Hiêu chỉnh phương trình toán tử loại I dựa trên toán tử tuyến tính đơn điệu mạnh

H
X
X
∗
X
R
n
n
∅
x := y x y
∀x x
∃x 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 : X → X
∗
f ∈ X
∗
x
0
∈ X
A(x
0
) = 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)
α = α(h, δ)
x
h,δ
α(h,δ)
h δ
A : X → X
∗
B : X → X
∗
h
A
h
(x) + αB(x) = f
δ
.
X
A : X → X
∗

X X
∗
D(A) ⊆ X D(A) ≡ X
R(A) X
∗
A
Ax − Ay, x − y ≥ 0, ∀x, y ∈ X.
A x = y
Gr(A) A X × X
∗
Gr(A) = {(x, y) : y = Ax}.
A
x
∗
− y
∗
, x − y ≥ 0, ∀x, y ∈ X, x
∗
∈ Ax, y
∗
∈ Ay.
Gr(A)
Gr(A)
X × X
∗
A
∀x ∈ X Ax, x ≥ 0 A
A ≥ 0
 A X
H A : H → H

Ax − Ay ≤ x − y, ∀x, y ∈ X.
I − A I
H
A : R
M
→ R
M
A = B
T
B,
B M
A
δ(t) t ≥ 0 δ(0) = 0
Ax − Ay, x − y ≥ δ(x − y), ∀x, y ∈ X.
δ(t) = c
A
t
2
c
A
A
A h
X A(x + ty)  Ax t → 0 x, y ∈ X A
d X x
n
→ x Ax
n
 Ax
n → ∞
ϕ(x, y) =






xy
2
(x
2
+ y
4
)
(x, y) = (0, 0)
0 (x, y) = (0, 0)
(0, 0) (0, 0)
h (0, 0).
 h X d
A : X → X
∗
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

H
I H
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
U
> 0,
U
s
(x) − U
s
(y) ≤ C(R)x − y
ν
, 0 < ν ≤ 1,
C(R) R = max{x, y}
X
∗
U : X → X
∗
d

X U
X f ∈ X
∗
A h X X
∗
A(x) − f, x − x
0
 ≥ 0, ∀x ∈ X,
A(x
0
) = f
A X
A(x
0
) − f, x − x
0
 ≥ 0, ∀x ∈ X.
X f : X → R
X
• f X
lim inf
y→x
f(y) ≥ f(x), ∀x ∈ X.
• ∂f(x)
∂f(x) = {x
∗
∈ X
∗
: f(x) ≤ f(y) + x
∗

, x − y, ∀y ∈ X}.
x
∗
∈ X
∗
f x ∂f(x)
f x
X
X
∗
X f : X → R ∪ {+∞}
X ∂f
X X
∗
f x ∈ X
x
∗
∈ X
∗
lim
λ→+0
f(x + λy) − f(x)
λ
= x
∗
, y, ∀y ∈ X,
x
∗
f x f


(x)
A : X → Y
X Y A
x ∈ X T ∈ L(X, Y )
A(x + h) = A(x) + T h + O( h ),
h θ T
A x A

(x) = T.
A(x) = f,
A : X → Y X
Y, f Y
A X
Y
A(x) = f f ∈ Y

x f x = R(f)
(X, Y ) ε > 0
δ(ε) > 0 ρ
Y
(f
1
, f
2
) ≤ δ(ε) ρ
X
(x
1
, x
2

) ≤ ε
x
1
= R(f
1
), x
2
= R(f
2
), f
1
, f
2
∈ Y, x
1
, x
2
∈ X.
f f
δ
f
δ
− f ≤ δ x
δ
f f
δ
δ → 0 f
δ
→ f
x

δ
x
A
A
x
n
 x
Ax
n
→ Ax
X Y
A 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











x
1
+ x
2
+ x
3
= 3
x
1
+ 1, 01x
2
+ x
3
= 3, 01
x
1

+ x
2
+ 1, 01x
3
= 3, 01
x
1
= 1; x
2
= 1 x
3
= 1











x
1
+ x
2
+ x
3
= 3

1, 01x
1
+ x
2
+ x
3
= 3, 05
x
1
+ 1, 03x
2
+ x
3
= 3, 06
x
1
= 205; x
2
=
206
3
x
3
=
−818
3
I
b

a

K(x, s)ϕ(s)ds = f
0
(x), x ∈ [a, b], −∞ < a < b < +∞
ϕ(s) f
0
(x)
K(x, s)
∂K
∂x
0 ≤ x, s ≤ 1 A
A : L
2
[a,b]
→ L
2
[a,b]
ϕ(s) → f
0
(x) =
b

a
K(x, s)ϕ(s)ds
L
2
[a,b]
f
1
(x) f
0

(x) L
2
[a,b]
ρ
L
2
[a,b]
(f
0
, f
1
) =

b

a
|f
0
(x) − f
1
(x)|
2
dx

1
2
.
ϕ
0
(s)

f
1
(x) = f
0
(x) + N
b

a
K(x, s)sin(ωs)ds
ϕ
1
(s) = ϕ
0
(s) + Nsin(ωs).
N ω f
0
f
1
L
2
[a,b]
ρ
L
2
[a,b]
(f
0
, f
1
) = |N|


b

a

b

a
K(x, s)sin(ωs)ds

2
dx

1
2
K
max
= max
a≤s,x≤b
|K(x, s)|,
ρ
L
2
[a,b]
(f
0
, f
1
) ≤ |N|


b

a

K
max
1
ω
cos(ωs)




b
a

2
dx

1
2
≤
|N|K
max
4(b − a)
ω
.
N ω
N
ω

ϕ
0
ϕ
1
L
2
[a,b]
ρ
L
2
[a,b]
(ϕ
0
, ϕ
1
) =

b

a
|ϕ
0
(s) − ϕ
1
(s)|
2
dx

1
2

= |N|

b

a
sin
2
(ωx)dx

1
2
= |N|

b − a
2
−
sin(ωb − ωa)cos(ωb + ωa)
2ω
.
N ω ρ
L
2
[a,b]
(f
0
, f
1
)
ρ
L

2
[a,b]
(ϕ
0
, ϕ
1
)
ϕ(y) = y
y = λ
0
x + y
0
Oxy
λ
0
y
0
y
0
> 0 λ
0
= 0
λ
0
λ
δ
: |λ
δ
− λ
0

| < δ
λ
δ
> 0
λ
δ
= λ
1
= λ
0
+ δ/2
y = y
0
d
1
: y = λ
1
x + y
0
ϕ(y) d
1
{x ≥ 0, y ≥ 0}
(0, y
0
) x = 0 ϕ(0) = y
0
y
0
x
2

(δ)
d
2
d
1
0
x
y
λ
δ
< 0
λ
δ
= λ
2
= λ
0
− δ/2
y = y
0
d
2
: y = λ
2
x + y
0
λ
δ
< 0
d

2
Ox x
2
(δ)
ϕ(y) d
2
{x ≥ 0, y ≥ 0}
(x
2
(δ), 0) x = x
2
(δ) ϕ(x
2
(δ)) = 0
|λ
1
− λ
2
| ≤ δ
| min
λ
1
ϕ(y) − min
λ
2
ϕ(y)| = |y
0
− 0| = y
0
> 0,

y
0
A h
X X
∗
A(x) = f
f ∈ X
∗
A
δ(t) : δ(t) → +∞ t → +∞ A(x), x ≥ ||x||δ(||x||).
a
f
(x) = A(x) − f f ∈ X
∗
a
f
a
f
(x), x = A(x), x − f, x ≥ ||x||(δ(||x||) − ||f||).
M
f
||x|| ≥ M
f
a
f
(x), x ≥ 0
x
0
A(x
0

) = f.
✷
A A + λU
X
∗
X X
∗
U : X → X
∗
X
A : X → X
∗
A
λ > 0 R(A + λU) X
∗
h
X X
∗
X
B : X → X
∗
h A : X → X
∗
A + B
B
X
X
A : X → X
∗
h X

A A
R(A) = X
∗
S
0
A : X −→ X
∗
S
0
x
0
∈ X x
0
Ax = f S
0
X
∗
.
f
1
, f
2
∈ Ax A
f
1
− g, x − y ≥ 0,
f
2
− g, x − y ≥ 0,
∀(y, g) ∈ GrA f = tf

1
+ (1 − t)f
2
t ∈ [0, 1] t
(1 − t)
tf
1
− g, x − y + (1 − t)f
2
− g, x − y ≥ 0, ∀(y, g) ∈ GrA
⇔ f − g, x − y ≥ 0, ∀(y, g) ∈ GrA.
f ∈ Ax S
0
f
n
∈ Ax, f
n
→ f
∗
f
∗
∈ Ax
f
n
− g, x − y ≥ 0, ∀(y, g) ∈ GrA.
n → ∞ f
∗
− g, x − y ≥ 0 f
∗
∈ Ax

S
0
✷
x
0
x
∗
x
0
x
∗
x
0
− x
∗
 = min
x∈S
0
x − x
∗
,
S
0
= {x ∈ X : A(x) = A(x
0
) = f}
A(x) = f, f ∈ X
∗
,
A h

X X
∗
A
f ∈ R(A) S
0
S
0
X
X B
S
0
⊂ D(B) D(B) ≡ X
A(x) + αBx = f
δ
,
f
δ
f
f − f
δ
 ≤ δ.
α > 0 f
δ
∈ X
∗
x
δ
α
α,
δ

α
→ 0, {x
δ
α
}
x
1
∈ S
0
Bx
1
, x − x
1
 ≥ 0, ∀x ∈ S
0
.
D(A) = X, A h
A B
B D(B) X
∗
B
h A + αB
D(B) X
∗
B α > 0 A + αB
(A + αB)(x), x = A(x) + αB(x), x
= A(x) − A(θ) + A(θ) + αB(x), x − θ
= A(x) − A(θ), x − θ + A(θ), x − θ
+ αB(x), x − θ.
A(x) − A(θ), x − θ ≥ 0 A

αB(x), x − θ = αB(x), x ≥ αm
B
x
2
, m
B
> 0;
A(θ), x − θ ≤ A(θ).x.
(A + αB)(x), x ≥ αm
B
x
2
− A(θ).x,
(A + αB)(x), x
x
≥
αm
B
x
2
− A(θ).x
x
= αm
B
x − A(θ).
lim
x→+∞
(A + αB)(x), x
x
= +∞.

α > 0
A+αB
(A + αB)(x) − (A + αB)(y), x − y
= Ax − Ay, x − y + αBx − By, x − y
≥ αC
B
x − y
2
,
C
B
x
δ
α
{x
δ
α
}
x
1
A(x
δ
α
) − A(x) + f − f
δ
, x − x
δ
α
 + αBx, x − x
δ

α

= αB(x − x
δ
α
), x − x
δ
α
, ∀x ∈ S
0
.
A B
αm
B
x − x
δ
α

2
≤ A(x
δ
α
) − A(x), x − x
δ
α
 + f − f
δ
, x − x
δ
α


+ αBx, x − x
δ
α

≤ f − f
δ
x − x
δ
α
 + αBx, x − x
δ
α
.
α
m
B
x − x
δ
α

2
≤
δ
α
x − x
δ
α
 + Bx, x − x
δ

α
.
{x
δ
α
} X
{x
δ
α
} x
1
X
x
δ
α
 x
1
α,
δ
α
→ 0
x
δ
α
∈ D(B)
A(x
δ
α
) + αBx
δ

α
− f
δ
, x − x
δ
α
 = 0, ∀x ∈ D(B).
A + αB
A(x) + αBx − f
δ
, x − x
δ
α
 ≥ 0, ∀x ∈ D(B).
α,
δ
α
→ 0
A(x) − f, x − x
1
 ≥ 0, ∀x ∈ D(B).
x tx + (1 − t)x
1
, 0 < t < 1
t t → 0 A
A(x
1
) − f, x − x
1
 ≥ 0, ∀x ∈ D(B).

D(B) = X x
1
∈ S
0
x
1
α,
δ
α
→ 0
0 ≤ m
B
x − x
1

2
≤ Bx, x − x
1
, ∀x ∈ S
0
.
S
0
tx + (1 − t)x
1
∈ S
0
, 0 < t < 1
t t → 0
Bx

1
, x − x
1
 ≥ 0, ∀x ∈ S
0
.
x
1
∈ S
0
{x
δ
α
}
x
1
.
✷
A A
h
A
A
h
(x) − A(x) ≤ hg(x)
A g(t)
α > 0, h > 0 f
δ
∈ X
∗
A

h
(x) + αBx = f
δ
x
τ
α
, τ = (h, δ). α,
δ
α
,
h
α
→ 0 {x
τ
α
}
x
1
∈ S
0
x
τ
α
, τ = (h, δ).
{x
τ
α
} x
1
α,

δ
α
,
h
α
→ 0
A
h
(x
τ
α
) − A(x) + f − f
δ
, x − x
τ
α
 + αBx, x − x
τ
α

= αB(x − x
τ
α
), x − x
τ
α
, ∀x ∈ S
0
.
B A A

h
m
B
x − x
τ
α

2
≤ Bx, x − x
τ
α

+
1
α

A
h
(x
τ
α
) − A
h
(x) + A
h
(x) − A(x) + f − f
δ
, x − x
τ
α



≤ Bx, x − x
τ
α
 +
hg(x) + δ
α
x − x
τ
α
.
✷
A, A
h
, B B
B
∗
= B
a, b, c ≥ 0, k > t, a
k
≤ ba
t
+ c =⇒ a
k
= O(b
k/(k−t)
+ c).
ρ(h)
h → 0 α > 0 M > 0

| ρ(h) |≤ M.h
α
ρ(h) = O(h
α
) ρ(h)
h → 0 ρ(h) Mh
α
A S
0
;
L > 0
A

(x) − A

(y) ≤ Lx − y, ∀x ∈ S
0
, y S
0
;
z ∈ D(B) A
∗
(x
1
)z = Bx
1
Lz ≤ 2m
B
α α ∼ (h + δ)
µ

, 0 < µ < 1
x
τ
α
− x
1
 = O((h + δ)
θ
), θ = min

1 − µ,
µ
2

.