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

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) ∈ X ×Y.
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

.
N(¯x; Ω) := ∅ ¯x ∈ Ω.
U X, f : U → 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.
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
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

.
X

N(¯x; Ω) =

T (¯x; Ω)

−
,
K
−
:=

x
∗
∈ X
∗
|x
∗
, v ≤ 0 ∀v ∈ K

K ⊂ X.

F : X ⇒ Y, F
DomF :=

x ∈ X |F (x) = ∅

gphF :=

(x, y) ∈ X ×Y |y ∈ F (x)

.
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)
F (¯x) = {¯y}, ¯y

D
∗
N
F (¯x) D
∗
N
F (¯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
¯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)); ϕ

.
∂ϕ(¯x) =

∂ϕ(¯x) := ∅ |ϕ(¯x)| = ∞.
g : X → Y
Θ ⊂ Y ¯y = g(¯x) ∈ Θ. g ¯x ∇g(¯x) : X → Y
N


¯x; g
−1
(Θ)

= ∇g(¯x)
∗
N(¯y; Θ)

N

¯x; g
−1
(Θ)

= ∇g(¯x)
∗

N(¯y; Θ).
f : X → Y ¯x ∈ X
F : X ⇒ Y ¯y − f(¯x) ∈ F (¯x), ¯y ∈ Y.
y
∗
∈ Y
∗
,
D
∗
N
(f + F )(¯x, ¯y)(y
∗

) = ∇f (¯x)
∗
y
∗
+ D
∗
N
F

¯x, ¯y − f(¯x)

(y
∗
)

D
∗
(f + F )(¯x, ¯y)(y
∗
) = ∇f (¯x)
∗
y
∗
+

D
∗
F

¯x, ¯y − f(¯x)


(y
∗
).
ψ : X → R ¯x ∈ X ϕ : X → R
¯x.
∂(ϕ + ψ)(¯x) = ∇ϕ(¯x) + ∂ψ(¯x)

∂(ϕ + ψ)(¯x) = ∇ϕ(¯x) +

∂ψ(¯x).
g : X → Y ¯x ∈ Ω, Ω := g
−1
(K)
K ⊂ Y. (P )

f(x) → inf,
x ∈ Ω,
f : X → R ¯x; f Ω
(P ).
¯x (P ) δ > 0
f(x) ≥ f(¯x) x ∈ Ω ∩B
δ
(¯x).
Y := R
m
, K := {0
R
p
} × R

m−p
−
(m, p ∈ N, p ≤ m)
g(x) =

g
1
(x), g
2
(x), , g
m
(x)

, (P )
¯x (P ),
λ = (λ
1
, , λ
m
) ∈ R
m



∇f(¯x) +
m

i=1
λ
i

∇g
i
(¯x) = 0, λ
j
≥ 0, ∀j ∈ I(¯x),
λ
i
g
i
(¯x) = 0, ∀i = 1, 2, , m.
dimX < ∞

T (¯x; Ω)

−
= ∇g(¯x)
∗
N

g(¯x); K


N

g(¯x); K

=

y
∗

∈ R
m
|λ
j
≥ 0, ∀j ∈ I(¯x), λ
i
g
i
(¯x) = 0, ∀i = 1, 2, , m

,
I(¯x) :=

j ∈ {p +1, , m}|g
j
(¯x) = 0

y
∗
:= (λ
1
, , λ
m
).
(P )
¯x y
∗
∈

N


g(¯x); K

∇f(¯x) + ∇g(¯x)
∗
y
∗
= 0.
dimX < ∞

N

¯x; Ω

= ∇g(¯x)
∗

N

g(¯x); K

.
g : X → Y ¯x ∈ Ω := g
−1
(K)
K ⊂ Y

N

¯x; Ω


= ∇g(¯x)
∗

N

g(¯x); K

f ¯x
(P ) ¯x ¯x (P ).
Y := R
m
, K := {0
R
p
} ×R
m−p
−
g : X → R
m
¯x ∈ Ω := g
−1
(K).
¯x.

N

¯x; Ω

= ∇g(¯x)

∗

N

g(¯x); K

.
X Y g : X → Y
K ⊂ Y ¯x ∈ Ω := g
−1
(K).
¯x 0 ∈ int

g(¯x) + ∇g(¯x)(X) −K

.
X Y g : X → Y
K ⊂ Y ¯x ∈ Ω := g
−1
(K).
¯x.

N

¯x; Ω

= ∇g(¯x)
∗

N


g(¯x); K

.
X
ϕ : X → R
a = b c ∈ [a, b) ψ(c) = min
x∈[a,b]
ψ(x),
ψ(x) := ϕ(x) +
ϕ(b) − ϕ(a)
b − a
x −b. x
k
ϕ
−→ c x
∗
k
∈

∂ϕ(x
k
)
lim inf
k→∞
x
∗
k
, b − x
k

 ≥
ϕ(b) − ϕ(a)
b − a
b − c, (2.1)
lim inf
k→∞
x
∗
k
, b − a ≥ ϕ(b) − ϕ(a), (2.2)
c = a
lim
k→∞
x
∗
k
, b − a = ϕ(b) − ϕ(a). (2.3)
X
X
ϕ : X → R x
k
ϕ
−→ c
x
∗
k
∈

∂ϕ(x
k

) (2.1) − (2.3)
X
X
T : X ⇒ X
∗
x
∗
1
− x
∗
2
, x
1
− x
2
 ≥ 0 (x
i
, x
∗
i
) ∈ gphT, i = 1, 2. T
T T
X X
∗
T : U ⇒ X
∗
T
e
: X ⇒ X
∗

U ⊂ X T
e
(x) := T (x)
x ∈ U T
e
(x) := ∅ x ∈ X\U.
X f : X → X
∗
u
∗
, u ≥ 0 x ∈ X u ∈ X ⊂ X
∗∗
u
∗
∈ D
∗
N
f(x)(u);
u
∗
, u ≥ 0 x ∈ X u ∈ X ⊂ X
∗∗
u
∗
∈ D
∗
M
f(x)(u);
u
∗

, u ≥ 0 x ∈ X u ∈ X ⊂ X
∗∗
u
∗
∈

D
∗
f(x)(u);
f
⇒ ⇒ ⇒ X ⇔ f
X ⇔ X
⇔ ⇔ ⇔
f : R → R f (x) = 0 x ∈ Q
f(x) = 1 x ∈ R\Q.

N

(x, f(x)); gphf

= {0} × R

D
∗
f(x)(u) = {0} x, u ∈ R. (c)

f(x
1
) − f(x
2

), x
1
− x
2

= x
2
− x
1
< 0, (x
1
, x
2
) ∈ Q ×

R\Q

,
x
1
> x
2
. f (c) ⇒ (d).
f
f : R
n
→ R
n
f
J

f
(x)
u
T
J
f
(x)u ≥ 0 x, u ∈ R
n
T : X ⇒ X X
T σ > 0
T − σI I X
X f : X → X
σ > 0 u
∗
, u ≥ σu
2
x, u ∈ X u
∗
∈ D
∗
M
f(x)(u);
σ > 0 u
∗
, u ≥ σu
2
x, u ∈ X u
∗
∈


D
∗
f(x)(u);
f
C
X f : X → X
∗
f C
f x ∈ C
z, u ≥ 0 ∀u ∈ C − C ⊂ X
∗∗
, ∀z ∈ D
∗
M
f(x)(u), ∀x ∈ C;
int C = ∅ f C x ∈ intC
z, u ≥ 0 ∀u ∈ intC −intC ⊂ X
∗∗
, ∀z ∈

D
∗
f(x)(u).
ϕ : X → R
¯x ∈ X ¯x
∗
∈ ∂ϕ(¯x) ϕ
(¯x, ¯x
∗
) ∂

2
N
ϕ(¯x, ¯x
∗
) : X
∗∗
⇒ X
∗
∂
2
N
ϕ(¯x, ¯x
∗
)(u) := D
∗
N

∂ϕ

(¯x, ¯x
∗
)(u) ∀u ∈ X
∗∗
.
ϕ (¯x, ¯x
∗
)
∂
2
M

ϕ(¯x, ¯x
∗
) : X
∗∗
⇒ X
∗
∂
2
M
ϕ(¯x, ¯x
∗
)(u) := D
∗
M

∂ϕ

(¯x, ¯x
∗
)(u) ∀u ∈ X
∗∗
.
¯x
∗
∈

∂ϕ(¯x)

∂
2

ϕ(¯x, ¯x
∗
) : X
∗∗
⇒ X
∗

∂
2
ϕ(¯x, ¯x
∗
)(u) :=

D
∗


∂ϕ

(¯x, ¯y)(u) ∀u ∈ X
∗∗
,
ϕ (¯x, ¯x
∗
).
¯x
∗
∈ ∂ϕ(¯x)
∂
2

N
ϕ(¯x, ¯x
∗
)(u) := ∅ ∂
2
M
ϕ(¯x, ¯x
∗
)(u) := ∅,
¯x
∗
∈

∂ϕ(¯x)

∂
2
ϕ(¯x, ¯x
∗
)(u) := ∅ u ∈ X
∗∗
.
∂ϕ(¯x) = {¯x
∗
}, ∂
2
N
ϕ(¯x) ∂
2
N

ϕ(¯x, ¯x
∗
) ∂
2
M
ϕ(¯x)
∂
2
M
ϕ(¯x, ¯x
∗
).

∂
2
ϕ(¯x)

∂
2
ϕ(¯x, ¯x
∗
)

∂ϕ(¯x) = {¯x
∗
}.
ϕ : X → R X
x ∈ X u
∗
, u ≥ 0 u ∈ X ⊂ X

∗∗
u
∗
∈ ∂
2
N
ϕ(x)(u).
x ∈ X u
∗
, u ≥ 0 u ∈ X ⊂ X
∗∗
u
∗
∈ ∂
2
M
ϕ(x)(u).
x ∈ X u
∗
, u ≥ 0 u ∈ X ⊂ X
∗∗
u
∗
∈

∂
2
ϕ(x)(u).
ϕ
⇒ ⇒ ⇒ X ⇔ ∇ϕ

X ⇔ X
⇔ ⇔ ⇔
ϕ : R
n
→ R ϕ
x ∈ R
n
H
ϕ
(x)
X
T :=

1, 2, , m

m ≥ 1

a
∗
i
∈ X
∗
| i ∈ T

X
X
X
b = (b
1
, b

2
, , b
m
) ∈ R
m
Θ(b) :=

x ∈ X | a
∗
i
, x ≤ b
i
, ∀i ∈ T

.
F : K → X
∗
K
X
∗
. x ∈ K

F (x), u −x

≥ 0 ∀u ∈ K,
(K, F ). F
K x ∈ K

F (x), u −x


≥ 0
u ∈ K VI(K, F )
K (K, F )
F
(K, F )
X = R
n
K = R
n
+
(K, F ) x ∈ R
n
0 ≤ F (x) ⊥ x ≥ 0.
f : Z ×X → X
∗
b ∈ R
m
p ∈ Z Z
x ∈ Θ(b)
f(p, x), u −x ≥ 0 u ∈ Θ(b),

f(p, ·); Θ(b)

x p, b
0 ∈ f(p, x) + N

x; Θ(b)

,
N


x; Θ(b)

Θ(b) x
N

x; Θ(b)

:=

x
∗
∈ X
∗
|x
∗
, u−x ≤ 0, ∀u ∈ Θ(b)

x ∈ Θ(b)
N

x; Θ(b)

:= ∅ x ∈ X\Θ(b) (p, b) ∈ Z ×R
m
,
S(p, b) :=

x ∈ X | 0 ∈ f(p, x) + N


x; Θ(b)


.
S : Z × R
m
⇒ X (p, b) → S(p, b),

f(p, ·); Θ(b)

X Y
F : X ⇒ Y
(¯x, ¯y) ∈ gphF  > 0 δ > 0
F (u) ∩B
δ
(¯y) ⊂ F (x) + u − xB
Y
u, x ∈ B
δ
(¯x).
{v
i
}
m
i=1
m

i=1
λ
i

v
i
= 0 λ
i
≥ 0,
i = 1, 2, , m, λ
i
= 0 i = 1, 2, , m.
(x, b) ∈ gphΘ
I(x, b) := {i ∈ T |a
∗
i
, x = b
i
},
b
i
i b ∈ R
m
∅ = I ⊂ T b
I
b
i
i ∈ I
¯
I := T \I
¯x
∗
∈ N


¯x, Θ(
¯
b)

, I := I(¯x,
¯
b),
Ξ(¯x,
¯
b, ¯x
∗
) :=

λ = (λ
j
)
j∈I
| λ
I
≥ 0, ¯x
∗
=

j∈I
λ
j
a
∗
j


I
1
(¯x,
¯
b, ¯x
∗
) :=

i ∈ I | ∃λ ∈ Ξ(¯x,
¯
b, ¯x
∗
) : λ
i
= 0

.
P Q P ⊂ Q ⊂ T
A
Q,P
:= span

a
∗
i
| i ∈ P } + pos{a
∗
j
| j ∈ Q\P


,
B
Q,P
:=

x ∈ X | a
∗
i
, x = 0 ∀ i ∈ P, a
∗
j
, x ≤ 0 ∀j ∈ Q\P

,
F
Q
:=

x ∈ X |a
∗
i
, x =
¯
b
i
, ∀i ∈ Q, a
∗
j
, x <
¯

b
j
, ∀j ∈ T \Q

,
span

a
∗
i
| i ∈ P } :=


i∈P
λ
i
a
i
|λ
i
∈ R ∀i ∈ P

pos

a
∗
i
| i ∈ Q\P } :=



i∈P
λ
i
a
i
|λ
i
≥ 0 ∀i ∈ Q\P

.
span ∅ = pos ∅ = {0}.
f : Z ×X → X
∗
(¯p, ¯x)
∇
p
f(¯p, ¯x) : Z → X
∗
S : Z ×R
m
⇒ X

f(p, ·); Θ(b)

. ¯x
∗
:= −f(¯p, ¯x) I := I(¯x,
¯
b)
J := I\I

1
(¯x,
¯
b, ¯x
∗
)
{a
∗
j
|j ∈ I} x
∗
∈ X
∗
,
D
∗
M
S(¯p,
¯
b, ¯x)(x
∗
) ⊂ D
∗
N
S(¯p,
¯
b, ¯x)(x
∗
)
⊂


(p
∗
, b
∗
) | −v ∈ B
Q,P
, p
∗
= ∇
p
f(¯p, ¯x)
∗
v, b
∗
∈ R
m
, b
∗
¯
Q
= 0,
b
∗
Q\P
≤ 0, x
∗
+ ∇
x
f(¯p, ¯x)

∗
v =

i∈Q
b
∗
i
a
∗
i
, J ⊂ P ⊂ Q ⊂ I

.
X x
∗
∈ X
∗
,
D
∗
N
S(¯p,
¯
b, ¯x)(x
∗
) = D
∗
M
S(¯p,
¯

b, ¯x)(x
∗
) ⊃

(p
∗
, b
∗
) | −v ∈ B
Q,P
,
p
∗
= ∇
p
f(¯p, ¯x)
∗
v, b
∗
∈ R
m
, b
∗
¯
Q
= 0, b
∗
Q\J
≤ 0, λ ∈ Ξ(¯x,
¯

b, ¯x
∗
),
x
∗
+ ∇
x
f(¯p, ¯x)
∗
v =

i∈Q
b
∗
i
a
∗
i
, J
1
(λ) ⊂ P ⊂ Q ⊂ I, F
Q
= ∅

.

f(p, ·); Θ(b)

S : Z × R
m

⇒ X

f(p, ·); Θ(b)

. (¯p,
¯
b, ¯x) ∈ gphS f : Z × X → X
∗
(¯p, ¯x) ∇
p
f(¯p, ¯x) : Z → X
∗
Z X ¯x
∗
:= −f(¯p, ¯x), I := I(¯x,
¯
b)
J := I\I
1
(¯x,
¯
b, ¯x
∗
)
(a) {a
∗
j
}
j∈I
S (¯p,

¯
b, ¯x);
(b) b
∗
∈ R
m
(P, Q) J ⊂ P ⊂ Q ⊂ I
−v ∈ B
Q,P
, b
∗
¯
Q
= 0, b
∗
Q\P
≤ 0 ∇
x
f(¯p, ¯x)
∗
v =

i∈Q
b
∗
i
a
∗
i
, (v, b

∗
) = (0, 0)
X S (¯p,
¯
b, ¯x)
λ ∈ Ξ(¯x,
¯
b, ¯x
∗
), b
∗
∈ R
m
, J
1
(λ) ⊂ P ⊂ Q ⊂ I, ∇
x
f(¯p, ¯x)
∗
v =

i∈Q
b
∗
i
a
∗
i
,
−v ∈ B

Q,P
, F
Q
= ∅, b
∗
¯
Q
= 0, b
∗
Q\J
≤ 0 (v, b
∗
) = (0, 0) S
(¯p,
¯
b, ¯x) {a
∗
j
}
j∈I
J
1
(λ) := {i ∈ I |λ
i
> 0}
¯
Q := T \Q.
X = Z = R, a
∗
1

= −1 f(x, p) := x
2
+ p ¯x = 0
¯
b = 0
S(p, b) =






√
−p

p < 0, −
√
−p < −b <
√
−p,

− b, −
√
−p,
√
−p

p ≤ 0, −b < −
√
−p,


− b

.
¯p = 1 {a
∗
j
}
j∈I(¯x,
¯
b)
S
(¯p,
¯
b, ¯x) ∈ gphS.
¯p = 0 {a
∗
j
}
j∈I(¯x,
¯
b)
S
(¯p,
¯
b, ¯x) ∈ gphS.
X = Z = R, a
∗
1
= −1, a

∗
2
= −2, f(x, p) := x
2
+ p ¯x = 0
¯p = 1
¯
b = (0, 0)
S(p, b) =
















√
−p

p < 0,
−

√
−p < max(−b
1
, −
b
2
2
) <
√
−p,

max(−b
1
, −
b
2
2
), −
√
−p,
√
−p

p ≤ 0,
max(−b
1
, −
b
2
2

) < −
√
−p,

max(−b
1
, −
b
2
2
)

.
S (¯p,
¯
b, ¯x) ∈ gphS.
{a
∗
j
}
j∈I(¯x,
¯
b)
X, Z S(p, b)

f(p, ·); Θ(b)

f : Z ×X → X
∗
b ∈ R

m
p ∈ Z
Θ(b) :=

x ∈ X | a
∗
i
, x ≤ b
i
, ∀i ∈ T

,
T :=

1, 2, , m

. S : Z × R
m
⇒ X, (p, b) → S(p, b),
(¯p,
¯
b, ¯x) ∈ gphS. {a
∗
j
|j ∈ I(¯x,
¯
b)}