tóm tắt luận án tiến sĩ một số vấn đề về phép tính vi phân và tích phân trong giải tích không trơn và lý thuyết tối ưu

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 ∈ λ
0
∂
Cl
f(¯x) +

i∈I
λ
i
∂
Cl
g
i
(¯x) +


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


i∈I
λ
i
∂g
i
(¯x) +

j∈J
∂(µ
j
h
j
)(¯x),
∂ λ
i
g
i
(¯x) = 0 ∀i ∈ I,
(P)
G(·)
L
1
(Ω; E)
f : X →
¯
R := [−∞, +∞] X.
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
f x v ∈ 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 v ∈ X, f

(x; v),
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).
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
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 : 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
X
f : U → R
U ⊂ X U
(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 =

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
).
X (X, A, µ)
A σ X.
f : U → R U ⊂ X Ω ⊂ U
µ(Ω) < ∞.

Ω
∂
Cl
f(x)dµ(x) = ∂
Cl

F (0)
=

x
∗
∈ X
∗
| x
∗
, v 

Ω
f
0
(x; v)dµ(x) ∀ v ∈ X

,
(2.1)
F (v) :=

Ω
f
0
(x; v)dµ(x).

Ω
∂
Cl
f(x)dµ(x) (2.1)
ξ

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

∗
x
, udµ(x).
f : X → Y X
Y.
f x
0
∈ X
D
s
f(x
0
) : X → Y
lim
x→x
0
, t→0
+
t
−1
(f(x + tv) − f(x)) = D
s
f(x
0
)(v)
v X D
s
f(x
0
)

f x
0
f

(x
0
) : X → Y
lim
x,x

x=x

−→x
0
f(x) − f(x
0
) − f

(x
0
)(x − x
0
)
x − x
0

= 0,
f x
0
f


(x
0
)
f x
0
f x
0
f

(x
0
) : X → Y
lim
x,x

x=x

−→x
0
f(x) − f(x

) − f

(x
0
)(x − x

)
x − x



= 0.
f

(x
0
) f x
0
f x
0
f
x
0
f

(x
0
) = D
s
f(x
0
). X
X
f : U → R
U R
n
Ω ⊂ U µ(Ω) < ∞.
(i)


Ω
∂
Cl
f(x)dµ(x)
(ii) v ∈ R
n
, f

(x), v = f
0
(x; v) Ω
(iii) f Ω
(iv) f Ω.
(i) (iv)

Ω
∂
Cl
f(x)dµ(x) =


Ω
f

(x)dµ(x)

.

b
a

f

(t)dt = f (b) − f (a)
f

(x)
∂
Cl
f(x)
f : [a, b] → R (a, b ∈ R, a < b)
f(b) − f(a) ∈
b

a
∂
Cl
f(x)dx (2.6)
b

a
∂
Cl
f(x)dx =

f(b) − f(a)

f [a, b].
(2.6)
{r
k

}
k∈N
(a, b) ⊂ R
a < b k ∈ N δ
k
> 0 (r
k
− δ
k
, r
k
+ δ
k
) ⊂ (a, b) δ
k
<
2
−(k+3)
(b − a). A = ∪
∞
k=1
(r
k
− δ
k
, r
k
+ δ
k
) P = [a, b]\A. A

R P A = ∪
∞
j=1
(a
j
, b
j
) {(a
j
, b
j
)}
j∈N
f : [a, b] → R
f(x) =



0 x ∈ P,
(x − a
j
)
2
(x − b
j
)
2
sin
1
(b

j
− a
j
)(x − a
j
)(x − b
j
)
x ∈ (a
j
, b
j
).
f [a, b] I :=

b
a
∂
Cl
f(t)dt
X f, g : X → R
f ∂
Cl
g(x) ⊂ ∂
Cl
f(x)
x ∈ X, α ∈ R f (x) = g(x) + α x ∈ X.
X f
∂
Cl

g(x) ⊂ ∂
Cl
f(x) x ∈ X
f, g : R
n
→ R f
∂
Cl
g(x) ⊂ ∂
Cl
f(x) R
n
α ∈ R
f(x) = g(x) + α x ∈ R
n
.
f : U → R
U ⊂ R
n
Ω ⊂ U µ(Ω) < ∞.

Ω
∂f(x)dµ(x) =

x
∗
∈ R
n
| x
∗

, v 

Ω
f
0
(x; v)dµ(x) ∀v ∈ R
n

.
f : [a, b] → R
f
0
(x; v) = max
ξ∈∂
Cl
f(x)
ξ, v =

|v| x ∈ P ∩ (a, b),

f

(x)v

x ∈ A,
v ∈ R
b

a
∂f(x)dx =


x
∗
∈ R | x
∗
, v  µ(P )|v| ∀v ∈ R

= [−µ(P ), µ(P )].
f : [a, b] → R (a, b ∈ R, a < b)
f(b) − f(a) ∈
b

a
∂f(x)dx
b

a
∂f(x)dx =

f(b) − f(a)

f [a, b].
F (x) =
x

a
f(t)dt, (3.1)
f [a, b] ⊂ R f
M > 0 |f(x)|  M [a, b]
L

∞
[a, b] [a, b].
f
+
(x) = inf

M | ∃ ε > 0 f(x

)  M [x − ε, x + ε]

,
f
+
+
(x) = inf

M | ∃ ε > 0 f(x

)  M [x, x + ε]

,
f
−
(x) = sup

M | ∃ ε > 0 f(x

)  M [x − ε, x + ε]

,

f
−
−
(x) = sup

M | ∃ ε > 0 f(x

)  M [x − ε, x]

.
f
−
(x)  f
−
−
(x)  f
+
(x) f
−
(x)  f
+
+
(x)  f
+
(x).

f
−
(x), f
+

+
(x)



f
−
−
(x), f
+
(x)

⊂

f
−
(x), f
+
(x)

.
f ∈ L
∞
[a, b], F (3.1)
x ∈ (a, b)
∂F (x) =

f
−
(x), f

+
+
(x)



f
−
−
(x), f
+
(x)

. (3.2)
∂F (x)
(3.2)
E [0, 1]
[0, 1] E [0, 1]\E
f(t) = 1 t ∈ E f(t) = 0 t ∈ [0, 1]\E
F (x) =
x

0
f(t)dt (x ∈ [0, 1]) f ∈ L
∞
[0, 1]
f
+
(x) = f
+

+
(x) = 1 f
−
(x) = f
−
−
(x) = 0 x ∈ (0, 1).
∂F (x) = [0, 1] x ∈ (0, 1).
E x
0
∈ E ∩ (0, 1)
f : [0, 1] → R
f(t) =











1 t ∈ [x
0
, 1]

E,
0 t ∈ [x

0
, 1]\E,
2 t ∈ [0, x
0
)

E,
3 t ∈ [0, x
0
)\E.
F (x) =

x
0
f(t)dt x ∈ [0, 1] f ∈ L
∞
[0, 1] f
+
(x
0
) = 3 f
+
+
(x
0
) =
1, f
−
(x
0

) = 0 f
−
−
(x
0
) = 2.
∂F (x
0
) =

0, 1

∪

2, 3

.

∂F (x) = ∅.
∂F (x) =

f
−
(x), f
+
(x)

,
∂F (x) = ∂
Cl

F (x).
∂F (x)

∂F (x) = ∅
ϕ : I → R
I R x ∈ I

∂ϕ(x) = ∅. ∂ϕ(x) = ∂
Cl
ϕ(x).
ϕ x
∂
0
ϕ(x) := ∂ϕ(x)∪[−∂(−ϕ)(x)] ∂ϕ(x) ⊂ ∂
0
ϕ(x) ⊂ ∂
Cl
ϕ(x) ϕ
x

∂ϕ(x) = {ϕ

(x)} = ∅
I R ϕ : I → R
∂ϕ(x) = ∂
Cl
ϕ(x) = ∂
0
ϕ(x).
L

1
(Ω; E)
(Ω, A, µ) σ−
E f : Ω×E →
¯
R A⊗B(E)−
F (u) =

Ω
f(ω, u(ω))dµ(ω) (u ∈ L
1
(Ω; E)). (3.16)
s : Ω → E
s =
m

i=1
c
i
χ
A
i
,
m ∈ N c
i
∈ E, A
i
∈ A i = 1, 2, , m Ω =
m


i=1
A
i
χ
A
(ω) = 1 ω ∈ A χ
A
(ω) = 0 ω ∈ X\A
u : Ω → E
s
k
: Ω → E
lim
k→∞
s
k
(ω) − u(ω)
E
= 0
s : Ω → E
s =
m

i=1
c
i
χ
A
i
,

m ∈ N c
i
∈ E, A
i
∈ A i = 1, 2, , m Ω =
m

i=1
A
i
c
i
= 0 µ(A
i
) = ∞. A ∈ A, s A

A
sdµ :=
m

i=1
µ(A ∩ A
i
)c
i
,
µ(A ∩ A
i
)c
i

:= 0 c
i
= 0 µ(A ∩ A
i
) = ∞
u : Ω → E
s
k
: Ω → E
lim
k→∞
s
k
(ω) − u(ω)
E
= 0
lim
k→∞

Ω
s
k
− udµ = 0.
A ∈ A u A

A
udµ := lim
k→∞

A

s
k
dµ.
L
1
(Ω; E) u : Ω → E
Ω u :=

Ω
u(ω)dµ u ∈ L
1
(Ω; E).
u ∈ L
1
(Ω; E),
I
f
(u) =

∗
Ω
f(ω, u(ω))dµ
:= inf


Ω
v(ω)dµ | v ∈ L
1
(Ω; R), v(ω) ≥ f(ω, u(ω))


.
(3.18)
ω → f(ω, u(ω)) Ω I
f
(u) = F (u),
F (u) (3.16)
v : Ω → E
∗ ∗
e ∈ E
Ω  ω → v(ω), e L
w
∞
(Ω; E
∗
)
∗
v : Ω → E
∗
Ω  ω → v(ω) L
∞
(Ω; R).
L
w
∞
(Ω; E
∗
) v
L
w
∞

(Ω;E
∗
)
= ess sup
ω∈Ω
v(ω),
ess sup
ω∈Ω
v(ω) = inf{α > 0 | v(ω) < α }.
L
p
(Ω, A, µ) σ− E
(i) T ∈ (L
1
(Ω; E))
∗
v ∈ L
w
∞
(Ω; E
∗
)
T (u) =

Ω
v(ω), u(ω)dµ (3.19)
u ∈ L
1
(Ω; E). T  = v
L

w
∞
(Ω;E
∗
)
(ii) T (3.19) v ∈ L
w
∞
(Ω; E
∗
),
L
1
(Ω; E)
I
f
(·) : L
1
(Ω; E) →
¯
R
(3.18) x ∈ L
1
(Ω; E) f (x) ∈ L
1
(Ω; R)

∂
ε
I

f
(x) =

x
∗
∈ L
w
∞
(Ω; E
∗
) | inf
e∈E
g
ε
(ω, e, x
∗
(ω)) ≥ 0 h.k.n.

=

x
∗
∈ L
w
∞
(Ω; E
∗
) | I
f
(u) − I

f
(x) − x
∗
, u − x
≥ −εu − x ∀u ∈ L
1
(Ω; E)

,
g
ε
(ω, e, e
∗
) := f (ω, e) − f(ω, x(ω)) − e
∗
, e − x(ω) + εe − x(ω), ω ∈ Ω,
e ∈ E, e
∗
∈ E
∗
, ε ≥ 0.
f : Ω × E →
¯
R A ⊗ B(E)−
f(u) ∈ L
1
(Ω; R) u ∈ L
1
(Ω; E) F (3.16)
∂F (x) =


∂F (x) = ∂
F en
F (x)
=

x
∗
∈ L
w
∞
(Ω; E
∗
) | inf
e∈E
g
0
(ω, e, x
∗
(ω)) ≥ 0 h.k.n.

,
g
0
(ω, e, e
∗
) := f (ω, e) − f(ω, x(ω)) − e
∗
, e − x(ω), ω ∈ Ω, e ∈ E, e
∗

∈ E
∗
x ∈ L
1
(Ω; E).
F
x
lim
k→∞
x
∗
k
− F

(x) = 0 x
∗
k
∈ ∂F (x
k
) lim
k→∞
x
k
= x.
F F
(P) min{F (x) | x ∈ L
1
(Ω; E)},
F (x) =


Ω
f(ω, x(ω))dµ(ω) (x ∈ L
1
(Ω; E))
x
(P)
min
e∈E
f(ω, e) = f (ω, x(ω))
f : X → R ∪ {+∞}
X
X
(i) X
(ii) f : X → R ∪ {+∞}
lim
x→∞
f(x)
x
= +∞, (4.1)

x∈X

∂f(x) = X
∗
.
(iii) Ω ⊂ X,

x∈Ω

N(x; Ω) = X

∗
.
X f : X → R ∪ {+∞}
f
(4.1)

x∈X

∂f(x) X
∗
X = 
2
e
n
:= (0, , 0, 1, 0, ),
n f : X → R ∪ {+∞}
f(x) :=











1
n

+
t
2

1
n + 1
−
1
n

x = (1 − t)e
n
+ te
n+1
(t ∈ [0, 1) n = 1, 2, ),
+∞ x ∈ X\

∞

n=1
[e
n
, e
n+1
)

,
[e
n
, e

n+1
) :=

(1− t)e
n
+te
n+1
| t ∈ [0, 1)

.

x∈X

∂f(x) = X
∗
.
X Ω
X.

x∈Ω

N(x; Ω) X
∗
.
X = 
2
e
n
:= (0, , 0, 1, 0, ) n.
Ω =

∞

n=1
[e
n
, e
n+1
], [e
n
, e
n+1
] :=

e
n
+ t(e
n+1
− e
n
) | t ∈ [0, 1]

. Ω
X

x∈Ω

N(x; Ω) = X
∗
.
(P

0
) min{f(x) | x ∈ X},
X f : X → R ∪ {+∞}
¯x ∈ X (P
0
) 0 ∈

∂f(¯x)
X f : X → R ∪ {+∞}
(4.1)
c ∈ X
∗
(P
c
) min{f(x) + c, x | x ∈ X}
X f : X → R ∪ {+∞}
(4.1)
c ∈ X
∗
(P
c
)
(P
c
)
(P
0
) c, x f(x)
(P
0

)
(P
0
)
X f : X → R ∪ {+∞}
X
(4.1) C X
∗
c ∈ C (P
c
)
X f : X → R ∪ {+∞}
X (4.1) C
X
∗
c ∈ C (P
c
)
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
∗
X
•
•
•
•
•
•
L
1
(Ω, E)