Báo cáo khoa học phương pháp hiệu chỉnh lặp giải hệ phương trình toán tử đơn điệu

•
•
1.
2.
3.
1.
2.
3.
4.
•
•
H
X
X
∗
X
R
n
n
R
n
+
R
n
∅

x := y x y
∀x x
∃x x
inf
x∈X
F (x) {F (x) : x ∈ X}
min
x∈X
F (x) F X
I
A ∩ B
A
T
A
a ∼ b a b
A
∗
A
D(A) A
R(A) A
x
k
→ x {x
k
} x
x
k
 x {x
k
} x

X X
∗
X . A
j
: X → X
∗
f
j
∈ X
∗
A
j
(x) = f
j
, j = 1, , N.
N = 1
A(x) = f.
A
H
x
h,δ
α
F
h,δ
α
(x) = A
h
(x) − f
δ


2
+ αx − x
∗

2
α > 0 h δ x
∗
(A
h
, f
δ
) (A, f)
x
h,δ
α(h,δ)
α = α(h, δ)
x
h,δ
α(h,δ)
h
δ
A : X → X
∗
M : X → X
∗
hemi U
s
X
A
h

(x) + αU
s
(x − x
∗
) = f
δ
α = α(δ)
A
h
≡ A
α δ
ρ(α) =
˜
Kδ
p
, 0 < p < 1,
˜
K ≥ 1,
ρ(α) = αx
δ
α

α = α(δ)
ρ(α) = δ
p
α
−q
, 0 < p ≤ q
A
h

≡ A
A
h
A
j
= I − T
j
T
j
H
T
j
X
N

j=1
α
λ
j
(A
h
j
(x) − f
j
) + αU
s
(x − x
∗
) = θ,
λ

1
= 0 < λ
i
< λ
i+1
< 1, i = 2, , N − 1,
A
h
j
A
j
H
z
n+1
z
n+1
= z
n
− β
n

N

j=1
α
λ
j
n
(A
j

(z
n
) − f
j
) + α
n
(z
n
− x
∗
)

, z
0
∈ H,
{α
n
} {β
n
}
A
j
{α
n
} {β
n
}
H
X X
∗

X . x
∗
, x
x
∗
(x) x
∗
∈ X
∗
x ∈ X
A(x) = f,
A : X → Y X
Y f Y f x
x f
A : X → Y X
Y
A(x) = f f ∈ Y
A : X → Y
X Y
A(x) = f
(A, f)
(A
h
, f
δ
)
A
h
≡ A
f

δ
f
δ
− f ≤ δ x
δ
f f
δ
δ → 0 f
δ
→ f x
δ
x
0
x
δ
A
A
{x
n
}
x x
n
 x x
n
→ x y
n
= A(x
n
) y = A(x)
A y

n
→ y
A(x) = f
D(A)
A
R(A) A
−1
A(x) = f
A A
A(x) − A(y), x − y ≥ 0, ∀x, y ∈ D(A).
X = Y = R
5
A
A =








1 1 1 1 1
1 1.0001 1 1 1
1 1 1.0001 1 1
1 1 1 1.0001 1
1 1 1 1 1.0001









f =

5 5.0001 5.0001 5.0001 5.0001

T
∈ R
5
.
Ax = f
x =

1 1 1 1 1

T
∈ R
5
.
A = A
h
1
=









1 1 1 1 1
1 1.0001 1 1 1
1 1 1.0001 1 1
1 1 1 1.0001 1
1 1 1 1 1








f = f
δ
1
=

5 5.0001 5.0001 5.0001 5

T
∈ R
5
Ax = f
A = A
h

2
=








1 1 1 1 1
1 1.0001 1 1 1
1 1 1.0001 1 1
1 1 1 1.0001 1
1 1 1 1 1








f = f
δ
2
=

5.0001 5.0001 5.0001 5.0001 5.0001


T
∈ R
5
Ax = f
x
0
x
∗
x
0
A(x
0
) = f
x
0
− x
∗
 = min{x − x
∗
 : A(x) = f}.
x
∗
x
∗
A
x
0
A(x
0
) = f

f
δ
f
δ
− f ≤ δ,
A
−1
x
δ
:= A
−1
f
δ
A : X → Y
X Y T (f, α)
α Y X
δ
1
α
1
T (f
δ
, α)
α ∈ (0, α
1
) f
δ
∈ Y
f
δ

− f ≤ δ, δ ∈ (0, δ
1
);
α = α(δ, f
δ
) δ ε > 0
δ(ε) ≤ δ
1
f
δ
∈ Y
f
δ
− f ≤ δ ≤ δ(ε)
x
δ
α
− x
0
 ≤ ε x
0
x
∗
x
δ
α
∈ T (f
δ
, α(δ, f
δ

))
T (f, α)
x
δ
α
∈ T (f
δ
, α(δ, f
δ
))
α = α(δ, f
δ
)
α(δ, f
δ
)
lim
δ→0
α(δ, f
δ
) = 0.
H T (f, α)
T (f
δ
, α) := arg min
x∈H
{A(x) − f
δ

2

+ αx − x
∗

2
}.
α
α x
δ
α
x
0
x
δ
α
− x
0

z ∈ X
x
0
− x
∗
= A

(x
0
)
∗
z.
• X Y

x
δ
α
F
δ
α
(x) = A(x) − f
δ

2
+ αx − x
∗

2
.
A α x
δ
α
x
0
•
H
A(x) + α(x − x
∗
) = f
δ
.
•
A : X → X
∗

D(A) ≡ X
A
Ax − Ay, x − y ≥ 0, ∀x, y ∈ X.
A x = y
A δ(t)
t ≥ 0 δ(0) = 0
A(x) − A(y), x − y ≥ δ(x − y), ∀x, y ∈ X.
δ(t) = c
A
t
2
c
A
A
A
A(x) − A(y), x − y ≥ m
A
Ax − Ay
2
, m
A
> 0.
A hemi X A(x + ty)  Ax
t → 0
+
x, y ∈ X A demi X
x
n
→ x Ax
n

 Ax n → ∞
A
lim
x→∞
Ax, x
x
= ∞, x ∈ X.
U
s
: X → X
∗
U
s
(x) =

x
∗
∈ X
∗
: x
∗
, x = x
∗

s−1
x = x
s

, s ≥ 2
X s = 2 U

s
U
X
X
U(x) U(λx) = λU(x) λ ∈ R
U X
∗
X U = I X
X
∗
U : X → X
∗
demi
X U
X = L
p
(Ω) 1 < p < ∞ Ω
R
n
U
(Ux)(t) = x
2−p
|x(t)|
p−2
x(t), t ∈ Ω.
L
p
(Ω) U
s
U

s
(x) − U
s
(y), x − y ≥ m
U
x − y
s
, m
s
> 0,
U
s
(x) − U
s
(y) ≤ C(R)x − y
ν
, 0 < ν ≤ 1,
C(R) R = max{x, y}
X = L
2
(Ω) U
s
= I s = 2 m
U
= 1
ν = 1 C(R) = 1 p = 2 L
p
(Ω) p > 1
U
s

1 < p < 2 s = 2 m
s
= p − 1 C(ρ) = p2
2p−1
f
p
K
p−1
ν = p − 1 f = max{2
p
, 2ρ} 1 < K < 3.18
2 < p s = p m
s
=
2
2−p
p
C(ρ) = 2
p
ρ
p−2

p

p − 1 +
max{ρ, K}

−1
ν = 1
X X

∗
X f ∈ X
∗
A : X → X
∗
hemi
x
0
∈ X
A(x) − f, x − x
0
 ≥ 0, ∀x ∈ X
x
0
A(x) = f
A X
A(x
0
) − f, x − x
0
 ≥ 0, ∀x ∈ X.
M : X → X
∗
hemi
M U
s
X
A(x) + αU
s
(x − x

∗
) = f
δ
,
A
h
≡ A
X
X

x
n
 x
 
x
n
 → x


x
n
− x → 0

X
∗
A : X → X
∗
hemi
α > 0 f
δ

∈ X
∗
x
δ
α
α, δ/α → 0 {x
δ
α
} x
∗
X
A
j
hemi
X X
∗
x
0
∈ X
A
j
(x
0
) = f
j
, ∀j = 1, , N.
S
j
= {¯x ∈ X : A
j

(¯x) = f
j
}, j = 1, , N. A
j
ϕ
j
: X → R ∪ {+∞}
S
j
inf
x∈X
ϕ
j
(x),
X j = 1, , N
F : X → R ∪ {+∞}
X
F

x
0
F (x) X
F

(x
0
), x − x
0
 ≥ 0 ∀x ∈ X
F


(x), x − x
0
 ≥ 0 ∀x ∈ X
ϕ : X → R ∪ {+∞}
X
inf
x∈X
ϕ(x)
F : X → R ∪ {+∞}
X F X
F

X X
∗
T
j
j = 1, , N
H
T
j
(x) − T
j
(y) ≤ x − y, ∀x, y ∈ H,
P
j
= F (T
j
) T
j

P
j
= F (T
j
) := {x ∈ H : T
j
x = x},
x
0
∈ P = ∩
N
j=1
P
j
P = F (T
N
T
N−1
T
1
) = F (T
N−1
T
1
T
2
) = = F (T
1
T
2

T
N
)
T
j
f
j
(x)
f

j
(x) = T
j
(x) j = 1, , N
ϕ
j
(x) =
x
2
2
− f
j
(x)
I − T
j
P
j
= {x ∈ H : ϕ

j

(x) = θ} ≡ S
j
.
j
F : X → R ∪ {+∞}
X
lim
x→∞
F (x) = +∞ x ∈ X.