Hiệu chỉnh tìm nghiệm chung của một họ hữu hạn phương trình với ánh xạ liên tục lipschitz và j đơn điệu luận văn thạc sĩ

E
∗
E
A
∗
: Y
∗
→ X
∗
A : X → Y
I
D(A) A
R(A) A
A
−1
A
x, y x y
x
E
x E
∅
x
n
 x x
n
x
x
n

→ x x
n
x
θ
. . .
J
J−
J
•
E
E
x, y
x, y ∈ E, x, y = y, x
x, y, z ∈ E, x + y, z = x, z + y, z
x, y ∈ E β βx, y = β x, y
x ∈ E, x, x ≥ 0 x, x = 0 x = 0
x = x, x
1/2
E
•
x ∈ E  x 
 x > 0, ∀x = 0,  x = 0 ⇔ x = 0,
 x + y ≤ x  +  y , ∀x, y ∈ E
 αx = |α|  x , ∀x ∈ E, α ∈ R.
L
p
[a, b] 1 ≤ p < ∞
 ϕ = (
b


a
|ϕ(x)|
p
dx)
1
p
, ϕ ∈ L
p
[a, b],
n R
n
R
n
x =

n

i=1
|x
i
|
2

1/2
,
d(x, y) = x − y,

x = (x
1
, x

2
, . . . , x
n
) ∈ R
n
, y = (y
1
, y
2
, . . . , y
n
) ∈ R
n

.
E
∗
E
∗
 .  x
∗
, x.
x
∗
∈ E
∗
x ∈ E J E
E
∗
E

x, j(x)  x   j(x) 
 x = j(x) , ∀x ∈ X, j(x) ∈ J(X).
•
E {x
n
} ⊂ E
x ∈ E x
∗
∈ E
∗
lim
n→∞
x
n
, x
∗
 = x, x
∗
 ,
x
n
 x n → ∞ {x
n
} ⊂ E
x ∈ E
x
n
− x → 0 n → ∞
•
E

∗
E
∗∗
= L(E
∗
, R)
x ∈ E
x
∗∗
E
∗∗
x
∗∗
, f = f, x, ∀f ∈ X
∗∗
,
f, x f ∈ E
∗
x ∈ E  x = x
∗∗
 h(x) = x
∗∗
h : E → E
∗∗
(R
n
)
∗
= R
n

(R
n
)
∗∗
= ((R
n
)
∗
)
∗
= (R
n
)
∗
= R
n
H : R
n
−→ (R
n
)
∗∗
R
n
L
p
[0, 1] p > 1
•
(x
n

 x)
( x
n
→ x ) ( x
n
− x → 0).
•
E E
∗
E ϕ : X → R

{∞}
E
ϕ(x) x ∈ E
ϕ(αx + (1 − α)y) ≤ αϕ(x) + (1 − α)ϕ(y), ∀α ∈ [0, 1], x, y ∈ E.
ϕ(x) E
E lim
y→x
ϕ(y) ≤ ϕ(x), ∀x ∈ E
ϕ(x) E
x
0
∀{x
n
} : ϕ(x
0
) ≤ ϕ(x
n
).
•

E
E
∗
A D(A) ⊆ E
R(A) ⊆ E
∗
.
A
A(x) − A(y), x − y ≥ 0, ∀x, y ∈ D(A),
A
x = y. A
A
d(t) t ≥ 0, d(0) = 0
A(x) − A(y), x − y ≥ [d( x ) − d( y )]( x  −  y ), ∀x, y ∈
D(A).
A
δ(t) t ≥ 0, δ(0) = 0
A(x) − A(y), x − y ≥ δ( x − y ), ∀x, y ∈ D(A).
δ(t) = C
A
t
2
C
A
A
A : R
M
→ R
M
A = B

T
B
B
A A
Ax, x ≥ m
A
 x 
2
, m
A
> 0, ∀x ∈ D(A).
f : R → R f(x) = 2012x
•
A h− A(x + ty)  Ax
t → 0
+
, ∀x, y ∈ X x
n
→ x
Ax
n
 Ax n → ∞.
ϕ(x, y) = xy
2
(x
2
+ y
4
)
−1

(0, 0)
(0, 0)
A : E −→ E
J− E j(x − y) ∈ J(x − y)
A(x) − A(y), j(x − y) ≥ 0, ∀x, y ∈ E
J− α
α 0
A(x) − A(y), j(x − y) ≥ α  x − y 
2
∀x, y ∈ E
E
A(x) − A(y) ≤ Lx − y, ∀x, y ∈ E,
L L = 1 A
J− λ E
λ
A(x) − A(y), j(x − y) ≥ λA(x) − A(y)
2
, ∀x, y ∈ E,
A J− λ
(1/λ)
m− J− E A J− R(A+λI) =
E ∀λ ≥ 0 R(A) A
I E
m J
•
T
E [7] λ ∈ [0, 1)
∀x, y ∈ E
T x − T y, j(x − y) ≤  x − y 
2

λ  x − y − (T x − T y) 
2
(I − T)x − (I − T )y, j(x − y) ≥ λ  (I − T)x − (I − T )y 
2
I − T J− λ λ = 0
T
T A := I − T J−
A J− T = I − A
f(x) = e
−x
x ∈ (−∞, +∞) → (−∞, 0)
f f
E
S
1
(0) := {x ∈ E : x = 1},
E
∃ lim
t→0
x + ty − x
t
, ∀x, y ∈ S
1
(0).
E
x ∈ S
1
(0)
∀x, y ∈ S
1

(0) x = y
 (1 − λ)x + λy  ∀λ ∈ (0, 1).
x, y ∈ E, x = y
 x = 1,  y = 1
||
x + y
2
|| < 1.
L
p
[a, b]
l
∞
(a
1
, a
2
, ) ∈ l
∞
 a 
∞
= sup
i∈N
|a
i
| µ
l
∞
µ
k

(a
k
) := µ((a
1
, a
2
, )) µ
µ
 µ  µ
k
(1) 1 µ
k
(a
k+1
) µ
k
(a
k
),
(a
1
, a
2
, ) ∈ l
∞
µ
lim inf
k→∞
a
k

≤ µ
k
(a
k
) ≤ lim sup
k→∞
a
k
(a
1
, a
2
, ) ∈ l
∞
a = (a
1
, a
2
, ) ∈ l
∞
b = (b
1
, b
2
, ) ∈ l
∞
a
k
−→ c a
k

− b
k
−→ 0 k −→ ∞ µ
k
(a
k
) =
µ(a) = c µ
k
(a
k
) = µ
k
(b
k
)
[6] C E
{x
k
}
E z ∈ C µ
µ
k
 x
k
− z 
2
min
u∈C
 x

k
− u 
2
,
µ
k
u − z, J(x
k
− z) ≤ u ∈ C
[7]
f ∈ E u = T
f
(x)
A(u) + u = f + x,
x ∈ E
T
f
• T
f
• F ix(T
f
) = S F ix(T
f
)
T
f
F ix(T
f
) {x ∈ E : x = T
f

(x)}.
∂
2
u
n
∂x
2
+
∂
2
u
n
∂y
2
= 0, −∞ < x < ∞, 0 < y,
u
n
(x, 0) = n
−2
sin nx, −∞ < x < ∞,
∂u
n
∂y
(x, 0) = n
−1
sin nx, −∞ < x < ∞.
u
n
(x, y) = n
−2

e
ny
sin nx
u
n
(x, 0),
∂u
n
∂y
(x, 0) → 0 n → ∞
u
n
(x, y) → ∞ n → ∞ y > 0
Ax = f, f ∈ Y,
x = R(f)
ρ
X
(x
1
, x
2
) ρ
Y
(f
1
, f
2
) x
1
, x

2
∈
X, f
1
, f
2
∈ Y
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

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

x
δ
ϕ
k
(t)
sup
t∈[a,b]
|ϕ
k
(t)| ≤ C
0
a = (a
1
, a
2
)
f(t) =
∞

k=1
a
k
ϕ
k
(t),
a
k
c
k
c

k
∞

k=1
(a
k
− c
k
)
2
≤ δ
2

f(t) =
∞

k=1
a
k
ϕ
k
(t)

f(t) = f(t), Max
t∈[a,b]
|ϕ
k
(t)| ≤ C
0
f(t

0
) =

f(t
0
)

f
n(δ)
(t
0
) =
n(δ)

k=1
c
k
ϕ
k
(t
0
),
n(δ) n(δ) → ∞, δ → ∞ n(δ) = [
η(δ)
δ
2
], η(δ) → 0, δ → 0,
|f(t
0
) −


f
n(δ)
(t
0
)| = |
n(δ)

k=1
a
k
ϕ
k
(t
0
) +
∞

k=1+n(δ)
a
k
ϕ
k
(t
0
) −
n(δ)

k=1
c

k
ϕ
k
(t
0
)| ≤
n(δ)

k=1
|a
k
− c
k
||ϕ
k
(t
0
)| + |
∞

k=1+n(δ)
a
k
ϕ
k
(t
0
)|,
∞


k=1
a
k
ϕ
k
(t
0
) |
n(δ)

k=1+n(δ)
a
k
ϕ
k
(t
0
)| → 0
n(δ) → ∞.
n(δ)

k=1
|a
k
− c
k
||ϕ
k
(t
0

)| ≤ (
n(δ)

k=1
|a
k
− c
k
|
2
.
n(δ)

k=1
|ϕ
k
(t
0
)|
2
)
1/2
≤ C
0
δ

n(δ) = C
0
δ.


[
η(δ)
δ
2
] = C
0

η(δ) → 0 δ → 0,
A X = Y = R
3
A : R
3
→ R
3
A =

1 0 0
0 1 0
0 0 0

Ax, x = x
2
1
+x
2
2
≥ 0, ∀x = (x
1
, x
2

, x
3
) ∈ R
3
A
x
1
= f
1
, x
2
= f
2
, 0x
1
+ 0x
2
+
0x
3
= f
3
f = (f
1
, f
2
, f
3
) ∈ R
3

f = (f
1
, f
2
, 0)
f
1
, f
2
f
δ
= (f
1
, f
2
, f
δ
3
) f
δ
3
= 0
x
0
A
−1
f
δ
: ρ
Y

(f
δ
, f) ≤
δ → 0 (A, f
δ
) δ
x
δ
x
0
x
δ
x
δ
= A
−1
f
δ
A
−1
f ∈ Y A
−1
A
−1
f
δ
A
−1
f
δ (1.2)

δ
δ → 0 x
0
f
δ
∈ Y
R(f, α) α
δ
1
α
1
R(f, α)
α ∈ (0, α
1
) f
δ
∈ Y : ρ
Y
(f
δ
, f) ≤ δ, δ ∈ (0, δ
1
)
α = α(f
δ
, δ) ε >
0, ∃δ(ε) ≤ δ
1
f
δ

∈ Y ρ
Y
(f
δ
, f) ≤ δ ≤ δ
1
ρ
X
(x
α
, x
0
) ≤ ε x
0
x
α
∈ R(f
δ
, α(f
δ
, δ));
x
α
∈ R(f
δ
, α)
α = α(f
δ
, δ) = α(δ)
α = δ

R(f, δ)
δ
1
R(f, δ)
0 ≤ δ ≤ δ
1
f ∈ Y ρ
Y
(f, f
0
) ≤ δ
ε > 0 δ
0
= δ
0
(ε, f
δ
) ≤ δ
1
ρ
Y
(f
δ
, f
0
) ≤
δ ≤ δ
0
ρ
X

(x
δ
, x
0
) ≤ ε x
δ
∈ R(f
δ
, δ)
J−
A
i
(x) = f
i
, i = 0, 1, , N,
{f
i
}
N
i=0
N + 1 E A
0
L
0
−
J− A
i
γ
i
J−

E i = 1, 2, , N
S
i
i (2.1)
S :=

N
i=0
S
i
= ∅
f
i
f
δ
i
 f
i
− f
δ
i
≤ δ, δ −→ 0.
(2.1)
f
i
2006 (2.1) f
i
= 0 A
i
: E −→ E

∗
i = 0, 1, , N N +1 h
E [2]
N

i=0
α
µ
i
A
h
i
(x) + αJ(x) = 0,
µ
0
= 0 µ
i
µ
i+1
A
h
i
h E
 A
i
(x) − A
h
i
(x) ≤ hg( x ).
g(t) t ≥ 0

N = 0 A
0
= A m J−
E
A
h
J− d
A
h
(x) αx f
δ
.
α > 0 x
τ
α
τ {δ, h} J
E (δ + h)/α −→ 0 x
τ
α
−→ y
∗
A(x) = f  f − f
δ
 ≤ δ
A
i
E ≡ H
[4]
N


i=1
 A
i
(x) − f
i

2
+α  x − x
+

2
,
x
+
∈ H
A
i
H (2.5)
N

i=1
A
∗
i
A
i
(x) + α(x − x
+
) =
N


i=1
A
∗
i
f
i
,
[5] A
∗
A
(2.1)
A
i
J− J−
A
i
f
i
A
h
i
f
δ
i
(2.2) (2.4) (2.1)
A
h
0
(x) + α


µ
N

i=1
(A
h
i
(x) − f
δ
i
) + α(x − x
+
) = f
δ
0
,
A
h
i
A
i
µ ∈ (0, 1)
A
0
(x) + α

µ
N


i=1
(A
i
(x) − f
δ
i
) + α(x − x
+
) = f
δ
0
,

µ ∈ [0, 1] α
E
A
0
A
i
γ
i
E
i = 1, 2, N
α > 0 f
δ
i
∈ E (2.8) x
δ
α
S = θ f

δ
i
(2.2) i = 0, N
α α, δ/α −→ 0 x
δ
α
p
∗
∈ S
p
∗
− x
+
, j(p
∗
− p) ≤ 0, ∀p ∈ S.
A
i
J− E
i = 0, 1, N A := A
0
+ α

µ
N

i=1
A
i
J−

E m − J−
(2.8) x
δ
α
α > 0
f
δ
i
∈ E
(A + α(I − x
+
))(.) J−
α
Nα
µ
≤ 1 (2.8)
A
0
(x
δ
α
) − A
0
(p) + α
µ
N

i=1
(A
i

(x
δ
α
) − A
i
(p) − (f
δ
i
− f
i
)) + α(x
δ
α
−
x
+
), J(x
δ
α
− p) f
δ
0
− f
0
, J(x
δ
α
− p) p ∈ S
x
δ

α
− x
+
, J(x
δ
α
− p) ≤
1
α
f
δ
0
− f
0
, J(x
δ
α
− p)
α
µ
α
N

i=1
f
δ
i
− f
i
, J(x

δ
α
− p)
A
i
J− i = 0, 1, N
 x
δ
α
− p 
2
≤ x
+
− p, Jx
δ
α
− p) + 2
δ
α
 x
δ
α
− p , ∀p ∈ S
x
δ
α
M
1
 x
δ

α
≤ M
1
∀α, δ > 0
 x
δ
α
− p 
2
≤ x
+
− p, J(x
δ
α
− p) + 2
δ
α
(M
1
+  p ),
(2.8) A
i
1
γ
i
i = 1, 2, N
 A
0
(x
δ

α
) − f
0
≤ α  x
δ
α
− x
+
 +α
µ
N

i=1
 A
i
(x
σ
α
) − A
i
(p)  +2δ
≤ α  x
δ
α
− x
+
 +α
µ
N


i=1
1
γ
i
(M
1
+  p ) + 2δ
lim
α,δ/α−→0
 A
0
(x
δ
α
) − f
0
= 0,
A
0
A
i
J−
γ
i
N

i=1
γ
i
 A

i
(x
δ
α
) − f
i

2
≤
N

i=1
A
i
(x
δ
α
) − f
i
, J(x
δ
α
− p)
≤ α
1−µ
x
+
− x
δ
α

, J(x
δ
α
− p) + (δ/α
µ
+ Nδ)  J(x
δ
α
− p) 
≤ (α
1−µ
 x
+
− p  +(α
1−µ
/α + Nδ))(M
1
+  p ),
lim
α,δ/α−→0
 A
i
(x
δ
α
) − f
i
= 0, i = 1, 2, N,
T
i

= I − A
i
T
f
i
= T
i
+ f
i
p ∈ S
p ∈

N
i=0
F ix(T
f
i
) A
i
J− T
i
E T
f
i
(2.11), (2.12)
 (I − T
f
i
)x
δ

α
−→ 0 α, δ/α −→ 0 i = 0, 1, N
Λ
i
= (2I − T
f
i
)
−1
2I − T
f
i
= I + I − T
f
i
= I + A
i
− f
i
J−
E
R(2I − T
f
i
) = E.
(1.1)
(2I − T
f
i
)x = (I + I − T

f
i
)x = (I + A
i
)x − f
i
,
A
i
(x) = A
i
(x)−f
i
m−J− (I +A
i
)
−1
Λ
i
F ix(Λ
i
) = F ix(T
f
i
) = S
i
x
δ
α
− T

f
i
x
δ
α
= (2I − T
f
i
)x
δ
α
− x
δ
α
= Λ
−1
i
x
δ
α
− x
δ
α
,
Λ
i
Λ
−1
i
x

δ
α
= x
δ
α
,
 x
δ
α
− Λ
i
x
δ
α
= Λ
i
Λ
−1
i
x
δ
α
− Λ
i
x
δ
α
≤ Λ
−1
i

x
δ
α
− x
δ
α
= (I − T
f
i
)x
δ
α
,
 x
δ
α
− Λ
i
x
δ
α
−→ 0 α, δ/α −→ 0.
{x
k
} {x
δ
α
} α
k
, δ

k
/α
k
→ 0 k → ∞
ϕ(x) = µ
k
 x
k
− x 
2
x ∈ E ϕ(x) −→ ∞
 x → ∞ ϕ