Một số định lý điểm bất động đối với các ánh xạ meir keeler cyclic trong không gian mêtric đầy đủ

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, d) , 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, d)
{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, d) → (Y, ρ) α ∈ [0; 1)
ρ (f(x), f(y)) ≤ α.d(x, y), x, y ∈ X.
(X, d)
f : X → X X
x
∗
∈ X f (x

∗
) = x
∗
x
∗
∈ X f (x
∗
) = x
∗
f
A B
(X, d) f : A ∪ B → A ∪ B
f(A) ⊂ B f(B) ⊂ A
d(f(x), f(y)) ≤ kd(x, y), x ∈ A, y ∈ B k ∈ (0; 1)
A ∩ B f A ∩ B
A B
X T : A∪B → A∪B T (A) ⊆ T (B)
T (B) ⊆ T (A)
A B
(X, d) f : A ∪ B → A ∪ B
x ∈ A k
x
∈ (0, 1)
d(f
2n
(x), f(y)) ≤ k
x
.d(f
2n−1
(x), y), y ∈ A n ∈ N.

(1.1)
f
A B
(X, d) f : A ∪ B → A ∪ B
f A ∩ B
{A
i
}
k
i=1
(X, d) A
k+1
= A
1
f :
k

i=1
A
i
→
k

i=1
A
i
f
f(A
i
) ⊂ A

i+1
i = 1, 2, . . . , k
{A
i
}
k
i=1
(X, d)
A
k+1
= A
1
f :
k

i=1
A
i
→
k

i=1
A
i
f(A
i
) ⊆ A
i+1
i = 1, 2, . . . , k
d(f(x), f(y)) ≤ d(x, y), x ∈ A

i
, y ∈ A
i+1
x = y, (i =
1, 2, , k)
f
S α : R
+
→ [0, 1)
α(t
n
) → 1 t
n
→ 0
(X, d) f : X → X
X α ∈ S
d (f(x), f(y)) ≤ α(d (x, y)).d (x, y) , x, y ∈ X.
f z ∈ X {f
n
(x)}
z x ∈ X
{A
i
}
k
i=1
(X, d) A
k+1
= A
1

α ∈ S f :
k

i=1
A
i
→
k

i=1
A
i
f(A
i
) ⊆ A
i+1
i = 1, 2, . . . , k
d (f(x), f(y)) ≤ α(d (x, y)) .d (x, y)) x ∈ A
i
, y ∈ A
i+1
i = 1, 2, . . . , k
f
X τ X
X
∅, X ∈ τ
G
i
∈ τ, i ∈ I


i∈I
G
i
∈ τ
G
1
, G
2
∈ τ G
1
∩ G
2
∈ τ
X τ
(X, τ) X X
X f : X → R
x
0
∈ X ε > 0 U
x
0
f (x) > f (x
0
) − ε x ∈ U
f X
X [f > α] = {x ∈ X : f(x) > α}
α ∈ R
f X −f
(−f)(x) = −f(x) x ∈ X
(X, d) f : (X, d) →

(X, d) Φ ψ :
[0, ∞) → [0, ∞)
d (f(x), f(y)) ≤ ψ(d(x, y)), x, y ∈ X.
Φ
{A
i
}
k
i=1
(X, d) A
k+1
= A
1
f :
k

i=1
A
i
→
k

i=1
A
i
f(A
i
) ⊆ A
i+1
i = 1, 2, . . . , k

d (f(x), f(y)) ≤ ψ(d (x, y)) x ∈ A
i
, y ∈ A
i+1
i = 1, 2, . . . , k ψ : R
+
→ R
+
0 ≤ ψ(t) < t t > 0
f
ψ : R
+
→ R
+
η ∈ R
+
δ > 0 t ∈ R
+
η ≤ t < η + δ ψ (t) < η
(X, d) ψ : R
+
→
[0, 1)
η > 0 δ > 0 γ
η
∈ [0, 1) x, y ∈ X
η ≤ d(x, y) < δ + η ψ(d(x, y)) < γ
η
X = R
2

d : X × X → R
+
d (x, y) =
|x
1
− y
1
| + |x
2
− y
2
| x = (x
1
, x
2
) y = (y
1
, y
2
) ∈ X ψ : R
+
→
[0; 1) ψ (d (x, y)) =
d(x,y)
d(x,y)+1
ψ
X = R
2
d : X × X → R
+

d (x, y) =
|x
1
− y
1
| + |x
2
− y
2
| x = (x
1
, x
2
) y = (y
1
, y
2
) ∈ X ψ : R
+
→
[0; 1)
ψ (d (x, y)) =

d(x, y) − 1 d(x, y) > 1,
0 d(x, y) ≤ 1
(X, d) φ : R
+
→
R
+

η > 0 δ > 0 x, y ∈ X η ≤ d(x, y) <
δ + η n
0
∈ N φ
n
0
(d(x, y)) < η
X = R
2
d : X × X → R
+
d (x, y) =
|x
1
− y
1
| + |x
2
− y
2
| x = (x
1
, x
2
) y = (y
1
, y
2
) ∈ X φ : R
+

→
[0; 1) φ (d (x, y)) =
d(x,y)
2
φ
X = R
2
d : X × X → R
+
d (x, y) =
|x
1
− y
1
| + |x
2
− y
2
| x = (x
1
, x
2
) y = (y
1
, y
2
) ∈ X φ : R
+
→
[0; 1)

φ (d (x, y)) =





0 d(x, y) ≤ 1,
2 1 < d(x, y) < 2,
1 d(x, y) ≥ 2
A B
(X, d) f : A ∪ B → A ∪ B
x ∈ X ψ : R
+
→ [0, 1)
X
d(f
2n
(x), f(y)) ≤ ψ(d(f
2n−1
(x), y).d(f
2n−1
(x), y), (2.1)
n ∈ N y ∈ A f ψ
A B
(X, d) ψ : R
+
→ [0, 1)
X f : A ∪ B → A ∪ B ψ
A ∩ B f
A ∩ B

f : A ∪ B → A ∪ B ψ
x ∈ A
n ∈ N
d(f
2n
(x), f
2n+1
(x)) ≤ ψ(d(f
2n−1
(x), f
2n
(x))).d(f
2n−1
(x), f
2n
(x))
≤ d(f
2n−1
(x), f
2n
(x)),
d(f
2n+1
(x), f
2n+2
(x)) = d(f
2n+2
(x), f
2n+1
(x))

≤ ψ(d(f
2n+1
(x), f
2n
(x))).d(f
2n+1
(x), f
2n
(x))
≤ d(f
2n+1
(x), f
2n
(x)).
d(f
n
(x), f
n+1
(x)) ≤ d(f
n−1
(x), f
n
(x)), n ∈ N.

d(f
n
(x), f
n+1
(x))


lim
n→∞
d(f
n
(x), f
n+1
(x)) = η k
0
∈ N
δ > 0 n ≥ k
0
η ≤ d(f
n
(x), f
n+1
(x)) < η + δ.
ψ
X η γ
η
∈ [0, 1)
ψ(d(f
k
0
+n
(x), f
k
0
+n+1
(x))) < γ
η

n ∈ N ∪ {0} .
n ∈ N
d(f
k
0
+n
(x), f
k
0
+n+1
(x)) ≤ ψ(d(f
k
0
+n−1
(x), f
k
0
+n
(x)).d(f
k
0
+n−1
(x), f
k
0
+n
(x))
< γ
n
.d(f

k
0
+n−1
(x), f
k
0
+n
(x)).
n ∈ N
d(f
k
0
+n
(x), f
k
0
+n+1
(x)) < γ
n
.d(f
k
0
+n−1
(x), f
k
0
+n
(x))
< · · ·
< γ

n
n
.d(f
k
0
(x), f
k
0
+1
(x)).
γ
n
∈ [0, 1)
lim
n→∞
d(f
k
0
+n
(x), f
k
0
+n+1
(x)) = 0.
lim
n→∞
d(f
k
0
+n

(x), f
k
0
+m
(x)) = 0 m > n
m, n ∈ N m > n
d(f
k
0
+n
(x), f
k
0
+m
(x)) ≤
m−1

i=n
d(f
k
0
+i
(x), f
k
0
+i+1
(x)) <
γ
n
η

1 − γ
n
η
.d(f
k
0
(x), f
k
0
+1
(x)).
0 < γ
n
< 1 d(f
n
(x), f
m
(x)) → 0
{f
n
(x)} (X, d)
v ∈ A ∩ B lim
n→∞
f
n
(x) = v x ∈ A f

f
2n
(x)


A

f
2n+1
(x)

B
v A B v ∈ A ∩ B A ∩ B
d(v, f(v)) = lim
n→∞
d(f
2n
(x), f(v))
≤ lim
n→∞

ψ(d(f
2n−1
(x), v).(d(f
2n−1
(x), v)

≤ lim
n→∞

γ
η
(d(f
2n−1

(x), v)

= 0,
v f
f
µ f f
ν, µ ∈ A ∩ B f ψ
d(v, µ) = d(v, f (µ)) = lim
n→∞
d(f
2n
(x), f(µ))
≤ lim
n→∞

ψ(d(f
2n−1
(x), µ).(d(f
2n−1
(x), µ)

≤ lim
n→∞

γ
η
(d(f
2n−1
(x), µ)


≤ γ
η
.d(v, µ).
γ
η
∈ (0, 1) d(v, µ) = 0 v = µ v
f
(X, d) φ : R
+
→
R
+
φ X φ
φ
1
φ X φ(0) = 0
φ
2
(a) lim
n→∞
t
n
= γ > 0 lim
n→∞
φ (t
n
) < γ
(b) lim
n→∞
t

n
= 0 lim
n→∞
φ (t
n
) = 0.
φ
3
{φ
n
(t)}
n∈N
A B
(X, d) f : A ∪ B → A ∪ B
x ∈ A φ φ : R
+
→ R
+
X
d(f
2n
(x), f(y)) ≤ φ

d(f
2n−1
(x), y)

n ∈ N, y ∈ A,
(2.2)
f φ

A B
(X, d) φ : R
+
→ R
+
φ X
f : A ∪ B → A ∪ B φ
A ∩ B f A ∩ B
f : A ∪ B → A ∪ B φ
x ∈ A
n ∈ N
d(f
2n+1
(x), f
2n+2
(x)) ≤ φ

d(f
2n+1
(x), f
2n
(x))

,
d(f
2n+1
(x), f
2n+2
(x)) = d(f
2n+2

(x), f
2n+1
(x)) ≤ φ

d(f
2n+1
(x), f
2n
(x))

.
d(f
n
(x), f
n+1
(x)) ≤ φ

d(f
n−1
(x), f
n
(x))

, n ∈ N.
n ∈ N
d(f
n
(x), f
n+1
(x)) ≤ φ


d(f
n−1
(x), f
n
(x))

≤ φ
2

d(f
n−2
(x), f
n−1
(x))

≤
≤ φ
n
(d(x, f(x))) .
{φ
n
(d(x, f(x)))}
n∈N
η ≥ 0
η = 0 η > 0
φ X δ > 0
x, y ∈ X η ≤ d(x, y) < δ + η n
0
∈ N

φ
n
0
(d(x, y)) < η
lim
n→∞
φ
n
(d(x, f(x))) = η m
0
∈ N η ≤ d(x, y) < δ + η
m > m
0
φ
n
0
+m
0
(d(x
0
, x
1
) < η
lim
n→∞
φ
n
(d(x, f(x))) = 0
lim
n→∞

d(f
n
(x), f
n+1
(x)) = 0
c
m
= d(f
n
(x), f
n+1
(x))
ε > 0 n
0
(ε) ∈ N m, n ≥ n
0
(ε)
d(f
m
(x), f
n
(x)) < ε. (∗)
(∗) (∗)
ε > 0 p ∈ N m
p
, n
p
∈ N
m
p

> n
p
≥ p
m
p
n
p
d(f
n
(x), f
m
(x)) ≥ ε
m
p
(i) (ii)
{c
m
} 0 lim
p→∞
d(f
m
(x), f
n
(x)) =
ε
ε ≤ d(f
m
p
(x), f
n

p
(x))
≤ d(f
m
p
(x), f
m
p
+1
(x)) + d(f
m
p
+1
(x), f
n
p
+1
(x)) + d(f
n
p
+1
(x), f
n
p
(x))
≤ d(f
m
p
(x), f
m

p
+1
(x)) + φ (d(f
m
p
(x), f
n
p
(x))) + d(f
n
p
+1
(x), f
n
p
(x)).
p φ
2
φ
ε ≤ 0 + lim
p→∞
φ (d(f
m
p
(x), f
n
p
(x))) + 0 ≤ ε.
{f
n

(x)} (X, d)
v ∈ A ∪ B lim
n→∞
f
n
(x) = v x ∈ A f

f
2n
(x)

A

f
2n+1
(x)

B
v A B
v ∈ A ∩ B A ∩ B φ
2
φ
d(v, f(v)) = lim
n→∞
d(f
2n
(x), f(v))
≤ lim
n→∞
φ


d(f
2n−1
(x), v

= 0.
v f µ f
v = µ f φ−
d(v, µ) = d(v, f (µ)) = lim
n→∞
d(f
2n
(x), f(µ))
≤ lim
n→∞
φ

d(f
2n−1
(x), µ

< d(v, µ).
v = µ v
f
{A
i
}
k
i=1
(X, d) A

k+1
= A
1
ψ : R
+
→ [0, 1)
X f :
k

i=1
A
i
→
k

i=1
A
i
f(A
i
) ⊆ A
i+1
i = 1, 2, . . . , k
d(f(x), f(y)) ≤ ψ (d(x, y)) .d(x, y) x ∈ A
i
y ∈ A
i+1
i =
1, 2, . . . , k
f ψ

{A
i
}
k
i=1
(X, d) ψ : R
+
→ [0, 1)
X f :
k

i=1
A
i
→
k

i=1
A
i
ψ
f
k

i=1
A
i
x
0
∈ X n ∈ N x

n
= f
n
(x
0
) f
ψ n ∈ N
d(x
n
, x
n+1
) = d

f
n
(x
0
), f
n+1
(x
0
)

≤ ψ

d

f
n−1
(x

0
), f
n
(x
0
)

.d

f
n−1
(x
0
), f
n
(x
0
)

≤ d

f
n−1
(x
0
), f
n
(x
0
)


= d(x
n−1
, x
n
).
{d(x
n
, x
n+1
)}
lim
n→∞
d(x
n
, x
n+1
) = η ≥ 0 k
0
∈ N δ > 0
n ≥ k
0
η ≤ d(x
n
, x
n+1
) < η + δ.
ψ
X η γ
η

∈ [0, 1)
ψ

d

f
k
0
+n
(x
0
), f
k
0
+n+1
(x
0
)

,
n ∈ N ∪ 0 n ∈ N
d(x
k
0
+n
, x
k
0
+n+1
) = d


f
k
0
+n
(x
0
), f
k
0
+n+1
(x
0
)

≤ ψ

d

f
k
0
+n−1
(x
0
), f
k
0
+n
(x

0
)

.d

f
k
0
+n−1
(x
0
), f
k
0
+n
(x
0
)

< γ
η
.d

f
k
0
+n−1
(x
0
), f

k
0
+n
(x
0
)

,
n ∈ N
d(x
k
0
+n
, x
k
0
+n+1
) < γ
η
.d

f
k
0
+n−1
(x
0
), f
k
0

+n
(x
0
)

< · · ·
< γ
η
n
.d

f
k
0
(x
0
), f
k
0
+1
(x
0
)

.
γ
η
< 1 lim
n→∞
d(x

k
0
+n
, x
k
0
+n+1
) = 0
lim
n→∞
d(x
k
0
+n
, x
k
0
+m
) = 0 m > n
m, n ∈ N m > n
d(x
k
0
+n
, x
k
0
+m
) = d


f
k
0
+n
(x
0
), f
k
0
+m
(x
0
)

≤
m−1

i=n
d

f
k
0
+i
(x
0
), f
k
0
+i+1

(x
0
)

<
γ
n
η
1−γ
η
.d

f
k
0
(x
0
), f
k
0
+1
(x
0
)

.
0 ≤ γ
η
< 1 d (f
n

(x
0
), f
m
(x
0
)) → 0
{f
n
(x
0
)} X
v ∈
k

i=1
A lim
n→∞
f
n
(x
0
) = v i = 1, 2, . . . , k − 1

f
kn−i
(x
0
)


A
i
v A
i
i = 1, 2, . . . , k v ∈
k

i=1
A
i
k

i=1
A
i
= φ
d(v, f(v)) = lim
n→∞
d(f
kn
(x), f(v))
≤ lim
n→∞

ψ

d(f
kn−1
(x), v)


.d(f
kn−1
(x), v)

≤ lim
n→∞

γ
η
.d(f
kn−1
(x), v)

= 0.
d(v, f(v)) = 0 v f
v
µ f v = µ f
µ ∈
k

i=1
A
i
f ψ
d(v, µ) = d(v, f (µ)) = lim
n→∞
d(f
kn
(x), f(µ))
≤ lim

n→∞

ψ

d(f
kn−1
(x), µ)

.d(f
kn−1
(x), µ)

≤ lim
n→∞

γ
η
.d(f
kn−1
(x), µ)

≤ γ
η
.d(v, µ) < d(v, µ).
v = µ v f
{A
i
}
k
i=1

(X, d) A
k+1
= A
1
φ : R
+
→ R
+
φ X
f :
k

i=1
A
i
→
k

i=1
A
i
f(A
i
) ⊆ A
i+1
i = 1, 2, . . . , k
d(f(x), f(y)) ≤ φ (d(x, y)) x ∈ A
i
y ∈ A
i+1

i = 1, 2, . . . , k
f φ
{A
i
}
k
i=1
(X, d) φ : R
+
→ R
+
φ X
f :
k

i=1
A
i
→
k

i=1
A
i
φ
f
k

i=1
A

i
x
0
∈ X n ∈ N x
n
= f
n
(x
0
) f
φ n ∈ N
d(x
n
, x
n+1
) = d

f
n
(x
0
), f
n+1
(x
0
)

≤ φ

d


f
n−1
(x
0
), f
n
(x
0
)

= φ (d(x
n−1
, x
n
))
≤ · · ·
≤ φ
n
(d(x
0
, x
1
)) .
{φ
n
(d(x
0
, x
1

))}
n∈N
η ≥ 0 η = 0 η > 0
φ X
δ > 0 x, y ∈ X η ≤ d(x, y) < δ + η
n
0
∈ N φ
n
0
(d(x, y)) < η lim
n→∞
φ
n
(d(x
0
, x
1
)) = η
m
0
∈ N η < φ
m
(d(x
0
, x
1
)) < δ + η m > m
0
φ

m
0
+n
0
(d(x
0
, x
1
)) < η lim
n→∞
φ
n
(d(x
0
, x
1
)) = 0
lim
n→∞
d (x
n
, x
n+1
) = 0
{x
n
}
ε > 0 n
0
(ε) ∈ N m, n ≥ n

0
(ε)
d (x
n
, x
m
) < ε. (∗∗)
(∗∗) (∗∗)
ε > 0 p ∈ N m
p
, n
p
∈ N m
p
>
n
p
≥ p
d

x
m
p
, x
n
p

≥ ε
m
p

n
p
ε ≤ d

x
m
p
, x
n
p

≤ d

x
m
p
, x
m
p−1

+ d(x
m
p−1
, x
n
p
)
≤ d

x

m
p
, x
m
p−1

+ ε.
lim
p→∞
d

x
m
p
, x
n
p

= ε
d

x
m
p
, x
n
p

− d


x
m
p
, x
m
p
+1

≤ d

x
m
p
+1
, x
n
p

≤ d

x
m
p
, x
m
p
+1

+ d


x
m
p
, x
n
p

,
lim
p→∞
d

x
m
p
, x
n
p

= ε i ∈ {0, 1, . . . , k−1}
m
p
− n
p
+ i = 1 k p i = 0 p
ε ≤ d(x
m
p
, x
n

p
)
≤ d(x
m
p
, x
m
p
+1
) + d(x
m
p
+1
, x
n
p
+1
) + d(x
n
p
+1
, x
n
p
)
≤ d(x
m
p
, x
m

p
+1
) + φ

d(x
m
p
, x
n
p
)

+ d(x
n
p
+1
, x
n
p
).
p → ∞ φ
2
φ
ε ≤ 0 + lim
p→∞
φ

d(x
m
p

, x
n
p
)

+ 0 < ε.
i = 0 {x
n
} X
v ∈
k

i=1
A
i
lim
n→∞
x
n
= v
i = 1, 2, . . . , k − 1

f
k
n
−i
(x)

A
i


f
k
n
−i
(x)

{x
n
} v A
i
i = 1, 2, . . . , k − 1
k

i=1
A
i
= φ
φ
2
φ
d(v, f(v)) = lim
n→∞
d

f
k
n
(x), f(v)


≤ lim
n→∞
φ

d

f
k
n
−1
x, v

= 0.
d(v, f(v)) = 0 v f µ
f f φ
d(v, µ) = d(v, f (µ)) = lim
n→∞
d

f
k
n
(x), f(µ)

≤ lim
n→∞
φ

d


f
k
n
−1
(x), µ

< d(v, µ).
v = µ v f
(ψ
x
, ϕ)
Θ ϕ : R
+
5
→ R
+
ϕ
1
ϕ
ϕ
2
t > 0 ϕ (t, t, t, 0, 2t) < t, ϕ (t, t, t, 2t, 0) < t, ϕ (t, 0, 0, t, t) < t
ϕ (0, 0, t, t, 0) < t ϕ (0, 0, 0, 0, 0) = 0
ϕ : R
+
5
→ R
+
ϕ (t
1

, t
2
, t
3
, t
4
, t
5
) =
2
3
. max

t
1
, t
2
, t
3
,
1
2
t
4
,
1
2
t
5


.
ϕ (ϕ
1
) (ϕ
2
)
A B
(X, d) f : A ∪ B → A ∪ B
x ∈ A ψ
x
: R
+
→ [0, 1) X
ϕ ∈ Θ
d

f
2n
(x), f (y)

≤ ψ
x

d

f
2n−1
(x), y

.θ,

θ = ϕ

d

f
2n−1
x, y

, d

f
2n−1
x, f
2n
x

, d (fy, y) , d

f
2n−1
x, f y

, d

f
2n
x, y

n ∈ N y ∈ A f (ψ
x

, ϕ)
fx f
n
x
f(x) f
n
(x)
A B
(X, d) ψ
x
: R
+
→ [0; 1)
X ϕ ∈ Θ f : A ∪ B → A ∪ B
(ψ
x
, ϕ) A ∩ B
f A ∩ B
f : A ∪ B → A ∪ B (ψ
x
, ϕ)
x ∈ A f
2n
x ∈ A
y = f
2n
x n ∈ N n ∈ N
d

f

2n
x, f
2n+1
x

≤ ψ
x

d

f
2n−1
x, f
2n
x

.θ,
θ = ϕ

d

f
2n−1
x, f
2n
x

, d

f

2n−1
x, f
2n
x

, d

f
2n+1
x, f
2n
x

,
d

f
2n−1
x, f
2n+1
x

, d

f
2n
x, f
2n
x


= ϕ

d

f
2n−1
x, f
2n
x

, d

f
2n−1
x, f
2n
x

, d

f
2n+1
x, f
2n
x

,
d

f

2n−1
x, f
2n
x

+ d

f
2n
x, f
2n+1
x

, 0

.
ϕ
θ < d

f
2n−1
x, f
2n
x

,
d

f
2n

x, f
2n+1
x

< ψ
x

d

f
2n−1
x, f
2n
x

.d

f
2n−1
x, f
2n
x

< d

f
2n−1
x, f
2n
x


.
(3.1)
y = f
2n
x n ∈ N
d

f
2n+1
x, f
2n+2
x

= d

f
2n+2
x, f
2n+1
x

≤ ψ
x

d

f
2n+1
x, f

2n
x

.θ,
θ = ϕ

d

f
2n+1
x, f
2n
x

, d

f
2n+1
x, f
2n+2
x

, d

f
2n+1
x, f
2n
x


,
d

f
2n+1
x, f
2n+1
x

, d

f
2n+2
x, f
2n
x

= ϕ

d

f
2n+1
x, f
2n
x

, d

f

2n+1
x, f
2n+2
x

, d

f
2n+1
x, f
2n
x

,
0, d

f
2n
x, f
2n+1
x

+ d

f
2n+1
x, f
2n+2
x


.
ϕ
θ < d

f
2n
x, f
2n+1
x

d

f
2n+1
x, f
2n+2
x

< ψ
x

d

f
2n
x, f
2n+1
x

.d


f
2n
x, f
2n+1
x

< d

f
2n
x, f
2n+1
x

.
(3.2)

d

f
n
x, f
n+1
x

lim
n→∞
d


f
n
x, f
n+1
x

= η
k
0
∈ N δ > 0 n ≥ k
0
η ≤ d

f
n
x, f
n+1
x

< η + δ.
ψ
x
X η γ
η
∈ [0, 1)
ψ
x

d


f
n
x, f
n+1
x

< γ
η
n ≥ k
0
.
(3.3)
n
0
=

k
0
+3
2
 
k
0
+3
2

k
0
+3
2

n ≥ n
0
d

f
2n
x, f
2n+1
x

≤ ψ
x

d

f
2n−1
x, f
2n
x

.d

f
2n−1
x, f
2n
x

< γ

η
.d

f
2n−1
x, f
2n
x

(3.4)
d

f
2n+1
x, f
2n+2
x

≤ ψ
x

d

f
2n+1
x, f
2n
x

.d


f
2n
x, f
2n+1
x

< γ
η
.d

f
2n
x, f
2n+1
x

.
(3.5)
n ∈ N ∪ {0}
d

f
2n
0
+n
x, f
2n
0
+n+1

x

≤ γ
n
η

d

f
2n
0
−1
x, f
2n
0
x

. (3.6)
γ
η
< 1
lim
n→∞
d

f
2n
0
+n
x, f

2n
0
+n+1
x

= 0.
m, n ∈ N m > n
d

f
2n
0
+n
x, f
2n
0
+m
x

≤
m−1

i=n
d

f
2n
0
+i
x, f

2n
0
+i+1
x

<
γ
n
η
1 − γ
η
.d

f
2n
0
x, f
2n
0
+1
x

.
0 < γ
n
< 1 d (f
n
x, f
m
x) → 0

{f
n
x} (X, d) A B
{f
n
x} ⊂ A ∪ B v ∈ A ∪ B lim
n→∞
f
n
x = v

f
2n
x

A

f
2n+1
x

B
v A B v ∈ A ∩ B
A ∩ B v f
v f d (v, fv) > 0
lim
n→∞
d

f

2n−1
x, v

= 0
d

f
2n
x, fv

≤ ψ
x

d

f
2n−1
x, v

.θ,
θ = ϕ

d

f
2n−1
x, v

, d


f
2n−1
x, f
2n
x

, d (fv, v) , d

f
2n−1
x, f v

, d

f
2n
x, v

.
d (v, fv) = lim
n→∞
d

f
2n
x, fv

≤ γ
n
.ϕ (d (v, v) , d (v, v) , d (fv, v) , d (v, fv) , d (v, v))

≤ ϕ (0, 0, d (v, fv) , d (v, fv) , 0)
< d (v, f v) .
d (v, fv) = 0 v f
u
f f u, v ∈ A ∩ B
f (ψ
x
, ϕ)
d (v, u) = d (v, fu) = lim
n→∞
d

f
2n
x, fu

, (3.7)
d

f
2n
x, fu

≤ ψ
x

d

f
2n−1

x, u

.θ < γ
η
.θ, (3.8)
θ = ϕ

d

f
2n−1
x, u

, d

f
2n−1
x, f
2n
x

, d (fu, u) , d

f
2n−1
x, f u

, d

f

2n
x, u

.
(ϕ
2
) ϕ
d (v, u) < γ
n
.ϕ (d (v, u) , d (v, v) , d (fu, u) , d (v, fu) , d (v, u))
< ϕ (d (v, u) , 0, 0, d (v, u) , d (v, u))
< d (v, u) .