G−
G−
G−
G−
G−
G−
G−
G−
G−
G−
G−
G−
G−
G−
2−
2− D−
G−
G−
G−
G−
G−
G−
G−
G−
G−
G−
2−
X R
+
d : X × X × X → R
+

x, y ∈ X, z ∈ X
d(x, y, z) = 0.
d(x, y, z) = 0 x, y, z ∈ X
d(x, y, z) = d(x, z, y) = d(y, z, x) x, y, z ∈ X.
d(x, y, z) ≤ d(x, y, a) + d(x, a, z) + d(a, y, z),
x, y, z, a ∈ X.
d 2− X (X, G)
2−
X = R
2
, x, y, z ∈ X d(x, y, z)
x, y, z. d 2− R
2
.
D−
D : X × X × X → R
+
D−
D(x, y, z) = 0 x = y = z,
D(x, y, z) ≤ D(x, z, z) + D(z, y, y), x, y, z ∈ X.
X = R
2
, x, y, z ∈ X d(x, y, z)
x, y, z. d D− R
2
.
(X, d)
(E
s
) D

s
(d)(x, y, z) =
1
3
[d(x, y) + d(y, z) + d(x, z) ]
(E
m
) D
m
(d)(x, y, z) = max{d(x, y), d(y, z), d(x, z)}
D−
X
G : X × X × X → R
+
G(x, y, z) = 0 x = y = z,
0 < G(x, x, y), x, y ∈ X, x = y,
G(x, x, y) ≤ G(x, y, z), x, y, z ∈ X, z = y,
G(x, y, z) = G(x, z, y) = G(y, z, x) = . . . ,
,
G(x, y, z) ≤ G(x, a, a) + G(a, y, z), x, y, z, a ∈ X
G G− X (X, G)
G− G− (X, G)
G(x, y, y) = G( x, x, y), x, y ∈ X.
2−
G− D− G−
G : R
3
−→ R
+
G(x, y, z) = |x − y| + |y −z| + |z − x|

x, y, z ∈ R. G G−
R (R, G) G−
G−
G−
(X, G) G−
x, y, z a ∈ X
G(x, y, z) = 0 x = y = z
G(x, y, z) ≤ G(x, x, y) + G(x, x, z)
G(x, y, y) ≤ 2G(y, x, x)
G(x, y, z) ≤ G(x, a, z) + G(a, y, z)
G(x, y, z) ≤
2
3
(G(x, y, a) + G(x, a, z) + G(a, y, z))
G(x, y, z) ≤ (G(x, a, a) + G(y, a, a) + G(z, a, a))
|G(x, y, z) − G(x, y, a)| ≤ max{G(a, z, z), G(z, a, a)}
|G(x, y, z) − G(x, y, a)| ≤ G(x, a, z)
|G(x, y, z) − G(y, z, z)| ≤ max{G(x, z, z), G(z, x, x)}
|G(x, y, y) −G(y, x, x)| ≤ max{G(y, x, x), G(x, y, y)}
G(x, y, z) = 0 x = y = z
x = y
G(x, y, z) ≥ G(x, x, y) > 0
x = y y = z x = y = z
G(x, y, z) ≤ G(x, x, y) + G(x, x, z)
G(x, y, z) = G(y, x, z)
(G5)
≤ G(y, x, x) + G(x, x, z) (
a = x)
= G(x, x, y) + G(x, x, z).
G(x, y, y) ≤ 2G(y, x, x) z = y

G(x, y, z) ≤ G(x, a, z) + G(a, y, z)
G(x, y, z) ≤ G(x, a, a) + G(a, y, z)
(G3)
≤ G(x, a, z) + G(a, y, z).
G(x, y, z) ≤
2
3
(G(x, y, a) + G(x, a, z) + G(a, y, z))
G(x, y, z)
(G5)
≤ G(x, a, a) + G(a, y, z) ≤ G( x, a, y) + G(a, y, z),
G(y, z, x) ≤ G(y, a, a) + G(a, z, x) ≤ G(y, a, z) + G(a, z, x),
G(z, x, y) ≤ G(z, a, a) + G(a, x, y) ≤ G(z, a, x) + G(a, x, y).
3G(x, y, z) ≤ 2(G(a, y, z) + G(x, y, a) + G(x, a, z)),
G(x, y, z) ≤ G(x, a, a) + G(y, a, a) + G(z, a, a)
G(x, y, z) ≤ G(x, a, a) + G(a, y, z)
G(x, y, z) ≤ G(x, a, a) + G(a, a, y) + G(a, a, z).
|G(x, y, z) − G(x, y, a)| ≤ max{G(a, z, z), G(z, a, a)}
G(x, y, z)
(G5)
≤ G(z, a, a) + G(x, y, a)
G(x, y, z) − G(x, y, a) ≤ G(z, a, a).
a z
G(x, y, a) −G(x, y, z) ≤ G(a, z, z).
|G(x, y, z) − G(x, y, a)| ≤ G(x, a, z)
G(x, y, z) − G(x, y, a) ≤ G(z, a, a) ≤ G(z, a, x),
G(x, y, a) − G(x, y, z) ≤ G(a, z, z) ≤ G(a, z, x).
|G(x, y, z) − G(y, z, z)| ≤ max{G(x, z, z), G(z, x, x)}
|G(x, y, z) − G(y, z, a)| ≤ max{G(a, x, x), G(x, a, a)}.
a = z

|G(x, y, y) −G(y, x, x)| ≤ max{G(y, x, x), G(x, y, y)}
z = y, a = x
G(x, y, y) − G(x, y, x) ≤ G(y, x, x).
x y
G(y, x, x) − G(y, x, y) ≤ G(x, y, y).

(X, G) G− k > 0
G
1
G
2
G− X
G
1
(x, y, z) = min{k, G(x, y, z)}
G
2
(x, y, z) =
G(x, y, z)
k + G(x, y, z)
·
X =
n

i=1
A
i
X
G
3

(x, y, z) =

G(x, y, z),
i x, y, z ∈ A
i
,
k + G(x, y, z),
G−
G
1
(x, y, z) = min{k, G(x, y, z)}
G
1
(x, y, z) = min{k, G(x, y, z)}
≤ min{k, G(x, a, a) + G(a, y, z)}
≤ min{k, G(x, a, a)}+ min{k, G(a, y, z)}
= G
1
(x, a, a) + G
1
(a, y, z).
G
2
(x, y, z) =
G(x, y, z)
k + G(x, y, z)
·
f(t) =
t
k + t

f
′
(t) =
1
(k + t)
2
> 0
f(t)
G(x, y, z)
G(x, x, y) ≤ G(x , y, z) f(t)
G
2
(x, x, y) =
G(x, x, y)
k + G(x, x, y)
≤
G(x, y, z)
k + G(x, y, z)
= G
2
(x, y, z).
G
2
(x, y, z) =
G(x, y, z)
k + G(x, y, z)
≤
G(x, a, a) + G(a, y, z)
k + G(x, a, a) + G(a, y, z)
≤

G(x, a, a)
k + G(x, a, a)
+
G(a, y, z)
k + G(a, y, z)
= G
2
(x, a, a) + G
2
(a, y, z).
G
3
(x, y, z) G−
(X, G) G−
(X, G)
G(x, y, y) ≤ G(x, y, a) x, y, a ∈ X
G(x, y, z) ≤ G(x, y, a) + G(z, y, b) x, y, z, a, b ∈ X
1) ⇒ 2) G(x, x, y) ≤ G(x, y, z), ∀x, y, z ∈ X, z = y
(X, G) G(x, x, y) = G(x, y, y)
G(x, y, y) ≤ G(x, y, z)
⇔ G(x, y, y) ≤ G(x, y, a), ∀x, y, a ∈ X.
2) ⇒ 3)
G(x, y, z) ≤ G(x, y, y) + G(z, y, y).
G(x, y, y) ≤ G(x, y, a)
G(z, y, y) ≤ G(z, y, b), ∀x, y, z, a, b ∈ X.
G(x, y, z) ≤ G(x, y, a) + G(z, y, b)
3) ⇒ 1) a = x, b = y
G(x, y, z) ≤ G(x, y, a) + G(z, y, b) z = y
⇔ G(x, y, y) ≤ G(x, y, x) + G(y, y, y)
⇔ G(x, y, y) ≤ G(x, x, y).

G(y, x, x) ≤ G(y, y, x) ⇔ G(x, x, y) ≤ G(x, y, y).
G(x, x, y) = G(x, y, y) 
G−
G−
X = ∅, X
G− X D
s
D
m
G−
G X
(E
d
) d
G
(x, y) = G(x, y, y) + G(x, x, y)
X. d
G
G,
G(x, y, z)  D
s
(d
G
)(x, y, z)  2G(x, y, z)
1
2
G(x, y, z)  D
m
(d
G

)(x, y, z)  2G(x, y, z).
d X
d
D
s
(d)
(x, y) =
4
3
d(x, y), d
D
m
(d)
(x, y) = 2d( x, y).
(X, G) G−
x
0
∈ X r > 0 G− x
0
r
B
G
(x
0
, r) = {y ∈ X : G(x
0
, y, y) < r}.
(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).
G(x
0
, x, y) < r x, y ∈ B
G
(x
0
, r)
G(x
0
, y, y) ≤ G(x

0
, x, y) < r,
G(x
0
, x, x) ≤ 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)
y ∈ B
G
(x
0
, r) G(x
0
, y, y) < r

r − G(x
0
, y, y) > 0.
δ = r −G(x
0
, y, y) > 0 x ∈ B
G
(y, δ) G(y, x, x) < δ
G(y, x, x) < r −G(x
0
, y, y) G(y, x, x) +G(x
0
, y, y) < r
G(x
0
, x, x) ≤ G(y, x, x) + G(x
0
, y, y) < r
x ∈ B
G
(x
0
, r)

B = {B
G
(x, r) : x ∈ X, r > 0 } ,
τ(G) X G−
(X, G) G−
x

0
∈ X r > 0
B
G

x
0
,
1
3
r

⊆ B
d
G
(x
0
, r) ⊆ B
G
(x
0
, r).
B
G

x
0
,
1
3

r

=

x ∈ X : G(x
0
, x, x) <
1
3
r

,
B
d
G
(x
0
, r) = {x ∈ X : d
G
(x
0
, x) < r}
= {x ∈ X : G(x
0
, x, x) + G(x
0
, x
0
, x) < r},
B

G
(x
0
, r) = {x ∈ X : G(x
0
, x, x) < r}.
x ∈ B
G

x
0
,
1
3
r

G(x
0
, x, x) <
1
3
r
G(x
0
, x, x) + G(x
0
, x
0
, x) ≤ G(x
0

, x, x) + G(x
0
, x, x) + G(x, x
0
, x)
= 3G(x
0
, x, x) < r.
d
G
(x
0
, x) < r x ∈ B
d
G
(x
0
, r) B
G

x
0
,
1
3
r

⊂ B
d
G

(x
0
, r)
x ∈ B
d
G
(x
0
, r)
G(x
0
, x, x) + G(x
0
, x
0
, x) < r ⇒ G(x
0
, x, x) < r ⇒ x ∈ B
G
(x
0
, r).
B
d
G
(x
0
, r) ⊂ B
G
(x

0
, r) 
G− τ(G)
d
G
G−
G−
(X, G) G− (x
n
) ⊆ X
G− x x G− τ(G)
(x
n
) G− x ∈ X G(x, x
n
, x
m
) → 0 m, n → ∞
ε > 0 N ∈ N G(x, x
n
, x
m
) < ε, ∀n, m >
N x (x
n
) x
n
→ x
(X, G) G−
(x

n
) ⊆ X x ∈ X
(x
n
) G− x
d
G
(x
n
, x) → 0 n → ∞ (x
n
) x d
G
G(x
n
, x
n
, x) → 0 n → ∞
G(x
n
, x, x) → 0 n → ∞
G(x
m
, x
n
, x) → 0 m, n → ∞
⇒ (x
n
) G− x lim
m,n→∞

G(x, x
n
, x
m
) = 0
G(x, x
n
, x
n
)
n→∞
−−−→ 0
d
G
(x
n
, x) = G(x
n
, x, x) + G(x
n
, x
n
, x)
= G(x, x
n
, x) + G(x, x
n
, x
n
)

≤ G(x, x
n
, x
n
) + G(x
n
, x
n
, x) + G(x, x
n
, x
n
)
= 3G(x, x
n
, x
n
)
n→∞
−−−→ 0.
x
n
→ x d
G
⇒ G(x
n
, x
n
, x) ≤ G(x
n

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

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

n,m→∞
−−−−→ 0
⇒ 
(X, G), (X
′
, G
′
) G−
f : X → X
′
G− x
0
∈ X
f
−1
(B
G
′
(f(x
0
), r)) ∈ τ(G),
r > 0 f G− X G−
X
X, τ(G) X
′
, τ(G
′
)
G−
(X, G), (X

′
, G
′
) G−
f : X → X
′
G− x ∈ X
G− x (x
n
) G− x (f(x
n
))
G−
f(x)
f : X → X
′
G
G
′
x
n
⊂ X x
n
G− x
0
G(x
0
, x
n
, x

n
) → 0
G
′
(f(x
0
), f(x
n
), f(x
n
)) → 0 ∀ε
B
G
′
(f(x
0
), ε) = {y ∈ X
′
: G
′
(f(x
0
), y, y) < ε}
f(x
0
) G
′
f
B
G

(x
0
, δ) = {x ∈ X : G(x
0
, x, x) < δ}
∀x ∈ B
G
(x
0
, δ) f(x) ∈ B
G
′
(f(x
0
), ε).
x
n
G− x
0
∃n
0
: ∀n ≥ n
0
: G(x
0
, x
n
, x
n
) < δ

x
n
∈ B
G
(x
0
, δ) f(x
n
) ∈ B
G
′
(f(x
0
), ε)
G
′
(f(x
0
), f(x
n
), f(x
n
)) < ε.
f(x
n
) G
′
− f(x
0
)

f x
0
B
G
′
(f(x
0
), ε
0
)
B
G
(x
0
, r)
f(B
G
(x
0
, r)) ⊂ B
G
′
(f(x
0
), ε
0
).
n ∈ N
∗
x

n
∈ B
G

x
0
,
1
n

f(x
n
) /∈ B
G
′
(f(x
0
), ε
0
).
{x
n
} ⊂ X x
n
G− x
0
f(x
n
)
G

′
− f(x
0
) f G
G
′

(X, G) G−
G(x, y, z)
(x
k
), (y
m
) (z
n
) G− x, y z
G(x, y, z) ≤ G(y, y
m
, y
m
) + G(y
m
, x, z),
G(z, x, y
m
) ≤ G(x, x
k
, x
k
) + G(x

k
, y
m
, z)
G(z, x
k
, y
m
) ≤ G(z, z
n
, z
n
) + G(z
n
, y
m
, x
k
),
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
k
, x, x) + G(y
m
, y, y) + G(z
n
, z, z).
|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 → ∞

G−
(X, G) G−
(x
n
) ⊆ X G− ε > 0 N ∈ N
G(x
n
, x
m
, x
l
) < ε n, m, l ≥ N

G− (X, G)
(x
n
) G−
ε > 0 N ∈ N G(x
n
, x
m
, x
m
) < ε
n, m ≥ N
(x
n
) (X, d
G
)
⇒ G(x
n
, x
m
, x
m
) ≤ G(x
n
, x
m
, x
l
)

lim
m,n,l→∞
G(x
n
, x
m
, x
l
) = 0
lim
m,n→∞
G(x
n
, x
m
, x
m
) = 0
⇒
d
G
(x
m
, x
n
) = G(x
m
, x
n
, x

n
) + G(x
m
, x
m
, x
n
) ≤ 3G(x
n
, x
m
, x
m
),
lim
m,n→∞
G(x
n
, x
m
, x
m
) = 0 lim
m,n→∞
d
G
(x
m
, x
n

) = 0
⇒
G(x
n
, x
m
, x
l
) ≤ G(x
n
, x
m
, x
m
) + G(x
m
, x
m
, x
l
).
(x
n
) (X, d
G
)
lim
m,n→∞
G(x
n

, x
m
, x
m
) = 0 lim
m,l→∞
G(x
m
, x
m
, x
l
) = 0.
lim
m,n,l→∞
G(x
n
, x
m
, x
l
) = 0 (x
n
) G− 
G− G−
G−
G− G−
(X, G) G− G−
G− (X, G) G−
G− (X, G) G− (X, G)

G− (X, G) G−
(X, d
G
)
Y G−
(X, G) (Y, G|
Y
) Y G−
(X, G) G|
Y
G Y
(X, G) G− (F
n
)
(F
1
⊇ F
2
⊇ F
3
⊇ . . .) G− X
sup{G(x, y, z) : x, y, z ∈ F
n
} → 0
n → ∞ (X, G) G−
∞

n=1
F
n

G−
(X, G) G−
ε > 0 A ⊆ X ε− (X, G)
x ∈ X a ∈ A x ∈ B
G
(a, ε) A
A ε− (X , G)
A ε− X =

a∈A
B
G
(a, ε)
G− (X, G) G−
ε > 0 ε−
G− (X, G)
G− G− G−
G− (X, G)
(X, G) G−
(X, τ(G)) G−
(X, d
G
)
(X, G) G− (x
n
) ⊆ X
sup{G(x
n
, x
m

, x
l
) : n, m, l ∈ N} < ∞,
(x
n
) G−
G−
G−
(X, G) G−
T : X → X
G(T (x), T (y), T (z)) ≤ {aG(x, y, z) + bG(x, x, T (x)) + cG(y, y, T(y))
+ dG(z, T (z), T (z))},
G(T (x), T (y), T (z)) ≤ {aG(x, y, z) + bG(x, x, T (x)) + cG(y, y, T(y))
+ dG(z, z, T (z))},
x, y, z ∈ X 0 ≤ a + b + c + d < 1 T
u T(u) = u T G− u
T x, y ∈ X
G(T (x), T (y), T (y)) ≤ aG(x, y, y) + bG(x, T (x), T (x) )
+ (c + d)G(y, T (y), T (y)),
G(T (y), T (x), T (x)) ≤ aG(y, x, x) + bG(y, T (y), T (y))
+ (c + d)G(x, T (x), T (x )).
(X, G) d
G
(x, y) = 2G(x, y, y) x, y ∈ X
(X, d
G
)
d
G
(T (x), T (y)) ≤ ad

G
(x, y) +
c + d + b
2
d
G
(x, T (x)) +
c + d + b
2
d
G
(y, T (y)),
x, y ∈ X 0 < a + b + c + d < 1
(X, d
G
)
(X, G)
3
2
G(x, y, y) ≤ d
G
(x, y) ≤ 3G(x, y, y),
x, y ∈ X,
(X, d
G
)
d
G
(T (x), T (y)) ≤ ad
G

(x, y) +
2(c + d + b)
3
d
G
(x, T (x))
+
2(c + d + b)
3
d
G
(y, T (y)),
x, y ∈ X
0 < a +
2(c + d + b)
3
+
2(c + d + b)
3
G−
x
0
∈ X (x
n
) x
n
= T
n
(x
0

)
G(x
n
, x
n+1
, x
n+1
) ≤ aG(x
n−1
, x
n
, x
n
) + bG(x
n−1
, x
n
, x
n
)
+ (c + d)G(x
n
, x
n+1
, x
n+1
),
G(x
n
, x

n+1
, x
n+1
) ≤
a + b
1 −(c + d)
G(x
n−1
, x
n
, x
n
)·
q =
a + b
1 −(c + d)
0 ≤ q < 1 0 ≤ a + b + c + d < 1
G(x
n
, x
n+1
, x
n+1
) ≤ qG(x
n−1
, x
n
, x
n
).

G(x
n
, x
n+1
, x
n+1
) ≤ q
n
G(x
0
, x
1
, x
1
).
n, m ∈ N, n < m
G(x
n
, x
m
, x
m
) ≤G(x
n
, x
n+1
, x
n+1
) + G(x
n+1

, x
n+2
, x
n+2
)
+ . . . + G(x
m−1
, x
m
, x
m
)
≤(q
n
+ q
n+1
+ . . . + q
m−1
)G(x
0
, x
1
, x
1
)
≤
q
n
1 −q
G(x

0
, x
1
, x
1
),
lim G(x
n
, x
m
, x
m
) = 0 n, m → ∞ (x
n
) G−
(X, G) u ∈ X (x
n
) G−
u
T (u) = u
G(x
n
, T (u), T (u)) ≤ aG(x
n−1
, u, u)+bG(x
n−1
, x
n
, x
n

)+(c+d)G(u, T (u), T (u)),
n → ∞ G
G(u, T (u), T (u) ) ≤ (c + d)G(u, T (u), T (u)) u = T (u)
u = v T (v) = v
G(u, v, v) ≤ aG(u, v, v) + bG(u, T (u), T (u)) + (c + d)G(v, T (v), T (v))
= aG(u, v, v).
u = v
T G− u (y
n
) ⊆ X
lim(y
n
) = u n → ∞
G(u, T (y
n
), T (y
n
)) ≤ aG(u, y
n
, y
n
) + bG(u, T (u), T (u))
+ (c + d)G(y
n
, T (y
n
), T (y
n
))
= aG(u, y

n
, y
n
) + (c + d)G(y
n
, T (y
n
), T (y
n
)).
G(y
n
, T (y
n
), T (y
n
)) ≤ G(y
n
, u, u) + G(u, T( y
n
), T (y
n
))
G(u, T (y
n
), T (y
n
)) ≤
a
1 −(c + d)

G(u, y
n
, y
n
) +
c + d
1 −(c + d)
G(y
n
, u, u).
n → ∞ G(u, T (y
n
), T (y
n
)) → 0
T (y
n
) → u = T (u) T
G− u
T
(x
n
) G−
G(x
n
, x
n
, x
n+1
) ≤ aG(x

n−1
, x
n−1
, x
n
) + (b + c)G(x
n−1
, x
n−1
, x
n
)
+ dG(x
n
, x
n
, x
n+1
),
G(x
n
, x
n
, x
n+1
) ≤
a + b + c
1 −d
G(x
n−1

, x
n−1
, x
n
)·
q =
a + b + c
1 −d
0 ≤ q < 1 0 ≤ a + b + c + d < 1
G(x
n
, x
n
, x
n+1
) ≤ q
n
G(x
0
, x
0
, x
1
).
n, m ∈ N, n < m
G(x
n
, x
n
, x

m
) ≤
q
n
1 −q
G(x
0
, x
0
, x
1
).

(X, G) G−
T : X → X
G(T
m
(x), T
m
(y), T
m
(z)) ≤{aG(x, y, y) + bG(x, T
m
(x), T
m
(x))
+ cG(y, T
m
(y), T
m

(y)) + dG(z, T
m
(z), T
m
(z))},
G(T
m
(x), T
m
(y), T
m
(z)) ≤{aG(x, y, y) + bG(x, x, T
m
(x)) + cG(y, y, T
m
(y))
+ dG(z, z, T
m
(z))},
x, y, z ∈ X 0 ≤ a + b + c + d < 1 T
u T
m
G− u
T
m
u T
m
(u) = u T (u) = T (T
m
(u)) =

T
m+1
(u) = T
m
(T (u))
T (u) T
m
T (u) = u 
(X, G) G−
T : X → X
G(T (x), T (y), T (z)) ≤ k max



G(x, T (x), T (x)),
G(y, T (y), T (y)),
G(z, T(z), T (z))



,
G(T (x), T (y), T (z)) ≤ k max



G(x, x, T (x)),
G(y, y, T(y)),
G(z, z, T (z))




,
x, y, z ∈ X 0 ≤ k < 1 T
u T G− u
T x, y ∈
X
G(T (x), T (y), T (y)) ≤ k max{G(x , T (x), T (x)), G(y, T(y), T(y))},
G(T (y), T (x), T (x)) ≤ k max{G(y, T (y), T (y)), G(x, T (x), T (x))}.
(X, G) (X, d
G
)
d
G
(x, y) = 2G(x , y, y) x, y ∈ X
d
G
(T (x), T (y)) ≤ k m ax{d
G
(x, T (x)), d
G
(y, T (y))}, x, y ∈ X.
k < 1
(X, d
G
)
(X, G)
3
2
G(x, y, y) ≤ d
G

(x, y) ≤ 3G(x, y, y),
x, y ∈ X,
(X, d
G
)
d
G
(T (x), T (y)) ≤
4k
3
max{d
G
(x, T (x)), d
G
(y, T (y))},
x, y ∈ X.
4k
3
G−
x
0
∈ X (x
n
) x
n
= T
n
(x
0
)

G(x
n
, x
n+1
, x
n+1
) ≤ k max{G(x
n−1
, x
n
, x
n
), G(x
n
, x
n+1
, x
n+1
)}
= kG(x
n−1
, x
n
, x
n
), (
0 ≤ k < 1).
G(x
n
, x

n+1
, x
n+1
) ≤ k
n
G(x
0
, x
1
, x
1
).
n, m ∈ N, n < m
G(x
n
, x
m
, x
m
) ≤G(x
n
, x
n+1
, x
n+1
) + G(x
n+1
, x
n+2
, x

n+2
)
+ . . . + G(x
m−1
, x
m
, x
m
)
≤(k
n
+ k
n+1
+ . . . + k
m−1
)G(x
0
, x
1
, x
1
)
≤
k
n
1 −k
G(x
0
, x
1

, x
1
),
lim G(x
n
, x
m
, x
m
) = 0 n, m → ∞ (x
n
) G−
(X, G) u ∈ X (x
n
) → u n → ∞
T (u) = u
G(x
n+1
, T (u), T (u)) ≤ k max{G(x
n+1
, x
n+2
, x
n+2
), G(u, T (u), T (u))},
n → ∞ G
G(u, T (u), T (u) ) ≤ kG(u, T (u), T (u)) u = T (u)
u = v T (v) = v
G(u, v, v) ≤ k max{G(v, v, v), G(u, u, u)} = 0 u = v
T G− u (y

n
) ⊆ X
lim(y
n
) = u n → ∞
G(u, T (y
n
), T (y
n
)) ≤ k max{G(u , T (u), T (u)), G(y
n
, T (y
n
), T (y
n
))}
= kG(y
n
, T (y
n
), T (y
n
)).
G(y
n
, T (y
n
), T (y
n
)) ≤ G(y

n
, u, u) + G(u, T (y
n
), T (y
n
))
G(u, T (y
n
), T (y
n
)) ≤
k
1 −k
G(y
n
, u, u).
n → ∞ G(u, T (y
n
), T (y
n
)) → 0
T (y
n
) → u = T (u) T G− u 