
n
− a

⊂ P P E −a ∈ P
−a ∈ P P a = 0
a ≤ λa λa − a ∈ P (λ − 1)a ∈ P 0 < λ < 1 1 − λ > 0
−a =
1
1−λ
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
) = y −x y −x ∈ P
x ≤ y 0 ≤ x
n
0 ≤ x 0 ≤ x ≤ y
P E {x
n
}
P x
n
→ 0 c ∈ intP n

0
∈ N x
n
 c
n ≥ n
0
{x
n
} P x
n
→ 0 c ∈ intP intP
δ > 0 c + B
E
(0, δ) ⊂ intP, B
E
(0, δ)
δ E x ∈ E x < δ
c − x ∈ intP δ > 0 n
0
∈ 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
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,
α, β
(X, d)
E = C
[a,b]
d : E × E −→ E
d(f, g) = |f − g| ∀f, g ∈ E,
|f − g|(x) = |f(x) − g(x)| x ∈ [a, b]
0 ≤ d(f, g) f, g ∈ E d(f, g) = 0 f(x) = g(x)
x ∈ [a, b] f = g
d(f, g) = d(g, f) = |f − g| f, g ∈ E
|f − g| = |f − h + h − g| ≤ |f − g| + |h − g| f, g, h ∈ E

d(f, g) ≤ d(f, h) + d(h, g), ∀f, g, h ∈ E.
E
(X, d) a ∈ X, c ∈ intP
B(a, c) = {x ∈ X : d(x, a)  c},
B(a, c)
 = {G ⊂ X : ∀x ∈ G, ∃c ∈ intP : B(x, c) ⊂ G}.
(X, d) 
 X
B(x, c) ∈  x ∈ X, c ∈ intP
 T
2
−

 = {U ⊂ X : ∀x ∈ U, ∃c ∈ intP : B(x, c) ⊂ U}
∅ ∈ , X ∈ 
U
i
∈  i ∈ I

i∈I
U
i
∈ 
x ∈

i∈I
U
i
i = i
0

∈ I x ∈ U
i
0
⊂  c ∈ intP
B(x, c) ⊂ U
i
0
B(x, c) ⊂

i∈I
U
i

i∈I
U
i
∈ 
U, V ∈  U ∩ V ∈  x ∈ U ∩ V
x ∈ U x ∈ V c
1
, c
2
∈ intP B(x, c
1
) ⊂ U
B(x, c
2
) ⊂ V c
1
, c

2
∈ intP c ∈ intP
c  c
1
c  c
2
B(x, c) ⊂ U ∩ V U ∩ V ∈ 
 
(X, )
∈ B(x, c) d(x, y)  c c − d(x, y) ∈ intP
c

= c−d(x, y) B(y, c

) ⊂ B(x, c) z ∈ B(y, c

)
d(z, y)  c

d(z, y)  c − d(x, y)
d(z, y)+d(y, x)  c d(z, x)  c z ∈ B(x, c)
B(y, c

) ⊂ B(x, c) B(x, c) ∈ 
x, y ∈ X x = y (X, ) T
2
−
c
1
, c

2
∈ intP B(x, c
1
) ∩ B(x, c
2
) = ∅
c
1
, c
2
∈ intP B(x, c
1
)∩B(x, c
2
) =
∅ c ∈ intP
B(x,
c
2n
) ∩ B(x,
c
2n
) = ∅, n = 1, 2
{z
n
} ⊂ X
z
n
∈ B(x,
c

2n
) ∩ B(x,
c
2n
), n = 1, 2
d(x, y) ≤ d(x, z
n
) + d(z
n
, y) 
c
n
, n = 1, 2
c
n
≤ c n = 1, 2, d(x, y)  c
intP d(x, y) = 0 x = y
x = y (X, ) T
2
−
x ∈ X
c ∈ intP
U = {B(x,
c
n
) : n = 1, 2, }.
U ∈  y ∈ intP
B(x, y) ⊂ V y ∈ intP ε > 0 B
E
(y, ε) ⊂ P

B
E
(y, ε) y ε n ∈ N
n >
c
ε
y − (y −
c
n
) =
c
n
< ε,
y −
c
n
∈ B
E
(y, ε) ⊂ P B
E
(y, ε) y −
c
n
∈ intP
c
n
 y
B(x,
c
n

) ⊂ B(x, y) ⊂ V.
U 
(X, )
 B(x, c)
(X, d)
(X, d) {x
n
} ⊂ X
x = y
(X, d) {x
n
} ⊂ X
x
n
→ x ∈ X c ∈ intP n
c
d(x, x
n
)  c n ≥ n
c
x
n
→ x ∈ X c ∈ intP B(x, c)
n
c
x
n
∈ B(x, c)
n ≥ n
c

d(x, x
n
)  c n ≥ n
c
.
c ∈ intP n
c
d(x, x
n
)  c
n ≥ n
c
c
0
∈ intP
B(x, c
0
) ⊂ U n
c
0
∈ N
x
n
∈ B(x, c
0
) ⊂ U ∀n ≥ n
c
0
.
x

n
→ 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 d(x
n
, x) 
c
2
n > N
m, n > N
d(x
m
, x

n
) ≤ d(x
n
, x) + d(x, x
m
) 
c
2
+
c
2
= c.
{x
n
}
{x
n
} X
{x
n
} {x
n
k
} x ∈ X {x
n
}
c ∈ intP {x
n
} {x
n

k
}
n
0
d(x
n
, x
m
) 
c
2
∀n, m ≥ n
0
d(x
n
k
, x) 
c
2
∀n
k
≥ n
0
.
d(x
n
, x) ≤ d(x
n
, x
n

k
) + d(x
n
k
, x)  c ∀n, n
k
≥ n
0
.
x
n
→ x
(X, d)
X
(X, d), (Y, d) f :
X → Y a ∈ X {x
n
} X
x
n
→ a f(x
n
) → f(a)
f a {x
n
} X x
n
→ a
f(x
n

) → f(a) V f(a) Y f
a U a X
f(U) ⊂ V x
n
→ a n
0
x
n
∈ U 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
) /∈
B(f(a), y
0
) x
n
∈ B(a,
c
n
) n = 1, 2,
c
n
→ 0 n → ∞
x
n
→ a
f(x
n
) → f(a) f(x
n
) /∈ B(f(a), y

0
) n = 1, 2,
f a
A
1
, A
2
, , A
p
, A
p+1
= A
1
X T :

p
i=1
A
i
→

p
i=1
A
i
. T
T (A
i
) ⊂ A
i+1

i = 1, 2, , p
T p− T
x ∈

p
i=1
A
i
.
X F : X → X
k ∈ [0, 1)
d(F x, F
2
x) ≤ kd(x, F x) ∀x ∈ X
F X x
0
∈ X, {F
n
x
0
}
F
F x F (x) F
2
x F
2
(x)
x
0
∈ X x

n
= Fx
n−1
n = 1, 2,
n = 1, 2,
d(x
n
, x
n+1
) = d(F x
n−1
, F
2
x
n−1
) ≤ kd(x
n−1
, F x
n−1
)
= kd(F x
n−2
, F
2
x
n−2
) ≤ k
2
d(x
n−2

, F x
n−2
)
≤ ≤ k
n
d(x
0
, F x
0
) = k
n
d(x
0
, x
1
).
d(x
n
, x
n+m
) ≤ d(x
n
, x
n+1
) + d(x
n+1
, x
n+2
) + + d(x
n+m−1

, x
n+m
)
≤ (k
n
+ k
n+1
+ + k
n+m−1
)d(x
0
, x
1
)
= k
n
1 − k
m
1 − k
d(x
0
, x
1
) ≤
k
n
1 − k
d(x
0
, x

1
),
n = 1, 2, m = 0, 1, 2, k ∈ [0, 1)
k
n
1 − k
d(x
0
, x
1
) → 0 → ∞.
c ∈ intP n
0
k
n
1 − k
d(x
0
, x
1
)  c ∀n ≥ n
0
,
n ≥ n
0
m = 0, 1, 2,
d(x
n
, x
n+m

)  c.
{x
n
} a ∈ X x
n
→ a ∈ X
n → ∞. F x
n
→ a
n → ∞ F x
n+1
= F (x
n
) → F (a).
X a = F (a) F
A, B
X F : X → X
F (A) ⊆ B F (B) ⊆ A
d(F x, F y) ≤ kd(x, y) ∀x ∈ A y ∈ B, k ∈ (0, 1).
A ∩ B
x ∈ A ∪ B
d(F x, F
2
x) ≤ kd(x, F x).
X A ∩ B A ∪ B X X
A ∪ B A ∩ B
{F
n
x} A ∪ B F
n

x → z ∈ A ∪ B
{F
n
x}
z ∈ A ∩ B
A ∩ B = ∅ F
|A∩B
A ∩ B
F
A ∩ B
{A
i
}
p
i=1
X T :

p
i=1
A
i
→

p
i=1
A
i
,
T (A
i

) ⊂ A
i+1
i = 1, 2, p,
A
p+1
= A
1
a ∈ [0,
1
2
)
d(T x, T y) ≤ a[d(x, T x) + d(y, T y)],
x ∈ A
i
, y ∈ A
i+1
, 1 ≤ i ≤ p.
x
∗

p
i=1
A
i
{x
n
}
x
n+1
= T x

n
, n ≥ 0
x
∗
x
0
∈

p
i=1
A
i
;
d(x
n
, x
∗
) ≤
λ
n
1 − λ
d(x
0
, x
1
), n ≥ 0;
d(x
n+1
, x
∗

) ≤
λ
1 − λ
d(x
n
, x
n+1
), n > 0;
λ =
a
1−a
.
d(x
n
, x
∗
) ≤
a
1 − a
d(x
n−1
, x
∗
), n = 1, 2,
x
0
∈

p
i=1

A
i
{x
n
} x
n+1
= Tx
n
n = 0, 1, i ∈ {1, 2, , p} x
0
∈ A
i
x
1
= T x
0
∈ A
i+1
.
d(x
1
, x
2
) ≤
a
1 − a
d(x
0
, x
1

).
λ :=
a
1−a
. a ∈ [0,
1
2
) 0 ≤ λ < 1 d(x
1
, x
2
) ≤
λd(x
0
, x
1
).
d(x
n
, x
n+1
) ≤ λ
n
d(x
0
, x
1
), n = 1, 2,
n, m ∈ N, m > 0
d(x

n
, x
n+m
) ≤
n+m−1

k=n
d(x
k
, x
k+1
) ≤ λ
n
1 − λ
m
1 − λ
d(x
1
, x
0
)
≤
λ
n
1 − λ
d(x
1
, x
0
).

λ ∈ [0, 1)
λ
n
1 − λ
d(x
0
, x
1
) → 0, n → ∞.
c ∈ intP n
0
λ
n
1 − λ
d(x
0
, x
1
)  c, ∀n ≥ n
0
n ≥ n
0
m = 0, 1, 2,
d(x
n
, x
n+m
)  c.
{x
n

}

p
i=1
A
i

p
i=1
A
i
{x
n
} x
∗
∈

p
i=1
A
i
.
{x
n
} A
i
i ∈
{1, 2, , p} x
∗
∈


p
i=1
A
i

p
i=1
A
i
= 0
x
∗
i ∈ {1, 2, , p} x
∗
∈ A
i
x
∗
∈ A
i+1
d(x
∗
, T x
∗
) ≤ d(x
∗
, x
n+1
) + d(x

n+1
, T x
∗
)
≤ d(x
∗
, x
n+1
) + a[d(x
n
, x
n+1
) + d(x
∗
, T x
∗
)]
(1 − a)d(x
∗
, T x
∗
) ≤ d(x
∗
, x
n+1
) + ad(x
n
, x
n+1
), ∀n ≥ 1.

x
n
→ x t ∈ intP n
t
n ≥ n
t
d(x
∗
, x
n+1
) + ad(x
n
, x
n+1
)  t.
(1 − a)d(x
∗
, T x
∗
)  t, ∀t ∈ intP. a ∈ [0,
1
2
]
d(x
∗
, T x
∗
) = 0 T x
∗
= x

∗
x
∗
T
x
∗
T
y
∗
∈
p

i=1
A
i
T y
∗
= y
∗
.
d(x
∗
, y
∗
) = d(T x
∗
, T y
∗
) ≤ a[d(x
∗

, T x
∗
) + d(y
∗
, T y
∗
)] = 0.
d(x
∗
, y
∗
) = 0 T
m → ∞ x := x
n−1
; y := x
n
d(x
n
, x
n+1
) ≤ λd(x
n−1
, x
n
).
d(x
n+k
, x
n+k+1
) ≤ λ

k+1
d(x
n−1
, x
n
), ∀k ≥ 0.
d(x
n
, x
n+m
) ≤
m−1

k=0
d(x
n+k
, x
n+k+1
) ≤
m−1

k=0
λ
k+1
d(x
n−1
, x
n
)
≤

λ
1 − λ
(1 − λ
m
)d(x
n−1
, x
n
).
m → ∞
i ∈ {1, 2, , p} x ∈ A
i
, y ∈ A
i+1
d(T x, T y) ≤ a[d(x, T x)+d(y, T y)] ≤ a{[d(x, y) +d(y, T y)+d(T y, T x)+d(y, T y)]},
d(T x, T y) ≤
a
1 − a
d(x, y) +
2a
1 − a
d(y, T y),
x ∈ A
i
, y ∈ A
i+1
, 1 ≤ i ≤ p x := x
n−1
y := x
∗

, 1 ≤ i ≤ p
x
∗
∈
p

i=1
A
i
{A
i
}
p
i=1
X T :

p
i=1
A
i
→

p
i=1
A
i
T (A
i
) ⊂ A
i+1

i = 1, 2, p, A
p+1
= A
1
α
1
, α
2
, α
3
, α
4
, α
5
α
1
+α
2
+α
3
+2α
4
< 1
d(T x, T y) ≤ α
1
d(x, y) + α
2
d(x, T x) + α
3
d(y, T y) + α

4
d(x, T y) + α
5
d(y, T x)
x ∈ A
i
, y ∈ A
i+1
, 1 ≤ i ≤ p
x
∗

p
i=1
A
i
{x
n
}
x
n+1
= T x
n
, n ≥ 0
x
∗
x
0
∈


p
i=1
A
i
;
d(x
n
, x
∗
) ≤
λ
n
1 − λ
d(x
0
, x
1
), n ≥ 0;
d(x
n+1
, x
∗
) ≤
λ
n
1 − λ
d(x
n
, x
n+1

), n ≥ 0,
λ =
α
1
+ α
2
+ α
4
1 − α
3
− α
4
.
d(x
n
, x
∗
) ≤
α
1
+ α
3
+ α
5
1 − α
3
− α
4
d(x
n−1

, x
∗
), n = 1, 2,
α
1
+ α
4
+ α
5
≤ 1
x
∗

p
i=1
A
i
x
0
∈ A
i
1 ≤ i ≤ p
x
1
= T x
0
, x
2
= T x
1

, , x
n
= T x
n−1
= T
n
x
0
,
x
n
∈ A
i
x
n+1
∈ A
i+1
, 1 ≤ i ≤ p
n = 1, 2,
d(x
n
, x
n+1
) = d(T x
n−1
, T x
n
) ≤ α
1
d(x

n−1
, x
n
) + α
2
d(x
n−1
, x
n
)
+ α
3
d(x
n
, x
n+1
) + α
4
d(x
n−1
, x
n+1
) + α
5
d(x
n
, x
n
)
≤ (α

1
+ α
2
)d(x
n−1
, x
n
) + α
3
d(x
n
, x
n+1
) + α
4
[d(x
n−1
, x
n
) + d(x
n
, x
n+1
)].
d(x
n
, x
n+1
) ≤
α

1
+ α
2
+ α
4
1 − α
3
− α
4
d(x
n−1
, x
n
)
:= λd(x
n−1
, x
n
) ∀n = 1, 2, ,
λ =
α
1
+ α
2
+ α
4
1 − α
3
− α
4

∈ [0, 1).
d(x
n
, x
n+1
) ≤ λd(x
n−1
, x
n
) ≤ λ
2
d(x
n−2
, x
n−1
) ≤ ≤ λ
n
d(x
0
, x
1
), ∀n = 1, 2,