Điểm bất động của các phép co yếu cyclic trong không gian g mêtric

G
G
(ψ, ϕ)
G
(ψ, ϕ)
G
G
G
2
D
D
G
G
G
G
(ψ, ϕ)
G
(ψ, ϕ) (ψ, ϕ)
(ψ, ϕ) (ψ, ϕ)
G
G
G
G
G
(ψ, ϕ)
G
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, y) ≤ d(x, z) + d(z, y) x, y, z ∈ X

X d
(X, d) X d (x, y)
x y
(X, d) {x
n
} ⊂ X
x ∈ X ε > 0 n
0
∈ N
∗
n ≥ n
0
d (x
n
, x) < ε lim
n→∞
x
n
= x x
n
→ x
n → ∞
(X, d) {x
n
} ⊂ X
ε > 0 n
0
∈ N
∗
n, m ≥ n

0
d(x
n
, x
m
) < ε {x
n
}
lim
n,m→+∞
d(x
n
, x
m
) = 0
(X, d)
M (X, d)
M
(X, d) (Y, ρ)
f : (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

∗
∈ X f (x
∗
) = x
∗
f
A
1
A
2
A
p
(X, d) T :
p

i=1
A
i
→
p

i=1
A
i
p
T (A
i
) ⊆ A
i+1
A

p+1
= A
1
i = 1, 2, 3, p.
A
1
A
2
A
p
(X, d) T :
p

i=1
A
i
→
p

i=1
A
i
k ∈ (0, 1) d(T x, Ty) ≤ kd(x, y) ∈ A
i
, y ∈ A
i+1
i = 1, 2, 3, p T
Φ ϕ : [0, 1) → [0, 1)
ϕ
ϕ(t) = 0 t = 0

X p
f : X → X X {X
i
}
p
i=1
X X f
X =
p

i=1
X
i
X
i
= φ i = 1, . . . , p
f (X
1
) ⊂ X
2
, f (X
2
) ⊂ X
3
, . . . , f (X
p−1
) ⊂ X
p
, f (X
p

) ⊂ X
1
A
1
A
2
A
p
(X, d) X =
p

i=1
A
i
T : X → X
(ψ, ϕ)
X =
p

i=1
A
i
X T
ψ(d(T x, Ty)) ≤ ψ(d(x, y)) − ϕ(d(x, y)) x ∈ A
i
y ∈
A
i+1
i = 1, 2, 3, p ψ, ϕ ∈ Φ A
p+1

.
= A
1
(X, d) A
1
, A
2
, A
p
X X =
p

i=1
A
i
T : X → X
(ψ, ϕ) T z ∈
p

i=1
A
i
X = φ G : X × X × X → R
+
G G X
G
1
G(x, y, z) = 0 x = y = z
G
2

0 < G(x, x, y) x, y ∈ X x = y
G
3
G(x, x, y) ≤ G(x, y, z) x, y, z ∈ X z = y
G
4
G(x, y, z) = G(p{x, y, z}) p x, y, z
G
5
G(x, y, z) ≤ G(x, a, a) + G(a, y, z) x, y, z, a ∈ X
X G G G
X = φ G : X × X × X → R
+
G(x, y, z) =

0 x = y = z,
1 .
(X, G) G G
G
1
G
2
G
3
G
4
G
G
5
x, y, z, a ∈ X

x = y = z G(x, y, z) = 0 G
G(x, y, z) = 0 ≤ G(x, a, a) + G(a, y, z)
x = y x = z
y = z G(x, y, z) = 1 a ∈ X 1 ≤ G(x, a, a)+
G(a, y, z) ≤ 1 + 1 = 2 G(x, y, z) ≤ G(x, a, a) + G(a, y, z)
(X, G) G
(X, d)
G : X × X × X → R
+
G(x, y, z) = max{d(x, y), d(y, z), d(x, z)} x, y, z ∈ X
(X, G) G G
G
(G
1
) d(x, y), d(y, z), d(x, z) x, y, z ∈
X G(x, y, z) = max{d(x, y), d(y, z), d(x, z)} ≥ 0 x, y, z ∈ X
x = y = z
(G
2
) G(x, x, y) = max{d(x, x), d(x, y), d(x, y)} = d(x, y) > 0
x, y ∈ X x = y
(G
3
) G(x, x, y) = max{d(x, x), d(x, y), d(x, y)} = d(x, y)
≤ max{d(x, y), d(y, z), d(x, z)} = G(x, y, z) x, y, z ∈ X
z = y
G
4
)
G

5
) G(x, y, z) = max{d(x, y), d(y, z), d(x, z)}
≤ max{d(x, a) + d(a, y), d(y, z), d(x, a) + d(a, z) + d(a, a)}
≤ max{d(x, a), d(x, a), d(a, a)} + max{d(a, y), d(a, z), d(y, z)}
= G(x, a, a) + G(a, y, z) x, y, z, a ∈ X
(X, d)
G : X × X × X → R
+
G(x, y, z) = d(x, y) + d(y, z) + d(x, z) x, y, z ∈ X
(X, G) G
d G(x, y, z) = d(x, y) + d(y, z) +
d(x, z) x, y, z ∈ X G G
1
G
2
G
3
G
4
G G
5
x, y, z, a ∈
X
G(x, y, z) = d(x, y) + d(y, z) + d(x, z)
≤ (d(x, a) + d(a, y) + d(y, z) + d(x, a) + d(a, z) + d(a, a))
= (d(x, a) + d(x, a) + d(a, a)] + [d(a, y) + d(y, z) + d(a, z))
= G(x, a, a) + G(a, y, z)
(X, G) G
(X, G) (X


, G

) G
f : (X, G) → (X

, G

) f G a ∈ X
ε > 0 δ > 0 x, y ∈ X
G(a, x, y) < δ G(f(a), f(x), f(y)) < ε f G
X f G a ∈ X
(X, G) G {x
n
}
X {x
n
} G x ∈ X
lim
n,m→∞
G(x, x
n
, x
m
) = 0 ε > 0
n
0
∈ N G(x, x
n
, x
m

) < ε n, m ≥ n
0
x
{x
n
} x
n
→ x lim x
n
= x
(X, G) G
{x
n
} G x
G(x
n
, x
n
, x) → 0 n → ∞
G(x
n
, x, x) → 0 n → ∞
⇒ {x
n
} G x G(x, x
m
, x
n
) → 0
n, m → ∞ G(x

n
, x
n
, x) = G(x, x
n
, x
n
) ≤ G(x, x
m
, x
n
)
G(x
n
, x
n
, x) → 0 n → ∞
⇒ G(x
n
, x, x) ≤ G(x, x
n
, x
n
)+G(x
n
, x, x
n
) = 2G(x
n
, x

n
, x)
G(x
n
, x
n
, x) → 0 n → ∞ G(x
n
, x, x) → 0
n → ∞
⇒ G(x
m
, x
n
, x) ≤ G(x
m
, x, x)+G(x, x
n
, x) =
G(x
m
, x, x) + G(x
n
, x, x) G(x
n
, x, x) → 0 n → ∞
G(x
m
, x
n

, x) → 0 n, m → ∞
(X, G) (X

, G

) G
f : (X, G) → (X

, G

) G x ∈ X
G x {x
n
} G
x {f(x
n
)} G f(x)
f G x ∈ X {x
n
}
G x {f(x
n
)} G f(x)
f G x ∈ X ε > 0
δ > 0 y, z ∈ X G(x, y, z) < δ
G(f(x), f(y), f(z)) < ε {x
n
} G x
N ∈ N G(x, x
n

, x
m
) < δ n, m ≥ N
G(f(x), f(x
n
), f(x
m
)) < ε n, m ≥ N {f(x
n
)} G
f(x)
{x
n
} G x ∈ X
{f(x
n
)} G f(x) f G x
ε
0
> 0 n ∈ N x
n
, y
n
∈ X
G(x, x
n
, y
n
) <
1

n
G(f(x), f(x
n
), f(y
n
)) ≥ ε
0
(G
3
)
G(x, x
n
, x
n
) ≤ G(x, x
n
, y
n
) <
1
n
G(x, y
n
, y
n
) ≤ G(x, x
n
, y
n
) <

1
n
G(x, x
n
, x
n
) → 0 G(x, y
n
, y
n
) → 0 n → ∞
G(f(x), f(x
n
), f(x
n
)) → 0 G(f(x), f(y
n
), f(y
n
)) → 0
n → ∞ G(f(x), f(x), f(x
n
)) → 0
G(f(x), f(x), f(y
n
)) → 0 n → ∞
G(f(x), f(x
n
), f(y
n

)) → 0 n → ∞ f
G x
(X, G) G {x
n
} ⊂
X G ε > 0 n
0
∈ N
G(x
m
, x
n
, x
l
) < ε n, m, l ≥ n
0
G(x
m
, x
n
, x
l
) → 0
n, m, l → ∞
(X, G) G
{x
n
} G
ε > 0 n
0

∈ N G(x
n
, x
m
, x
m
) < ε
n, m ≥ n
0
⇒ {x
n
} G ε > 0
n
0
∈ N G(x
m
, x
n
, x
l
) < ε n, m, l ≥ n
0
G(x
n
, x
m
, x
m
) < G(x
m

, x
n
, x
l
) m = l G(x
n
, x
m
, x
m
) < ε
n, m ≥ n
0
⇒ G
5
) G
G(x
m
, x
n
, x
l
) < G(x
m
, x
n
, x
n
) + G(x
n

, x
n
, x
l
).
G(x
m
, x
n
, x
n
) < ε n, m ≥ n
0
n, m, l ≥ n
0
G(x
m
, x
n
, x
l
) < 2ε {x
n
} G 
(X, G) G
x
0
∈ X r > 0 G x
0
r B

G
(x
0
, r)
B
G
(x
0
, r) = {y ∈ X : G(x
0
, y, y) < r}.
G τ(G) X
τ(G) G
G (X, G) G
G(x, y, y) = G(y, x, x) x, y ∈ X
G (X, G)
(X, d
G
) d
G
d
G
(x, y) =
G(x, y, y) +G(y, x, x) x, y ∈ X (X, G) G
d
G
(x, y) = 2G(x, y, y) x, y ∈ X
(X, G)
G
3

2
G(x, y, y) ≤ d
G
(x, y) ≤ 3G(x, y, y) x, y ∈ X
(X, G) G
G(x, y, z) x, y, z τ (G)
{x
n
} {y
n
} {z
n
} G
x, y, z ∈ X G
5
) G
G(x, y, z) ≤ G(y, y
m
, y
m
) + G(y
m
, x, z)
= G(y, y
m
, y
m
) + G(x, y
m
, z)

≤ G(y, y
m
, y
m
) + G(x, x
k
, x
k
) + G(x
k
, y
m
, z)
= G(y, y
m
, y
m
) + G(x, x
k
, x
k
) + G(z, x
k
, y
m
)
≤ G(y, y
m
, y
m

) + G(x, x
k
, x
k
) + G(z, z
n
, z
n
) + G(x
k
, y
m
, z
n
).
G(x, y, z) − G(x
k
, y
m
, z
n
) ≤ G(y, y
m
, y
m
) + G(x, x
k
, x
k
) + G(z, z

n
, z
n
).
G(x
k
, y
m
, z
n
) − G(x, y, z) ≤ G(x, x, x
k
) + G(y, y, y
m
) + G(z, z, z
n
).
G
5
) G G(x, x, x
k
) ≤
G(x, x
k
, x
k
) + G(x
k
, x, x
k

) = 2G(x, x
k
, x
k
) G(y, y, y
m
) ≤ 2G(y, y
m
, y
m
)
G(z, z, z
n
) ≤ 2G(z, z
n
, z
n
).
G(x
k
, y
m
, z
n
) − G(x, y, z) ≤
≤ G(x, x, x
k
) + G(y, y, y
m
) + G(z, z, z

n
)
≤ 2{G(x, x
k
, x
k
) + G(y, y
m
, y
m
) + G(z, z
n
, z
n
)}.
|G(x
k
, y
m
, z
n
)−G(x, y, z)| ≤ 2{G(x, x
k
, x
k
)+G(y, y
m
, y
m
)+G(z, z

n
, z
n
)}.
G(x
k
, y
m
, z
n
) → G(x, y, z) k, m, n → ∞ 
(X, G) G x
0
∈ X
r > 0
G(x
0
, x, y) < r x, y ∈ B
G
(x
0
, r)
y ∈ B
G
(x
0
, r), δ > 0 B
G
(y, δ) ⊆ B
G

(x
0
, r)
x, y ∈ X G(x
0
, x, x) ≤ G(x
0
, x, y) < r
G(x
0
, y, y) ≤ G(x
0
, x, y) < r x, y ∈ B
G
(x
0
, r).
y ∈ B
G
(x
0
, r), G(x
0
, y, y) < r δ = r − G(x
0
, y, y)
δ > 0 B
G
(y, δ) ⊆ B
G

(x
0
, r)
z ∈ B
G
(y, δ) G(y, z, z) < δ = r − G(x
0
, y, y) G(y, z, z) +
G(x
0
, y, y) < r. G(x
0
, z, z) ≤ G(y, z, z) + G(x
0
, y, y) < r
z ∈ B
G
(x
0
, r) δ > 0 B
G
(y, δ) ⊆ B
G
(x
0
, r).

(X, d) T : (X, d) →
(X, d) O(x, ∞) = {x, T x, T
2

x, T
3
x, . . .} X
T {x
n
} ⊂ O(x, ∞) x X
X
A
1
, A
2
, A
p
G (X, G) T :
p

i=1
A
i
→
p

i=1
A
i
G T(A
i
) ⊆ A
i+1
A

p+1
= A
1
i = 1, 2, , p
A
1
, A
2
, A
p
G (X, G) X =
p

i=1
A
i
T : X → X
(ψ, ϕ)
X =
p

i=1
A
i
X T
ψ(G(T x, Tx, T y)) ≤ ψ(G(x, x, y)) − ϕ(G(x, x, y)) x ∈ A
i
y ∈ A
i+1
i = 1, 2, , p ψ, ϕ ∈ Φ A

p+1
.
= A
1
A
1
, A
2
, , A
p
G (X, G) T :
p

i=1
A
i
→
p

i=1
A
i
(ψ, ϕ)
ψ(G(T x, Tx, T y)) ≤ ψ(M(x, x, y)) − ϕ(M(x, x, y)), (1.1)
x ∈ A
i
y ∈ A
i+1
i = 1, 2, , p ψ, ϕ ∈ Φ, A
p+1

= A
1
M(x, x, y) = max

G(x, x, y), G(x, x, T x), G(y, y, T y),
G(x, x, T y) + G(y, y, Tx)
2

.
(ψ, ϕ)
G
(ψ, ϕ) G
A
1
, A
2
, , A
p
G (X, G) A
i
i =
1, 2, . . . , p T :
p

i=1
A
i
→
p


i=1
A
i
k ∈ (0, 1) i = 1, 2, . . . , p
G(T x, Tx, T y) ≤ kG(x, x, y)
x ∈ A
i
y ∈ A
i+1
A
1
d = dist(A
1
, A
p
) = inf{G(x, x, y) : x ∈ A
1
, y ∈ A
p
}
A
1
x
0
∈ A
1
{u
n
} ∈ A
p

lim
n→∝
G(x
0
, x
0
, u
n
) = d.
d > 0
G(T
p+1
x
0
, T
p+1
x
0
, T
p+1
u
n
) < · · · < G(T x
0
, T x
0
, T u
n
) < G(x
0

, x
0
, u
n
).
(1.2)
{T
{p+1}
n
}
∞
n=1
⊂ A
1
A
1
z ∈ A
1
G(z, z, T
p+1
x
0
) ≤
d G(T
p−1
z, T
p−1
z, T
2p
x

0
) ≤ d T
p−1
z ∈ A
p
T
2p
x
0
∈ A
1
d d = 0
A
1

A
p
= φ A
1

A
2
= φ
A

1
= A
1

A

2
A

2
= A
2

A
3
A

p
= A
p

A
1
A

1
T {A

i
}
p
i=1
A
1
A
p

A

1

A

p
= φ A
1

A
2

A
3
=
φ
A =
p

i=1
A
i
= φ
A T A T
A T
A 
(X, G) G A
1
, A

2
, A
p
X X =

p
i=1
A
i
. T : X →
X (ψ, ϕ) T
z ∈
p

i=1
A
i
x
0
∈ X {x
n
}
x
n+1
= T x
n
, n = 0, 1, 2, . . .
n
0
∈ N x

n
0
+1
= x
n
0
x
n
0
+1
= T x
n
0
= x
n
0
x
n
0
T x
n+1
= x
n
n = 0, 1, 2,
X =
p

i
A
i

n > 0 i
n
∈ {1, 2, , p} x
n−1
∈ A
i
n
x
n
∈ A
i
n
+1
T (ψ, ϕ)
ψ(G(x
n
, x
n
, x
n+1
)) = ψ(G(T x
n−1
, T x
n−1
, T x
n
))
≤ ψ(G(x
n−1
, x

n−1
, x
n
)) − ϕ(G(x
n−1
, x
n−1
, x
n
))
≤ ψ(G(x
n−1
, x
n−1
, x
n
)).
(1.3)
(1.3) ϕ
G(x
n
, x
n
, x
n+1
) ≤ G(x
n−1
, x
n−1
, x

n
) n = 1, 2,
{G(x
n
, x
n
, x
n+1
)}
λ ≥ 0 lim
n→∞
G(x
n
, x
n
, x
n+1
) = λ n →
∞ (1.3) ψ ϕ ψ(λ) ≤ ψ(λ)−ϕ(λ)
ϕ(λ) = 0
ϕ ∈ Φ λ = 0
lim
n→∞
G(x
n
, x
n
, x
n+1
) = 0. (1.4)

x
n
G
ε > 0
n
1
∈ N n, m ≥ n
1
G(x
n
, x
n
, x
m
) ≤
ε
2
. (1.5)
lim
n→∞
G(x
n
, x
n
, x
n+1
) = 0 ε > 0 n
2
∈ N
n ≥ n

2
G(x
n
, x
n
, x
n+1
≤
ε
2p
. (1.6)
n
0
= max{n
1
, n
2
} a, b ≥ n
0
a > b
j ∈ {1, 2, 3, , p} b − a ≡ j p b − a + k ≡ 1 p
k = p − j + 1
G(x
a
, x
a
, x
b
) ≤ G(x
a

, x
a
, x
b+k
)+G(x
a+k
, x
a+k
, x
a+k−1
)+ +G(x
b+1
, x
b+1
, x
b
).
(1.7)
(1.5), (1.6) (1.7)
G(x
a
, x
a
, x
b
) ≤
ε
2
+ k
ε

2p
≤
ε
2
+ p.
ε
2p
= ε. (1.8)
x
n
G X
x ∈ X lim
n→∞
x
n
= x.
x T X
T x
n
A
i
i ∈ {1, 2, 3, , p} x ∈ A
i
T x ∈ A
i+1
{x
n
k
}
{x

n
} {x
n
k
} ⊂ A
i−1
(ψ, ϕ
ψ(G(x
n
k+1
, x
n
k+1
, T x)) = ψ(G(T x
n
k
, T x
n
k
, T x))
≤ ψ(G(T x
n
k
, T x
n
k
, x)) − ϕ(G(x
n
k
, x

n
k
, x))
≤ ψ(G(x
n
k
, x
n
k
, x)).
(1.9)
x
n
k
→ x ψ, ϕ ∈ Φ k → ∞ (1.9)
ψ(G(x, x, T x)) ≤ ψ(G(x, x, x)) = ψ(0) = 0.
ψ(G(x, x, T x)) = 0 x T
y ∈ X
T T y, x
T y, x ∈
m

i=1
A
i
(ψ, ϕ
ψ(G(y, y, x)) ≤ ψ(G(T y, T y, T x))
≤ ψ(G(y, y, x)) − ϕ(G(y, y, x)).
(1.10)
ϕ(G(y, y, x)) = 0 ϕ G(y, y, x) =

0 y = x T
A
1
, A
2
, , A
p
G (X, G) T :
p

i=1
A
i
→
p

i=1
A
i
(ψ, ϕ X T
z ∈
p

i=1
A
i
.
i ∈ {1, 2, , p} x ∈ A
i
n ∈ N n n + 1

ψ(G(T
n
x, T
n
x, T
n+1
x)) ≤ ψ(M(T
n−1
x, T
n−1
x, T
n
x)) − ϕ(M(T
n−1
x, T
n−1
x, T
n
x))
≤ ψ(M(T
n−1
x, T
n−1
x, T
n
x)).
(1.11)
ψ
G(T
n

x, T
n
x, T
n+1
x) ≤ max{G(T
n−1
x, T
n−1
x, T
n
x), G(T
n−1
x, T
n−1
x, T
n
x),
G(T
n
x, T
n
x, T
n+1
x),
G(T
n−1
x, T
n−1
x, T
n+1

x) + G((T
n+1
x, T
n+1
x, T
n+1
x)
2
}
≤ G(T
n−1
x, T
n−1
x, T
n
x), (1.12)
n ∈ N {G(T
n
x, T
n
x, T
n+1
x)}
lim
n→∞
G(T
n
x, T
n
x, T

n+1
x) = r r ≥ 0 r > 0
n → ∞ ψ ϕ
ϕ(r) > 0
ψ(r) ≤ ψ(r) − ϕ(r) < ψ(r).
lim
n→∞
G(T
n
x, T
n
x, T
n+1
x) = 0
{T
n
x} G {T
n
x}
G µ > 0
{m
k
} {n
k
} k ≤ m
k
< n
k
k ∈ N
G(T

m
k
x, T
m
k
x, T
n
k
x) ≥ µ,
G(T
m
k
x, T
m
k
x, T
n
k
−1
x) < µ.
G(T
m
k
x, T
m
k
x, T
n
k
x) ≤ G(T

m
k
x, T
m
k
x, T
n
k
−1
x) + G(T
n
k
−1
x, T
n
k
−1
x, T
n
k
x).
lim
n→∞
G(T
n
x, T
n
x, T
n+1
x) = 0

G(T
m
k
x, T
m
k
x, T
n
k
x) = µ.
(ψ, ϕ
ψ(G(T
m
k
+1
x, T
m
k
+1
x, T
n
k
+1
x)) = ψ(G(T T
m
k
x, T T
m
k
x, T T

n
k
x))
≤ ψ(M(T
m
k
x, T
m
k
x, T
n
k
x))−ϕ(M(T
m
k
x, T
m
k
x, T
n
k
x))
≤ ψ(M(T
m
k
x, T
m
k
x, T
n

k
x)).
k → ∞
ψ(µ) ≤ ψ(µ) − ϕ(µ) ≤ ψ(µ).
µ = 0 µ = 0
µ > 0 {T
n
x} G
X G z ∈
p

i=1
A
i
{T
n
x} z i ∈ {1, 2, p} X =
p

i=1
A
i
T {T
2n
x} {T
2n−1
x}
A
i
A

i+1
A
p+1
= A
1
z
(ψ, ϕ
ψ(G(T
2n
x, T
2n
x, T z)) = ψ(G(T T
2n−1
x, T T
2n−1
x, T z))
≤ ψ(M(T
2n−1
x, T
2n−1
x, z))−ϕ(M(T
2n−1
x, T
2n−1
x, z))
≤ ψ(M(T
2n−1
x, T
2n−1
x, z)).

n → ∞
ψ(G(z, z, T z)) ≤ ψ(G(z, z, z)) = ψ(0),
ψ(G(z, z, T z)) = 0 ψ ∈ Φ G(z, z, T z) = 0 z = T z
z T (ψ, ϕ
ψ, ϕ ∈ Φ

G
G
(X, G) G p
A
1
A
2
A
p
φ X Y =
p

i=1
A
i
T : Y → Y
Y =

p
i=1
A
i
Y T
(x, y, z) ∈ A

i
× A
i+1
× A
i+1
, i = 1, 2, p A
p+1
= A
1
ψ(G(T x, T y, T z)) ≤ ψ(Θ(x, y, z))−ϕ(θ(x, y, z)), (2.1)
Θ(x, y, z)
= max{G(x, y, z), G(x, T x, T x), G(y, Ty, T y), G(z, T z, T z),
1
2
[G(x, T y, T y) + G(y, T x, T x)],
1
2
[G(y, T z, T z) + G(z, T y, T y)],
1
2
[G(x, T z, T z) + G(z, T x, T x)],
1
3
[G(x, T y, T y) + G(y, T z, T z) + G(z, T x, T x)]} (2.2)
θ(x, y, z) = max{G(x, y, z), G(x, T x, T x), G(y, T y, Ty), G(z, T z, T z)}; (2.3)
ψ : [0, +∞) → [0, +∞) ϕ : [0, +∞) →
[0, +∞) ϕ(t) = 0 t = 0
(X, G) p ∈ N A
1
, A

2
, A
p
X Y =
p

i=1
A
i
T : Y → Y
X T
z ∈
p

i=1
A
i
.
x
0
∈ A
1
x
0
A
1
= φ
(x
n
) ⊂ X x

n+1
= T x
n
, n = 0, 1, 2
x
n
= x
n+1
, n ∈ N ∪ {0}. (2.4)
lim
n→∞
G(x
n
, x
n+1
, x
n+1
) = 0. (2.5)
G(x
n
, x
n+1
, x
n+1
) > 0 n ∈ N
n ∈ N i = i(n) ∈ {1, 2, , p}
(x
n
, x
n+1

, x
n+1
) ∈ A
1
× A
i+1
× A
i+1
x = x
n
y = x
n+1
Θ(x
n
, x
n+1
, x
n+1
)
= max{G(x
n
, x
n+1
, x
n+1
), G(x
n
, T x
n
, T x

n
), G(x
n+1
, T x
n+1
, T x
n+1
),
G(x
n+1
, T x
n+1
, T x
n+1
),
1
2
[G(x
n
, T x
n+1
, T x
n+1
) + G(x
n+1
, T x
n
, T x
n
)],

1
2
[G(x
n+1
, T x
n+1
, T x
n+1
) + G(x
n+1
, T x
n+1
, T x
n+1
)],
1
2
[G(x
n
, T x
n+1
, T x
n+1
) + G(x
n+1
, T x
n
, T x
n
)],

1
3
[G(x
n
, T x
n+1
, T x
n+1
) + G(x
n+1
, T x
n+1
, T x
n+1
) + G(x
n+1
, T x
n
, T x
n
)]}
= max{G(x
n
, x
n+1
, x
n+1
), G(x
n+1
, x

n+2
, x
n+2
),
1
2
G(x
n
, x
n+2
, x
n+2
),
1
3
[G(x
n
, x
n+2
, x
n+2
) + G(x
n+1
, x
n+2
, x
n+2
)]}.
(G5) G
G(x

n
, x
n+2
, x
n+2
) ≤ G(x
n
, x
n+1
, x
n+1
) + G(x
n+1
, x
n+2
, x
n+2
).
Θ(x
n
, x
n+1
, x
n+1
) = θ(x
n
, x
n+1
, x
n+1

)
= max{G(x
n
, x
n+1
, x
n+1
), G(x
n+1
, x
n+2
, x
n+2
)}.
(2.1)
ψ(G(x
n+1
, x
n+2
, x
n+2
) = ψ(G(T x
n
, T x
n+1
, T x
n+1
))
≤ ψ(Θ(x
n

, x
n+1
, x
n+1
)) − ϕ(θ(x
n
, x
n+1
, x
n+1
))
= ψ(max{G(x
n
, x
n+1
, x
n+1
), G(x
n+1
, x
n+2
, x
n+2
))
−ϕ(max{G(x
n
, x
n+1
, x
n+1

), G(x
n+1
, x
n+2
, x
n+2
)}). (2.6)
G(x
n+1
, x
n+2
, x
n+2
) ≤ G(x
n
, x
n+1
, x
n+1
) n ≥ 0. (2.7)
n
0
≥ 0
G(x
n
0
+1
, x
n
0

+2
, x
n
0
+2
) > G(x
n
0
, x
n
0
+1
, x
n
0
+1
)
ψ(G(x
n
0
+1
, x
n
0
+2
, x
n
0
+2
)

≤ ψ(max{G(x
n
0
, x
n
0
+1
, x
n
0
+1
), G(x
n
0
+1
, x
n
0
+2
, x
n
0
+2
))
−ϕ(max{G(x
n
0
, x
n
0

+1
, x
n
0
+1
), G(x
n
0
+1
, x
n
0
+2
, x
n
0
+2
)})
= ψ(G(x
n
0
+1
, x
n
0
+2
, x
n
0
+2

)) − ϕ(G(x
n
0
+1
, x
n
0
+2
, x
n
0
+2
)).
ϕ(G(x
n
0
+1
, x
n
0
+2
, x
n
0
+2
)) = 0 ϕ
G(x
n
0
+1

, x
n
0
+2
, x
n
0
+2
) = 0
(2.4) (2.7) (G(x
n
, x
n+1
, x
n+1
))
ρ ≥ 0
lim
n→∞
G(x
n
, x
n+1
, x
n+1
) = ρ. (2.8)
ρ > 0 n → ∞ (2.6)
(2.8) ψ, ϕ ψ(ρ) ≤ ψ(ρ) − ϕ(ρ) ≤
ψ(ρ) ϕ(ρ) = 0 ρ = 0
lim

n→∞
G(x
n
, x
n+1
, x
n+1
) = 0.
(x
n
) G X
(x
n
) G X ε > 0
(x
m(k)
) (x
n(k)
) (x
n
) n(k)
n(k) > m(k) > k, G(x
m(k)
, x
n(k)
, x
n(k)
) ≥ ε. (2.9)
G(x
m(k)

, x
n(k)−1
, x
n(k)−1
) ≤ ε. (2.10)
(2.9), (2.10) G
ε ≤ G(x
m(k)
, x
n(k)
, x
n(k)
)
≤ G(x
m(k)
, x
n(k)−1
, x
n(k)−1
) + G(x
n(k)−1
, x
n(k)
, x
n(k)
)
< ε + G(x
n(k)−1
, x
n(k)

, x
n(k)
).
k → ∞ (2.5)
lim
k→∞
G(x
m(k)
, x
n(k)
, x
n(k)
) = ε. (2.11)
k j(k) ∈ {1, , p}
n(k) − m(k) + j(k) ≡ 1 ( p).
k m(k) > j(k) x
m(k)−j(k)
x
n(k)
A
i
A
i+1
i
i ∈ {1, , p} G
|G(x, y, y) − G(x, z, z)| ≤ 2G(y, z, z) x, y, z ∈ X,
(2.5)
|G(x
m(k)−j(k)
, x

m(k)−j(k)
, x
n(k)
) − G(x
n(k)
, x
m(k)
, x
m(k)
)|
≤ 2G(x
m(k)−j(k)
, x
m(k)−j(k)
, x
m(k)
)
≤ 2
j(k)−1

l=0
G(x
m(k)−j(k)+l
, x
m(k)−j(k)+l
, x
m(k)−j(k)+l+1
)
≤ 2
p−1


l=0
G(x
m(k)−j(k)+l
, x
m(k)−j(k)+l
, x
m(k)−j(k)+l+1
) → 0,
k → ∞ (2.11)
lim
k→∞
G(x
m(k)−j(k)
, x
m(k)−j(k)
, x
n(k)
) = ε. (2.12)
(2.5)
lim
k→∞
G(x
m(k)−j(k)+1
, x
m(k)−j(k)+1
, x
m(k)−j(k)
) = 0
lim

k→∞
G(x
n(k)+1
, x
n(k)+1
, x
n(k)
) = 0. (2.13)
|G(x, y, y) − G(x, z, z)| ≤ 2G(y, z, z) x, y, z ∈ X
|G(x
m(k)−j(k)
, x
m(k)−j(k)
, x
n(k)+1
) − G(x
m(k)−j(k)
, x
m(k)
, x
n(k)
)|
≤ 2G(x
n(k)
, x
n(k)+1
, x
n(k)+1
).
k → ∞ (2.13), (2.12)

lim
k→∞
G(x
m(k)−j(k)
, x
m(k)−j(k)
, x
n(k)+1
) = ε.
|G(x
n(k)
, x
m(k)−j(k)+1
, x
n(k)+1
) − G(x
m(k)−j(k)
, x
m(k)−j(k)
, x
n(k)
)|
≤ 2G(x
m(k)−j(k)
, x
m(k)−j(k)+1
, x
m(k)−j(k)+1
).
k → ∞ (2.5), (2.12)

lim
k→∞
G(x
n(k)
, x
m(k)−j(k)+1
, x
m(k)−j(k)+1
) = ε.
lim
k→∞
G(x
m(k)−j(k)+1
, x
n(k)+1
, x
n(k)+1
) = ε.
Θ(x, y, z), θ(x, y, z)
lim
k→∞
Θ(x
n(k)+1
, x
m(k)−j(k)+1
, x
m(k)−j(k)+1
)
= lim
k→∞

θ(x
n(k)+1
, x
m(k)−j(k)+1
, x
m(k)−j(k)+1
) = ε.
(2.14)
x = x
n(k)+1
, y = x
m(k)−j(k)+1
, z = x
m(k)−j(k)+1
ψ(G(x
n(k)+1
, x
m(k)−j(k)+1
, x
m(k)−j(k)+1
))
= ψ(G(T x
n(k)
, T x
m(k)−j(k)
, T x
m(k)−j(k)
))
≤ ψ(Θ(x
n(k)+1

, x
m(k)−j(k)+1
, x
m(k)−j(k)+1
))
−ϕ(θ(x
n(k)+1
, x
m(k)−j(k)+1
, x
m(k)−j(k)+1
)).