Một số định lý điểm bất động đối với các phép co cyclic

ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
φ
φ
φ
φ
ϕ
ϕ
0
ϕ ϕ
ϕ
ϕ
ϕ
ϕ ϕ ϕ
0 0
ϕ
ϕ
ϕ
ϕ
ϕ
0 0
ϕ
ϕ
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, z) ≤ d (x, y) + d (y, z) x, y, z ∈ X
X d
(X, d) X
(X, d) x
i
∈ X, i =
1, 2, . . . , n
d(x
1
, x
n
) ≤ d(x
1
, x
2
) + d(x
2
, x
3
) + · · · + d(x
n−1
, x
n
).
{x
n
} (X, d)
x ∈ X x
n

→ x lim
n→∞
x
n
= x
d (x, x
n
) → 0 n → ∞
(X, d)
{x
n
} ⊂ X x
n
→ x x
n
→ y x = y
{x
n
}, {y
n
} X x
n
→ x, y
n
→ y d (x
n
, y
n
) →
d (x, y)

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

∗
) = x
∗
f
f F
f
X p : X ×X → [0, +∞)
p (x, y) = p (y, x)
p (x, x) = p (x, y) = p (y, y) x = y
p (x, x) ≤ p (x, y)
p (x, z) ≤ p (x, y) + p (y, z) − p (y, y)
x, y, z ∈ X
(X, p)
p X τ
p
X
{B
p
(x, ε) : x ∈ X, ε > 0} B
p
(x, ε) = {y ∈
X : p (x, y) < p (x, x) + ε} x ∈ X ε > 0
(X, p)
{x
n
} ⊂ X x ∈ X
lim
n→∞
p(x, x
n

) = p(x, x)
{x
n
} (X, p)
lim
n,m→∞
p(x
n
, x
m
)
(X, p)
{x
n
} X τ
p
x ∈ X
lim
n,m→∞
p(x
n
, x
m
) = p(x, x)
f : X → X x
0
∈ X
ε > 0 δ > 0 f(B
p
(x

0
, δ)) ⊂ B
p
(f(x
0
), )
f : X → X X f
x ∈ X
(X, p)
{x
n
} (X, p) 0
lim
n,m→∞
p(x
n
, x
m
) = 0
(X, p) 0 0
X τ
p
x ∈ X
p(x, x) = 0
(X, p) {x
n
} ⊂ X
x
n
→ x ∈ X p(x, x) = 0 lim

n→∞
p(x
n
, z) = p(x, z) z ∈ X
z ∈ X
p(x, z) − p(x
n
, x) ≤ p(x
n
, z) ≤ p(x, z) + p(x
n
, x).
n → ∞ x
n
→ x ∈ X
p (x, x) lim
n→∞
p(x
n
, z) = p(x, z) 
X m
f : X → X X {X
i
}
m
i=1
X X f X =
m

i=1

X
i
X
i
= φ i = 1, . . . , m
f (X
1
) ⊂ X
2
, f (X
2
) ⊂ X
3
, . . . , f (X
m−1
) ⊂ X
m
, f (X
m
) ⊂ X
1
ϕ : R
+
→ R
+
ϕ
ϕ t
1
, t
2

∈ R
+
t
1
≤ t
2
ϕ (t
1
) ≤ ϕ (t
2
)
ϕ
{ϕ
n
(t)}
n∈N
0 n → +∞ t ∈ R
+
(X, d) ϕ :
R
+
→ R
+
f : X → X ϕ
d(f(x), f(y)) ≤ ϕ(d(x, y)) x, y ∈ X.
(X, d) m
A
1
, A
2

, , A
m
, A
m+1
X
A
1
= A
m+1
Y =
m

i=1
A
i
f : Y → Y
{A
i
}
m
i=1
Y f
ϕ : R
+
→ R
+
d (f (x) , f (y)) ≤ ϕ (d (x, y)) (1.1)
x ∈ A
i
y ∈ A

i+1
f ϕ
ϕ : R
+
→ R
+
ϕ
ϕ
ϕ
k
0
∈ N α ∈ (0, 1)
∞

k=1
v
k
ϕ
k+1
(t) ≤ αϕ
k
(t) + v
k
(1.2)
k ≥ k
0
t ∈ R
+
ϕ : R
+

→ R
+
(c)
ϕ
ϕ (t) < t t ∈ R
+
ϕ
∞

k=0
ϕ
k
(t) t ∈ R
+
ϕ : R
+
→ R
+
(c)
s : R
+
→ R
+
s (t) =
∞

k=0
ϕ
k
(t), t ∈ R

+
, (1.3)
0
ϕ : R
+
→ R
+
(c) ϕ
ϕ : R
+
→ R
+
c
{b
n
} ⊂ R
+
b
n
→ 0 n → ∞
n

k=0
ϕ
n−k
(b
k
) → 0, khi n → ∞. (1.4)
ϕ
X x

0
∈ X F : X → X
F : X → X
d(F (x), F (y)) < d(x, y) x, y ∈ X x = y.
{x
n
} X x
1
= F (x
0
), x
2
= F (x
1
), . . . , x
n
= F (x
n−1
) =
F
n
(x
0
), . . .
X F : X → X
X x ∈ X {F
n
(x)}
X F
(X, d) m

A
1
, A
2
, , A
m
X Y =
m

i=1
A
i
ϕ : R
+
→ R
+
(c) f : Y → Y Y
Y
{A
i
}
m
i=1
Y f
f ϕ
f x
∗
∈
m


i=1
A
i
x
0
∈ Y
{x
n
}
n≥0
x
∗
d(x
n
, x
∗
) ≤ s(ϕ
n
(d(x
0
, x
1
))), n ≥ 1.
(2.1)
d(x
n
, x
∗
) ≤ s(ϕ
n

(d(x
n
, x
n+1
))), n ≥ 1.
(2.2)
y ∈ Y
d(x, x
∗
) ≤ s(ϕ
n
(d(x, f(x))),
(2.3)
s
x
0
∈ Y =
m

i=1
A
i
{x
n
}
n≥0
x
0
n ≥ 1
d (x

n
, x
n+1
) = d (f (x
n−1
) , f (x
n
)) , n ≥ 1. (2.4)
x
n
= f (x
n−1
) n ≥ 1
n ≥ 0 i
n
∈ {1, 2, , m}
x
n
∈ A
i
n
x
n+1
∈ A
i
n+1
d (x
n
, x
n+1

) ≤ ϕ (d (x
n−1
, x
n
)) n ≥ 1.
ϕ
d (x
n
, x
n+1
) ≤ ϕ
n
(d (x
0
, x
1
)) , n ≥ 1. (2.5)
p ≥ 1
d (x
n
, x
n+p
) ≤ ϕ
n
(d (x
0
, x
1
)) + · · · + ϕ
n+p−1

(d (x
0
, x
1
)) . (2.6)
n ≥ 0 s
n
=
n

k=0
ϕ
k
(d (x
0
, x
1
))
d (x
n
, x
n+p
) ≤ s
n+p−1
− s
n−1
. (2.7)
ϕ d (x
0
, x

1
) > 0
∞

k=0
ϕ
k
(d (x
0
, x
1
)) < ∞ s ∈ R
+
lim
n→∞
s
n
= s
d (x
n
, x
n+p
) → 0 n → ∞
{x
n
}
n≥0
Y =
m


i=1
A
i
lim
n→∞
x
n
= p ∈ Y
m

i=1
A
i
Y f
i = 1, . . . , m {x
n
}
n≥0
A
i
i = 1, . . . , m A
i
{x
i
n
k
}
{x
n
} {x

i
n
k
} p ∈ Y A
i
p ∈ A
i
i = 1, . . . , m p ∈
m

i=1
A
i
m

i=1
A
i
= φ. f
m

i=1
A
i
f|
m

i=1
A
i

:
m

i=1
A
i
→
m

i=1
A
i
.
ϕ (c) f ϕ
d (f (x) , f (y)) ≤ ϕ (d (x, y)) < d (x, y) x, y ∈
m

i=1
A
i
.
x
∗
∈
m

i=1
A
i
f

m

i=1
A
i
x ∈ Y =
m

i=1
A
i
x x
∗
i
0
∈ {1, 2, , m}
x A
i
0
x
∗
∈
m

i=1
A
i
x
∗
∈ A

i
0
+1
d (f (x) , f (x
∗
)) ≤ ϕ (d (x, x
∗
)) .
ϕ
d (f
n
(x) , x
∗
) ≤ ϕ
n
(d (x, x
∗
)) n ≥ 0. (2.8)
ϕ x = x
∗
f
n
(x) → x
∗
n → ∞ x ∈ Y
x
∗
f
n ≥ 1 p → ∞
n ≥ 0 k ≥ 1 ϕ f

d (x
n+k
, x
n+k+1
) = d (f (x
n+k−1
) , f (x
n+k
)) ≤ ϕ (d (x
n+k−1
, x
n+k
)) ,
(2.9)
k ≥ 2
d (x
n+k−1
, x
n+k
) = d (f (x
n+k−2
) , f (x
n+k−1
)) ≤ ϕ (d (x
n+k−2
, x
n+k−1
))
(2.10).
ϕ

n ≥ 0 k ≥ 2
d (x
n+k
, x
n+k+1
) ≤ ϕ
2
(d (x
n+k−2
, x
n+k−1
)) .
n ≥ 0 k ≥ 2
d (x
n+k
, x
n+k+1
) ≤ ϕ
n
(d (x
n
, x
n+1
)) (2.11).
d (x
n
, x
n+p
) ≤ d (x
n

, x
n+1
) + + d (x
n+p−1
, x
n+p
) ,
n ≥ 0 p ≥ 1
d (x
n
, x
n+p
) ≤
p−1

k=0
ϕ
k
(d (x
n
, x
n+1
)). (2.12)
p → ∞ n ≥ 0
d (x
n
, x
∗
) ≤
∞


k=0
ϕ
k
(d (x
n
, x
n+1
)). (2.13)
s
x ∈ Y n = 0
x
0
= x
d (x, x
∗
) ≤
∞

k=0
ϕ
k
(d (x, f (x))) = s (d (x, f (x))) . 
f : Y → Y
∞

n=0
d

f

n
(x) , f
n+1
(x)

< ∞ x ∈ Y
x = x
0
∈ Y f ϕ
n ≥ 0
d

f
n
(x
0
) , f
n+1
(x
0
)

= d (x
n
, x
n+1
) ≤ ϕ
n
(d (x
0

, x
1
)) .
∞

n=0
d

f
n
(x
0
) , f
n+1
(x
0
)

≤
∞

n=0
ϕ
n
(d (x
0
, x
1
)) = s (d (x
0

, x
1
)) .
∞

n=0
d

f
n
(x
0
) , f
n+1
(x
0
)

< ∞. 
f : Y → Y
∞

n=0
d (f
n
(x) , x
∗
) < ∞ x ∈ Y
f : Y → Y
x

∗
∈ Y f(x
∗
) = x
∗
f ϕ
x ∈ Y d (f (x) , f (x
∗
)) ≤ ϕ (d (x, x
∗
))
ϕ n ≥ 0
d (f
n
(x) , x
∗
) ≤ ϕ
n
(d (x, x
∗
)) .
∞

n=0
d (f
n
(x) , x
∗
) ≤
∞


n=0
ϕ
n
(d (x, x
∗
)) = s(d (x, x
∗
)).
∞

n=0
d (f
n
(x) , x
∗
) < ∞ 
f : Y → Y
f
n ∈ N z
n
∈ Y
d (z
n
, f (z
n
)) → 0 n → ∞ z
n
→ x
∗

n → ∞ F
f
= {x
∗
}
f : Y → Y
x
∗
∈ Y f(x
∗
) = x
∗
n ∈ N z
n
∈ Y
d (z
n
, f (z
n
)) → 0, n → ∞. (2.14)
z
n
n ∈ N
d (z
n
, x
∗
) ≤ s (d (z
n
, f (z

n
))) . (2.15)
s s (t) =
∞

k=0
ϕ
k
(t) t R
+
0 n → ∞
d (z
n
, x
∗
) → 0 f

f : Y → Y
ϕ f
n ∈ N z
n
∈
Y d (z
n+1
, f (z
n
)) → 0 n → ∞ x ∈ Y
d (z
n+1
, f

n
(x)) → 0 n → ∞
f : Y → Y
x
∗
∈ Y f(x
∗
) = x
∗
n ∈ N z
n
∈ Y
d (z
n
, f (z
n
)) → 0, n → ∞. (2.16)
n ≥ 0
d (z
n+1
, x
∗
) ≤ d (z
n+1
, f (z
n
)) + d (f (z
n
) , f (x
∗

)) . (2.17)
z
n
∈ Y =
m

i=1
A
i
n ≥ 0 n ∈ N i
n
∈
{1, 2, , m} z
n
∈ A
i
n
x
∗
∈
m

i=1
A
i
ϕ
f n ≥ 0
d (z
n+1
, x

∗
) ≤ d (z
n+1
, f (z
n
)) + ϕ (d (z
n
, x
∗
)) . (2.18)
n ≥ 1
d (z
n
, x
∗
) ≤ d (z
n
, f (z
n−1
)) + ϕ (d (z
n−1
, x
∗
)) .
ϕ
d (z
n+1
, x
∗
) ≤ d (z

n+1
, f (z
n
)) + ϕ (d (z
n
, f (z
n−1
))) + ϕ
2
(d (z
n−1
, x
∗
)) .
d (z
n+1
, x
∗
) ≤ d (z
n+1
, f (z
n
)) + ϕ (d (z
n
, f (z
n−1
))) + · · ·
· · · + ϕ
n
(d (z

1
, f (z
0
))) + ϕ
n+1
(d (z
0
, x
∗
)) .
(2.19)
ϕ b
n
= d (z
n+1
, f (z
n
))
d (z
n+1
, f (z
n
))+ϕ (d (z
n
, f (z
n−1
)))+ +ϕ
n
(d (z
1

, f (z
0
))) → 0, n → ∞.
z
0
= x∗ ϕ
ϕ
n+1
(d (z
0
, x∗)) → 0 n → ∞.
n → ∞
d (z
n+1
, x
∗
) → 0 n → ∞. (2.20)
x ∈ Y {f
n
(x)}
n≥0
x
∗
x ∈ Y n ≥ 0
d (z
n+1
, f
n
(x)) ≤ d (z
n+1

, x
∗
) + d (x
∗
, f
n
(x)) (2.21)
n → ∞
d (z
n+1
, f
n
(x)) → 0 n → ∞.
f 
f : Y → Y
g : Y → Y
g x
∗
g
∈ F
g
η > 0
d (f (x) , g (x)) ≤ η x ∈ X. (2.22)
d

x
∗
f
, x
∗

g

≤ s (η) F
g
=

x
∗
g

s
f : Y → Y
x = x
∗
g
x
∗
f
f
d

x
∗
f
, x
∗
g

≤ s


d

x
∗
g
, f

x
∗
g

= s

d

g

x
∗
g

, f

x
∗
g

.
s
d


x
∗
f
, x
∗
g

≤ s (η) 
f : Y → Y
n ∈ N f
n
: Y → Y
x
∗
n
∈ F
f
n
{f
n
}
∞
i=1
f
x
∗
n
→ x
∗

n → ∞ F
f
= {x
∗
}
{f
n
}
n≥0
f
n ∈ N η
n
∈ R
+
η
n
→ 0 n → ∞
d (f
n
(x) , f (x)) ≤ η
n
x ∈ Y.
f f
n
x
∗
x
∗
n
n ∈ N

d (x
∗
n
, x
∗
) ≤ s(η
n
) n ∈ N. (2.23)
η
n
→ 0 n → ∞ s 0

X d ρ
X m A
1
, A
2
, , A
m
X Y =
m

i=1
A
i
f : Y → Y Y Y
{A
i
}
m

i=1
Y f
d (x, y) ≤ ρ (x, y) x, y ∈ Y
(Y, d)
f : (Y, d) → (Y, d)
f : (Y, ρ) → (Y, ρ) ϕ ϕ : R
+
→ R
+
(c)
F
f
= {x
∗
}
{f
n
(x
0
)}
n≥0
x
∗
∈ (Y, d) x
0
∈ X
x
0
∈ Y
{f

n
(x
0
)}
n≥0
(X, ρ)
{f
n
(x
0
)}
n≥0
(X, d)
{f
n
(x
0
)}
n≥0
(X, d) x
∗
f 
ϕ
ϕ
ϕ
ϕ
(X, d)
f : X → X X ϕ
d(f(x), f(y)) ≤ d(x, y) − ϕ(d(x, y)), x, y ∈ X,
ϕ : [0, ∞) → [0, ∞) ϕ(t) = 0

t = 0
(X, d)
f : X → X X f
(X, d) m
A
1
, A
2
, , A
m
X A
m+1
= A
1
Y =
m

i=1
A
i
f : Y → Y Y Y ϕ
{A
i
}
m
i=1
Y f
ϕ : [0, ∞) → [0, ∞)
ϕ (t) > 0 t ∈ (0, ∞) ϕ (0) = 0
d (f(x), f(y)) ≤ d (x, y) − ϕ (d (x, y)) (3.1)

x ∈ A
i
y ∈ A
i+1
i = 1, . . . , m
X = R
R A
1
= [−1, 0] = A
3
A
2
= [0, 1] = A
4
Y =
4

i=1
A
i
f : Y → Y f(x) = −
x
3
x ∈ Y
4

i=1
A
i
Y f

ϕ : [0, ∞) → [0, ∞) ϕ (t) =
t
2
t ∈ [0, +∞) ϕ
|f(x) − f(y)| = | −
x
3
− (−
y
3
)| =
1
3
|x − y|
≤
1
2
|x − y| = |x − y| −
1
2
|x − y|
= |x − y| − ϕ(|x − y|).
f ϕ
X = R
A
1
= A
2
= = A
m

= [0, 1] Y =
m

i=1
A
i
f : Y → Y
f(x) =
x
1+x
x ∈ Y
m

i=1
A
i
Y f ϕ : [0, ∞) → [0, ∞) ϕ (t) =
t
2
t+2
t ∈ [0, ∞) ϕ
|f(x) − f(y)| = |
x
1 + x
−
y
1 + y
| = |
x − y
(1 + x)(1 + y)

|
≤ |x − y| −
|x − y|
2
|x − y| + 2
.
f ϕ
(X, d) m
A
1
, A
2
, , A
m
X Y =
m

i=1
A
i
f ϕ f
z ∈
m

i=1
A
i
x
0
∈ Y =

m

i=1
A
i
x
n+1
= f(x
n
) n ≥ 0
n ≥ 0 i
n
∈ {1, 2, , m} x
n
∈ A
i
n
x
n+1
∈ A
i
n+1
d (x
n+1
, x
n+2
) = d (f(x
n
), f(x
n+1

)) ≤ d (x
n
, x
n+1
) − ϕ (d (x
n
, x
n+1
)) .
t
n
= d (x
n
, x
n+1
)
t
n+1
≤ t
n
− ϕ (t
n
) ≤ t
n
(3.2)
{t
n
} {t
n
}

L L ≥ 0 L > 0 L = 0
L > 0 ϕ 0 < ϕ (L) ≤
ϕ (t
n
)
t
n+1
≤ t
n
− ϕ (t
n
) ≤ t
n
− ϕ (L) .
t
n+2
≤ t
n+1
− ϕ (t
n+1
) ≤ t
n
− ϕ (t
n
) − ϕ (t
n+1
) ≤ t
n
− 2ϕ (L) .
t

n+p
≤ t
n
−pϕ (L)
p ∈ N p L = 0
ε > 0 N
0
∈ N d (x
N
0
, x
N
0
+1
) ≤ min

ε
2
, ϕ

ε
2

f B (x
N
0
, ε)
x ∈ B (x
N
0

, ε) d (x, x
N
0
) ≤
ε
2
d (f(x), x
N
0
) ≤ d (f(x), f(x
N
0
)) + d (f (x
N
0
) , x
N
0
)
≤ d (x, x
N
0
) − ϕ (d (x, x
N
0
)) + d (x
N
0
+1
, x

N
0
)
<
ε
2
+
ε
2
= ε. (3.3)
d (x, x
N
0
) >
ε
2
ε
2
< d (x, x
N
0
) ≤ ε.
ϕ ϕ (t) > 0 ϕ

ε
2

≤ ϕ (d (x, x
N
0

))
d (f(x), x
N
0
) ≤ d (f(x), f(x
N
0
)) + d (f (x
N
0
) , x
N
0
)
≤ d (x, x
N
0
) − ϕ (d (x, x
N
0
)) + d (x
N
0
+1
, x
N
0
)
≤ ε − ϕ


ε
2

+ ϕ

ε
2

≤ ε.
f(x) ∈ B (x
N
0
, ε) f
B (x
N
0
, ε) x
n
∈ B (x
N
0
, ε)
n > N
0
{x
n
} Y
B (x
N
0

, ε) {x
n
} Y y ∈ Y
{x
n
} A
i
i = 1, . . . , m
i = 1, . . . , m A
i
{x
i
n
k
}
{x
n
} {x
i
n
k
} y ∈ Y A
i
y ∈ A
i
i = 1, . . . , m y ∈
m

i=1
A

i
m

i=1
A
i
= φ.
Z =
m

i=1
A
i
Z
f Z f
|Z
: Z → Z
f
|Z
f
|Z
z ∈ Z x
0
x ∈ Y
z ∈ Z =
m

i=1
A
i

x ∈ Y =
m

i=1
A
i
x f
|Z
w ∈
m

i=1
A
i
z, w ∈
m

i=1
A
i
z, w ∈ A
i
i d (z, w) d (f(z), f(w))
d (z, w) = d (f(z), f(w)) ≤ d (z, w) − ϕ (d (z, w)) .
ϕ (d (z, w)) = 0 d (z, w) = 0
z = w z f
x ∈ Y 
X (X, d) (X, ρ)
m A
1

, A
2
, , A
m
X Y =
m

i=1
A
i
{A
i
}
m
i=1
Y f
d (x, y) ≤ ρ (x, y) x, y ∈ Y
(Y, d)
f : (Y, d) → (Y, ρ)
f : (Y, d) → (Y, ρ) ϕ ϕ : [0, ∞) →
[0, ∞) ϕ(t) > 0
t > 0 ϕ (0) = 0
{f
n
(x
0
)} z ∈ Y (Y, d)
x
0
∈ Y z f

x
0
∈ Y
{f
n
(x
0
)}
(Y, ρ)
{f
n
(x
0
)} (Y, d)
(Y, d) {f
n
(x
0
)}
(Y, d) z ∈ Y
f : (Y, d) → (Y, ρ) {f
n+1
(x
0
)}
f(z) (Y, ρ)
{f
n+1
(x
0

)} f(z) (Y, d)
f(z) = z z
f
z 