Điểm bất động chung của các ánh xạ hầu co suy rộng trong không gian mêtric thứ tự

(ψ, ϕ)
(ψ, ϕ)
(ψ, ϕ)
(ψ, ϕ)
(ψ, ϕ)
(ψ, ϕ)
(ψ, ϕ)
(ψ, ϕ)
f
(ψ, ϕ)
(ψ, ϕ) f
(ψ, ϕ)
(ψ, ϕ)
(ψ, ϕ)
g
(f, g) f
(ψ, ϕ) g
f
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 = R d : R × R → R d (x, y) = |x − y|
x, y ∈ R d R
X = R
n

x = (x
1
, . . . , x
n
), y = (y
1
, . . . , y
n
) ∈ R
n
d
1
(x, y) =

n

i=1
|x
i
− y
i
|
2

1

2
d
2
(x, y) =

n

i=1
|x
i
− y
i
| d
1
, d
2
R
n
(X, d) x, y, u, v ∈
X
|d (x, y) − d (u, v)| ≤ d (x, u) + d (y, v) .
(X, d) A ⊂ X x ∈ X
d(x, A) = inf
y∈A
d (x, y) d(x, A) x
A
(X, d) A ⊂ X
x, y ∈
|d (x, A) − d (y, A)| ≤ d (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) E ⊂ X x ∈ X
E {x
n
} ⊂ E x
n
→ x
x ∈ E
x ∈ E {x
n
} ⊂ E x
n
→ x
(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
R d (x, y) = |x−y|
R
n
n d
1
(x, y) d
2
(x, y)
(X, d) M ⊂ X

M M
M X 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
(X, d) g : X → X
f : X → X δ ∈ [0; 1)
L ≥ 0
d (g (x) , g (y)) ≤ δd (f (x) , f (y)) + Ld (f (y) , g (x)) x, y ∈ X.
f =
X X
X g
f, g : X → X
(X, d)
y ∈ X f g X

x ∈ X y = g (x) = f (x) x ∈ X
f g
(f, g) f g
fg (x) = gf (x)
x ∈ X f (x) = g (x)
(X, ≤)
≤ x, y ∈ X x ≤ y
y ≤ x
x, y ∈ X x ≤ y x = y x < y y > x
X = φ (X, d)
(X, ≤) ≤ (X, d, ≤)
(X, ≤)
f : X → X
f x, y ∈ X x ≤ y
f (x) ≤ f (y)
f x, y ∈ X x ≤ y
f (x) ≥ f (y)
f x, y ∈ X
x < y f (x) < f (y)
f x, y ∈ X
x < y f (x) > f (y)
f f
(X, ≤)
f, g : X → X f (x) ≤ gf (x)
g (x) ≤ fg (x) x ∈ X
f, g : X → X f (x) < gf (x)
g (x) < fg (x) x ∈ X
X = (0, ∞) X
f, g : X → X
f (x) =


3x + 2 0 < x < 1
2x + 1 1 ≤ x < ∞,
g (x) =

4x + 1 0 < x < 1
3x 1 ≤ x < ∞.
0 < x < 1 f (x) = 3x + 2 ≤ 4 (3x + 2) + 1 = gf (x)
g (x) = 4x + 1 ≤ 12x + 5 = 3(4x + 1) + 2 = fg (x) 1 < x < ∞
f (x) = 2x + 1 ≤ 3 (2x + 1) = gf (x) g (x) = 3x ≤ 2 (3x) + 1 = fg (x)
f g
X = [0, ∞) × [0; ∞) X
(x, y) ≤ (z, w) x ≤ z y ≤ w
f, g : X → X
f (x, y) =

(x, y) max {x, y} ≤ 1
(0, 0) max {x, y} > 1,
g (x, y) =

(
√
x,
√
y) max {x, y} ≤ 1
(0, 0) max {x, y} > 1.
max (x, y) ≤ 1 f (x, y) = (x, y) ≤

√
x,

√
y

= gf (x, y)
g (x, y) =

√
x,
√
y

≤

√
x,
√
y

= f g (x, y) max {x, y} > 1
f (x, y) = g (x, y) = (0; 0) ≤ f g (x, y) = gf (x, y) f g
f g
(
1
2
, 1) ≤ (1, 2) f(
1
2
, 1) = (
1
2

, 1) ≤ (0, 0) = f(1, 2)
(X, ≤)
x, y ∈ X d
X (X, d) f : X → X
δ ∈ [0, 1)
d (f (x) , f (y)) ≤ δd (x, y) x, y ∈ X x, y
x
0
∈ X x
0
≤ f (x
0
) f p
X
(X, d) f : X → X
δ ∈ (0; 1) L ≥ 0
d (f (x) , f (y)) ≤ δd (x, y) +L min {d (x, f (x)) , d (x, f (y)) , d (y, f (x))},
x, y ∈ X
f, g : X → X
(X, d) f g
δ ∈ (0, 1) L ≥ 0
d (f (x) , g (y)) ≤δ max

d (x, y) , d (x, f (x)) , d (y, g (y)) ,
d (x, g (y)) + d (y, f (x))
2

+
+ L min {d (x, f (x)) , d (y, g (y)) , d (x, g (y)) , d (y, f (x))}.
(1)

x, y ∈ X
(X, ≤)
d X (X, d)
f : X → X
≤ δ ∈ (0, 1) L ≥ 0
d (f (x) , f (y)) ≤δM (x, y) +
+ L min {d (x, f (x)) , d (y, f (y)) , d (x, f (y)) , d (y, f (x))},
(2)
x, y ∈ X
M (x, y) = max

d (x, y) , d (x, f (x)) , d (y, f (y)) ,
d (x, f (x)) + d (y, f (x))
2

.
x
0
∈ X x
0
≤ f (x
0
) f X
f (x
0
) = x
0
f (x
0
) = x

0
n ≥ 0 x
n+1
= f
n+1
(x
0
) = f (x
n
)
{x
n
} X f x
1
= f (x
0
) <
f
2
(x
0
) = x
2
< f
3
(x
1
) = x
3
x

1
<
x
2
< < x
n
< x
n+1
< {x
n
}
M(x
n
, x
n+1
) = max {d(x
n
, x
n+1
), d(x
n
, f(x
n
)), d(x
n+1
, f(x
n+1
)),
d(x
n

, f(x
n+1
)) + d(x
n+1
, f(x
n
))
2

,
= max

d (x
n
, x
n+1
) , d (x
n
, x
n+1
) , d (x
n+1
, x
n+2
)
d (x
n
, x
n+2
) + d (x

n+1
, x
n+1
)
2

,
= max

d (x
n
, x
n+1
) , d (x
n
, x
n+2
) ,
d (x
n
, x
n+2
)
2

,
≤ max

d (x
n

, x
n+1
) , d (x
n+1
, x
n+2
) ,
d (x
n
, x
n+1
) + d (x
n+1
, x
n+2
)
2

= max {d (x
n
, x
n+1
) , d (x
n+1
, x
n+2
)}.
x
n
x

n+1
x x
2n
y x
2n+1
d (x
n+1
, x
n+2
) = d (f (x
n
) , f (x
n+1
))
≤ δM (x
n
, x
n+1
) +
+ L min {d (x
n
, f (x
n
)) , d (x
n+1
, f (x
n+1
)) , d (x
n
, f (x

n+1
)) , d (x
n+1
, f (x
n
))}
≤ δ max {d (x
n
, x
n+1
) , d (x
n+1
, x
n+2
)} +
+ L min {d (x
n
, x
n+1
) , d (x
n+1
, x
n+2
) , d (x
n
, x
n+2
) , d (x
n+1
, x

n+1
)}.
d (x
n+1
, x
n+2
) ≤ δ max {d (x
n
, x
n+1
) , d (x
n+1
, x
n+2
)}.
max {d (x
n
, x
n+1
) , d (x
n+1
, x
n+2
)} = d (x
n
, x
n+1
)
d (x
n+1

, x
n+2
) ≤ δd (x
n
, x
n+1
) .
n
max {d (x
n
, x
n+1
) , d (x
n+1
, x
n+2
)} = d (x
n+1
, x
n+2
)
d (x
n+1
, x
n+2
) ≤ δd (x
n+1
, x
n+2
) .

δ ∈ (0, 1)
d (x
n+1
, x
n+2
) ≤ δd (x
n
, x
n+1
) .
d (x
n+1
, x
n+2
) ≤ δd (x
n
, x
n+1
) ≤ ≤ δ
n+1
d (x
0
, x
1
) ,
n ≥ 1 m n m ≥ n
d(x
m
, x
n

) ≤ d(x
n
, x
n+1
) + · · +d(x
m−1
, x
m
) ≤
δ
n
1 − δ
d(x
0
, x
1
).
{x
n
} X
p ∈ X x
n
→ p n → ∞
f f(x
n
) = f (f
n
(x
0
)) = f

n+1
(x
0
) = x
n+1
f (p) = p p f 
(X, ≤)
d X (X, d)
f : X → X ≤
x
0
∈ X x
0
≤ f (x
0
)
{x
n
} ⊂ X x X x
n
≤ x n ∈ N f
X
f (x) = x x ∈ X
{x
n
}
x
n
→ p n → ∞ x
n

≤ p n ∈ N
d (x
n+1
, f (p)) = d (f (x
n
) , f (p))
≤ δ max

d (x
n
, p) , d (x
n
, f (x
n
)) , d (p, f (p)) ,
d (x
n
, f (p)) + d (p, f (x
n
))
2

+
+ L min {d (x
n
, f (x
n
)) , d (p, f (p)) , d (x
n
, f (p)) , d (p, f (x

n
))}
≤ δ max

d (x
n
, p) , d (x
n
, x
n+1
) , d (p, f (p))
d (x
n
, p) + d (p, x
n+1
)
2

+
+ L min {d (x
n
, x
n+1
) , d (p, f (p)) , d (x
n+1
, f (p)) , d (p, x
n+1
)}.
n → ∞ d (p, f (p)) ≤ δd (p, f (p)) δ ∈ (0, 1)
x

0
∈ X f(x
0
) = x
0

(X, ≤)
d X (X, d)
f, g : X → X ≤
x, y ∈ X
f g f g
X
x
0
∈ X
{x
n
} ⊂ X x
2n+1
= f(x
2n
) x
2n+2
= g(x
2n+1
) n ≥ 0
f g x
1
= f(x
0

) < g(f(x
0
)) =
g(x
1
) = x
2
= g(x
1
) < f (g(x
1
)) = f (x
2
) = x
3
x
1
< x
2
< ··· < x
n
< x
n+1
< ··· {x
n
}
M (x
2n
, x
2n+1

) = max{d (x
2n
, x
2n+1
) , d (x
2n
, f (x
2n
)) , d (x
2n+1
, g (x
2n+1
)) ,
d (x
2n
, g (x
2n+1
)) + d (x
2n+1
, f (x
2n
))
2

,
= max{d (x
2n
, x
2n+1
) , d (x

2n
, x
2n+1
) , d (x
2n+1
, x
2n+2
) ,
d (x
2n
, x
2n+2
) + d (x
2n+1
, x
2n+1
)
2

,
= max

d (x
2n
, x
2n+1
) , d (x
2n+1
, x
2n+2

) ,
d (x
2n
, x
2n+2
)
2

,
≤ max

d (x
2n
, x
2n+1
) , d (x
2n+1
, x
2n+2
) ,
d (x
2n
, x
2n+1
) + d (x
2n+1
, x
2n+2
)
2


= max {d (x
2n
, x
2n+1
) , d (x
2n+1
, x
2n+2
)}.
x
n
x
n+1
x x
2n
y x
2n+1
d (x
2n+1
, x
2n+2
) = d (f (x
2n
) , g (x
2n+1
))
≤ δM (x
2n
, x

2n+1
) +
+ L min {d(x
2n
, f(x
2n
)), d(x
2n+1
, g(x
2n+1
)), d(x
2n
, g(x
2n+1
)),
d(x
2n+1
, f(x
2n
)), }
≤ δ max {d (x
2n
, x
2n+1
) , d (x
2n+1
, x
2n+2
)}+
+ L min {d (x

2n
, x
2n+1
) , d (x
2n+1
, x
2n+2
) , d (x
2n
, x
2n+12
, 0)}.
d (x
2n+1
, x
2n+2
) ≤ δ max {d (x
2n
, x
2n+1
) , d (x
2n+1
, x
2n+2
)}.
max {d (x
2n
, x
2n+1
) , d (x

2n+1
, x
2n+2
)} = d (x
2n+1
, x
2n+2
)
d (x
2n+1
, x
2n+2
) ≤ δd (x
2n
, x
2n+1
) .
max {d (x
2n
, x
2n+1
) , d (x
2n+1
, x
2n+2
)} = d (x
2n+1
, x
2n+2
)

n ∈ N
d (x
2n+1
, x
2n+2
) ≤ δd (x
2n+1
, x
2n+2
) .
δ ∈ (0, 1)
d (x
2n+1
, x
2n+2
) ≤ δd (x
2n
, x
2n+1
) .
d (x
2n+3
, x
2n+2
) ≤ δd (x
2n+2
, x
2n+1
) .
d (x

n
, x
n+1
) ≤ d (x
n−1
, x
n
) ≤ ≤ δ
n
d (x
0
, x
1
)
n ≥ 1 m, n m ≥ n
d (x
m
, x
n
) ≤ d (x
n
, x
n+1
) + d (x
n+1
, x
n+2
) + + d (x
m−1
, x

m
) ≤
δ
n
1 − δ
d (f (x
0
) , f (x
1
)) .
{x
n
} X
p ∈ X x
n
→ p n → ∞ f
f(x
n
) = f(f
n
(x
0
)) = f
n+1
(x
0
) f
x
n
→ p f(p) = p p f

g(p) = p p ≤ p
d (f (p) , g (p)) ≤ δ max

d(p, p), d (p, f (p)) , d (p, g (p)) ,
d (p, g (p)) + d (p, f (p))
2

+L min {d (p, f (p)) , d (p, g (p)) , d (p, g (p)) , d (p, f (p))}.
d (p, g (p)) ≤ δd (p, g (p)) .
δ ∈ (0, 1) g(p) = p p
f g 
(X, ≤)
d X (X, d)
f, g : X → X ≤
δ ∈ (0, 1) L ≥ 0
d (f (x) , g (y)) ≤ δ max

d (x, y) ,
d (x, f (x)) + d (y, g (y))
2
,
d (x, g (y)) + d (y, f (x))
2

+L min {d (x, f (x)) , d (y, g (y)) , d (x, g (y)) , d (y, f (x))}.
(3)
x, y ∈ X
f g f g
(X, ≤)
d X (X, d)

f, g : X → X ≤
δ ∈ (0, 1) L ≥ 0
d (f (x) , g (y)) ≤ δd (x, y) + L min {d (x, f (x)) d (y, g (y)) , d (x, g (y)) , d (y, f (x))}.
(4)
x, y ∈ X
f g f g p ∈ X

(X, ≤)
d X (X, d)
f, g : X → X
x, y ∈ X
F (f, g) = φ {x
n
} X x
n
→ x X
x
n
≤ x n ∈ N f g p ∈ F (f, g)
p ∈ F (f, g) {x
n
} X
p y x
n
x p
d (f (p) , g (x
n
)) ≤ δ max

d (p, x

n
) , d (p, f (p)) , d (x
n
, g (x
n
)) ,
d (p, g (x
n
)) + d (x
n
, f (p))
2

+L min {d (p, f (p)) , d (x
n
, g (x
n
)) d (p, g (x
n
)) , d (x
n
, f (p))}.
f(p) = p
d (p, g (x
n
)) ≤ δ max

d (p, x
n
) , d (x

n
, g (x
n
)) ,
d (p, g (x
n
)) + d (x
n
, p)
2

n = 1, 2 n → ∞ g(x
n
) → p = g(p)
g p ∈ X f
p ∈ F (f, g) 
(ψ, ϕ)
(ψ, ϕ)
(X, d) f : X → X
α ∈ [0; 1) L ≥ 0
d (f (x) , f (y)) ≤ αM (x, y) + Ld (y, f (x))
x, y ∈ X
M (x, y) = max

d (x, y) , d (x, f (x)) , d (y, f (y)) ,
d (x, f (y)) + d (y, f (x))
2

.
(X, ≤)

d X (X, d)
f : X → X ≤
δ ∈ (0, 1) L ≥ 0
d (f (x) , f (y)) ≤ δM (x, y) + L min {d (x, f (x)) , d (x, f (y)) , d (y, f (x))},
x, y ∈ X
M (x, y) = max

d (x, y) , d (x, f (x)) , d (y, f (y)) ,
d (x, f (y)) + d (y, f (x))
2

.
x
0
∈ X x
0
≤ f (x
0
) f
X
(X, ≤)
d X (X, d)
f, g : X → X
≤ x, y ∈ X
f g f g
X
ϕ : [0, +∞) → [0, +∞)
ϕ
ϕ (t) = 0 t = 0
(X, ≤)

f : X → X X ψ, ϕ
M (x, y) = max

d (x, y) , d (x, f (x)) , d (y, f (y)) ,
d (x, f (y)) + d (y, f (x))
2

N (x, y) = min {d (x, f (x)) , d (y, f (y))}.
(X, d) ψ, ϕ
f : X → X (ψ, ϕ)
L ≥ 0
ψ (d (f (x) , f (y))) ≤ ψ (M (x, y)) − ϕ (M (x, y)) + Lψ (N (x, y)) ,
x, y ∈ X
(X, ≤)
d X (X, d)
f : X → X ≤
f (ψ, ϕ) x
0
∈ X x
0
≤ f (x
0
)
f
x
0
∈ X {x
n
} X
x

n+1
= f (x
n
) x
0
≤ f (x
0
) = x
1
f
x
1
= f (x
0
) ≤ x
2
= f (x
1
) x
1
≤ x
2
f
x
2
= f (x
1
) ≤ x
3
= f (x

2
)
{x
n
}
x
0
≤ x
1
≤ ≤ x
n
≤ x
n+1
≤ ···
x
n
= x
n+1
n ∈ N x
n
= f (x
n
) x
n
f x
n
= x
n+1
n ∈ N
ψ (d (x

n
, x
n+1
)) = ψ (d (f (x
n−1
) , f (x
n
)))
≤ ψ (M (x
n−1
, x
n
)) −ϕ (M (x
n−1
, x
n
)) + Lψ (N (x
n−1
, x
n
)) ,
M(x
n−1
, x
n
) = max {d(x
n−1
, x
n
), d(x

n−1
, f(x
n−1
)), d(x
n
, f(x
n
)),
d(x
n−1
, f(x
n
)) + d(x
n
, f(x
n−1
))
2

= max

d(x
n−1
, x
n
), d(x
n
, x
n+1
),

d(x
n−1
, x
n+1
)
2

≤ max

d(x
n−1
, x
n
), d(x
n
, x
n+1
),
d(x
n−1
, x
n
) + d(x
n
, x
n+1
)
2

= max {d(x

n−1
, x
n
), d(x
n
, x
n+1
)}
N (x
n−1
, x
n
) = min {d (x
n−1
, f (x
n−1
)) , d (x
n
, f (x
n−1
))}
= min {d (x
n−1
, x
n
) , 0}
= 0.
ϕ ψ
ψ (d (x
n

, x
n+1
)) = ψ (max {d (x
n−1
, x
n
) , d (x
n
, x
n+1
)}) − ϕ (max {d (x
n−1
, x
n
) , d (x
n
, x
n+1
)})
≤ ψ (max {d (x
n−1
, x
n
) , d (x
n
, x
n+1
)}) .
max {d (x
n−1

, x
n
) , d (x
n
, x
n+1
)} = d (x
n
, x
n+1
) ,
ψ (d (x
n
, x
n+1
)) ≤ ψ (d (x
n
, x
n+1
)) − ψ (d (x
n
, x
n+1
)) < ψ (d (x
n
, x
n+1
)) .
max {d (x
n−1

, x
n
) , d (x
n
, x
n+1
)} = d (x
n−1
, x
n
) .
ψ (d (x
n
, x
n+1
)) ≤ ψ (d (x
n
, x
n−1
)) −ϕ (d (x
n−1
, x
n
)) < ψ(d (x
n
, x
n−1
)).
ψ {d (x
n

, x
n+1
) : n ∈ N

{0}}
r > 0 lim
n→∞
d (x
n
, x
n+1
) = r.
n → ∞ ψ (r) ≤ ψ (r) −ϕ (r) ≤ ψ (r)
ϕ (r) = 0 r = 0
lim
n→∞
d (x
n
, x
n+1
) = 0.
{x
n
} X
{x
n
} ε > 0

x
m(i)

 
x
n(i)

(x
n
) n (i)
n (i) > m (i) > i, d

x
m(i)
, x
n(i)

≥ ε.
d

x
m(i)
, x
n(i)−1

< ε.
ε ≤ d

x
m(i)
, x
n(i)


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

x
n(i)−1
, x
n(i)

≤ 2d

x
m(i)
, x
m(i)−1

+ d

x
m(i)
, x
n(i)−1

+ d

x
n(i)−1
, x
n(i)

< 2d

x
m(i)

, x
m(i)−1

+ ε + d

x
n(i)−1
, x
n(i)

.
i → ∞
ε ≤ lim
i→∞
d

x
m(i)
, x
n(i)

≤ 2 lim
i→∞
d

x
m(i)
, x
m(i)−1


+ ε + lim
i→∞
d

x
n(i)−1
, x
n(i)

.
lim
i→∞
d

x
m(i)
, x
n(i)

= ε.
ψ(d(x
m(i)
, x
n(i)
)) = ψ(d(f (x
m(i)−1
), f(x
n(i)
))
≤ ψ


M

x
m(i)−1
, x
n(i)−1

− ϕ

N

x
m(i)−1
, x
n(i)−1

+ Lψ

N

x
m(i)−1
, x
n(i)−1

,
M

x

m(i)−1
, x
n(i)−1

= max{d

x
m(i)−1
, x
n(i)−1

, d

x
m(i)−1
, f

x
m(i)−1

, d

x
n(i)−1
, f

x
n(i)−1

,

d

x
m(i)−1
, f

x
n(i)−1

+ d

f

x
m(i)−1

, x
n(i)−1

2
}
= max{d

x
m(i)−1
, x
n(i)−1

, d


x
m(i)−1
, x
m(i)

, d

x
n(i)−1
, x
n(i)

,
d

x
m(i)−1
, x
n(i)

+ d

x
m(i)
, x
n(i)−1

2
}
N


x
m(i)−1
, x
n(i)−1

= min

d

x
m(i)−1
, f

x
m(i)−1

, d

f

x
m(i)−1

, x
n(i)−1

= min

d


x
m(i)−1
, x
m(i)

, d

x
m(i)
, x
n(i)−1

.
i → +∞
lim
n→+∞
M

x
m(i)−1
, x
n(i)−1

= ε
lim
n→+∞
N

x

m(i)−1
, x
n(i)−1

= 0.
i → +∞
ψ (ε) ≤ ψ (ε) − ϕ (ε) < ψ (ε) .
{x
n
} X {x
n
} x
n+1
=
f(x
n
), n ≥ 1 X X
u ∈ X
lim
n→+∞
x
n+1
= lim
n→+∞
f (x
n
) = u.
f x
n
→ u x

n+1
= f (x
n
) → f (u)
f (u) = u u
f 
f (x
n
)
X x
n
→ x ∈ X x
n
≤ x
n ∈ N f X
(x
n
) X (x
n
) → u u ∈ X
x
n
≤ x n ∈ N
f (u) = u
ψ (d (x
n+1
) , f (u)) = ψ (d (f (x
n
)) , f (u))
≤ ψ (M (x

n
, u) − ϕM (x
n
, u) + Lψ (x
n
, u)) ,
M (x
n
, u) = max

d (x
n
, u) , d (x
n
, f (x
n
)) , d (u, f (u)) ,
d (x
n
, f (u)) + d (f (x
n
) , u)
2

= max

d (x
n
, u) , d (x
n

, x
n+1
) , d (u, f (u)) ,
d (x
n
, f (u)) + d (x
n+1
, u)
2

N (x
n
, u) = min {d (x
n
, f (x
n
)) , d (u, f (x
n
))}
= min {d (x
n
, x
n+1
) , d (u, x
n+1
)}.
n → ∞ M (x
n
, u) → d (u, f (u))
N (x

n
, u) → 0 n → +∞