Về sự tồn tại điểm giả bất động của ánh xạ liên tục trên tập s lồi trong không gian p định chuẩn


S
P

S
P
p s
p
s p
s p
s
s−
p
p
p
E
p
p p
s p 0 < s  p  1.
s
p
p s p
s− p
p s
p
s p
s
p
s−
p

P S
p s
p
U X αU ⊂ U
α ∈ K |α| < 1 U x ∈ X
δ > 0 αx ∈ U |α| < δ.
U 0
U ∈ U V ∈ U V + V ⊂ U
U X
x, y ∈ U 0  λ  1 λx + (1 − λ)y ∈ U
U 0
U E
V 0 s > 0 U ⊂ tV
t > s
0
E A ⊂ E
A A
A (A) A
A
(A) = {
n

i
λ
i
a
i
: a
i
∈ A, λ

i
> 0,
n

i=1
λ
i
= 1, n = 1, 2, }.
0
0
E F
d E d(x, y) = d(x + z, y + z)
x, y, z ∈ E (E, d) d E
F
F
E R
. : E → R E
x  0 x ∈ E x = 0 ⇔ x = 0
λx = |λ|x λ ∈ R x ∈ E
x + y  x + y, x, y ∈ E
(E, .)
d(x, y) = x−y, ∀x, y ∈ E E
E
E
B
n
= {x ∈ E : x <
1
n
}, n = 1, 2,

E
X, Y
F : X → Y
F F (X)
Y
F F (X)
Y
C
E F : C → C
p
p p
K = R
C.
p− p > 0 E
. : E → R
+
x = 0 x = 0
λx = |λ|
p
x, λ ∈ K x ∈ E
x + y  x + y, x, y ∈ E
(E, .) p
p
E
K . : E → R
+
x = 0 x = 0
λx = |λ|x, λ ∈ K x ∈ E
x + y  σ(x + y), x, y ∈ E σ  1
x, y

σ
(E, .)
(E, .)
B
E
(0, ε) = {x ∈ E : x < ε}, ε > 0 0 E
. p E 0 < p  1 .
1
p
d
p
(x, y) = x − y
1
p
E
E
p . E d
p
(x, y) = x−y
1
p
E
p
p
p− E p
p
p F
0 < p < 1 l
p
p

x =
∞

n=1
|x
n
|
p
x ∈ l
p
p
. p− E
|x − y|  x − y
x, y ∈ E
x, y ∈ E
x = x − y + y  x − y + y.
x − y  x − y.
y = y −x+x  y−x+x = |−1|
p
x−y+x = x−y+x.
− x − y  x − y.
|x − y|  x − y.
p
E F p
q A : E → F
A(tx + y) = tA(x) + A(y) x, y ∈ E
t ∈ K
p
E = C(I, K) p
I = [0, 1] K

p 0 < p  1
f
p
= sup
x∈I
|f(x)|
p
.
F E f ∈ E f
df I
D : F → E D(f) = df f ∈ F
D D
{f
n
} ⊂ F f
n
(x) =
sin nx
n
n = 1, 2,
x ∈ I
f
n

1
p
p
=

sup

x∈I



sin nx
n



p

1
p

1
n
.
f
n

p
p
→ 0 n → ∞ {f
n
} 0 F
Df
n
(x) = df
n
(x) = cos nx,

Df
n

1
p
=

sup
x∈I


cos nx


p

1
p
= 1
n Df
n
0 E D
E F p
q 0 < p, q  1 A : U ⊂ E → F
U C > 0
A(x)
1
q
 Cx
1

p
x ∈ U U = E A
E
A(x)  C
1
x
q
p
,
C
1
= C
q
p
E F p
q 0 < p, q  1 A : E → F
A
A 0
M > 0 A(x)  Mx
q
p
, x ∈ E
A E F
p = q A(x)  Mx
E F p q
0 < p, q  1 L(E, F )
E F L(E, F )
A ∈ L(E, F )
A = inf


M : A(x)  Mx
q
p
x ∈ E

.
A
A(x)  Ax
q
p
x ∈ E
E F p
q 0 < p, q  1 A ∈ L(E, F )
A = sup
x∈E\{0}
A(x)
x
q
p
= sup
x1,x=0
A(x)
x
q
p
= sup
x=1
A(x).
L(E, F ) q
F

q L(E, F )
p
K
n
0 < p  1
x
p
= |x
1
|
p
+ |x
2
|
p
+ + |x
n
|
p
, ∀x = (x
1
, , x
n
) ∈ K
n
.
p K
n
K
n

p l
p
(n) l
p
(n)
n
X p n
L X l
1
(n)
mx  Lx
p
 Mx
x ∈ X m, M
B
p
B
1
l
p
(n) l
1
(n) H l
p
(n) l
1
(n)
H(B
p
) = B

1
α ∈ K α α|α| = α
H : l
p
(n) → l
1
(n)
H(α
1
, , α
n
) = (β
1
, , β
n
),
β
i
= (α
i
)|α
i
|
p
, i = 1, , n. H

n
i=1
|α
i

|
p
 1

n
i=1
|β
i
|  1 H(B
p
) = B
1
f
i
(α
i
) = ( α
i
)|α
i
|
p
g
i
(β
i
) = ( β
i
)|β
i

|
1/p
K H H
−1
H
B
p
l
p
(n) (0 < p 
1) T : B
p
→ B
p
u ∈ B
p
T u = u
H B
1
l
1
(n) H ◦ T ◦ H
−1
: B
1
→ B
1
l
1
(p)

x ∈ B
1
H ◦ T ◦ H
−1
x = x u = H
−1
x
u ∈ B
p
T u = u.
s p
s p
A p E
0 < s  p
A s x, y ∈ A t ∈ [0, 1]
(1 − t)
1
s
x + t
1
s
y ∈ A
A s x, y ∈ A
tx + r ∈ A
t, r ∈ R |t|
s
+ |r|
s
= 1
A s x, y ∈ A

tx + ry ∈ A
t, r  0 t
s
+r
s
= 1 s = p = 1 s−
s s
l
p
(0 < p < 1) p− p−
x =
∞

n=1
|x
n
|
p
x = (x
n
) ∈ l
p
B(0, r) = {x ∈ l
p
: x < r}
s− x, y ∈ B(0, r) t ∈ [0, 1]
(1 − t)
1/s
x + t
1/s

y − 0  (1 − t)
1/s
x + t
1/s
y
= (1 − t)
p/s
x + (t)
p/s
y
< (1 − t)
p/s
r + (t)
p/s
r  (1 − t + t)r = r,
p/s  1 t ∈ [0, 1] (1 − t)
1/s
x +
t
1/s
y ∈ B(0, r) B(0, r) s−
B(0, r) l
p
(0 < p < 1)
A p E
s− A s A
s
A
x
1

, x
2
, , x
n
⊂ A t
1
, t
2
, , t
n
 0

n
i=1
t
s
i
= 1

n
i=1
t
i
x
i
s {x
1
, x
2
, , x

n
}
s
A s A
s
X p (0 < p  1)
0 < s  p
C ⊂ X s αC s α ∈ K.
C
1
, C
2
s C
1
+ C
2
s
{C
i
: i ∈ I} s X ∩
i∈I
C
i
s
A ⊂ X 0 ∈ A
s
A ⊂ A A
A
u, v ∈ αC x, y ∈ C
u = αx, v = αy t, r ∈ [0, 1] t

s
+ r
s
= 1
tx + ry ∈ C
tu + rv = t(αx) + r(αy) = α[tx + ry] ∈ αC.
αC s−
C
1
, C
2
s− z
1
, z
2
∈ C
1
, C
2
x
1
, x
2
∈ C
1
y
1
, y
2
∈ C

2
z
1
= x
1
+ y
1
, z
2
= x
2
+ y
2
C
1
, C
2
s t, r ∈ [0, 1] t
s
+ r
s
= 1
tx
1
+ rx
2
∈ C
1
ty
1

+ ry
2
∈ C
2
tz
1
+ rz
2
= tx
1
+ rx
2
+ ty
1
+ ry
2
∈ C
1
+ C
2
,
C
1
+ C
2
s
{C
i
: i ∈ I} s x, y ∈ ∩
i∈I

C
i
x, y ∈ C
i
i ∈ I C
i
s t, r ∈ [0, 1]
t
s
+r
s
= 1 tx+ry ∈ C
i
y ∈ I tx+ry ∈ ∩
i∈I
C
i
∩
i∈I
C
i
s
z ∈
s
A
z =
n

i=1
t

i
x
i
x
i
∈ A

n
i=1
t
s
i
= 1 0 ∈ A
t
1−s
i
x
i
= t
1−s
i
x
i
+ (1 − t
1−s
i
)0 ∈ A
i = 1, 2, , n

n

i=1
t
s
i
= 1
z =
n

i=1
t
i
x
i
=
n

i=1
t
s
i

t
1−s
i
x
i

∈ A.
s
A ⊂ A.

E ρ : E → R
p
ρ(x)  0 x ∈ E
ρ(λx) = |λ|
p
ρ(x) x ∈ E λ ∈ K
ρ(x + y)  ρ(x) + ρ(y) x, y ∈ E
C s 0
p X V q
C
: X → R
p
V
(x) = inf{t > 0 : x ∈ t
1/s
V }
x ∈ X
p
V
(x) = inf{t
s
> 0 : x ∈ tV }
X p 0 < p  1
C s− X 0 ∈ C q
C
C
q
C
(0) = 0
q

C
(rx) = r
s
q
C
(x) x ∈ X t > 0 s
q
C
(x + y)  q
C
(x) + q
C
(y)
C q
C
(x) > 0 x ∈ X x = 0
C q
C
(x) < ∞ x ∈ X
0 ∈ C q
C
(0) = 0
r = 0 p
C
(0x) = r
s
p
C
(x) = 0 r > 0
p

C
(rx) = inf{t
s
: rx ∈ tC} = r
s
inf{

t
r

s
: x ∈
t
r
C} = r
p
p
C
(x).
x, y ∈ X λ, µ ∈ R λ > p
C
(x)
µ > p
C
(y)
x
λ
s
∈ V
y

µ
s
∈ µV
x + y
(λ + µ)
s
=
λ
s
(λ + µ)
s
x
λ
p
+
µ
p
(λ + µ)
s
x
µ
s
s− C
x+y
(λ+µ)
s
∈ C
p
C
(x + y)  λ + µ.

λ > p
C
(x) µ > p
C
(y) p
C
(x + y)  p
C
(x) + p
C
(y)
C R > 0 C ⊂ B(0, R)
x = 0 q
C
(x) = 0 n = 1, 2,
x ∈
1
n
C.
nx ∈ C n nx < R n
C x ∈ X r > 0
rx ∈ C
p
C
(x) = inf{t
s
> 0 : x ∈ tV } 
1
r
s

< ∞.
C s 0
p E q
C
{x ∈ E : q
C
(x) < 1} ⊂ C ⊂ {x ∈ E : q
C
(x)  1}.
C
V = {x ∈ E : q
C
(x) < 1}.
p
C
C ⊂ {x ∈ E : p
C
(x)  1}
q
C
(x) < 1 λ q
C
(x) < λ < 1
x
λ
∈ V. C
x = λ
x
λ
∈ C {x ∈ E : q

C
(x) < 1} ⊂ V
s s
E p K, A ⊂ E
K K ∩ A = ∅ s
V K + V ∩ A = ∅.
K ∩ A = ∅ E
x ∈ K y ∈ A U ∈ V x + U
x
∩ y + U
x
= ∅
{x + U
x
: x ∈ K} K K
x
1
, , x
n
∈ K K ⊂

n
i=1
(x
i
+ U
x
i
) W = ∩
n

i=1
U
x
i
W ∈ V V ∈ V V +V ⊂ W K +V ∩A = ∅.
X p K ⊂ X
co
s
(K) X
X p
co
s
(K)
U 0 s V 0
V + V ⊂ U K F
K K ⊂ F + V
K
1
= C
s
(K), F
1
= C
s
(F ).
F = {y
1
, y
2
, , y

n
} F
1
{(t
1
, t
2
, , t
n
) : t
i
 0,

t
s
i
= 1} ⊂ R
n
R
n
X
(t
1
, , t
n
) →
n

i=1
t

i
y
i
F
1
x ∈ K
1
x =

k
i=1
r
i
x
i
r
i
 0

k
i=1
r
s
i
= 1
x
i
∈ K i z
i
∈ F x

i
− z
i
∈ V x
x =
k

i=1
r
i
x
i
=
k

i=1
r
i
z
i
+
k

i=1
r
i
(x
i
− z
i

) := x
1
+ x
2
.
x
1
∈ F
1
x
2
∈ V V s K
1
⊂ F
1
+ V F
1
F
2
F
1
⊂ F
2
+ V
K
1
⊂ F
1
+ V ⊂ F
2

+ V + V ⊂ F
2
+ U,
K
1
= co
s
(K)
s−
X p K
X K {x
n
} ⊂ X
x
n

p
→ 0 K ⊂
s
{x
n
: n = 1, 2, }.
{x
n
} ⊂ X 0 E =
{x
n
: n = 1, 2, } ∪ {0}
s
E K K ⊂

s
{x
n
: n = 1, 2, }
K
K {x
1
i
: i =
1, , n
1
} ⊂ K
2K ⊂
n
1

i=1
B(x
1
i
,
1
4
).
K
2
=
n
1


i=1

B(x
1
i
,
1
4
) − x
1
i

.
K
2
B(0,
1
4
).
{x
2
i
: i = 1, , n
2
} ⊂ B(0,
1
2
)
2K
2

⊂
n
2

i=1
B(x
2
i
,
1
4
2
).
K
3
=
n
2

i=1

B(x
2
i
,
1
4
2
∩ 2K
2

) − x
2
i

.
{x
j
i
}
n
j
i=1
, j = 1, 2,
x ∈ K n
k
x −

x
1
i
1
2
+
x
2
i
2
2
2
+ +

x
k
i
k
2
k

∈ 2
−k
K
k+1
,
1  i
1
 n
1
1  i
2
 n
2
1  i
k
 n
k
x ∈
s
{x
j
i
: 1  i  n

j
, j = 1, 2, },
x
j
i
 
2
4
j−1
→ 0 j → ∞.
A
1
, A
2
, , A
n
s
X
s
(K
1
∪ ∪ K
n
)
S = {(t
1
, , t
n
) ∈ R
n

: t
i
 0, t
s
1
+ + t
s
n
= 1}
A = A
1
× × A
n
.
f : S × A → X
f(t, a) = t
1
a
1
+ + t
n
a
n
t = (t
1
, , t
n
) ∈ S a = (a
1
, , a

n
) ∈ A
f K := f(S × A)
S K ⊂
s
(A
1
∪ ∪ A
n
)
s
(A
1
∪ ∪A
n
) ⊂ K (t, a), (r, b) ∈ S ×K α, β  0
α
s
+ β
s
= 1
αf(t, a) + βf(r, b) = f(u, c)
u = (α
s
t
s
+ β
s
r
s

)
1/p
∈ S
c
i
=
αt
i
a
i
+ βr
i
b
i
((αt
i
)
p
+ (βr
i
)
p
)
1/p
∈ A
i
.
K s A
j
⊂ K j s K

s
(A
1
∪ ∪A
n
) ⊂ K
s
(A
1
∪ ∪A
n
) = K
s
(A
1
∪ ∪A
n
)
S P
s p
s
s
X p C
s− X 0 < s < p  1 0 = x
0
/∈ C
D = C −
s
{x
0

} D s 0 ∈ D C ⊂ D
C
s
{x
0
} s
D = C −
s
{x
0
} s x
0
∈ C
0 ∈ C −
s
{x
0
} = D 0 ∈
s
{x
0
}
C = C − 0 ⊂ C −
s
{x
0
} = D.
X C x
0
D

x ∈ D
C
x
= {y ∈ C : x = y − tx
0
}
t ∈ [0, 1]
α(x) = min
y∈C
x
x − y
x
0

, x ∈ D
α [0, 1] x ∈ D
x = y(x) − α(x)x
0
, y(x) ∈ C.
x ∈ D C
x
C
x = y − tx
0
t =
x − y
x
0

C

x
p
α(x) := min
y∈C
x
x − y
x
0

y(x) = x + α(x)x
0
,
x
α(x) = min
y∈C
x
x − y
x
0

.
α(x) (x
n
) ⊂ D
x
n
→ x y ∈ C
x
f
y

(x) =
x − y
x
0

α(x)
lim sup
n→∞
α(x
n
)  α(x).
lim inf
n→∞
α(x
n
) = β(x) C
(x
n
j
) (x
n
) lim inf
j→∞
α(x
n
j
) = β(x) lim
j→∞
y(x
n

j
) = y
0
∈ C
x
n
j
= y(x
n
j
) + α(x
n
j
)x
0