tóm tắt luận án tiên sĩ tìm điểm bất động chung cho một họ các ánh xạ giả co chặt

C
i
i = 1, 2, . . .
C
i
T
i
A
i
G
i
(u, v)
cardG = 2 C
1
C
2
H
x
{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
(x) C C
1
C
2
H
P
C

(x)
cardG ≥ 2 C
i
T
i
{T
i
}
i≥2
λ
i
{T
i
}
∞
i=1
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
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 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
{u
k
}
∞
k=0
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.
x
∗
∈ C
V I(F, C)
C H F : C →
H
C 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
}
x
∗
•
F
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 h
ϕ : H −→ R ϕ

n ∈ N

z
n+1
{
n
}
∞
n=0
{α
n
}
∞
n=0
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) 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)
 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 (x) − T(y) ≤ x − y  ∀x, y ∈ C.
H C
H T : C −→ C λ
T
•
•
•
•
1
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 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

{
n
}
n≥0
{α
n
}
n≥0
Ψ
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
{
n
}
n≥0
{α
n
}
n≥0
Ψ
lim
n→∞
z
n
= u
∗
∈ F.
B =

∞
i=1

γ
i
A
i
W
W
n
W
• 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
W

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 L 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 0 µ ∈ (0, 1)
ϕ : H → R ϕ

{
n
}

n≥1
{α
n
}
n≥1
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 h {T
i
}
∞
i=1
C H 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 C {T
i

}
∞
i=1
C H S := V I(C, A)

F = ∅
ϕ : H −→ R
ϕ

n ≥ 0
z
n+1
{α
n
}
n≥1
{
n
}
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 {T
i
}
N
i=1
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) ∀x ∈ H,
α ∈ (γ, 1) H F ix(

T ) = F ix(S)
F =

N
i=1
F ix(S
i
) S
i
γ
i
{S
i
}

N
i=1
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 {S
i
}
N
i=1
γ
i
H 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
T
i
T
h
i