Bài toán tựa cân bằng tổng quát và một số ứng dụng

BỘ GIÁO DỤC VÀ ĐÀO TẠO
ĐẠI HỌC THÁI NGUYÊN



NGUYỄN THỊ QUỲNH ANH



BÀI TOÁN TỰA CÂN BẰNG TỔNG QUÁT
VÀ MỘT SỐ ỨNG DỤNG




LUẬN ÁN TIẾN SĨ TOÁN HỌC





THÁI NGUYÊN - 2015

BỘ GIÁO DỤC VÀ ĐÀO TẠO
ĐẠI HỌC THÁI NGUYÊN



NGUYỄN THỊ QUỲNH ANH

BÀI TOÁN TỰA CÂN BẰNG TỔNG QUÁT
VÀ MỘT SỐ ỨNG DỤNG
Chuyên ngành: Toán Giải tích
Mã số: 62 46 01 02


LUẬN ÁN TIẾN SĨ TOÁN HỌC


Người hướng dẫn khoa học: GS. TSKH. Nguyễn Xuân Tấn



THÁI NGUYÊN - 2015
N
∗
Q
R
R
+
R
−
R
n
n−

R
n
+
R
n
R
n
−
R
n
X
∗
X
2
X
X
T, K ξ ∈ T ⊆ L(X, Y ) x ∈ K ⊆ X
i = 1, n i = 1, 2, , n
{x
α
}
x
n
 x x
n
x
∅
F : X → 2
Y
X Y

F F
F F
C

C
C
+
C
C
−
C
A ⊆ B A B
A ⊆ B A B
A ∪ B A B
A ∩ B A B
A \ B A B
A + B A B
A × B A B
A A
A A
A A
f D ¯x ∈ D
f(¯x) ≤ f(x), x ∈ D,
D X f : D → R
D
R
n
, G : D → R
n

¯x ∈ D
G(x), x − x ≥ 0 x ∈ D.
f D,
G(x) = ∇f(x)
ϕ : D → R
D X
∗
, G : D → X
∗
ϕ : D → R ¯x ∈ D
G(x), x − x + ϕ(x) − ϕ(¯x) ≥ 0 x ∈ D.
¯x ∈ D ⊆ R
n
G(x), x − x ≥ 0 x ∈ D.
D
G
T : D → X
¯x ∈ D
¯x = T (¯x).
T G := I − T I
D
D
X, ϕ : D × D → R. ¯x ∈ D
ϕ(t, ¯x) ≥ 0 t ∈ D.
X, Z
D ⊆ X, K ⊆ Z
S : D × K → 2
D
, T : D × K → 2
K

F : K × D × D → R
(¯x, ¯y) ∈ D × K
1) ¯x ∈ S(¯x, ¯y), ¯y ∈ T(¯x, ¯y),
2) F (¯y, ¯x, ¯x) = min
t∈S(x,y)
F (¯y, ¯x, t).
F y
F (x, x) = 0 x ∈ D, S(x, y) ≡ D ϕ(t, x) = F (x, t)
x, t ∈ D. 0 = F (¯x, ¯x) ≤ F(¯x, t), ∀t ∈ D,
ϕ(t, ¯x) ≥ 0 t ∈ D
X, Y
D X C
Y C Y : x  y
x − y ∈ C.
A ⊆ Y,
αMin(A/C) α A C, α
¯x ∈ D
F (¯x) ∈ αMin(F (D)/C),
F : D → Y α ¯x
F (¯x) α
D S.
D
X X
∗
. S : D → 2
D
, P : D → 2
X
∗
ϕ : D → R ¯x ∈ D, ¯x ∈ S(¯x) ¯y ∈ P(¯x)

y, x − x + ϕ(x) − ϕ(x) ≥ 0 x ∈ S(x),
X, Z, Y
D ⊆ X, K ⊆ Z C ⊆ Y
S : D × K → 2
D
, T : D × K → 2
K
, P
i
: D → 2
D
, i = 1, 2, Q : D × D → 2
K
, F :
K × D × D → 2
Y
,
(¯x, ¯y) ∈
D × K
1) ¯x ∈ S(¯x, ¯y), ¯y ∈ T(¯x, ¯y),
2) F (¯y, ¯x, t) ⊆ F (¯y, ¯x, ¯x) + C t ∈ S(¯x, ¯y),

F (¯y, ¯x, t) ∩ F (¯y, ¯x, ¯x) + C = ∅ t ∈ S(¯x, ¯y)

.
(¯x, ¯y) ∈ D × K
1) ¯x ∈ S(¯x, ¯y), ¯y ∈ T(¯x, ¯y),
2) F (¯y, ¯x, t) ⊆ F (¯y, ¯x, ¯x) − (C \ {0}) t ∈ S(¯x, ¯y),

F (¯y, ¯x, t) ∩ F (¯y, ¯x, ¯x) − (C \ {0}) = ∅ t ∈ S(¯x, ¯y)


.
¯x ∈ D ¯x ∈ P (¯x)
F (y, ¯x, t) ⊆ F (y, ¯x, ¯x) + C t ∈ P (¯x), y ∈ Q(¯x, t),

F (y, ¯x, t) ∩ F (y, ¯x, ¯x) + C = ∅ t ∈ P (¯x), y ∈ Q(¯x, t)

,
¯x ∈ D ¯x ∈ P
1
(¯x)
F (y, ¯x, t) ⊆ F (y, ¯x, ¯x) − (C \ {0}) t ∈ P
2
(¯x), y ∈ Q(¯x, t),

F (y, ¯x, ¯x) ⊆ F (y, ¯x, t) + (C \ {0} t ∈ P
2
(¯x), y ∈ Q(¯x, t)

.
¯x ∈ D ¯x ∈ P
1
(¯x)
0 ∈ F (y, ¯x, t) t ∈ P
2
(¯x) y ∈ Q(¯x, t).
(¯x, ¯y) ∈ D × K
1) ¯x ∈ S(¯x, ¯y), ¯y ∈ T (¯x, ¯y),
2) 0 ∈ F (¯y, ¯y, ¯x, t) t ∈ S(¯x, ¯y),
3) 0 ∈ G(y, ¯x, t) t ∈ P (¯x), y ∈ Q(¯x, t).

X, Y
1
, Y
2
, Z
F : K × K ×D ×D → 2
Y
, G : K × D × D → 2
Y
P, Q, S, T
X
G X X
∅, X G
G
G G;
G
G G.
X G X (X, G)
X
G
G, G

X, G ⊆ G

G
G

G


G.
(X, d), τ X
X, d,
(X, G), A ⊆ X.
U X A U
A;
x ∈ X {x}.
X, Y
f : X → Y x ∈ X
U f(x) Y, V x X
f(V ) ⊆ U.
f X f
X.
(X, G),
x ∈ X, V
x
x
G x x
U x V ∈ V
x
x ∈ V ⊆ U.
V G G X
G V.
M G G
X M
G.
(X, G)
x, y ∈ X U
x, V y U ∩ V = ∅.
X K.

τ X X
(+) : X × X → X,
(x, y) → x + y;
(.) : K × X → X,
(λ, x) → λx,
K
(X, τ), X K,
τ X.
X
X
X X
X
x + y
2
≤ x
2
+ 2y, x + y, ∀x, y ∈ X.
t ∈ [0, 1] (1 − t)x + ty
2
= (1 − t)x
2
+ ty
2
− (1 − t)tx −
y
2
, x, y ∈ X.
T X, F : X → X L−
η− T
t

= T x − tµF(T x), x ∈ H, t ∈ [0, 1]
T
t
x − T
t
y ≤ (1 − λ
t
τ)x − y, x, y ∈ X,
µ ∈ (0, 2η/L
2
) τ = 1 −

1 − µ(2η − µL
2
) ∈ (0, 1).
D
X. T : D → D T
I − T {x
k
} D
x ∈ D {(I − T )x
k
} y
(I − T)x = y
Y C ⊆ Y. C
Y tc ∈ C, ∀c ∈ C, t ≥ 0.
Y C Y clC, intC,
convC C.
X, Y C :
X → 2

Y
C(x) Y x ∈ X ∩domC
C C
l(C) = C ∩ (−C) C C
l(C) = {0}
C Y
∀x, y ∈ Y, x 
C
y x − y ∈ C, x  y
∀x, y ∈ Y, x  y x − y ∈ C\l(C)
∀x, y ∈ Y, x  y x − y ∈ intC
C
Y. C
x  y x  y x = y
Y Y
∗
Y < ξ, y > ξ ∈ Y
∗
y ∈ Y C

C
+
C
C

= {ξ ∈ Y
∗
|ξ, c ≥ 0, c ∈ C},
C
+

= {ξ ∈ Y
∗
|ξ, c > 0, c ∈ C \ l(C)}.
Y
C, A Y.
x ∈ A A C
y − x ∈ C y ∈ A.
A C IMin(A|C)
x ∈ A A
C y ∈ A, y = x x − y ∈ C \ l(C).
A C PMin(A|C)
Min(A|C).
x ∈ A A C
intC = ∅ C = Y ) x ∈ Min(A| (intC ∪ {0}) x
A (intC ∪ {0}) .
A C WMin(A|C)
WMin(A).
x ∈ A A C
˜
C C \ l(C)
x ∈ PMin(A|
˜
C)
A C PrMin(A|C)
IMin(A|C) ⊆ PrMin(A|C) ⊆ Min(A|C) ⊆ WMin(A|C).
X, Y, D ⊆ X
F : D → Y x ∈ D
F (x) Y F (x)
2
Y

Y F : D → 2
Y
D Y
x ∈ X F (x) F
F : X → Y.
D ⊆ X,
G : D → 2
Y
G = {x ∈ D| G(x) = ∅} ,
(G) = {(x, y) ∈ D × Y | y ∈ G(x)} .
G Gr(G)
X × Y.
G clG(D) G(D)
Y
G y ∈ Y, G
−1
(y) = {x ∈
D | y ∈ G(x)}
G(x) x ∈ D G
G {x
α
} ⊆ D, {y
α
} ⊆
Y, x
α
→ x, y
α
∈ G(x
α

) y ∈ G(x)
X, Y F : X → 2
Y
X coF : X → 2
Y
( F )(x) = F (x),
X, Y D ⊆ X f
D Y x ∈ X V f(x)
U x f(x

) ∈ V x

∈ U ∩ D.
f(x) ∈ V F (x) ⊆ V F (x) ∩ V = ∅.
D ⊆ X, F : D → 2
Y
F
¯x ∈ D V F (¯x) F (¯x) ∩ V = ∅)
U ¯x F (x) ⊆ V F (x) ∩ V = ∅)
x ∈ U ∩ D.
F D
x ∈ D.
X = Y = R, D = [−a, a], a ∈ R, a > 0.
F : D → 2
R
, F (x) =



{0}, x = 0,

[ − a, a], x = 0.
F x = 0. V V ∩ F (0) = ∅
V 0 = F (0)
U x = 0 x

∈ U, x

= 0 F (x

) = [−a, a] ∩ V = ∅
0
F x = 0.
V =

−
a
2
,
a
2

, F (0) = {0} ⊂ V. U 0,
x

∈ U, x

= 0 F(x

) = [−a, a] ⊆ V.
H : D → 2

R
, H(x) =



[ − a, a], x = 0,
{0}, x = 0,
x = 0.
X Y
D ⊆ X, K ⊆ Y
F : D → 2
Y
F x ∈ D y ∈ F (x)
{x
α
} D x {y
α
}, y
α
∈ F (x
α
) α
y
α
→ y.
F