tìm điểm bất động chung cho một họ các ánh xạ giả co chặt bản tóm tắt tiếng anh

C
i
C
i
i = 1, 2, . . .
T
i
A
i
G
i
(u, v)
cardG = 2 C
1
C
2
H
x ∈ H {x
k
}
∞
k=1
{y
k
}
∞
k=1
y
0
= x, x
k

= P
C
1
(y
k−1
), y
k
= P
C
2
(x
k
), k = 1, 2, ,
P
C
(x) k → ∞
C = C
1
∩ C
2
P
C
H C C
1
C
2
H
P
C
(x) k → ∞

cardG ≥ 2 C
i
T
i
{T
i
}
i≥2
{T
i
}
∞
i=1
λ
i
C H H
F =

∞
i=1
F ix(T
i
) = ∅ F ix(T
i
)
T
i
u
∗
∈ F

u
∗
∈ C
F (u
∗
), v − u
∗
 ≥ 0 v ∈ C,
F : C → H
C H F
u
α
∈ C
u
α
∈ C

∞

i=1
γ
i
A
i
(u
α
) + αu
α
, v − u
α


≥ 0 ∀v ∈ C,
A
i
= I − T
i
α > 0
{γ
i
}
γ
i
> 0;
∞

i=1
γ
i

λ
i
= γ < ∞,

λ
i
=
1 − λ
i
2
.

ϕ : H −→ R
ϕ

{
n
}
∞
n=0
{α
n
}
∞
n=0
•
(i) k = 0 z
0
∈ C 
0
α
0
(ii) k = n z ∈ C
min
z∈C

ϕ(z) +


n

∞


i=1
γ
i
A
i
(z
n
) + α
n
z
n

− ϕ

(z
n
), z

.
z
n+1
(iii) z
n+1
− z
n
 n ← n + 1
(ii)
{
n

}
∞
n=0
{α
n
}
∞
n=0
0 < 
n
≤ 1; 0 < α
n+1
≤ α
n
≤ 1; α
n
→ 0 n → ∞;
∞

n=0

n
α
n
= ∞;
∞

n=0

2

n
< ∞;
∞

n=0
(α
n
− α
n+1
)
2
α
3
n

n
< ∞,
{u
α
} {z
n
}
u
∗
B =

∞
i=1
γ
i

A
i
W
T
n
T
n−1
T
1
γ
n
γ
n−1
γ
1
{T
i
}
N
i=1
u
0
∈ H, u
k+1
= T
[k+1]
u
k
− λ
k+1

µF (T
[k+1]
u
k
),
T
[n]
= T
n mod N
µ ∈

0,
2η
L
2

η L
F
{u
k
}
∞
k=0
u
∗
C =

N
i=1
F ix(T

i
)
x
0
∈ H {x
n
}
∞
n=0
x
0
∈ H, y
0
0
= x
0
,
y
i
k
= (1 − β
i
k
)y
i−1
k
+ β
i
k
T

i
(y
i−1
k
), i = 1, 2, · · · , N,
x
k+1
= (1 − β
0
k
)x
k
+ β
0
k
(I − λ
k
µF )y
N
k
, k ≥ 0.
λ
k
β
i
k
i = 0, . . . , N
λ
k
∈ (0, 1) β

i
k
∈ (α, β) α, β ∈ (0, 1) k ≥ 0
lim
k→∞
λ
k
= 0;
∞

k=0
λ
k
= ∞; lim
k→∞


β
i
k+1
− β
i
k


= 0.
u
∗
W
n

H C H
F : C −→ H
x
∗
∈ C
F (x
∗
), x − x
∗
 ≥ 0 ∀x ∈ C.
V I(F, C)
C H F :
C −→ H
C
H C
C H
F : C −→ H
U C u ∈ C \U v ∈ U
F (u), u − v > 0.
•
x
∗
∈ C x
∗
∈ C
x
∗
= P
C
(x

∗
− λF (x
∗
))
λ > 0
x
0
∈ C
x
n+1
= P
C
(x
n
− λF (x
n
)), n = 0, 1, 2, · · ·
{x
n
}
•
F
x
0
= x ∈ C,
y
n
= P
C
(x

n
− λF (x
n
)),
x
n+1
= P
C
(x
n
− λF (y
n
)), n = 0, 1, 2, · · ·
λ ∈ (0, 1/L) L F
•
J H
u
∗
∈ C
J(u
∗
) = min
u∈C
J(u).
ϕ : H → R
v ∈ C  > 0
G : u → ϕ(u) + J

(v) − ϕ


(v), u .
G

(v) = J

(v).
v ∈ C v
min
u∈C
{ϕ(u) + J

(v) − ϕ

(v), u}.
{
n
}
n∈N
(i) k = 0 
0
u
0
∈ C
(ii) k = n u ∈ C
min
u∈C
{ϕ(u) + 
n
J


(u
n
) − ϕ

(u
n
), u}.
u
n+1
(iii) u
n+1
− u
n
 n ←
n + 1 (ii)
J

F
u
0
∈ C

0
> 0
min
u∈C
{ϕ(u) + 
0
F (u
0

) − ϕ

(u
0
), u}.
u
1
u
0

0
u
1

1
u
2
z
n
∈ C z
n+1
min
u∈C
{ϕ(u) + 
n
F (u
n
) − ϕ

(u

n
), u};
C H
F : C −→ H a
C u
∗
ϕ : C −→ R
ϕ

b C
u
n+1
F L
C 0 < 
n
< 2ab/L
2
{u
n
}
u
∗
F
{
n
}
∞
n=0
{α
n

}
∞
n=0
•
(i) k = 0 z
0
∈ C 
0
α
0
(ii) k = n z ∈ C
min
z∈C
{ϕ(z) + 
n
(F (z
n
) + α
n
z
n
) − ϕ

(z
n
), z}.
z
n+1
(iii) z
n+1

− z
n
 n ← n + 1
(ii)
{
n
}
∞
n=0
{α
n
}
∞
n=0
Ψ
(i) 0 < 
n
≤ 1 0 < α
n+1
≤ α
n
≤ 1 α
n
→ 0 n → ∞
(ii)

∞
n=0

n

α
n
= ∞

∞
n=0

2
n
< ∞
(iii)

∞
n=0
(α
n
− α
n+1
)
2
α
3
n

n
< ∞
H C
H F : C −→ H
ϕ : H −→ R
ϕ


n ∈ N z
n+1
Ψ F
lim z
n+1
− u
∗
 = 0,
u
∗
E C
E T C E k
x y ∈ D(T ) T k > 0
j(x − y) ∈ J(x − y)
T (x) − T (y), j(x − y) ≤ x − y
2
− k(x − y) − (T (x) − T (y))
2
,
j(x) E I
E
(I − T )(x) − (I − T )(y), j(x − y) ≥ k(I − T )(x) − (I − T )(y)
2
,
x, y ∈ D(T ) j(x − y) ∈ J(x − y)
H
 T (x) − T (y) 
2
≤ x − y 

2
+λ  (I − T )(x) − (I − T )(y) 
2
,
x, y ∈ D(T ) λ = 1 − k T λ
0 ≤ λ < 1
λ = 0 T
T (x) − T (y) ≤ x − y  ∀x, y ∈ D(T ),
H C
H T : C −→ C λ
T
•
•
•
•
H C
H {T
i
}
∞
i=1
λ
i
C H F =

∞
i=1
F ix(T
i
) = ∅

u
∗
∈ F.
u
α
u
α
∈ C

∞

i=1
γ
i
A
i
(u
α
) + αu
α
, v − u
α

≥ 0 ∀v ∈ C,
A
i
= I − T
i
, i ≥ 1.
α > 0 {γ

i
}
∞
i=1
γ
i
> 0;
∞

i=1
γ
i

λ
i
= γ < ∞,

λ
i
=
1 − λ
i
2
.
ϕ : H −→ R
ϕ

Ψ {
n
}

n≥0
{α
n
}
n≥0
z
0
∈ C α
0
> 0 
0
> 0
z ∈ C
min
z∈C
{ϕ(z) + 
0
(B(z
0
) + α
0
z
0
) − ϕ

(z
0
), z}, B =
∞


i=1
γ
i
A
i
.
z
1
z
0
α
0

0
z
1
α
1

1
z
2
•
(i) k = 0 z
0

0
α
0
(ii) k = n z ∈ C

min
z∈C
{ϕ(z) + 
n
(B(z
n
) + α
n
z
n
) − ϕ

(z
n
), z}.
z
n+1
(iii) z
n+1
− z
n
 n ← n + 1
(ii)
C
H {T
i
}
∞
i=1
λ

i
C H F :=

∞
i=1
F ix(T
i
) = ∅ {γ
i
}
∞
i=1
(i) α > 0 u
α
(ii)
lim
α→0
u
α
= u
∗
, u
∗
∈ F, u
∗
 ≤ y y ∈ F.
(iii)
u
α
− u

β
 ≤
|α − β|
α
u
∗
 , α, β > 0.
C H
{T
i
}
∞
i=1
λ
i
C H F :=

∞
i=1
F ix(T
i
) = ∅ {γ
i
}
∞
i=1
ϕ : H −→ R
H ϕ

n ≥ 0 z

n+1
Ψ
lim
n→∞
z
n
= u
∗
∈ F.
B =

∞
i=1
γ
i
A
i
W
W
n
W
n
• W
n
H C
H T
1
, T
2
, . . . C γ

1
, γ
2
, . . .
0 < γ
i
< 1 i = 1, 2, . . . n ∈ N
W
n
: C −→ C
U
n,n+1
= I,
U
n,n
= γ
n
T
n
U
n,n+1
+ (1 − γ
n
)I,
U
n,n−1
= γ
n−1
T
n−1

U
n,n
+ (1 − γ
n−1
)I,
U
n,2
= γ
2
T
2
U
n,3
+ (1 − γ
2
)I,
W
n
= U
n,1
= γ
1
T
1
U
n,2
+ (1 − γ
1
)I.
W

n
W T
n
T
n−1
T
1
γ
n
γ
n−1
γ
1
C H
T
1
, T
2
, . . . C F =

∞
i=1
F (T
i
) = ∅
γ
1
, γ
2
, . . . 0 < γ

i
< 1 i = 1, 2, . . .
W
n
: C −→ C W T
n
T
n−1
T
1
γ
n
γ
n−1
γ
1
F ix(W
n
) =
n

i=1
F ix(T
i
).
C
H {T
i
}
∞

i=1
C
F =

∞
i=1
F ix(T
i
) = ∅ {γ
i
} (0, γ] γ ∈ (0, 1)
x ∈ C i ≥ 1 lim
n→∞
U
n,i
x
i ∈ N x ∈ C
U
∞,i
: C −→ C W : C −→ C
U
∞,i
x := lim
n→∞
U
n,i
x,
W x := lim
n→∞
W

n
x.
W W T
1
T
2
, γ
1
γ
2
, . . .
C
H {T
i
}
∞
i=1
C
F =

∞
i=1
F ix(T
i
) = ∅ {γ
i
} (0, γ] γ ∈ (0, 1)
F ix(W ) = F.
A : C −→ H C
H V I(A, C) u ∈ C

A(u), v − u ≥ 0 ∀v ∈ C.
{T
i
}
∞
i=1
C
S = V I(A, C) ∩ F = ∅.
u
∗
∈ S.
u
n
u
n
∈ C
A(u
n
) + α
µ
n
A
n
(u
n
) + α
n
u
n
, v − u

n
 ≥ 0 ∀v ∈ C,
A
n
= I − W
n
.
α
n
> 0 µ ∈ (0, 1)
ϕ : H −→ R
ϕ

Ψ
z
1
∈ C α
1
> 0 
1
> 0
z ∈ C
min
z∈C
(ϕ(z) + 
1
(A
1
(z
1

) + α
1
z
1
) − ϕ

(z
1
), z),
A
1
= A + α
µ
1
A
1
, A
1
= I − W
1
.
z
2
z
1
α
1

1
z

2
α
2

2
•
(i) k = 1 z
1

1
α
1
(ii) k = n z ∈ C
min
z∈C
(ϕ(z) + 
n
(A
n
(z
n
) + α
n
z
n
) − ϕ

(z
n
), z),

A
n
= A + α
µ
n
A
n
, A
n
= I − W
n
.
z
n+1
(iii) z
n+1
− z
n
 n ← n + 1
(ii)
H C
H A : C −→ H
{T
i
}
∞
i=1
S := V I(C, A)

F = ∅

(i) α
n
> 0 u
n
(ii)
lim
n→∞
u
n
= u
∗
, u
∗
∈ S, u
∗
 ≤ y ∀y ∈ S.
(iii)
u
n
− u
m
 ≤
|α
n
− α
m
|
α
n
u

∗
 , α
n
, α
m
> 0.
H C
H A : C −→ H
{T
i
}
∞
i=1
C
S := V I(C, A)

F = ∅ ϕ : H −→ R
ϕ

n ≥ 1 z
n+1
Ψ
lim
n→∞
z
n
= u
∗
.
W

n
H F : H −→ H
{T
i
}
N
i=1
H F =

N
i=1
F ix(T
i
) = ∅
p
∗
∈ F
F (p
∗
), p − p
∗
 ≥ 0 ∀p ∈ F.
x
0
∈ H
{x
n
}
x
0

∈ H, y
0
0
= x
0
,
y
i
k
= (1 − β
i
k
)y
i−1
k
+ β
i
k
T
i
(y
i−1
k
), i = 1, 2, · · · , N,
x
k+1
= (1 − β
0
k
)x

k
+ β
0
k
(I − λ
k
µF )y
N
k
, k ≥ 0,
λ
k
β
i
k
i = 0, 1, . . . , N
λ
k
∈ (0, 1) β
i
k
∈ (α, β) α, β ∈ (0, 1) k ≥ 0
lim
k→∞
λ
k
= 0;
∞

k=0

λ
k
= ∞; lim
k→∞


β
i
k+1
− β
i
k


= 0.
x
k+1
= (1 − β
0
k
)x
k
+ β
0
k
T
k
0
T
k

N
· · · T
k
1
x
k
T
k
i
:= (1 − β
i
k
)I + β
i
k
T
i
i = 1, 2, . . . , N T
k
0
= I − λ
k
µF
H F : H → H
L η F L
η {T
i
}
N
i=1

N H
F =

N
i=1
F ix(T
i
) = ∅ {x
k
}
k∈N
−
p
∗
S : H −→ H γ H

T

T (x) = αx + (1 − α)S(x),
α ∈ (γ, 1) x ∈ H H
H F ix(

T ) = F ix(S) F =

N
i=1
F ix(S
i
) S
i

γ
i
{S
i
}
N
i=1
N γ
i
α
i
∈ (γ
i
, 1)
F =

N
i=1
F ix(

T
i
)

T
i
= α
i
I + (1 − α
i

)S
i
,
i = 1, 2, . . . , N
H F : H −→ H
L η F L
η {S
i
}
N
i=1
N γ
i
F =

N
i=1
F ix(S
i
) = ∅ α
i
∈ (γ
i
, 1) i = 1, 2, . . . , N
µ ∈ (0,
2η
L
2
) {λ
k

}
k∈N
⊂ (0, 1) {β
i
k
}
k∈N
⊂ (α, β) α
β ∈ (0, 1) i = 1, 2, . . . , N {x
k
}
k∈N
T
i

T
i
p
∗
S
n
=

n
i=1
γ
i
A
i

B =

∞
i=1
γ
i
A
i
A
i
= I − T
i
T
i
T
h
i