Toán tử đơn điệu cực đại và một số ứng dụng luận văn thạc sĩ toán học

BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI 2








VŨ VĂN QUYỀN







TOÁN TỬ ĐƠN ĐIỆU CỰC ĐẠI VÀ
MỘT SỐ ỨNG DỤNG

Chuyên ngành: Toán Giải Tích
Mã số: 60 46 01 02



TÓM TẮT LUẬN VĂN THẠC SĨ TOÁN HỌC

Người hướng dẫn khoa học: TS. NGUYỄN VĂN HÀO









HÀ NỘI, 2014
E E
∗
G ⊂ E × E
∗
(x, x
∗
) (y, y
∗
)
G
(x
∗
− y
∗

, x − y) ≥ 0.
G ⊂ E × E
∗
T : E ⇒ E
∗
•
•
X
C ⊆ X x, y ∈ C (x, y) ⊆ C
(x, y) := {z = λx + (1 − λ)y : λ ∈ (0, 1)}
x, y ∈ C λ ∈ (0, 1) λx + (1 − λ)y ∈ C
A ⊆ X A
A A
A
A = {x|x A}
C C = C
A ⊆ X |λ| ≤ 1
λA ⊆ A A = −A A = ∅ 0 ∈ A
A, B ⊆ X α ∈ R A+B, αA
E C E
C C C\E E
C C
∀x, y ∈ C, ∀λ ∈ (0, 1) : (λx + (1 − λ)y ∈ E =⇒ x, y ∈ E).
C C
E = {¯x} ¯x
C C
ext(C) C
L ext(C) = L C
L C

C C
C X
C C
C C
E ⊆ D ⊆ C D C E D
E C
E C ext(E) = ext(C) ∩ E
A ⊆ X A := {λa : λ > 0, a ∈ A}
A ⊆ X k ∈ ( A)\{0}
{a
1
, a
2
, , a
m
} ⊆ A λ
1
, , λ
m
k = λ
1
a
1
+ λ
2
a
2
+ + λ
m
a

m
dim X = n < ∞ A ⊆ X
x ∈ A x n + 1
A {a
0
, a
1
, , a
m
} ⊆ A m ≤ n
λ
0
, , λ
m
≥ 0
m

i=0
λ
i
= 1 x =
m

i=0
λ
i
a
i
.
X

X
#
:= L(X, R) X
ϕ : X → R
ϕ(x + y) ≤ ϕ(x) + ϕ(y) x, y ∈ X
ϕ(λx) = λϕ(x) λ > 0 x ∈ X
ϕ
X M X f ∈ M
#
f(m) ≤ ϕ(m); ∀m ∈ M.
F ∈ X
#
F |
M
= f
F (x) ≤ ϕ(x) x ∈ X
X M
X f M
F X
F |
M
= f ||F || = ||f||.
X x
0
∈ X\{0}
f X
||f|| = 1 f(x
0
) = ||x
0

||.
A B X
f A B
f(a) ≤ f(b) ( f(a) ≥ f(b)), ∀a ∈ A, b ∈ B.
α ∈ R
f(a) ≤ α ≤ f(b), ∀a ∈ A, b ∈ B.
H(f; α) := f
−1
(α) = {x ∈ X : f(x) = α}
A B B B = {x
0
}
H(f; α) A x
0
H(f; α)
A ⊆ X A
H
+
(f; α) := {x ∈ X|f(x) ≥ α}; H
−
(f; α) := {x ∈ X|f(x) ≤ α}.
H(f; α) A B
A ⊆ H
−
(f; α) B ⊆ H
+
(f; α)
f(a) = f(b) = α a ∈ A b ∈ B
H(f; α) A B H(f; α)
A B A B

sup{f(a), a ∈ A} ≤ inf{f(b), b ∈ B};
inf{f(a), a ∈ A} < sup{f(b), b ∈ B}
R
2
O(0; 0), M(1; 1), N(2; 2), P (3; 3),
Q(4; 4) A, B, C ON, MP, P Q
x − y = 0 f(x, y) = x − y α = 0 A B
A C
A C g(x, y) = x + y
A B
A X
∀v ∈ X, ∃ > 0, (−v, v) ⊂ A.
x
0
A A −x
0
A A
A
x
0
∈ A ⇐⇒ ∀v ∈ X, ∃ > 0, ∀λ ∈ (−, ) : x
0
+ λv ∈ A.
C A
H(f; α) A
H(f; α) ∩ A = ∅
C x
0
∈ X\C
C /∈ ∅

X
C x
0
A B
A ∪ B = ∅
A B
(C
i
)
1≤i≤m
C
m+1
m+1

i=1
C
i
= ∅.
f
i
, 1 ≤ i ≤ m + 1
f
1
+ f
2
+ + f
m+1
= 0
f
1

(x
1
) + f
2
(x
2
) + + f
m+1
(x
m+1
) ≤ 0, ∀x
i
∈ C
i
.
H
−
0
(f; α) := {x ∈ X|f(x) < α}; H
+
0
(f; α) := {x ∈ X|f(x) > α},
H(f; α) A B A ⊆ H
−
0
(f; α)
B ⊆ H
+
0
(f; α)

X  · 
X X
∗ ∗
X
∗
w
∗
X X
∗
B
X
B
X
∗
A : X → Y
A
∗
x ρ B
ρ
(x)
Ω ⊂ X Ω
Ω Ω Ω ⊂ X ¯x ∈ Ω
U ¯x Ω ∩ clU
Ω ⊂ X ¯x ∈ Ω ε  0
Ω ¯x

N(¯x; Ω) :=

x
∗

∈ X
∗
| lim sup
x
Ω
→¯x
x
∗
, x − ¯x
 x − ¯x 
 0

,
x
Ω
−→ ¯x x → ¯x x ∈ Ω

N(¯x; Ω) Ω ¯x
f : X →
¯
R
¯
R := R ∪ {∞} ¯x
ˆ
∂f(¯x) :=

x
∗
∈ X
∗

| lim inf
x→¯x
f(x) − f(¯x) − x
∗
, x − ¯x
x − ¯x
 0

.
f ¯x
f ¯x
F : X ⇒ Y D(F ) := {x ∈ X :
F (x) = ∅} = ∅ (x, y) ∈ X×Y
F (x, y)

D
∗
F (x, y) : Y
∗
⇒ X
∗

D
∗
F (x, y)(y
∗
) :=

x
∗

∈ X
∗
|(x
∗
, −y
∗
) ∈

N((x, y); G(F ))

,
G(F ) := {(x, y) ∈ X × Y : y ∈ F(x)}

D
∗
F (x, y)(y
∗
) = ∅ y
∗
∈ Y
∗
(x, y) /∈
G(F )
X Y Ω ⊂ X
∆ : X → Y Ω
Y
∆(x; Ω) :=




0 ∈ Y x ∈ Ω,
∅ x /∈ Ω.
¯x ∈ Ω y
∗
∈ Y
∗

D
∗
∆(¯x; Ω)(y
∗
) =

N(¯x; Ω).
E E
∗
T : E ⇒ E
∗
x
∗
− y
∗
, x − y ≥ 0 ∀x, y ∈ E x
∗
∈ T(x), y
∗
∈ T(y).
T (x)
T D(T ) = {x ∈ E : T (x) = ∅}.
H

T : H → H
∗
≡ H T
T T x, x ≥ 0 ∀x ∈ H.
D R.
ϕ : D → R
∗
≡ R ϕ
[ϕ(t
2
) − ϕ(t
1
)](t
2
− t
1
) ≥ 0 ∀t
1
, t
2
∈ D
ϕ(t
1
) ≤ ϕ(t
2
) t
1
< t
2
.

R R
ϕ(x) =











0 x < 0
1 x > 0
[0, 1] x = 0.
C
H
U C
U(x) − U(y) ≤ x − y x, y ∈ C I
H T = I −U D(T ) = C.
x, y ∈ C
T (x) − T (y), x − y = x − y − (U(x) − U(y)), x − y
= x − y
2
− U(x) − U(y), x − y
≥ x − y
2
− U(x) − U(y) · x − y ≥ 0.
0 T U

C
C
P H C; P (x)
C x−P (x) = inf{x−y : y ∈ C}.
P
x ∈ H
x − P (x), z − P(x) ≤ 0 z ∈ C.
z ∈ C 0 < t < 1 z
t
≡ tz + (1 − t)P (x) ∈ C
x − P (x) ≤ x − z
t
 = (x − P(x)) − t(z − P (x)).
x −
P (x)
2
0 ≤ −2tx − P(x), z − P (x) + t
2
z − P(x)
2
.
t t → 0
y ∈ H z = P (y)
x − P (x), P (y) − P (x) ≥ 0.
x y z = P (x)
y − P (y), P (x) − P (y) ≥ 0.
x − y, P (x) − P(y) ≥ P (x) − P (y)
2
∀x, y ∈ H.
P

P
x − y, P (x) − P(y) ≤ x − y · P (x) − P (y).
P (x) − P (y) ≤ x − y ∀x, y ∈ H.
E E
∗
. x ∈ E
J(x) = {x
∗
∈ E
∗
: x
∗
, x = x
∗
 · x x
∗
 = x}.
J(x) x D(J) =
E x
∗
∈ J(x) y
∗
∈ J(y).
x
∗
− y
∗
, x − y = x
∗


2
− x
∗
, y − y
∗
, x + y
∗

2
≥ x
∗

2
− x
∗
.y − y
∗
.x + y
∗

2
= x
∗

2
− 2x
∗
.y
∗
 + y

∗

2
= (x
∗
 + y
∗
)
2
,
G E × E
∗
x
∗
− y
∗
, x − y ≥ 0
(x, x
∗
), (y, y
∗
) ∈ G. T : E ⇒ E
∗
G(T ) = {(x, x
∗
) ∈ E × E
∗
: x
∗
∈ T(x)}

E × E
∗
(x, x
∗
) ∈ E × E
∗
G
x
∗
− y
∗
, x − y ≥ 0 ∀(y, y
∗
) ∈ G.
T
T
(x, x
∗
) ∈ E × E
∗
G(T )
x ∈ D(T) x
∗
∈ T(x).
T T ,
G(T ) ⊂ G(T ).
T : E ⇒ E
∗
T
−1

E
∗
E
T
−1
(x
∗
) = {x ∈ E : x
∗
∈ T(x)}.
G(T
−1
) = {(x
∗
, x) ∈ E
∗
× E : x
∗
∈ T(x)},
G(T )
T
−1
T
ϕ
ϕ(0) = [0, 1]
ϕ R ϕ(x) =
[ϕ(x
−
), ϕ(x
+

)] x ∈ R ϕ(x
−
) := lim
t→x
−
ϕ(t) ϕ(x
+
) :=
lim
t→x
+
ϕ(t)
T
(x, x
∗
) ∈ H × H
G(T ) z ∈ H λ > 0
0 ≤ T(x ± λz) − x
∗
, (x ± λz) − x = ±λT (x) ± λT (z) − x
∗
, z
= ±λT (x) − x
∗
, z + λ
2
T (z), z.
λ λ → 0 T (x) − x
∗
, z = 0 z ∈ H

x
∗
= T (x).
T : E ⇒ E
∗
T T
−1
T G(T ) T
T T T
−1
D(T ) R(T )
D(T ) = E R(T ) = E
∗
,
E E
∗
T
C D(T ) = E.
C
E
∗
, φ : C → E
M ⊂ E ×C x
∗
0
∈ C
{(φ(x
∗
0
), x

∗
0
)} ∪ M
(y, y
∗
) ∈ M
U(y, y
∗
) = {x
∗
∈ C : x
∗
− y
∗
, φ(x
∗
) − y < 0}.
x
∗
→ x
∗
− y
∗
, φ(x
∗
) − y C
C = ∪{U(y, y
∗
) : (y, y
∗

) ∈ M}. C
(y
1
, y
∗
1
), (y, y
∗
2
), . . . , (y, y
∗
n
) ∈ M C =
n

i=1
{U(y
i
, y
∗
i
)}.
β
1
, β
2
, . . . , β
n
C; β
i

C, 0 ≤ β
i
≤ 1,

β
i
= 1 {x
∗
∈ C : β
i
(x
∗
) > 0} ⊂ U(y
i
, y
∗
i
)
i. K = {y
∗
i
} ⊂ C p
K
p(x
∗
) =

β
i
(x

∗
)y
∗
i
, x
∗
∈ K.
K
z
∗
∈ K p(z
∗
) = z
∗
.
0 = p(z
∗
) − z
∗
,

β
j
(z
∗
)(y
j
− φ(z
∗
))

= 

β
j
(z
∗
)(y
∗
i
− z
∗
),

β
j
(z
∗
)(y
j
− φ(z
∗
))
=

i,j
β
i
(z
∗
)β

j
(z
∗
)y
∗
i
− z
∗
, y
j
− φ(z
∗
).
α
ij
:= y
∗
i
− z
∗
, y
j
− φ(z
∗
).
α
ij
+ α
ji
= α

ii
+ α
jj
+ y
∗
i
− y
∗
j
, y
j
− y
i
 ≤ α
ii
+ α
jj
,
M.
β
i
(z
∗
) = β
i
. i, j
β
i
β
j

α
ij
+ β
j
β
i
α
ji
= β
i
β
j
(
α
ij
+ α
ji
2
) + β
j
β
i
(
α
ij
+ α
ji
2
).
0 =


β
i
β
j
α
ij
=

β
i
β
j
(
α
ij
+ α
ji
2
) ≤

β
i
β
j
(
α
ii
+ α
jj

2
).
β
i
β
j
= 0 i, j.
i, j β
i
β
j
> 0 z
∗
∈ U(y
i
, y
∗
i
)∩ U(y
j
, y
∗
j
),
α
ii
< 0 α
jj
< 0


β
i
β
j
(
α
ii
+α
jj
2
) < 0,
β
i
≡ β
i
(z
∗
) = 0 i

β
i
(z
∗
) = 1.
T : E ⇒ E
∗
x ∈ E U x T(U)
x ∈ D(T).
T E \ D(T ).
T

D(T ).
D(T )
D ⊂ E, D D
D ⊂ D, D ⊂ ( D).
( D) ⊂ D. ( D) D
( D) = D, D
( D) D
( D) = D D = ( D) ( D)
C C ⊂ C
D ⊂ D ⊂ ( D) D ⊂ ( D) = D ⊂ D
{C
n
}
∪ C
n
⊂ ∪ C
n
E
∗