Sự tồn tại điểm bất động chung của các ánh xạ t CO trong không gian metric nón

BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC VINH




ĐẶNG THÀNH TRUNG



SỰ TỒN TẠI ĐIỂM BẤT ĐỘNG CHUNG
CỦA CÁC ÁNH XẠ T – CO
TRONG KHÔNG GIAN MÊTRIC NÓN



LUẬN VĂN THẠC SĨ TOÁN HỌC










NGHỆ AN – 2014
BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC VINH

ĐẶNG THÀNH TRUNG


SỰ TỒN TẠI ĐIỂM BẤT ĐỘNG CHUNG
CỦA CÁC ÁNH XẠ T-CO
TRONG KHÔNG GIAN MÊTRIC NÓN

CHUYÊN NGÀNH: TOÁN GIẢI TÍCH
MÃ SỐ: 60.46.01

LUẬN VĂN THẠC SĨ TOÁN HỌC



Cán bộ hướng dẫn khoa học
PGS. TS. ĐINH HUY HOÀNG


NGHỆ AN - 2014
MU
.
C LU
.
C
Trang
MU
.
C LU

.
C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
X τ X
X
T
1
∅ X ∈ τ
T
2
G
i
∈ τ, i ∈ I

i∈I
G
i
∈ τ
T
3
G
1
, G
2
∈ τ G
1

G
2

∈ τ
X τ
X τ X
X
τ
A ⊂ X A X \ A
A ⊂ X
intA
X A X
x ∈ X V ⊂ X x ∈ V ⊂ A
X x ∈ X U(x) x
B(x) ⊂ U(x) x U ∈ U(x)
V ∈ B(x) V ⊂ U
(x
n
) X
x ∈ X U x n
0
∈ N
x
n
∈ U n  n
0
x
n
→ x lim
n→∞
x
n
= x

X
x ∈ X B(x)
X T
2
x, y ∈ X, x = y U
x
, V
y
x y U
x

V
y
= ∅
X X
X, Y f : X → Y
f x ∈ X V f(x)
U x f(U) ⊂ V f
x ∈ X
X d : X X → R d
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
(X, d) X
X (x
n
) X
ε > 0 n

0
∈ N m
n  n
0
d(x
m
, x
n
) < ε
X X
A ⊂ X
A A
E K = R
K = C p : E → R E
p(x) ≥ 0 x ∈ E p(x) = 0 x = 0
p(kx) = |k|p(x) x ∈ E k ∈ K
p(x + y) ≤ p(x) + p(y) x, y ∈ E
p(x) x ∈ E x
 x  E
E
d(x, y) = x − y , ∀x, y ∈ E
E
E
x → x , ∀x ∈ E
(x, y) → x + y, ∀(x, y) ∈ E E
(λ, x) → λx, ∀(λ, x) ∈ K E
E
a ∈ E λ ∈ K λ = 0
x → x + a, x → λx, ∀x ∈ E
E E

X ≤
X ≤ X
x ≤ x, ∀x ∈ X
x ≤ y y ≤ x x = y, ∀x, y ∈ X
x ≤ y; y ≤ z x ≤ z, ∀x, y, z ∈ X
X
(X, ≤) X
≤ X A ⊂ X
x ∈ X A
a ≤ x x ≤ a a ∈ A
x ∈ X
A x A y
A x ≤ y y ≤ x
x = sup A x = inf A
E
R P E E
P P = ∅, P =
a, b ∈ R, a, b ≥ 0 x, y ∈ P ax + by ∈ P
x ∈ P −x ∈ P x = 0
R
P = x ∈ R : x ≥ 0
E = R
2
, P = {(x, y) ∈ E : x, y ≥ 0} ⊂ R
2
P
P P = ∅, P =
(x, y), (u, v) ∈ P a, b ∈ R; a, b ≥ 0 a(x, y)+b(u, v) ∈
P
(x, y) ∈ P (−x, −y) ∈ P (x, y) = (0, 0)

E
C
[a,b]
[a, b]
C
[a,b]
f = sup
x∈[a,b]
|f(x)| , ∀f ∈ C
[a,b]
C
[a,b]
≤
f, g ∈ C
[a,b]
f ≤ g ⇔ f(x) ≤ g(x), ∀x ∈ [a, b]
P = {f ∈ C
[a,b]
: 0 ≤ f(x)} ∀x ∈ [a, b]
P
P P = ∅, P =
∀a, b ∈ R; a, b ≥ 0 f, g ∈ P
0 ≤ af(x) + bg(x)∀x ∈ [a, b]
af(x) + bg(x) ∈ P
f ∈ P −f ∈ P f = 0
P E
P E
E ” ≤ ” P x ≤ y
y − x ∈ P x < y x ≤ y x = y x  y
y − x ∈ intP intP P

P E
P K > 0
x, y ∈ E 0 ≤ x ≤ y x ≤ Ky K
P
P E a, b, c ∈ E
{x
n
}, {y
n
} E α
a  b b  c a  c
a ≤ b b  c a  c
a  b, c  d a + c  b + d
αintP ⊂ intP
δ > 0 x ∈ intP 0 < γ < 1 γx < δ
c
1
∈ intP c
2
∈ P d ∈ intP c
1
 d
c
2
 d
c
1
, c
2
∈ intP e ∈ intP e  c

1
e  c
2
a ∈ P a ≤ x x ∈ intP a = 0
a ≤ λa a ∈ P, 0 < λ < 1 a = 0
0 ≤ x
n
≤ y
n
n ∈ N lim
n→∞
x
n
= x lim
n→∞
y
n
= y
0 ≤ x ≤ y
intP + intP ⊂ intP a  b
b  c b − a ∈ intP c − b ∈ intP c − a = c − b + b − a ∈
intP + intP ⊂ intP a  c
intP +P =

x∈P
(x + intP) P x+intP ⊂ P
intP + P ⊂ intP a ≤ b b  c b − a ∈ P c − b ∈ intP
c − a = c − b + b − a ∈ intP + P ⊂ intP c − a ∈ intP a  c
a  b, c  d b − a ∈ intP d − c ∈ intP
b − a + d − c ∈ intP (b + d) − (a + c) ∈ intP a + c  b + d

αintP ⊂ intP
δ > 0 x ∈ intP n > 1
δ
nx
< 1
γ =
δ
nx
0 < γ < 1
γx ≤ γx ≤
δ
nx
x ≤
δ
n
< δ
δ > 0 c
1
+ B(0, δ) ⊂ intP B(0, δ) = {x ∈ E :
x < δ} B(0, δ) m > 1 c
2
∈ mB(0, δ)
−c
2
∈ mB(0, δ) mc
1
− c
2
∈ intP d = mc
1

− c
2
c
1
 d
c
2
 d
δ

> 0 c
1
+ B(0, δ

) ⊂ intP, c
2
+ B(0, δ

) ⊂ intP
B(0, δ

) = {x ∈ E : x < δ

} B(0, δ

) m > 0
c
1
∈ mB(0, δ


) c
2
∈ mB(0, δ

) −c
1
∈ mB(0, δ

) −c
2
∈ mB(0, δ

)
mc
1
− c
1
∈ intP mc
2
− c
2
∈ intP e = mc
1
− c
1
+ mc
2
− c
2
e  c

1
e  c
2
x ∈ intP a ≤
x
n
n = 1, 2,
x
n
− a ∈ P n = 1, 2, 
x
n
 =
x
n
→ 0
x
n
→ 0
x
n
− a → −a {
x
n
− a} ⊂ P P E −a ∈ P
a −a ∈ P P a = 0
a ≤ λa λa − a ∈ P (λ − 1)a ∈ P 0 < λ < 1
1 − λ > 0 −a =
1
1 − λ

a ∈ P −a ∈ P a −a ∈ P
P a = 0
x
n
≤ y
n
y
n
− x
n
∈ P P lim
n→∞
(y
n
− x
n
) ∈ P
lim
n→∞
x
n
= x lim
n→∞
y
n
= y lim
n→∞
(y
n
− x

n
) = y − x y − x ∈ P
x ≤ y 0 ≤ x
n
0 ≤ x 0 ≤ x ≤ y
P E {x
n
}
P x
n
→ 0 c ∈ intP n
0
∈ N
x
n
 c n ≥ n
0
x
n
P x
n
→ 0 c ∈ intP intP
δ > 0 c + B
E
(0, δ) ⊂ intP B
E
(0, δ)
δ E x ∈ E x < δ
c − x ∈ intP δ > 0 n
0

∈ N
x
n
 < δ, ∀n > n
0
c − x
n
∈ intP ∀n > n
0
x
n
 c, ∀n > n
0
P E
intP = 0 ≤ E P
X d : X × X → E
d 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
(X, d) X
E = R P = {x ∈ R : x ≥ 0}
E = R
2
P = {(x, y) ∈ R
2
: x, y ≥ 0}
X = R d : X × X → E
d(x, y) = (α|x − y|, β|x − y|), ∀x, y ∈ X

α, β
d (X, d)
E = C
[a,b]
P
d : X × X → E
d(f, g) = |f − g| ∀f, g ∈ E
|f − g|(x) = |f(x) − g(x)| x ∈ [a, b] d
d(f, g) ≥ 0 ∀f, g ∈ E d(f, g) = 0 f(x) = g(x)
x ∈ [a, b] f = g
d(f, g) = d(g, f) = |f − g| ∀f, g ∈ E
|f − g| = |f − h + h − g| ≤ |f − h| + |h − g| f, g, h ∈ E
d(f, g) ≤ d(f, h) + d(h, g) ∀f, g, h ∈ E
d E
(X, d) a ∈ X c ∈ P
B(a, c) = {x ∈ X : d(x, a)  c}
B(a, c) a c
F = {G ⊂ X : ∀x ∈ G, ∃c ∈ P B(x, c) ⊂ G}
(X, d) F
F X
B(x, c) ∈ F x ∈ X, c ∈ intP
(X, F) T
2
(X, F)
F = {U ⊂ X : ∀x ∈ U, ∃c ∈ intP : B(x, c) ⊂ U}
∅ ∈ F, X ∈ F
U
i
∈ F i ∈ I


i∈I
U
i
∈ F
x ∈

i∈I
U
i
i = i
0
∈ I x ∈ U
i
0
⊂ F c ∈ intP
B(x, c) ⊂ U
i
0
B(x, c) ⊂

i∈I
U
i

i∈I
U
i
∈ F
U, V ∈ F U ∩ V ∈ F x ∈ U ∩ V
x ∈ U x ∈ V c

1
, c
2
∈ P B(x, c
1
) ⊂ U
B(x, c
2
) ⊂ V c
1
∈ P c
2
∈ P
c ∈ intP c  c
1
c  c
2
B(x, c) ⊂ U ∩ V
U ∩ V ∈ F
F X
y ∈ B(x, c) d(y, x)  c c−d(y, x) ∈ intP c

=
c − d(x, y) B(y, c

) ⊂ B(x, c) z ∈ B(y, c

)
d(z, y)  c


d(z, y)  c − d(y, x) d(z, y) + d(y, x)  c
d(z, x)  c z ∈ B(x, c) B(y, c

) ⊂ B(x, c) B(x, c) ∈ F
x, y ∈ X x = y
(X, F) T
2
c
1
, c
2
∈ intP
B(x, c
1
)

B(y, c
2
) = ∅ c
1
, c
2
∈
intP B(x, c
1
)

B(y, c
2
) = ∅ c ∈ intP

B(x,
c
2n
)

B(y,
c
2n
) = ∅
{z
n
} ⊂ X
z
n
∈ B(x,
c
2n
)

B(y,
c
2n
), n = 1, 2,
d(x, y) ≤ d(x, z
n
) + d(z
n
, y) 
c
n

c
n
≤ c n = 1, 2, d(x, y)  c c
intP d(x, y) = 0 x = y
x = y (X, F) T
2
x ∈ X x
c ∈ intP
U = {B(x,
c
n
) : n = 1, 2, }
U ⊂ F V x y ∈ intP
B(x, y) ⊂ V y ∈ intP  > 0 B
E
(y, ) ⊂ P
B
E
(y, ) y  n ∈ N n >
c

y − (y −
c
n
) =
c
n
< 
y −
c

n
∈ B
E
(y, ) ⊂ P B
E
(y, ) E y −
c
n
∈ intP
c
n
 y
B(x,
c
n
) ⊂ B(x, y) ⊂ V
U x F U
(X, F)
F B(x, c)
(X, d)
(X, d) {x
n
} ⊂
X x ∈ X {x
n
} x c ∈ P
N d(x
n
, x)  c n ≥ n
0

lim
n→∞
x
n
= x
x
n
→ x n → ∞
(X, d) {x
n
} ⊂
X x y x = y
(X, d) {x
n
} ⊂ X
x
n
→ x ∈ X c ∈ intP n
c
d(x
n
, x)  c n ≥ n
c
{x
n
} ⊂ X c ∈ intP B(x, c)
x n
c
x
n

∈ B(x, c) n ≥ n
c
d(x
n
, x)  c n ≥ n
c
c ∈ intP n
c
d(x
n
, x)  c
n ≥ n
c
U x c
0
∈ intP
B(x, c
0
) ⊂ U n
c
0
∈ N
x
n
∈ B(x, c
0
) ⊂ U, ∀n ≥ n
c
0
x

n
→ x ∈ X
(X, d) {x
n
} ⊂
X c ∈ intP N
d(x
m
, x
n
)  c m, n > N
(X, d)
{x
n
} (X, d)
{x
n
} X x
n
→ x ∈ X
0  c ∈ E N d(x
n
, x) 
c
2
n > N
m, n > N
d(x
m
, x

n
) ≤ d(x
m
, x) + d(x, x
n
) 
c
2
+
c
2
= c
x
n
{x
n
}
X {x
n
} {x
n
k
} x ∈ X {x
n
} x
c ∈ intP {x
n
} {x
n
k

} x
N
d(x
n
, x
m
) 
c
2
, ∀m, n ≥ N
d(x
n
k
, x) 
c
2
, ∀n
k
≥ N
d(x
n
, x) ≤ d(x
n
, x
n
k
) + d(x
n
k
, x)  c, ∀n, n

k
≥ N
x
n
→ x
(X, d)
X
(X, d) (Y, d)
f : X → Y f a ∈ X {x
n
}
X x
n
→ a f(x
n
) → f(a)
f a x
n
X x
n
→ a
V f(a) Y f a
U a X f(U) ⊂ V x
n
→ a
n
0
x
n
∈ U n ≥ n

0
f(x
n
) ∈ f(U) ⊂ V, ∀n ≥ n
0
f(x
n
) → f(a)
{x
n
} X x
n
→ a f(x
n
) → f(a)
f a
f a
y
0
∈ intP c ∈ intP
f(B(a, c)) ⊂ B(f(a), y
0
)
n = 1, 2, x
n
∈ B(a,
c
n
) f(x
n

) ∈
B(f(a), y
0
) x
n
∈ B(a,
c
n
) n = 1, 2,
c
n
→ 0 n → ∞
x
n
→ a
f(x
n
) → f(a) f(x
n
) ∈ B(f(a), y
0
)
n = 1, 2, f a
(M, d) S, T :
M → M
T (y
n
) T (y
n
)

(y
n
)
T (y
n
) T (y
n
)
(y
n
)
T (x
n
) lim
n→∞
x
n
= x
lim
n→∞
T x
n
= T x
x ∈ M S Sx = x
x ∈ M S T Sx =
T x = x
(M, d) T, S :
M → M
S T T K
1

b ∈ [0,
1
2
)
d(T Sx, T Sy) ≤ b[d(T x, T Sx) + d(T y, TSy)]
x, y ∈ M
S T T K
2
c ∈ [0,
1
2
)
d(T Sx, T Sy) ≤ c[d(T x, T Sy) + d(T y, TSx)]
x, y ∈ M
E = (C
[0,1]
, R) P = {ϕ ∈ E : ϕ ≥ 0} ⊂ E M = R
d : M × M → E d(x, y) = |x − y|e
t
e
t
∈ E
(M, d) T, S : M → M
T x = x
2
Sx =
x
2
d(T Sx, T Sy) = |T Sx − T Sy|e
t

= |
x
2
4
−
y
2
4
|e
t
≤
1
3
[|T x − T Sx| + |T y − T Sy|]e
t
=
1
3
[d(T x, T Sx) + d(T y, TSy)]
S T K
1
T : M → M
S : M → M T K
1
x
0
∈ M
lim
n→∞
d


T S
n
x
0
, T S
n+1
x
0

= 0;
v ∈ M
lim
n→∞
T S
n
x
0
= v;
(S
n
x
0
)
u ∈ M
Su = u;
x
0
∈ M (S
n

x
0
)
x
0
M (x
n
)
x
n.+1
= Sx
n
= S
n+1
x
0
S T K
1
d(T x
n
, T x
n+1
) = d(T Sx
n−1
, T Sx
n
)
≤ b[d(T x
n−1
, T Sx

n−1
) + d(T x
n
, T Sx
n
)] ∀n = 1, 2,
d(T x
n
, T x
n+1
) ≤
b
1 − b
d(T x
n−1
, T x
n
)
d

T S
n
x
0
, T S
n+1
x
0

≤


b
1 − b

n
d (T x
0
, T Sx
0
) ∀n = 1, 2,


d

T S
n
x
0
, T S
n+1
x
0



≤

b
1 − b


n
K d (T x
0
, T Sx
0
) ∀n = 1, 2,
lim
n→∞


d

T S
n
x
0
, T S
n+1
x
0



= 0.
lim
n→∞
d

T S
n

x
0
, T S
n+1
x
0

= 0.
m, n ∈ N m > n
d(T x
n
, T x
m
) ≤ d(T x
n
, T x
n+1
) + + d(T x
m−1
, T x
m
)
≤


b
1 − b

n
+ +


b
1 − b

m−1

d (T x
0
, T Sx
0
)
≤

b
1 − b

n
1
1 −

b
1 − b

d (T x
0
, T Sx
0
)
d (T S
n

x
0
, T S
m
x
0
) ≤

b
1 − b

n
1
1 −

b
1 − b

d (T x
0
, T Sx
0
) .
d (T S
n
x
0
, T S
m
x

0
) ≤

b
1 − b

n
K
1 −

b
1 − b

d (T x
0
, T Sx
0
) .
lim
m,n→∞
d (T S
n
x
0
, T S
m
x
0
) = 0
lim

m,n→∞
d (T S
n
x
0
, T S
m
x
0
) = 0
(T S
n
x
0
) M M
v ∈ M
lim
n→∞
T S
n
x
0
= v.
T (S
n
x
0
)
u ∈ M (x
n

i
)
lim
i→∞
S
n
i
x
0
= u.
T
lim
i→∞
T S
n
i
x
0
= T u.
T u = v.
d(T Su, T u) ≤ d (T Su, T S
n
i
x
0
) + d

T S
n
i

x
0
, T S
n
i
+1
x
0

+ d

T S
n
i
+1
x
0
, T u

≤ b

d (T u, T Su) + d

T S
n
i
−1
x
0
, T S

n
i
x
0

+

b
1 − b

n
i
d (T x
0
, T Sx
0
) + d

T S
n
i
+1
x
0
, T u

d(T Su, T u) ≤
b
1 − b
d


T S
n
i
−1
x
0
, T S
n
i
x
0

+
1
1 − b

b
1 − b

n
i
d (T x
0
, T Sx
0
)
+
1
1 − b

d

T S
n
i
+1
x
0
, T u

d(T Su, T u) ≤
bK
1 − b


d

T S
n
i
−1
x
0
, T S
n
i
x
0




+
K
1 − b

b
1−b

n
i
d (T x
0
, T Sx
0
)
+
K
1 − b


d

T S
n
i
+1
x
0
, T u




→ 0 (i → ∞)
T S
n
i
x
0
→ T u (
b
1 − b
)
n
i
→ 0
d(T Su, T u) = 0 T Su = Tu T Su = u
u S S TK
1
v
S
d(T Su, T Sv) ≤ b[d(T u, T Su) + d(T v, TSv)] = 0
T Su = T Sv T Su = Sv u = v
S
T (n
i
) (n)
lim
n→∞
S
n

x
0
= u
(S
n
x
0
) S
T : M → M
S : M → M T K
2
x
0
∈ M
lim
n→∞
d

T S
n
x
0
, T S
n+1
x
0

= 0;
v ∈ M
lim

n→∞
T S
n
x
0
= v;
(S
n
x
0
)
u ∈ M
Su = u;
x
0
∈ M (S
n
x
0
)
x
0
M (x
n
)
x
n.+1
= Sx
n
= S

n
x
0
S T K
2
d(T Sx
n
, T Sx
n+1
) ≤ c[d(T x
n
, T Sx
n+1
) + d(T x
n+1
, T Sx
n
)]
≤ c[d(T Sx
n−1
, T Sx
n
) + d(T Sx
n
, T Sx
n+1
)]
d(T Sx
n
, T Sx

n+1
) ≤
c
1 − c
d(T Sx
n−1
, T Sx
n
) = hd(T Sx
n−1
, T Sx
n
) ∀n =
1, 2,
h :=
c
1 − c
d(T Sx
n
, T Sx
n+1
) ≤ h
n
d(T Sx
0
, T Sx
1
) ∀n = 1, 2,
d(T Sx
n

, T Sx
n+1
) ≤ h
n
Kd(T Sx
0
, T Sx
1
)
h ∈ [0, 1)
lim
n→∞
d(T Sx
n
, T Sx
n+1
) = 0
lim
n→∞


d(T S
n
x
0
, T S
n+1
x
0
)



= 0
m, n ∈ N m > n
d(T Sx
n
, T Sx
m
) ≤ d(T Sx
n
, T Sx
n+1
) + + d(T Sx
m−1
, T Sx
m
)
≤ [h
n
+ h
n+1
+ + h
m−1
]d(T Sx
0
, T Sx
1
)
≤
h

n
1 − h
d(T Sx
0
, T Sx
1
)
d(T Sx
n
, T Sx
m
) ≤
h
n
1 − h
Kd(T Sx
0
, T Sx
1
)
lim
m,n→∞
d(T Sx
n
, T Sx
m
) = 0.
(T S
n
x

0
) M M
v ∈ M
lim
n→∞
T S
n
x
0
= v.
(X, d)
f : X → X λ x, y ∈ X
q(x, y), r(x, y), s(x, y) t(x, y) X × X → [0, ∞)
sup
x,y∈X
{q(x, y) + r(x, y) + s(x, y) + 2t(x, y)} ≤ λ < 1
d(fx, fy) ≤ q(x, y)d(fx, fy) + r(x, y)d(x, fx) + s(x, y)d(y, fy)
+2t(x, y)[d(x, fy) + d(y, fx)]
x, y ∈ X.
(X, d)
T : X → X f g : X → X
d(T fx, Tgy) ≤ q(x, y)d(T x, T y) + r(x, y)d(T x, T fx) + s(x, y)d(Ty, T gy)
+ t(x, y)[d(T x, T gy) + d(T y, Tfx)]
x, y ∈ X q, r, s t
sup
x,y∈X
{q(x, y) + r(x, y) + s(x, y) + 2t(x, y)} ≤ λ < 1.
z
x
∈ X lim

n→∞
T fx
2n
= lim
n→∞
T gx
2n+1
= z
x
.
{fx
2n
} {gx
2n+1
}
w
x
∈ X fw
x
= gw
x
= w
x
{fx
2n
} {gx
2n+1
} w
x
x

0
X {x
n
} x
1
=
fx
0
, x
2
= gx
1
, , x
2n+1
= fx
2n
, x
2n+2
= gx
2n+1

{T x
n
}
d(T x
2n+1
, T x
2n+2
) = d(T fx
2n

, T gx
2n+1
)
≤ q(x
2n
, x
2n+1
)d(T x
2n
, T x
2n+1
) + r(x
2n
, x
2n+1
)d(T x
2n
, T fx
2n
)
+s(x
2n
, x
2n+1
)d(T x
2n+1
, T gx
2n+1
)
+t(x

2n
, x
2n+1
)[d(T x
2n
, T gx
2n+1
) + d(T x
2n+1
, T fx
2n
)]
= q(x
2n
, x
2n+1
)d(T x
2n
, T x
2n+1
) + r(x
2n
, x
2n+1
)d(T x
2n
, T x
2n+1
)
+s(x

2n
, x
2n+1
)d(T x
2n+1
, T x
2n+2
)
+t(x
2n
, x
2n+1
)[d(T x
2n
, T x
2n+2
) + d(T x
2n+1
, T x
2n+1
)]
≤ (q + r + t)(x
2n
, x
2n+1
)d(T x
2n
, T x
2n+1
)

+(s + t)(x
2n
, x
2n+1
)d(T x
2n+1
, T x
2n+2
).
d(T x
2n+1
, T x
2n+2
) ≤
q(x
2n
, x
2n+1
) + r(x
2n
, x
2n+1
) + t(x
2n
, x
2n+1
)
1 − s(x
2n
, x

2n+1
) − t(x
2n
, x
2n+1
)
d(T x
2n
, T x
2n+1
).
q(x, y) + r(x, y) + t(x, y)
1 − s(x, y) − t(x, y)
≤ λ
x, y ∈ X
d(T x
2n+1
, T x
2n+2
) ≤ λd(T x
2n
, T x
2n+1
).
d(T x
2n+2
, T x
2n+3
) ≤ λd(T x
2n+1

, T x
2n+2
)
q(x, y) + s(x, y) + t(x, y)
1 − r(x, y) − t(x, y)
≤ λ