Một số vấn đề về phép tính vi phân và tích phân trong giải tích không trơn và lý thuyết tối ưu

L
1
(Ω; E)
X
∗
L
1
(Ω; E)
X
∗
L
1
(Ω; E)
F : X ⇒ Y X Y
R
¯
R := R ∪ {±∞}
Q
N
X
∗
X
x
∗
, x X
∗
X
x x

x
X
x X
|x| x ∈ R
{x
i
}
σ(K, v) K v
∅
∃x x
∀x x
¯
A A
coM M
f

(x) f x
f

(x; v) v f x
f
0
(x; v) v f x
∂
Cl
f(x) f x
∂f(x) f x

∂f(x) f x
∂

F en
f(x) f x
x
∗
k
w
∗
→ x
∗
{x
∗
k
} x
∗ ∗
w
∗
x := y x y
✷
f : [a, b] → R a, b ∈ R

b
a
f

(t)dt = f(b) − f(a)
f

(·)
∂

Cl
f(·) ( ∂f(·))
G(x) =

Ω
g(ω, x)dµ(ω),
g Ω ×U U
(Ω, µ)
(P) min{f(x) | x ∈ X, g
i
(x) ≤ 0 ∀i ∈ I, h
j
(x) = 0 ∀j ∈ J},
X I J f, g
i
, h
j
X
¯x
(P) f, g
i
(i ∈ I), h
j
(j ∈ J)
¯x λ
0
≥ 0 λ
i
≥ 0 (i ∈ I) µ
j

∈ R
(j ∈ J)
0 ∈ ∂
Cl
x
L(¯x, λ, µ)
λ
i
g
i
(¯x) = 0 ∀i ∈ I,
L(x, λ, µ) := λ
0
f(x) +

i∈I
λ
i
g
i
(x) +

j∈J
µ
j
h
j
(x)
(P) ∂
Cl

x
L(¯x, λ, µ)
L(·, λ, µ) ¯x
X ¯x (P)
f, g
i
(i ∈ I), h
j
(j ∈ J) ¯x
λ
0
≥ 0 λ
i
≥ 0 (i ∈ I) µ
j
∈ R (j ∈ J)
0 ∈ ∂
x
L(¯x, λ, µ),
∂
x
L(¯x, λ, µ)
L(·, λ, µ) ¯x
f, g
i
(i ∈ I), h
j
(j ∈ J) ¯x
∂
Cl

x
L(¯x, λ, µ) ⊂ λ
0
∂
Cl
f(¯x) +

i∈I
λ
i
∂
Cl
g
i
(¯x) +

j∈J
µ
j
∂
Cl
h
j
(¯x).
0 ∈ λ
0
∂
Cl
f(¯x) +


i∈I
λ
i
∂
Cl
g
i
(¯x) +

j∈J
µ
j
∂
Cl
h
j
(¯x).
f, g
i
(i ∈ I), h
j
(j ∈ J) ¯x
∂
x
L(¯x, λ, µ) ⊂ λ
0
∂f(¯x) +

i∈I
λ

i
∂g
i
(¯x) +

j∈J
∂(µ
j
h
j
)(¯x).
0 ∈ λ
0
∂f(¯x) +

i∈I
λ
i
∂g
i
(¯x) +

j∈J
∂(µ
j
h
j
)(¯x).
(P)
G(·)

∂F (¯x)
F (x) =

x
a
f(t)dt,
f
F (u) =

Ω
f(ω, u(ω))dµ(ω) (u ∈ L
1
(Ω; E)),
(Ω, A, µ) σ−
E f : Ω × E →
¯
R
A ⊗ B(E)−
X f : X →
¯
R := [−∞, +∞]
X X
∗
X
∗
X x
∗
, x. X

X
∗
B
X
B
X
∗
G: X ⇒ X
∗
Lim sup
u→x
G(x) :=

x
∗
∈ X
∗



∃u
k
→ x, x
∗
k
w
∗
−→ x
∗
,

x
∗
k
∈ G(u
k
) ∀k = 1, 2, . . .

X
∗
w
∗
X
∗
u
f
→ x f : X →
¯
R u
Ω
→ x
Ω ⊂ X u → x f(u) → f(x)
u → x u ∈ Ω. t → t
+
0
t ↓ t
0
t → t
0
t > t
0

t → t
0
t ≥ t
0
f
x ∈ X  > 0 f
x U x
|f(x
1
) −f(x
2
)| ≤ x
1
− x
2
 ∀x
1
, x
2
∈ U.
v ∈ X f x
f
0
(x; v) := lim sup
x

→x, t→0
+
f(x


+ tv) −f(x

)
t
.
f x
∂
Cl
f(x) :=

ξ
∗
∈ X
∗
| ξ
∗
, v  f
0
(x; v) ∀v ∈ X

.
f
x 
(i) v → f
0
(x; v) |f
0
(x; v)|  v v ∈ X;
(ii) (u, v) → f
0

(u; v) (x, v) v → f
0
(x; v)
X 
(iii) ∂
Cl
f(x)
∗
X
∗
ξ
∗
   ξ
∗
∈ ∂
Cl
f(x);
(iv) v ∈ X, f
0
(x; v) = max{ξ
∗
, v | ξ
∗
∈ ∂
Cl
f(x)};
(v) X = R
n
∂
Cl

f(·) x
∂
Cl
f(x) = co

lim f

(x
k
) | x
k
→ x, x
k
∈ S, x
k
∈ Ω
f

,
Ω
f
:= {u ∈ R
n
| f u}, S R
n
F (·) := ∂
Cl
f(·)
W ⊂ R
n

F (x) ⊂ W U x F (u) ⊂ W
u ∈ U.
v ∈ X f x
f

(x; v) := lim
t→0
+
f(x + tv) − f(x)
t
,
f
x ∈ X. f x v ∈ X
f

(x; v) f

(x; v) = f
0
(x; v).
f : R → R f(0) = 0 f(x) = x
2
sin
1
x
x ∈ R\{0} 0
0.
(Ω, A, µ)
U X g
ω

: U → R
ω ∈ Ω
(i) v ∈ U, ω → g
ω
(v)
(ii) k(·) ∈ L
1
(Ω, R)
|g
ω
(v
1
) −g
ω
(v
2
)|  k(ω)v
1
− v
2
 ∀v
1
, v
2
∈ U, ∀ω ∈ Ω.
F (v) :=

Ω
g
ω

(v)dµ(ω)
v
0
∈ U. F U
∂
Cl
F (v) ⊂

Ω
∂
Cl
g
ω
(v)dµ(ω) ∀v ∈ X.
ω ∈ Ω g
ω
(·) v, F
v (1.1)

Ω
∂
Cl
g
ω
(v)dµ(ω)
ξ
∗
∈

Ω

∂
Cl
g
ω
(v)dµ(ω)
ξ
∗
∈ X
∗
ω → ξ
∗
ω
Ω X
∗
ξ
∗
ω
∈ ∂
Cl
g
ω
(v) x ∈ X ω → ξ
∗
ω
, x
Ω ξ
∗
, x =

Ω

ξ
∗
ω
, xdµ(ω).
ε ≥ 0, ε f
x ∈ X f(x) ∈ R

∂
ε
f(x) :=

x
∗
∈ X
∗



lim inf
u→x
f(u) −f(x) −x
∗
, u −x
u −x
≥ −ε

.
|f(x)| = ∞

∂

ε
f(x) = ∅. ε = 0

∂
0
f(x)

∂f(x) f x.
∂f(x) := Lim sup
u
f
−→x
ε↓0

∂
ε
f(u)
f x x
∗
∈ ∂f(x) u
k
f
→ x,
ε
k
↓ 0, x
∗
k
∈


∂
ε
k
f(u
k
) x
∗
k
w
∗
→ x
∗
.

∂f(x)
∗
∂f(x)

∂f(x) ⊂ ∂f(x).
f g
X
¯
R, x. f x,

∂(f + g)(x) = f

(x) +

∂g(x).
g = 0


∂g(x) = {0}

∂f(x) = {f

(x)} f x

∂f(x) f x.
f(x) = |x|(sin(ln |x|) + 1) x = 0 f(0) = 0;

∂f(0) = {0}, f x = 0.
Ω ⊂ X δ(x; Ω) = 0 x ∈ Ω
δ(x; Ω) = +∞ x ∈ X\Ω
Ω x ∈ X

N(x; Ω) :=

∂δ(x; Ω) N(x; Ω) := ∂δ(x; Ω).
f x ∈ X f(x) ∈ R
∂
F en
f(x) := {x
∗
∈ X
∗
| f(u) − f(x) ≥ x
∗
, u −x ∀u ∈ X}.
f ∂f(x) =


∂f(x) = ∂
F en
f(x).
f : X →
¯
R x ∈ X
f(x)  lim inf
u→x
f(u) lim inf
u→x
f(u) := sup
U∈N (x)
inf
u∈U
f(u) N(x)
X x f
x U ∈ N(x) f u ∈ U
X X
X
f : U → R
U ⊂ X U

1
C[0, 1] L
1
[0, 1]
B
X
∗
X

∗
X
∗
LS(x) (f
1
, f
2
) f
i
: X →
¯
R
(i = 1, 2) f
1
x ∈ domf
1
∩ domf
2
f
2
x.
X
(i) Ω ⊂ X x ∈ Ω
N(x; Ω) = Lim sup
u → x

N(u; Ω);
(ii) f : X →
¯
R x ∈ domf

∂f(x) = Lim sup
f
u → x

∂f(u);
(iii) (f
1
, f
2
) ∈ LS(x) ε  0 γ > 0

∂
ε
(f
1
+ f
2
)(x) ⊂



∂f
1
(x
1
) +

∂f
2
(x

2
) | x
i
∈ x + γB,
|f
i
(x
i
) −f
i
(x)|  γ, i = 1, 2

+ (ε + γ)B
∗
.
(iii)
X f :
X →
¯
R x ∂
Cl
f(x) = co
∗
∂f(x),
co
∗
∗
X
∂
Cl

f(x)
∂f(x)
(Ω, A, µ) σ− G : Ω ⇒ R
n
Ω R
n
G
G
−1
(W ) := {ω ∈ Ω | G(ω) ∩ W = ∅} ∈ A
W ⊂ R
n
G k(·) ∈ L
1
(Ω)
G(ω) ⊂ k(ω)B
R
n
Ω L
1
(Ω)
Ω R
G : Ω ⇒ R
n
, Ω
R
n
,
G
G G,

G =

g ∈ L
1
(Ω; R
n
) | g(ω) ∈ G(ω) Ω

.
G Ω
G :

Ω
Gdµ :=


Ω
gdµ | g ∈ G

,

Ω
gdµ =


Ω
g
1
dµ, ,


Ω
g
n
dµ

g = (g
1
, , g
n
).