Một số quy tắc tính toán trong giải tích biến phân và ứng dụng (FULL)


f(p, ·); Θ(b)

∃x x
∀x x
f : X → Y X Y
F : X ⇒ Y X Y
gphF F : X ⇒ Y
DomF F : X ⇒ Y
domf f : X → R
B
r
(x) x r > 0
B B
X
X
X
∗
X
σ(X
∗
, X)
∗
X
∗
τ
·

x
∗
, x x
∗
x x
∗
∈ X
∗
x ∈ X
∇f(x) : X → Y f x
T
∗
: Y
∗
→ X
∗
T : X → Y
δ
Ω
(·) Ω
R
R
−
R R ∪{±∞}
Q
N
∅
int(Ω) Ω
x ∈ R
n

x R
n
x = (x
1
, , x
n
) x =


x
1
x
n


{x
i
}
x
∗
k
w
∗
→ x
∗
{x
∗
k
} x
∗

σ(X
∗
, X)
x
ϕ
→ ¯x x → ¯x ϕ(x) → ϕ(¯x)
x
Ω
→ ¯x x → ¯x x ∈ Ω
ε ↓ 0 ε → 0 ε ≥ 0
o(t) t lim
t→0
o(t)
t
= 0
P := Q P Q
✷
lim inf ϕ ϕ
lim sup ϕ ϕ
Lim sup F
F

N(x; Ω) Ω x
N(x; Ω) Ω x
T (x; Ω) Ω x
T
w
(x; Ω) Ω x

D

∗
F F
D
∗
N
F F
D
∗
M
F F

∂ϕ ϕ
∂ϕ ϕ

∂
2
ϕ ϕ
∂
2
N
ϕ ϕ
∂
2
M
ϕ ϕ
VI(K; F ) K
F

f X
f
X
X
•
•
•
•
•
•
•
X Y
X
∗
Y
∗
. (x
∗
, x) ∈ X
∗
×X
x
∗
, x := x
∗

(x). τ
·
∗
X
∗
σ

X
∗
, X

. X × Y
(x, y) := x + y. ϕ : X →
¯
R := R ∪ {±∞},
ϕ x ∈ X
lim sup
u→x
ϕ(u) := inf
δ>0
sup
u∈B
δ
(x)\{x}
ϕ(u) lim inf
u→x
ϕ(u) := sup
δ>0
inf
u∈B

δ
(x)\{x}
ϕ(u).
Ω X.
(i) ε ≥ 0 ε Ω ¯x ∈ Ω

N
ε
(¯x; Ω) X
∗

N
ε
(¯x; Ω) :=

x
∗
∈ X
∗


lim sup
x
Ω
−→¯x
x
∗
, x − ¯x
x − ¯x
≤ ε


,
x
Ω
−→ ¯x x → ¯x x ∈ Ω ¯x ∈ Ω

N
ε
(¯x; Ω) := ∅
ε = 0

N(¯x; Ω) :=

N
0
(¯x; Ω)
Ω ¯x
(ii) Ω ¯x ∈ Ω N(¯x; Ω)
X
∗
N(¯x; Ω) := Lim sup
x → ¯x
ε↓0

N
ε
(x; Ω),
“ Lim sup ”
x
∗

∈ N(¯x; Ω) ε
k
↓ 0 x
k
→ ¯x x
∗
k
∈

N
ε
k
(x
k
; Ω)
x
∗
k
w
∗
−→ x
∗
x
∗
k
w
∗
−→ x
∗
x

∗
k
→ x
∗
σ

X
∗
, X

,
lim
k→∞
x
∗
k
, x = x
∗
, x x ∈ X N(¯x; Ω) := ∅ ¯x ∈ Ω.

N
ε
(¯x; Ω) X
∗
τ
·
N(¯x; Ω)
X N(¯x; Ω)
Ω ⊂ X N(¯x; Ω)


N(¯x; Ω)
Ω ¯x
N(¯x; Ω) =

N(¯x; Ω) =


x
∗
∈ X
∗
|x
∗
, x − ¯x ≤ 0, ∀x ∈ Ω

¯x ∈ Ω,
∅ ¯x ∈ X\Ω.
f : U → Y
U ⊂ X Y ¯x ∈ U.
(i) f ¯x
∇f(¯x) : X → Y
lim
x,u→¯x
x=u
f(x) −f(u) −

∇f(¯x), x − u

x − u
= 0.

∇f(¯x) f ¯x.
(ii) ∇f(¯x) : X → Y
u = ¯x, f ¯x ∇f(¯x) : X → Y
f ¯x.
(iii) f ¯x δ > 0 f
x ∈ B
δ
(¯x) ∇f : B
δ
(¯x) → L(X; Y ), x → ∇f(x)
¯x B
δ
(¯x) ¯x δ L(X; Y )
X Y.
(iv) f U
U.
f ¯x f ¯x f
¯x f ¯x.
f(x) =

x
2
x ∈ Q,
0 x ∈ R\Q
0 0 f : [−1, 1] → R
f(x) =



x

2
x =
1
k
, k ∈ N,
0 x = 0,
0 0
f ¯x f ¯x f
¯x
x ∈ X ϕ
x
: X
∗
→ R, x
∗
→ ϕ
x
(x
∗
) := x
∗
, x
X
∗
, ϕ
x
∈ X
∗∗
:=


X
∗

∗
.
Φ : X → X
∗∗
, x → Φ(x) := ϕ
x
Φ(x) = ϕ
x
 x ∈ X. Φ X X
∗∗
.
x ϕ
x
X
X
∗∗
X ⊂ X
∗∗
. Φ(X) = X
∗∗
, X
(i) (X, ·)
x → x 0
(ii) (X, ·)
 · 
1
X  · 

1
 ·  (X,  · 
1
)
(iii) X ϕ : U → R
U ⊂ X
U.
X
X
C[a, b] L
1
[a, b] L
∞
[a, b] c
0
ε = 0 X Ω
¯x, δ > 0 Ω ∩B
δ
(¯x)
X, B
δ
(¯x) ¯x δ.
Ω X ¯x ∈ Ω.
Ω ¯x T (¯x; Ω)
X
T (¯x; Ω) :=

v ∈ X |∃t
k
↓ 0, ∃v

k
∈ X : v
k
→ v, ¯x + t
k
v
k
∈ Ω, ∀k

.

N(¯x; Ω) =

T (¯x; Ω)

−
X
K
−
:=

x
∗
∈ X
∗
|x
∗
, v ≤ 0, ∀v ∈ K

K ⊂ X.

X Ω ⊂ X ¯x ∈ Ω

N(¯x; Ω) =

T (¯x; Ω)

−
.
F : X ⇒ Y, F
DomF :=

x ∈ X |F (x) = ∅

gphF :=

(x, y) ∈ X ×Y |y ∈ F (x)

.
F : X ⇒ Y,
gphF.
F (¯x, ¯y) ∈ gphF
DF : X ⇒ Y gphDF gphF (¯x, ¯y)
gphF
F : X ⇒ Y (¯x, ¯y) ∈ X ×Y
(i) F (¯x, ¯y) D
∗
N
F (¯x, ¯y) : Y
∗
⇒ X

∗
D
∗
N
F (¯x, ¯y)(y
∗
) :=

x
∗
∈ X
∗
|(x
∗
, −y
∗
) ∈ N

(¯x, ¯y); gph F

.
(ii) F (¯x, ¯y)

D
∗
F (¯x, ¯y) : Y
∗
⇒ X
∗


D
∗
F (¯x, ¯y)(y
∗
) :=

x
∗
∈ X
∗
|(x
∗
, −y
∗
) ∈

N

(¯x, ¯y); gph F

.
(iii) F (¯x, ¯y) D
∗
M
F (¯x, ¯y) : Y
∗
⇒ X
∗
D
∗

M
F (¯x, ¯y)(y
∗
) :=

x
∗
∈ X
∗
|∃ε
k
↓ 0, (x
k
, y
k
) → (¯x, ¯y), x
∗
k
w
∗
−→ x
∗
,
y
∗
k
·
−→ y
∗
: (x

∗
k
, −y
∗
k
) ∈

N
ε
k

(x
k
, y
k
); gph F

, ∀k

.
D
∗
M
F (¯x, ¯y) D
∗
N
F (¯x, ¯y),
D
∗
F (¯x, ¯y)


D
∗
F (¯x, ¯y)(y
∗
) ⊂ D
∗
M
F (¯x, ¯y)(y
∗
) ⊂ D
∗
N
F (¯x, ¯y)(y
∗
), ∀y
∗
∈ Y
∗
.
Y
F (¯x) = {¯y}, ¯y

D
∗
N
F (¯x) D
∗
N
F (¯x, ¯y)


F : X → Y
¯x ∇F (¯x)
D
∗
F (¯x)(y
∗
) =

D
∗
F (¯x)(y
∗
) =

∇F (¯x)
∗
y
∗

, ∀y
∗
∈ Y
∗
,
F ¯x ∈ X,
∇F (¯x)
∗
∇F (¯x) : X → Y
ϕ : X → R := R ∪{±∞}.

(i) ϕ
dom ϕ :=

x ∈ X | ϕ(x) < ∞

epi ϕ :=

(x, α) ∈ X ×R | α ≥ ϕ(x)

.
(ii) ϕ ϕ = ∅ ϕ(x) > −∞ x ∈ X.
(iii) ϕ x lim inf
u→x
ϕ(u) ≥ ϕ(x).
(iv) δ > 0 ϕ u ∈ B
δ
(x)
ϕ x.
(v) ϕ x ϕ
ϕ epi ϕ
X × R. ϕ(x) ∈ R ϕ x epi ϕ

x, ϕ(x)

∈ X ×R.
¯x ∈ X ϕ(¯x) ∈ R.
(i) ϕ ¯x

∂ϕ(¯x) ⊂ X
∗


∂ϕ(¯x) :=

x
∗
∈ X
∗
| (x
∗
, −1) ∈

N

(¯x, ϕ(¯x)); epiϕ

.
(ii) ϕ ¯x ∂ϕ(¯x) ⊂ X
∗
∂ϕ(¯x) :=

x
∗
∈ X
∗
| (x
∗
, −1) ∈ N

(¯x, ϕ(¯x)); epiϕ


.
∂ϕ(¯x) =

∂ϕ(¯x) := ∅ |ϕ(¯x)| = ∞

∂ϕ(¯x) ⊂ ∂ϕ(¯x)

∂ϕ(¯x)
∂ϕ(¯x) ϕ : R → R,
x → ϕ(x) := −|x| ¯x := 0 X ∂ϕ(¯x)
X ∂ϕ(¯x)
ϕ(¯x) ∈ R

∂ϕ(¯x) =

x
∗
∈ X
∗
| lim inf
x→¯x
ϕ(x) − ϕ(¯x) − x
∗
, x − ¯x
x − ¯x
 0

.
ϕ ϕ(¯x) ∈ R
∂ϕ(¯x) =


∂ϕ(¯x) =

x
∗
∈ X
∗
| x
∗
, x − ¯x ≤ ϕ(x) − ϕ(¯x), ∀x ∈ X

.
ϕ ¯x ∂ϕ(¯x) =

∇ϕ(¯x)

ϕ ¯x

∂ϕ(¯x) =

∇ϕ(¯x)

(i) Ω ⊂ X, δ
Ω
: X → R δ
Ω
(x) := 0
x ∈ Ω δ
Ω
(x) := ∞ x ∈ X\Ω, Ω.

(ii) ϕ : X → R E
ϕ
: X ⇒ R
E
ϕ
(x) :=

α ∈ R |α ≥ ϕ(x)

x ∈ X,
ϕ.
N(x; Ω) = ∂δ
Ω
(x)

N(x; Ω) =

∂δ
Ω
(x).
d
Ω
(x) = d(x; Ω) := inf
u∈Ω
u − x
x ∈ Ω,

N(x; Ω) =

λ>0

λ

∂d
Ω
(x) Ω
x ∈ Ω N(x; Ω) =

λ>0
λ∂d
Ω
(x). ∂ϕ(¯x) = D
∗
E
ϕ

¯x, ϕ(¯x)

(1)

∂ϕ(¯x) =

D
∗
E
ϕ

¯x, ϕ(¯x)

(1). A(¯x)
ϕ : X → R ¯x ∈ domϕ. X

¯x ∈ X ϕ ∈ A(¯x), ∂ϕ(¯x) = Lim sup
x
ϕ
−→¯x

∂ϕ(x).