ФАН - конспект лекций

f [a, b] & & [a, b]

f '# @ f (a) f (x) f (b)

! C AC = {x|f (x) < C} % f %

4 & '# * @

8 @ %

x0 [a, b] xn → x0 & xn < x0 9 {f (xn)} & + f (a)

f (b) 5 ' & & ' * * %

& & f 9 f (x0 −0) 4 & f (x0 +0)

, 8 % & * @ & &		1	f [a, b] |f (b) −
! * ' & & &
	*
− f (a)| n & & & " &		n & 5 n =
= 1, 2, ... & & # & & & &
[a, b] & & & & x1, . . . , xn			, . . . *
$ %
& & hn > f [a, b] f (x) =				hn f (x)

xn<x

& '# xn

* * & & hn

?! / ' ' ' @ ' *

* @ @ &

f % '# @ x1, x2, . . . % & h1, h2, . . .

& & H(x) = xn<x hn

+ϕ = f − H '# @

(x )

−

H(x )) x < x

&

ϕ(x ) − ϕ(x ) = (f (x ) − f (x )) − (H

ϕ % '#

! x

ϕ(x − 0) = f (x − 0) − H(x − 0) = f (x − 0) −

xn<x

hn

ϕ(x + 0) = f (x + 0)

−

H(x + 0) = f (x + 0)

h

−

−

− xn<x

n

−

>

−

x

−

0)

h = 0 h & @ H x > '

ϕ(x + 0) ϕ(x 0) = f (x + 0) f (

f, H ϕ

:! 2! // , " /, 3 / @

[a, b] & ' & ' ' [a, b]

5 ( 7 A O

x

( f [a, b] f (t) dµ ' [a, b]

a

% x f (t) dµ @

a

7 f [a, b] % % & &
Vab[f ] = sup k=1 |f (xk ) − f (xk−1)| < +∞ & [a, b]
n
a = x0 < x1 < . . . < xn = b
V b[f ] $ f [a, b]
a
/ & & R '# . VR[f ] = lim	V b[f ]
b→+∞	a

a→−∞

! 7 f & ' ' f = g − α g, α '#

@

	( 7 & * ' & ' & ' '
	5 9
	5
		x
	(	f (t) dµ ' * * [a, b] @ f & ' '
		a
		5 9
	: ; ( /, "
	! f [a, b]			d	x f (t) dt	. .	f (x) [a, b]
				dx

a

7 f [a, b] & [a, b] ε > 0 δ >

> 0 : & * '# (ak , bk ), k = 1, n [a, b] * *

< δ % n |f (bk ) − f (ak )| < ε

k=1

f ' [a, b] [a, b]

ε > 0 δ > 0 : x , x [a, b] : |x − x | < δ |f (x ) − f (x )| < ε + (x , x )

>

7 [0, 1] ' % (

f (x) = 2k−1 , k = 1, 2n−1 k% n% % [0, 1] + %

2n

' 9 f (x) = 12 , x [ 13 , 23 ] f (x) = 14 , x [ 19 , 29 ] f (x) = 34 , x [ 79 , 89 ]

! @ ' " & '# * + '# * %

# & & @ * *

[0, 1]

			f	' * @	F			[a, b]		x
!		x					[a, b]
+=		%
[a, b]		f (t) dµ(t) = F (x) − F (a)

a

.

X = ' X ρ : X × X → R *

(x, y) X × X * & & ' .

1)ρ(x, y) = 0 x = y

2)ρ(x, y) = ρ(y, x) x, y X

3)ρ(x, y) ρ(x, z) + ρ(z, y) x, y, z X

& % , ρ(x, y) 0 x, y X ! * , ρ(x, x) ρ(x, y) + ρ(x, y)

0 2ρ(x, y) ρ(x, y) 0

X = * * %

A X ρA(x, y) = ρ(x, y) x, y A (A, ρA) %

(X, ρ)

X = R, ρ(x, y) = |x − y|

n

X = Rn, ρ(x, y) = i=1(xi − yi)2

p, q

R : 1 < p < ∞,

1

+

1

= 1

a, b R

: a, b 0

#

p

q

p

q

ab ap

+ bq

+

0 < α < 1 ϕ(x) = xα − αx, 0 < x < +∞; ϕ (x) = αxα−1 − α = α(xα−1 − 1), ϕ (x) > 0 0 < x < 1

x > 1 ϕ < 0

x = 1 ϕ(x) < ϕ(1), x

(0,

)

1

}

ϕ(1) = 1

∞a \{

aα

a

−

α

xα

αx + 1

−

α, 0 < x < +

∞

x =

a, b = 0

α

+ 1

−

α

aαb1−α

α

1

1

b

b

b

αa1+ (11− α)b. α = p

p > 1 0 < p

< 1

p

1−p

a

1

a 1 b

1

p + (1 − p )b

1

a p b q

ap + qb

2 % % + (a = (a p )p) P QRD % + &

n

#1

= ($ 1

p, q R 1 < p < +∞,

p1 + q1 = 1 a, b Rn

n

p

n

q

k=1 |ak bk|

k=1 |ak |p

k=1 |bk|q

+

n

K & + * . +

a, b R

λa

µb

λ, µ R.

n

|ak |p =

n

|bk|q = 1 +,

+ & % & .

=1

=1

& λ =

n

1

1

, µ =

1

1

k

k

|p p

n

q

k=1 |ak

k=1 |bk |q

+, % + n |ak bk| 1 +? ! +? +, /

% .

k=1

n

p

q

n

n

|

p|

+

|

q|

n

|ak|p + q

n

|bk|q = p

+ q

= 1

k=1 |akbk | k=1

= p k=1

k=1

ak

bk

1

1

1

1

#1

.1

1 p R. 1 p

+∞

a, b

R

1

k=1 |ak + bk|p

k=1 |ak |p

+ k=1

|bk |p

+:

1

n

p

n

p

n

p

n

n

q

q

n

q

n

n

n

n

p

n

q

k=1 (|ak + bk |)p = k=1 (|ak | + |bk |)p−1

|ak |+k=1 (|ak | + |bk |)p−1 |bk |

k=1 |ak |p

k=1

(|ak | + |bk |)p−1

+

1

1

1

1

n

1

n

1

k=1 |ak |p

1

1

=

+

k=1 |bk |p

p

(|ak| + |bk|)p−1

q

=

(|ak | + |bk |)p−1

q

p

+

k=1 |bk |p

p

k=1

k=1

=

|bk|p

n

(|ak | +

|bk|)p

q

n

|an|p p +

n

p

k=1

k=1

k=1

k=1 (|ak | + |bk |)

k=1

|ak |p

+ k=1

p + q = 1 q = p−1 & (p − 1)q = p .

1

1

/ % % %

1−q

1 1

p

n

p

n

p

n

% 8

a = x − z = (x1

− z1

. . , x

n −

z

), b = z

−

y = (z

1 −

y

, . . . , z

n −

y

)

, .n

n

1

y

& % % * 9 (R

, ρ) &

X = Rn

ρ(x, y) =

|xi − yi|

n

1

p

|bk |p

p = 2

i=1

max x

y

i|

ρ∞(x, y) = i=1,n |

i −

1

n

p

, ρ(x, y) = i=1 |xi − yi|p

X = C .

|

−

|

ρ(x, y) = x

y

X = C

[a,b]

= {

* & @

[a, b]}

max

|x(t) −

y(t)

ρ(x, y) = a t b

|

x = x(t) y = y(t).

.

[a, b]

ρ(x, y) = 0

max

x = y

> & 2

ρ(x, y) = a t b |x(t) − y(t)| = 0 x(t) = y(t) t

%

% &

= max

max

max

=

, ρ(x, y) = max

|x(t) − y(t)|

− y(t)|

a t b

a t b |x(t) − z(t) + z(t) − y(t)| a t b

|x(t) − z(t)| + a t b |z(t)

= ρ(x, z) + ρ(z, y) 2 ,%

8 % C[a,b] * *

∞

p

< +∞

X = lp p 1 % 8 % % * {xn} k=1 |xk |

x

= (y1, . . . , yn, . . .)

% @ .

ρ(x, y) =

= (x1, . . . , xn, . . .) y

1

∞

p

= k=1 |ak − bk |p

.

ρ(x, y) = 0

∞

|ak − bk |p = 0 nlim

n

|ak − bk |p = 0 xk = yk k N &

k=1

→∞ k=1

) % &

ρ(x, y) = ρ(y, x

, % +: + % 8 n → ∞ + .

p

p

p

∞

p

∞

p

< +∞ & % .

k=1

|ak | < +∞ k=1 |bk|

#

k=1 |ak + bk |p

1

k=1 |ak |p

1

+ k=1

|bk|p

1

2

lp

R

∞

∞

∞

% %

+ &

n

% lp

(X, Σ, µ) % % X Σ % σ% < % M µ% σ%

σ% & Σ

> & & Lp(x)% % M @% * % <

% M

> & &

. .

M}

Lp(x) = {x(t)|

|x(t)|p

dµ(t) < +∞, x(t)

X

f = {g Lp(x)|g

X}

f Lp(x)

f

Lp(x) % % @% * ' "

% Lp(x) Lp(x) = {f |f Lp(x)} 9 . ρ(f , g) =

1

p

p

/ *"

= X |f (t) − g(t)|

dµ(t)

p 1

Lp & "

f

L & f %

@ *

p

.

ρ(f, g) = 0

X

|x(t) − g(t)|pdµ(t) = 0 % * @% |f (t) −

− g(t)|

p

M

. .

g

M

f = g

M + > &

= 0

f

>&

,

* ($ - = ($ x

L

(X) y

L

(X) 1 < p < +

∞

1

+ 1

= 1

1

p

1

q

p

q

xy L1(X) . X |x(t)y(t)|dµ(t) X |x(t)|pdµ(t) p

X |y(t)|q dµ(t) q

+F

x(t)

p

y(t)

q

p

q

t X

X |x(t)|

dµ(t) = X |y(t)| dµ(t) = 1+I / % |x(t)y(t)| |

p

|

+

|

q

|

2 % & .

1

1

+ I

t

|x(t)y(t)|dµ(t) p + q = 1

X

Ix =

|x(t)|

p

q

Ix = 0

Iy

= 0

% &

dµ(t), Iy = |y(t)| dµ(t)

X

X

I

=

0

I

=

0

x (t)

=

x(t)

,

y

(t)

=

x(t)

&

&

@%

1

1

x

y

0

(Ix ) p

0

(Ix ) q

1

1

X |

x(t)y(t) dµ(t)

1

1

1

X |x(t)|pdµ(t)

p

X |y(t)|q dµ(t)

q

|

> ' %

* ($ - .

x,

y

L

(X), 1

p

+

∞

x + y

L

(X)

1

1

1

p

p

X |x(t) + y(t)|pdµ(t) p

X |x(t)|pdµ(t) p

+

X |y(t)|pdµ(t) p

+J

p

p

p

p

p

p

2

p

p

t X

|x(t) +py(t)|

(2 max{|x(t)|, |y(t)|})

= 2

max{|x(t)| , |y(t)| }

(|x(t)|

+

|y(t)|

)

q =

(|x(t) + y(t)|p−1)q = |x(t) + y(t)|p

p−1

/ % G . X |x(t) + y(t)|pdµ(t) X |x(t) + y(t)|p−1|x(t)|dµ(t) + X |x(t) + y(t)|p−1|y(t)|dµ(t)

1

1

1

1

p

q

p

q

p

p 1 q dµ(t)

y(t) pdµ(t)

( x(t) + y(t) p 1)q dµ(t)

=

X |x(t)|

dµ(t)

X (|x(t1) + y(t)|

− )

1

+ X |

1

|

X

|

| −

q

p

p

=

X |x(t) + y(t)|pdµ(t)

X |x(t)|p

+

X |y(t)|p

1

6 % . X |x(t) + y(t)|pdµ(t) q

& % +J

2 % 8 % % Lp(X)

|x(t) − y(t)|2dt)

1

b

2

1

X = C2[a,b] % % @% * [a, b] * ρ(x, y) = a

8

%

%

&

% *

*

=

sup x

y

x =

= l∞

ρ(x, y)

−

k |

X

k

| k

= (x1, ..., xn, ...), y = (y1, .., yn, ...).

9

X % %

ρ(x, y) =

0, x = y

1, x = y

! ( 4

4- (- 4 4

(X, ρ)% & % % {xn} & &

( (X, ρ) a X : ρ(xn, a) → 0 n → ∞ ε > 0 Nε N : n > Nε ρ(xn, a) < ε

% {xn} (X, ρ) & $ ) & * +

ε > 0 Nε : n > Nε, m > Nε ρ(xn, xm) < ε ε > 0 Nε : n > Nε p N ρ(xn, xn+p) <

<ε

% & @

! * ρ(xn, xm) ρ(xm, a) + ρ(xn, a) a % {xn} > &

, -

5 % τ % % X X .

Uα τ α Uα τ
,X, τ			τ
U1, U2	τ U1	U2

(X, ρ)% & %, ) + & x0 r

% . B(x0, r) = {x X|ρ(x, x0) < r} (B[x0 , r] = {x X|ρ(x, x0) r})

B[	1 X	= R+ = [0, +∞] ρ(x, y) = |x − y| x, y X . B[3; 3] = [0; 6] B[ 21 ; 5, 5]
	2 ; 5, 5] = [0; 5, 5]
	(X, ρ)% & % 8 % A X ' * & x0 A r > 0
& B(x0, r) A
	> * " B(x0, r) & % %
	x1 B(x0, r)	r1 = r − ρ(x0, x1) x B(x1, r1) ρ(x, x0) ρ(x, x1) + ρ(x1, x0) < r1 + ρ(x1, x0) = r

> * " %

9 & x0 % A B(x0 , r) : B(x0, r) ∩ A =

9 " % ' & " A " *

A % " &

9 & x0 % % A ' " B(x0, r) ∩ A =

8 %

! (X, ρ) & % A X 5 '# .

T %
¯	¯
,A	A A % 4 % & 4
¯
A = A

? 2 & xn → x0, xn A x0 A= K !>(%/4

# 4 4

> f : (X, ρX ) → (Y, ρY ) x0 X ε > 0 δε > 0 : x

X, ρX (x, x0 ) < δε, ρY (f (x), f (x0)) < ε

> f M + A X f x0 X( A X)

! f : (X, ρX ) → (Y, ρY ) 5 '# .

U x0 X

{xn} : xn → x0, x0 X, f (xn) → f (x0)

+1 2 f % x0 xn → x0 δε

Nδε : n > Nδε ρX (x, xn) < δε ρY (f (xn), f (x0)) < ε

+2 1+ f x0 ε0 : δ > 0 xδ X : ρx(xδ , x0) < δ,

ρy (f (xδ ), f (x0)) ε0 (1)

{δn} : δn → 0, n N xn = xδn : ρ(xn, x0) < δn xn → x0 (x, ρx )

9 f (xn) → f (x0) (Y, ρy ) & & '

( ' * {xn} & xn → x0, xn X '# {f (xn)}

(Y, ρy ) f : (x, ρx) → (y, ρy ) % x0

x

n →

x

0

(X, ρ

x

) f (x )

→

y = f (x )

x

& x

= x , x

= x

n

0

{ n}

2n

n

2n+1

0

) =

n →

x (X, ρ)

/

f (x )

(Y, ρ

y

) f (x

0

n

n2+1

= f (x0) → f (x0) & K & y = f (x0) f % x0

> f : (X, ρx) → (Y, ρy ) .

f %

ρx(x, y) = ρy (f (x), f (y)), x, y X

! & (X, ρx), (Y, ρy ) % #

(X, ρx) (Y, ρy )

% C[0,1] C[a,b] &

x = a − (a − b)t f C[a,b] f → f (a − (a − b)t) 9 C[a,b] C[0,1]

* &

; > , ( 4 4

8 A X

& ¯

¯

A = X

Q ' R

Q = R

n

X = C[0,1] % P = {

ak tk |ak R, n N} * @ .

=1

n

k

: |f0

k

ε > 0 Pn(t) = k=1 akt

(t)

− Pn(t)| < ε, t [0, 1]

, f ) < ε ' @ & *

ρ(f, P

max f

(t)

P (t)

< ε B(f

, ε) % " ρ(f

n) =

0

−

n

|

0

0

t [0,1] |

P 8 % & ' % @ *

(

( ( 4 ( 4 4

8 & % (X, ρ) @ %

X =

R, ρ(x, y) = x

−

y

|

(R, ρ) %

( "

n

|

n

, ρ) %

X = R

, ρ(x, y) % (R

X = (−1, 1), ρ(x, y) = |x − y|

xn =

n

, {xn}

n+1

X

X = Q, ρ(x, y) = |x − y|. xn = (1 + n1 )n (Q, ρ) %

4 -

1

1

1

X = C1[0, 1], ρ(x, y) = |x(t) − y(t)| dt

xn =

n(t

21 ),

21

0

21 + n1

lim xn =

2

1

xn % @ ρ(xn

xn+p) =

t <

1,

2

+ n

≤ t

≤ 1

1, x

[ 1

, 1]

−

1

≤

1

2

1

& *

−

t <

1≤

n→∞

0, x

[0,

2 )

0, 0

=

|

dt =

(x

−

) dt

n

1

1

| −

x

x

.

xn

n+p

n+p

n

0

1

0

1

2

x

C [0,

1] * @ y

C

[0, 1]. x

=

1, x

[ 2 , 1]

y

n

1

0

1

0

0, x

[0, 1 )

0

| 0−

− x0| dt

0

0

. .

|y0 − xn| dt +

|xn

− x0| dt → 0 y0

x0

/ & C1[0, 1]

0

0

@ * y0 = x0 4 x0

/ C1[0, 1] C1[0, 1]

( C[a,b]

! C[a,b] %

{

x

% @ C[a,b]

ε > 0

N

ε

:

n, m > N

ε

max x (t)

−

x

(t) < ε H

n}

t

[a,b] |

n

m

|

' |xn(t) − xm(t)| < ε

t [a, b]

(1) K @ t0 [a, b] H

{xn(t0)} @ R R nlim

xn(t0) = x(t0) 9

@ x(t) = nlim

→∞

xn(t) & xn → x C[a,b]. +

→∞

max x ( t )

x(t)

< ε & ρ(x , x) < ε

@

t

&

−

|

|xn(t) − x(t)| < ε t [a,b]

| n

n

& xn x & 3 x(t)

* xn * # *

( lp

! lp (p 1) %

∞

1

{xm} % @ lp ε > 0 Nε : m, n > Nε

n

m p

p

ρ(xn, xm) = k=1 |xk

− xk |

<

< ε (2)

n

m

2 % &

|

< ε k N

7

& & & %

n

n

|xk

− xk

k

xk @

R xk → xk n →n

∞

2 + &

&

! &

x

→ x

lp

x = (x1, . . . , xk , . . .)

1

1

∞

m

n p

p

∞

n p

p

< ε (4) +?

m → ∞ % k=1 |xk

− xk |

< ε (3) & k=1 |xk − xk |

n → ∞ & xn → x

& x lp

+ xk |

k=1 |xk − xk |

1

k=1 |xk |

1

k=1 |xk |p

1

= k=1 |xk − xk

+

< +∞

∞

p

∞

n

n p

p −

∞

n p

p

∞

n p

p

ε

. .

lp

<ε n>N

xn

( - Lp(X)

! % L1(x) %

{f } % @ % L (x)

n 1

ε > 0 Nε : n, m > Nε ρ(fn, fm) = |fn −fm| dµ < ε k N nk :

X

|fnk −fnk+1| dµ < 21k

X

/ 9 = %< & & {nk} '#

6 |fn|+ |fn+1 −fn|+ . . . X fn + fn+1 −fn + . . . X {fnk } %

X

*

7 &

→∞

klim fnk

= f X @ {fnk

} ε > 0

k0

N : k, l > k0

|fnk

−

& & {fnk } L1(X)

X

− fnl | dµ < ε (4 )

l → ∞

X

|fnk − f | dµ < ε

& % @ * % * %

|fn − f |dµ

|fn − fnk + fnk − f |dµ

|fn − fnk |dµ + |fnk − f |dµ < 2ε

+ 2ε

= ε

X

x

X

X

! % Lp(X) %

A

1q dµ

= (µA) q

A |fn − fm|pdµ

A X : µ(A) < +∞ A |fn −fm|dµ A |fn − fm|pdµ

f

@ L (A)

1

1

1

L (A)

q

p

p

1

{ n}

p

@

1

9 , A X : µ(A) < +∞ % fnk

→ A Lp(X)

/ " *

f 1

k X 2 %

f 2

∞

& µ σ% & X =

Xk : µ(Xk ) < +∞

=1

{ nk1 } →

1

{ nk2 } →

X2 / fn11 , fn22 , . . . , fnkk , . . . + % & % # '
lim fnk = f ! *" &
∞
% Xk, k = 1, 2, . . . µ % σ% µ(X) =	µ(Xk ) X
k=1

k	9 , V % 8
k→∞

"

! (X, ρ) % & ' %

" D &

(X, ρ) % Bn[xn, rn] : Bn Bn+1, rn → 0 n → ∞ / " . mn xm Bn[xn, rn] ρ(xm, xn) rn rn → 0 {xn} % @ % {xn}

x X

ρ(xn, x ) ρ(xn, xm) + ρ(xm, x ) rn + ρ(xm, x ) m → ∞ & ρ(xn, x )

rn, n N ∞ Bn[xn, rn] =

n=1

& '

" D &

' % " D

& {xn} % @ %

9

n1

N : ρ(xn,1xn1 ) <

1

, n n1

1

2

" B[x

, 1]

/

n2 > n1

: ρ(x , x

) <

,

n > n

B[x

,

]

= B

n

> n

k−1

:

n1

1 = B1

1

n n2

22

2

n2

2

2

k

ρ(xn, xnk ) <

, n > nk & B[xnk

,

] = Bk

k

k

2

2

x

%1

% '

" ! *

1

1

1

Bk+1 ρ(xnk , x)

ρ(xnk , x) ρ(xnk , xnk+1 ) + ρ(xnk+1 , x)

+

=

x Bk

2k

2k

2k

2k−1

/

% &

x

Bk k N

9

x Bk ρ(xnk , x) <

1

→ 0 k → ∞ xnk → x (X, ρ)

<

2k−1

x

xn

% @ ε > 0

k1 : ρ(xn

, x) <

ε

ρ(xn, x)

ρ(xn, xn

) + ρ(xn

, x) ( ) 9 xn

k →

{

}

εk

k

k

2

k2 : ρ(xn, xnk ) <

2

k = max{k1, k2} & & ( ) < ε > ' xn → x

&

&

(Y, ρy )

&

(X, ρx) .

X

= Y ρx(x, y) = ρy (x, y) x, y X

& (Y, ρy ) ' * %

X = Q, Y = R ρ(x, y) = |x − y| (R, ρ) (Q, ρ)

! <' &

X = Y ' @ % {xn}, xn X Y Y %

+ %

+ (X, ρ) % & % Z @ % *

# X = & & %

{

z

n}

{

z

lim ρ(z

, z

) = 0 +6 @ %

n}

n→∞

n

n

& & & %

" Z '# (

{

z

n}

& [z

n

] 8 % [z

n

] & Z ( x

X

>

Y

% * [xn] : xn → x & X

Y = Z + X

'# . ρY ([xn], [yn]) = lim ρX (xn, yn)

(3)

n→∞

!

& +, # {xn}

[xn] {yn} [yn] / & . |ρ(xn, yn) − ρ(xm, ym)| ρ(xn, xm) + ρ(yn, ym) (4)

& " m n. |ρ(xn, yn) − ρ(xm, ym)| ε {xn} {yn} @ 9 {ρ(xn, yn)}

% @ &

{xn} [xn] {yn} [yn] ! * {xn}, {xn} [xn]

{yn}, {yn} [yn]

|

ρ(x , y

n

)

−

ρ(x , y

)

ρ(x , x ) + ρ(y

n

, y ) /

& lim

ρ(x , y

n

) =

n

n n

|

n

n

n

n

→∞

n

= lim ρ(xn, yn)

n→∞

! & Y

& % 4

@ % * 4 & , 9

X

ρ(xn, zn) ρ(xn, yn) + ρ(yn, zn) n → ∞ +, &

& ρ([xn], [zn]) ρ([xn], [yn]) + ρ([yn], [zn])

! & X % Y ( x X

* # x ' %

& * x 5 x [xn] & XY 9 X % Y

& X = Y +X ' Y [xn] Y ε > 0 / [xn]

N : ρ(xn, xm) < ε n, m > N	9	lim		ρ(xn, xm) ε	9
		ρ(xn, [xn]) = m	→∞

& [xn] ' & X 9 X = Y

K & ' Y ' @ % & X Y %

'& & Y %

>? 4 "

>? (

(X, ρ) &

f : (X, ρ) → (X, ρ) &( α R : α [0; 1) ρ(f (x), f (y))

αρ(x, y) x, y X

/ '#

2 ρ(f (x), f (y)) < ρ(x, y)

x X ' % f : (X, ρ) → (X, ρ) f (x) = x

! 2 >? 3 / '# * '#

& ' &

x0 X x1 = f (x0), x2 = f (x1), . . . , xn = f (xn−1), . . .! & (xn)@

!	ρ(xm, xn) m > n f n(x) = f (f n−1(x)) f 1(x) = f (x), f 0(x) = x
ρ(xn, xm) = ρ(f n(x0), f m(x0)) 0 W%X% 1 ρ(f n(x0), f n+1		(x0)) +
+ρ(f n−1(x0), f n(x0))+. . .+ρ(f m−1(x0), f m(x0)) = ρ(f (f n−1(x0)), f (f n(x0)))+. . .+ρ(f (f m−2(x0), f (f m−1		(x0))
	0 1 αρ(f n−1(x0), f n(x0)) + . . . + αρ(f m−2(x0), f m−1(x0)) αnρ(f 0(x0), f (x0)) +

+αn+1ρ(x0, f (x0)) + . . . + αm−1ρ(x0, f (x0)) = ρ(x0, f (x0))(αn + αn+1 + . . . + αm−1) ρ(x0, f (x0))(αn + αn+1 +

+. . . + αm−1 + . . .) = ρ(x0, f (x0)) 1α−nα → 0 n → ∞ > ' (xn) @

9 X (xn) lim xn = x f f (x) = f ( lim xn) =
n→∞	n→∞

= lim f (xn) = x 9 x &

n→∞

! . & y = x : f (y) = y ρ(x, y) = ρ(f (x), f (y)) < ρ(x, y) %

& 3 &

( 9 ? * & x

		αn			&
	ρ(xn, x0) ρ(x0, f (x0)) 1−α , n N /							x → ρ(xn, x)
					n
	xn @ ρ(xm, xn) ρ(x0, f (x0))			α		! . %
				1−α
	|ρ(xn, x) − ρ(xn, x0)| < ρ(x, x0) + 0.					ε > 0 δε > 0 : x X	ρ(x, x0 ) < δε |f (x) − f (x0)| < ε

ρ(x0, x) < αn ρ(x0, f (x0))

1−α

") >? 4 "

6 x = f (x) f ' < " [a; b] M > 0 : |f (x) − f (y)| < < M |x − y| x, y [a; b] 5& & f : [a; b] → [a; b]

M < 1 f '# / '# * ' %

' & " x = f (x) / x0 [a; b]

f (x0), f (f (x0)), . . . & " ' * & '

6 F (x) = 0 f (x) = x − λF (x) 9 f ' x = f (x) λ = 0

F ' 0 k1 F (x) k2 n

1 − λk2 f (x) 1 − k1λ

x [a; b]

6 x = Ax

A : Rn → Rn

yi =

=1 aij xj + bi K A = (aij )

ρ2(y , y ) =

n

(y

y )2 =

n (

n a

(x x ) )2

n

j n

( n

(a )2

(x

x ))2

0 G) 1

n

n

i

i

−

n

ij j −

n

ij

j

− j

=1

i

i=1

| j=1

j |

i=1 j=1

x

x 2)) = ρ2(x , x )

(a

)2 = ρ2(x

, x )α

((

a )2

·

(

i=1

j=1 |

ij

|

j=1 |

j −

j |

i=1 j=1

ij

1

'# %

i

'#

n

n

)

α =

(a2

=1 j=1

ij

√

ρ(y , y ) ρ(Ax , Ax ) √

< 1

|aij | < n

i, j

α

α

*

>? 4 " @

dxdy = f (x, y) + & y0 = f (x0) + = & & f (x, y) %

G R2 ' < " y M : (x, y1), (x, y2) G

|f (x, y1) −

− f (x, y2)| M (y1 − y2)

# d y = ϕ(x) ) {x|

|x − x0| d}

" + +

|f (x, y)|

G G G f G M > 0 : (x, y) G

M

) K &

(x, y) G

(x, y) : |x − x0| < d |y − y0| < Kd

M d < 1

C [x0 − d, x0 + d] @ * ϕ & |ϕ(x) − y0| < Kd ρ(ϕ1, ϕ2) =

=

max

ϕ (t)

−

ϕ (t) 9 (C

[a;b], ρ) 6 %

t [x0

−

d,x0+d] |

1

2

|

x

x

A : C

→ C

ψ = Aϕ = y0 + x0 x

|Aϕ − y0| = |x0 f (t, ϕ(t))dt|

Kd &

A

f (t, ϕ(t))dt

ρ(Aϕ

, Aϕ ) = t [x0−d,x0+d] |x0

f (t, ϕ

(t)) − f (t, ϕ (t))dt| M · x [x0−d,x0+d] |x − x0

| · x0 t x |

−

'#

max

max

max

ϕ (t)

− ϕ (t)| M dρ(ϕ , ϕ )

9 M d < 1 A '# 9 A ' ' & : Aϕ = ϕ

!Y + & + ϕ(x) = y0

+ x0

f (t, ϕ(t))dt

x

&

+, 2 +,

* ($ , 5 ($

[c, d]

+ ϕ(x) +? f @ K(x, y) Π = [a, b]

b

- . // 0 f (x) = λ

K(x, y)f (y)dy +

a

b

b

b

×

A(f )(x) := λ

K(x, y)f (y)dy + ϕ(x)

| |

|

a

a y b

1

− f2(y)| · (b − a) = |λ|M (b −

1 2

(y)

f

(y))dy

f

(y)

f

(y) dy

A : C

[a;b] →

C

max

K(x, y)(f

−

|

λ M max

−

[a;b]

ρ(A(f1), A(f2)) = λ a x b | a

1

2

| |

a x b a

| 1

2

|

λ M max

f (y)

a)ρ(f , f )

λ : |λ| <

1

A '# 2 '# *

M (b−a)

' ' & & " +? )

b

* @ fn(x) = λ K(x, y)fn−1(y)dy + ϕ(x) 9 & )

a

' @ 9 ?

; ($

x

9 ' f (x) = λ K(x, y)f (y)dy + ϕ(x) '

7 K(x, y) = 0

a

y > x

!

2 ? >? 4 "3

A : (X, ρ) →

(X, ρ)

(X, ρ)

n

'#

n

N

!x

X : Ax = x

n

A

n

n

x = A x

A(x) = A(A (x)) = A (A(x)) & & & x & A

6 A :

x

max

x

(y)

−

−

2

| | |

a x b

| 1

−

2

A(f (x)) = λ a K(x, y)f (y)dy + ϕ(x) ρ(Af1

, Af2) = a x b |λ a K(x, y)(f1

(y))dy

(x)

|

−

a)

| | ·

(b

−

1

2

)

f

λ M max

f

f

(x) (x

λ M

a)ρ(f

, f

ρ(Anf1, Anf2) M n|λ|n(b − a)nρ(f1, f2) & " n An '# An

' ' &

A$ 4

2 3 B

8 & (X, ρ) * % &

% # # %

! + % Y

(X, ρ) % B

X &

&

! (

{xn} % @ & (X, ρ) 2

{xnk }

ρ(xn, x) − . ρ(xn, xnk )					+
<ε				−

: lim xnk = x X

k→∞

ρ(xnk , x) xn → x

<ε −

>

R %

8 & (X, ρ) ' * % &

% # @ %

8 & % (X, ρ) (X, ρ) (X, ρ)

(X, ρ) 9

% {xn} % % (X, ρ) /

@ % {xnk }

8 % M & % (X, ρ) ) + (M, ρ) %

+

8 % M & % (X, ρ) %

" + x0 X r R : r > 0 M B(x0 , r)

& R % +9 / *"

% & % &

M (X, ρ) & M & x0 X r > 0 xr

M : ρ(xr , x0) r

x1 X n N xn M : ρ(xn, x1) n lim ρ(xn, x1) = +∞ 5 *

n→∞

M {xn} @ % {xnk } / & |ρ(xnk , x1)−ρ(xnl , x1)| ρ(xnk , xnl )+ρ(x1, x1) < ε + {xnk } @ ' & &

{ρ(xnk , x1)} @ R R nlim ρ(xnk , x1) < +∞ 3 %
								→∞
% ' % '# * &
/ 9 / *" Rn &
& % #
			∞
M l1 : M = {x = (x, . . . , xk , . . .)| k=1 |xk | 1} =							B	(0, 1)B
	n	n m		n)		@ >& & '
	x = (0, . . . , 0,
		1 , 0, . . .) ρ(x , x ) = 2 (m =
		n−

% @ *