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(, , .) . u = 0 . u G , . , . , . . ­ . . . . . , . , . . ( ), 10-01-00297, 10-01-91219-- 2.1.1-3828

3


.. , .. , ..

, . . . . , . . 1. u = 0, ( x, y ) G
u = ( x, y ),

(1.1)

(1.2)



2 2 + 2 - , ( x, y ) - x 2 y

=

{(

x, y ) : 0 x 1; y = 0;1}

{(

x, y ) : 0 y 1; x = 0;1}



G = ( 0, 1) â ( 0, 1) .

. (1.1), (1.2) ( x, y ) [1-6];

. (1.1), (1.2)
u(M ) = d dnQ 1 1 ln RMQ ( Q ) dsQ , (1.3)

2 ( Q ) = ( , ) ­ , M = M ( x, y ) G , RMQ = ( x -

)

2

+ ( y - ) ,
2

nQ ­ Q ( ).

4




, (1.1), (1.2) , , ( Q ) , [1-6]. , . . (1.1), (1.2). , , . , (1.1), (1.2) Lp H


,

[7-9]. ( x) ( -, ) , y > 0 :
u ( x, y ) = y




-



( )d y2 + ( x -

)

2

.

[1,10] . (1.1), (1.2) ( x) , [ 0, 1] . ( x, y ) C ( ) , C ( ) ­
C(

)

= max ( Q ) .
Q

1 ( x ) = ( x, 0 ) ,
, i C

( y) [0, 1],
2

= (1, y ) ,

3

i = 1, 4 , C

( x) = [0, 1]

( x,1) ,

4

( y)

= (0, y ) .

­

[ 0, 1] C[0, 1] = max ( t ) . t[ 0 , 1] 5


.. , .. , ..

(1.1), (1.2) (1.3)
u ( x, y ) =

u (
i =1 i

4

x, y

)

(1.*)

:
u1 u 2 u 3 u4

( ( ( (

x, y ) = x, y ) = x, y ) = x, y ) =

y




0

1

1 ( )d , 2 y2 + ( x - )

1- x


1- y

(1 - x ) + (
2 0 1 2 0

1

2 ( )d
y - x -

) )

2

, (1.4) ,


x

(1 - y ) + (

3 ( )d

2

4 ( )d , 0 x 2 + ( y - )2



1

i ( x), i = 1, 4 ­ [ 0;1] . :
( ( ( (
1 ) ( x, y ) = ( 0 2 ) ( x, y 3 ) ( x, y 4 ) ( x, y

) ( y, x ) , ) = ( 0 ) (1 - x, y ) ) = ( 0 ) (1 - y, x ) ) = ( 0 ) ( x, y ) ,
x2 + ( y - 0
x
1

, ,

(1.5)



( 0 ) (
(

x, y ) =

( ) d

(1.*) u
) ( x, y ) =

) ( x, y ) = (
2

.

(1.6) ) ( x , y ) , (1.7)

(
i =1

4



i

)(

x, y ) .

, , C ( G ) L ( G ) . 1.1. : ) G u ( x , y ) ;
6




) G u ( x , y ) G = G ; ) u ( x , y

)

u =


i =1

4

ui , ,

ui = ui ( x , y ) , i = 1, 4 (1.4), -

- = (1 , 2 , 3 ,

4

)



-

,

1 ( 0 ) = 4 ( 0 ) , 1 ( 1 ) = 2 ( 0 ) , 3 ( 1 ) = 2 ( 1) , 3 ( 0 ) = 4 ( 1 ) .
1.2. : ) G u ( x , y ) ­ ; ) u ( x , y
4

)

u =


i =1

ui

ui = ui ( x , y ) , i = 1, 4 (1.4), i ( x ) , i = 1, 4

[ 0, 1] : i L [ 0, 1] , i = 1, 4 . 1.3. (1.1), (1.2) G , , ( x , y

)



G . 1.1 ­ 1.3 . 9. 1.4. (1.1), (1.2) G , , ( x , y ) G . . (1.2) 1.4 § 11. 1.1 - 1.3 . 9, 1.4 § 9.

7


.. , .. , ..

2. y > 0
[1,10] u ( x, y ) = y




-



( )d y2 + ( x -

)

2

,

(2.1)

( x

)



(

-, ) , L (( ).
P ( x, y ) = y x2 + y

(

2

)

(2.2)

y > 0 : P = 0 . , u ( x, y ) , ( x ) [11], , (2.1) y > 0 . x
1 lim ( + x )d , h > 0 . h 0 2 h -h
h

(2.3)

x (2.3),

( x):
1 ( x) = lim ( + x )d . h 0 2 h -h
h

[12], D

()

( x

)

. Dc ( )

( x ) . , Dc ( ) (2.3) , .. Dc ( ) ( x ) :
Dc ( ) D , , x Dc ( ) ,

()

( x) = ( x) .

(2.4)

8




u ( x, y ) ( x, 0 ) y > 0 :
u


( (

u ( x, y ) , x, y ) = ( x ) , u ( x, y ) , x, y ) = ( x ) ,


y > 0, - < x < , y = 0, x Dc ( ) ; y = 0, x D .


y > 0, - < x < ,

u



()

, u ,

(

x, y ) u

(

x, y ) ,

x Dc ( ) , , (2.4),

u



(

x, y ) = u



(

x, y ) .

( x ) ,

u



(

x, y ) u



(

x, y ) y 0



u



(

x, y ) u



(

x, y ) ,

y 0, - < x < .

, ( x ) ­ , Dc ( ) D , Dc ( ) , D .

()

()

2.1. ( x x0 Dc

) ( )

­
. u ( x , y ) -

y > 0 ( x0 , 0 ) . . , u ( x, y ) y > 0 u ( x, y )
L [ 0, 1]

.

x0 ( x ) , > 0 > 0 ,

( ) - ( x0 ) <


2

- x0 < 2 .

(2.5)

9


.. , .. , ..


y
-


:



y2 + ( x -

( x0 ) d

)

2

( x0 ) ,

y > 0, - < x < ,

u ( x, y ) - ( x0 ) = =
y
-

y





-



( ) - ( x0
y2 + ( x -

)

2

)d

=





( x + t ) - ( x0
y2 + t
2

)

dt =

y



t <



+

y



t



I1 + I 2 .

I1 :
I1 = y



t<



( x + t ) - ( x0
y +t
2 2

)

dt

sup ( x + t ) - ( x0 )
t <

y





-



dt = sup ( x + t ) - ( x0 ) . y 2 + t 2 t <

(2.5) :
I1 sup ( x + t ) - ( x0 ) <
t <


2



x - x0 < .

(2.6)

I
I
2

2

:



y



t



( x + t ) - ( x0
y2 + t
2

)

dt

4 sup ( t )
-
y



t



1 dt = 4 sup ( t ) - arctg . 2 y +t y - 2

(2.7)



y 0 +

lim arctg


y

=


2



y0 > 0 , 0 < y < y0
0< 1 - arctg < . 2 y 8 sup ( t )
-
(2.7) :
I
2

<


2

.

(2.8)

10




,
u ( x, y ) - ( x0 ) <

0 < y < y0 , x - x0 < .


(2.9)

( x, y ) ­ . y > 0 , u x ( -, ) . y = 0 , u


(

x, y ) = u ( x, y

)

(

x, 0 ) = ( x ) x Dc ( ) . -

, x Dc ( ) x - x0 < , (2.5)
u ( x, 0 ) - ( x0 ) = ( x ) - ( x0 ) <


2

.

(2.10)

(2.9) (2.10)
u


(

x, y ) - u



(

x, 0 ) < 0 y < y0 ,

x - x0 < .

2.1 . 2.1. ( x ) , u ( x , y )
y 0
- < x < y0

sup u ( x , y ) = sup u ( x , y ) = sup ( x ) .
- < x < y>0 - < x <

(2.11)

, u y 0 2.1.
u ( x, y ) sup ( x )
- < x <



(

x, y

)

-

y



-



y2 + ( x -

d

)

2

= sup ( x
- < x <

)

u (2.11).



(

x, y ) y 0 -

1 ( x ) 4 ( y ) [ 0,1] , ; .
u1 ( x, y ) = ( 11 ) ( x, y ) y




0

1

y2 + ( x -

1 ( ) d

)

2

y > 0 , 11


.. , .. , ..

u4 ( x, y ) = ( 4

4

)(

x, y )

x

0 x2 + ( y -



1

4 ( ) d

)

2

­ x > 0 .

u ( x, y ) = u1 ( x, y ) + u4 ( x, y

)

(2.12)

, .. : x > 0, y > 0 . 2.1 , u
11

(

x, y ) 4
4

y 0 , , , ( 0, 0 ) (1, 0 ) , u ( 0, 0 ) ( 0,1) . ,
u ( x, y ) = u
11

(

x, y ) ­

(

x, y ) + u

4

4

(

x, y

)
)

x 0, y 0 , , , ( 0, 0 ) , (1, 0 ) , ( 0,1) . ( 0, 0 (2.12), ( x, y ) ( 0+, 0 + ) . 2.2. (2.12) . -

(

x , y ) ( 0+ , 0 + ) ,

1 ( 0 ) = 4 ( 0 ) .
, ,
(
x , y ) ( 0 + , 0 +

(2.13)
31 ( 0 ) . 2

lim

)

u ( x, y ) =

(2.14)

. . u1 ( x, y ) , u4 ( x, y )

1 ( 0 ) y 1 d u1 ( x, y ) = v1 ( x, y ) + y2 + ( x - 0 4 ( 0 ) x 1 d u4 ( x, y ) = v4 ( x, y ) + x2 + ( y - 0
v1 ( x, y ) =
y

)

2

,




0

1

(1 ( )

y2 + ( x -

- 1 ( 0

)

))

2

d , v4 ( x, y

) 1 x ( 4 ( ) - 4 ( 0 ) ) d )= 2 0 x + ( y - )2
2

,

.

12




v4 ( x, y ) ( x, y ) ( 0, 0 ) , x > 0, y > 0 ,
(
x , y ) ( 0 + , 0 +

2.1 v1 ( x, y )
lim v1 ( x, y ) = lim v4 ( x, y ) = 0 .

)

(

x , y ) ( 0 + , 0 +

)

(

, (2.12) x, y ) ( 0+, 0 + )

1 ( 0 ) y 1 d w ( x, y ) = y2 + ( x - 0
w ( x, y ) =

)

2

4 ( 0 ) x 1 d + x2 + ( y - 0

)

2

.

(2.15)

, (2.13) :

1 ( 0 ) 1- x x (0) 1- y y + arctg + 4 + arctg = arctg arctg y y x x 1 ( 0 ) 1- x 1- y x + arctg + arctg + arctg arctg y x y
y x .

=

arctg

x y + arctg = x > 0, y > 0 y x2
(
x , y ) ( 0 + , 0 +

lim

)

arctg

1- x = y

(

x , y ) ( 0 + , 0 +

lim

)

arctg

1- y =, x 2

:

31 ( 0 ) . ( x , y ) ( 0 + , 0 + ) 2 (2.14) . . , (2.12) ( x, y ) ( 0+, 0 + ) lim w ( x, y ) =


(
x , y ) ( 0 + , 0 +

lim

)

w ( x, y ) .

(2.16)

(2.16).
(2.13).

w ( x, y )

w ( x, y ) =
+

1 ( 0 ) 1 - x 4 ( 0 ) 1 - y 1 ( 0 ) x y arctg + arctg + arctg + arctg + y x y x

4 ( 0 ) - 1 ( 0 ) y ( 0) 1 - x 4 ( 0 ) 1- y + + arctg = 1 arctg arctg x y x
13


.. , .. , ..

+

1 ( 0
2

) + 4 ( 0 ) - 14 ( 0 )


arctg

y . x

y = x
y 0 + , arctg

2

y = x

y x 0 + , arctg = arctg x ( 0) y arctg 2 , 4 y 2

x2 0 , , x = y 2 x - 1 ( 0 ) y arctg x

( x, y ) ( 0+, 0 + ) , , 4 ( 0 ) = 1 ( 0 ) . 2.2 . 2.1. ( x , y

)



G . , (1.3), G . . 1 ( x ) = ( x, 0 ) ,

3 ( x ) = ( x, 1) ,

4 ( y ) = ( 0, y ) ,

2 ( y ) = (1, y ) ,

,

(2.17)

(1.3)
u ( x, y ) =

u (
i =1 i

4

x, y ) ,

(2.18)

ui ( x, y ) , i = 1, 4 (1.4). ,
ui ( x, y ) , i = 1, 4 G . 2.1 -

, , (2.18) . (2.18) . ( 0, 0 ) ( (1, 0 ) , ( 0, 1) , (1, 1) ). ( 0, 0 )
u2 ( x, y ) u3 ( x, y ) , u2 ( x, y ) + u3 ( x, y ) -

. , (2.18) ( 0, 0 )
u1 ( x, y ) + u4 ( x, y

)

(2.19)

.

14




(2.17) ( x, y ) ( 0, 0 ) : 1 ( 0 ) = 4 ( 0 ) . 2.2 , (2.19) ( 0, 0 ) . 2.1 . , , , ( x, y ) , u ( x, y ) , (2.18), G . , ( x, y
1- x
1

)



(2.18), .. u = ( x, y ) ,

1 ( x ) +
y
1



(1 - x )
0

2 ( ) d
2

+

2

+
1

1+ (x - 0 3 ( )d
2

1

1

3 ( ) d

)

2

1 x 4 ( ) d +2 =1 ( x), 0 x +2

1 ( )d y 2 + (1 - 0
1
1

)

2

+ 2 ( y ) +
1

1- y


2

(1 - y ) + (1 - )
0 2

2

+

1



1+ (
0

1

4 ( )d = 2 ( y ), 2 y - )

( )d 1- x 1 +1( x - )2 + 0
1 1

(1 - x ) + (1 - )
0

2 ( )d

+ 3 ( x ) +

4 ( )d 0 x 2 + (1 -
x



1

)

2

= 3 ( x),

y 1 ( )d 1 2 ( )d 1- y + + 2 y2 + 2 0 1 + ( y - ) 0


0

1

3 ( )d + 4 ( y ) = 4 ( y ). 2 2 (1 - y ) +








i ( x), i = 1, 4 i ( x), i = 1, 4 .
:
x 1 ( ) d , 0 < x 1, 2 2 0 x + x) = ( 0) , x = 0, 2 =

( K ) (

(2.20)

( K 0 ) ( x )

1+ (x - 0

1

1

( ) d

)

2

, 0 x 1,

(2.21) (2.22)

( S ) ( x ) = (1

- x), 0 x 1.

15


.. , .. , ..


1 ( x ) + ( SK 2 ) ( x ) + ( K 03 ) ( x ) + ( K4 ) ( x ) = 1 ( x ) , ( KS1 ) ( y ) + 2 ( y ) + ( SKS3 ) ( y ) + ( K 0 4 ) ( y ) = 2 ( y ) , (2.23) ( K 01 ) ( x ) + ( SKS 2 ) ( x ) + 3 ( x ) + ( KS4 ) ( x ) = 3 ( x ) , ( K1 ) ( y ) + ( K 02 ) ( y ) + ( SK3 ) ( y ) + 4 ( y ) = 4 ( y ) . (2.23) .

3. K

( K ) ( x )


=

( ) d x2 + 2 0
x
1

L1 [ 0,1]









( K ) ( x ) = ( x ) ( x )

, ( 0,1] . x = 0 -

, , ,

. , ( x ) x 0 . ( x ( 0,1] . 3.1. ( x ) L1 [ 0, 1] . > 0 C m ( ) > 0 , , ( x ) = ( K ) ( x ) C


)



[ ,1]

, -



(m)

( x)

C m ( ) L [ 0 ,1] 1

(

x 1) .

. m = 0

( K )

( x)

1




0

1

x ( ) d . x +2
2

16






x x + ,
2

2

[ 0,1] ,

max
0 1

x x 11 =2 =, x 2 + 2 x + 02 x

( x 1) .



( K )
1

( x)

1


2


0

1

x ( ) d x +2
2

x max 2 0 1 x +


0

1

( ) d =

1





L1

.

( K ) ( x) . { xk } , x : < x < 1 . ,

( K )


( xk ) - ( K ) ( x) 1 1 2 - xk x ( )d . =2 xk - x 0 ( xk + 2 )( x 2 + 2 )

2 - xk x f k ( ) = 2 ( ) , ( xk + 2 )( x 2 + 2 ) 2 - x2 f ( ) = 2 ( ) , ( x + 2 )2

F ( ) = 2 ( )



4

.

{ f k ( )}

k [ 0,1]

f ( ) . , k ,
f k ( ) F ( ) . ,

[9,10],
( K )( x) = lim

( K )

k

( xk ) - ( K ) ( x) 1 1 2 - x 2 ( )d . =2 xk - x 0 ( x + 2 )2

(3.1)


( K )( x)

( x2 + 2 ) 0
1 = x
2

1

1

2 - x
1

2 2

( ) d

1




0

1

x2 ( ) d = x4 .


0

( ) d

1 x2

L1

17


.. , .. , ..

(. (3.1)),

( K )

( x) =

1





0

1

x ( )d x +2
2

, x . ,
1 1 1 x 1 2 - x2 ( K )( x) = 2 ( )d . ( )d = 2 0 x + 2 x 0 ( x + 2 )2

m - ( K )
( K )
(m) 1 1 P ( x, 1 x 1 ( x) = 2 ( )d = 2m 2 2 0 x + x 0 (x + ) (m) m +1

(m)

( x) -

) ( )d

,

Pm ( x,

Pm ( x, ) [ ,1] ( x 2 + 2 ) m+1 , 2 2 m +1 2 ( m +1) (m) (x + ) > 0 , ( K ) ( x)

)

- m .

[ ,1]

, , . 3.1 . 3.1. ( x ) L1 [ 0, 1] .

( x ) = ( K


( x ) = ( K ) ( x
x0+. 3.2. ( ) L1 [ 0, 1]

)( x) ( 0,1] .

-

)



1 lim ( )d = A . x 0+ x 0

x

(3.2)


A . x 0+ 2 . , : lim ( K ) ( x ) =
1

(

x ( )d x = K ) ( x ) = 2 2 0 x + x2 +
1

(

2

)
0

( s)ds +

2x




00

1

( s)ds

d . (3.3) ( x + 2 )2
2

18




,
x ( )d x 2x x 2 + 2 = (1 + x 2 ) (s)ds + 0 0
1 1


00 1

1

( s)ds

(

d
x +
2

22

)

.

(3.4)

( ) 1 , :
1x d x 2x arctg = 2 = + 2 2 (1 + x ) x 0 x + 1
1


0

(

2 d
x +
2

22

)

,

x > 0 .

(3.5)

arctg

1 = - arctgx , x 0 (3.5) x2 2x




0

1

(
(

2 d
x +
2

22

)
1

=

1 x 1 . - arctgx - 2 2 2 (1 + x )
1 ( s )ds - Ad + 0

(3.6)

(3.3):

(

x K ) ( x ) = 1+ x

2

)
0

( s )ds +

2x




0

1

(



2 22

x2 +

)

+

2 xA




0

1

(

2 d
x +
2

22

)

I 0 + I1 + I 2 .

(3.7)

(3.6),
A . x 0 2 I1 . limI 2 = (3.8)

( ) =

1




0 1



( s)ds - A > 0;

(0) = 0 .

(3.2) , ( ) [ 0,1] . I1 = = 0 < x < x0 < 1. 2 2x




0

(

2 ( )
x +
2 22

22

)

d = 2
1/ x

2

1/ x




0

u 2 ( xu ) du = (1 + u 2 ) 2

1 / x0




0

(1

u 2 ( xu ) +u

)

du +





1 / x0

(1

u 2 ( xu ) +u
22

)

du I11 + I1

2

(3.9)

19


.. , .. , ..

M = sup ( ) (, M < ) ,
0< 1

I12

2

1/ x





1 / x0

u 2 ( xu ) 2M du (1 + u 2 ) 2
1/ x

1/ x



1 / x0

u2 du . (1 + u 2 ) 2
1/ x

(3.10)


2x




0

1

(

2 d
x2 +

22

)

= xu 2 = = d = xdu
1/ x


0

u 2 du 2 = 22 (1 + u )


0

(

2d

1+

22

)

,

(3.5) ,
2




0

(1

2d
+

22

)

=

1 1 x arctg - . x 1 + x2

(3.11)

(3.11), (3.10):
I12 = 2M
1/ x





1 / x0

(1

u 2 du +u

22

)

1/ x 2M u 2 du = - (1 + u 2 ) 2 0

1 / x0


0

u 2 du = (1 + u 2 ) 2

x0 M x M x0 - . (3.12) ( arctg x0 - arctg x ) + < arctg x0 + 2 2 1 + x0 1 + x 2 1 + x0 x0 [ 0,1] arctg x0 < x0 < x0 , 2 1 + x0

I12

M



[

x0 + x0 ] <

2M



x0 .

> 0 - . x0
2M



x0 = , x : 0 < x < x0 I12 < . (3.13)

x0 I11 :
I11 2
1 / x0




0

u 2 ( xu )

(1

+u

22

)

du

2

1 / x0




0

(1


u 2 du +u

2 2 0u 1 / x0

)

max ( xu ) max ( xu ) , (3.14)
0u 1 / x0


1/ x

M1 =

2



0

0

(1

u 2 du +u

22

)

2





0

(1

u 2 du +u

22

)



2





0

du = 1. 1 + u2

20






0u 1/

max ( xu ) = max
x0
0

x / x0

( ) 0 x 0 , , -

> 0 > 0 , , 0 < x < max ( ) < . , (3.14)
1 / x
0

I11 < .

(3.15)

(3.9) (3.13) (3.15), I1 < 2 . (3.16) (3.16), (3.8) (3.7) . 3.2 . 3.2. ( x ) L1 [ 0, 1] x = 0 .
x ( )d 1 lim 2 = ( 0) . x0 x +2 2 0
1

. 3.2
1 lim ( ) d = ( 0 ) . x 0 x 0
x

(3.17)


1 1 ( ) d = x ( ) - ( 0 ) d + ( 0 ) . x0 0
x x

(3.18)

x = 0 ( x ) ,

> 0 > 0 , 0 < x < ( x ) - ( 0) < .

1 1 [ ( ) - (0)]d x ( ) - (0) d < x d = . x0 0 0
x x x

(3.19)

> 0 - , (3.18) (3.19) (3.17). 3.2 . 3.3. K : ( x ( x ) = ( K

)

)( x)

[ 0, 1] . 21

[ 0, 1] ,


.. , .. , ..

, K . [12,13]. [ a , b ] :
fk ( ) f ( ) , k ,

f k ( ) F ( ) , k = 1, 2, ...



F ( ) L1 [ a , b ] . f ( ) L1 [ a , b ] ,

,
lim f k ( )d =
k a b


a

b

f ( )d .
k

3.3. { :
lim
k
0

( x )}

C [ 0, 1] -

k

( x)

= ( x ) , x [ 0, 1] , lim K
k

k

( x)



0

( x)

, k = 1, 2, ...,

x [ 0, 1] ,



( x)

L1 [ 0, 1] .
k

( x)

= K ( x ) , x ( 0, 1] .

(3.20)

( x ) x = 0 , (3.20) [ 0, 1] . . .
f k ( ) =

( x2 +

x k ( x

)
2

)

,

f ( ) =

( x2 +

x ( x

)

2

)

,

f 0 ( ) =

( x2 +

x0 ( x

)
2

)

,

x > 0 . ,
1 1 x k ( ) d x ( ) d lim f k ( ) d = f ( ) d lim 2 =2 , k k x +2 0 x +2 0 0 0 1 1

0 < x 1,


1 1 lim K k ( 0 ) = lim k ( 0 ) = ( 0 ) = K ( 0 k 2 k 2 3.3 .

)

x = 0 .

22




3.4. { :
lim
k 0 x k

k

( x )}

L1 [ 0, 1] -

( s)

ds = ( s ) ds
0

x

x [ 0, 1] ,

(3.21)

L1 [ 0, 1]
s up
k 0 x 1 0 x k

( s)

ds C .


lim K
k k

( x)

= K ( x ) ,

0 < x 1.

. ,
1 x ( ) d x 1 K ( x ) = 2 = 2 2 0 x + x +


2

( s )
0

ds


1

=0

+

2x



( s )
0 0

1



ds

(

d x +
2

22

)

=

=

( x 2 + 1)
x

x


0

1

( s ) ds +

2x




0

1

( s ) ds
0

(

d x +
2

22

)

,
d

(3.22)

K k ( x ) =

( x 2 + 1)


0

1

k ( s ) ds ++

2x




0

1

k ( s ) ds
0



(

x2 +

22

)

.

(3.23)

.
f k ( ) =

(


x +
2


22

)

( s )
k
0

ds , f ( ) =

(


x +
2

22

)

( s )
0



ds , f 0 ( ) =

(

C x +
2

22

)

,

x > 0 . ,
lim 2x
k




0

1

k ( s ) ds
0



(

d x +
2 1

22

)

=

2x


1


0

1

( s ) ds
0



(

d x +
2

22

)

.

(3.24)

, (3.21) x = 1
k

lim k ( s ) ds = ( s ) ds .
0 0

(3.25)

23


.. , .. , ..

(3.22) (3.23), (3.24) (3.25), :
lim Kk ( x ) = K ( x
k

)



0 < x 1. 3.4 . 3.3 .

3.5. { :
lim
k k

k

( x )}

C [ 0, 1] -

( x)
( x)

= ( x ) , x [ 0, 1] , x [ 0, 1] ,
x [ 0, 1] .



0

k

( x)



0

, k = 1, 2, ...,
K 0

( x)

L1 [ 0, 1] . lim ( K 0
k k

)( x) = (

)( x)

,

4. , ,

, .

( K 0 ) ( x )

=

1+ ( x - 0

1

1

( ) d

K 0 C

) [0,1]
2

.

-

. . 4.1. n K 0 : L1 [ 0, 1] C n [ 0, 1] , L1 [ 0, 1] ­ , C n [ 0, 1] ­ n

[

0, 1] , .
24




4.1 .

( K ) (

1 x ( ) d x) = 2 , 0 < x 1; 0 x +2

( K ) ( 0 )

=

(0
2

)

.

4.2. K C [ 0, 1] . . K C [ 0,1] -

3.3. , K . C [ 0,1] n ( x ) = x , n = 1, 2, ... .
1 n

( K n ) ( x )

=

d x t dt = = , x2 + 2 1+ t2 0 0
x
1

1 n

11 nx

1 n

x > 0,

( K n ) ( 0 )
lim ( K

= 0 , n = 1, 2, ... .

x > 0 , n ,

n

n

)( x)

=

dt 1 11 = arctg 0 < x 1 . 2 x4 0 1+ t

1



1 x



( K n ) ( 0 )

= 0,

n = 1, 2, ... ,

{ K n }

. ,

[13] . 4.2 . 4.3. 1 ( K ) = K C C = , 2 ( K ) ­
K : C [ 0, 1] C [ 0, 1] .

25


.. , .. , ..

.

1 1 x ( ) d x ( ) d K ( x ) = 2 0 x + 2 x2 + 2 0



x



C


0

1

d 11 = C arctg C , 2 2 x + x2
1 K ( 0 ) = ( 0 ) . 2 (4.1)

0 < x 1

,
1 . (4.2) C 2C (4.2) K : C [ 0,1] C [ 0,1] K


1 . 2 ( t ) 1 . K (4.3)
0C

= 1 (4.1) -

K K
C C

0C

1 . (4.3) 2

1 =. 2

K K n
0C

n C C

K

n C C

=

1 , 2n

(4.4) (4.5)

K n0 ( 0 ) =
n

1 1 ( 0) = n . n0 2 2 1 . 2n
C C

(4.5) ,
K
n C C

(4.6) = 1 . 2n

(4.4) (4.6) : K

( K ) = lim

n

k

K

n

=

1 . 2

4.3 . (2.23). x , y ( 0,1)
26




, SK 2 0 KS 0 A= K 01 SKS 2 K 0 2 K1

, x .
K 0
3

K4 SKS3 K 0 4 , I 0 KS 4 SK3 0 K , K 0 , S (
J - : ( 0 ) ( x ) = 0,

=

J 0 0 0

0 J 0 0

0 0 J 0

0 0 , 0 J

(4.7)

2.20)-(2.22), 0 ­ ,

( J ) ( x ) = ( x )

-

C [ 0,1] . (2.23)

(


I + A) ( x ) = ( x ) ,

(4.8)

( x ) = (1 ( x ) , 2 ( x ) ,3 ( x ) , 4 ( x

))



, ( x ) = ( 1 ( x ) ,

2

( x ) , 3 ( x ) , 4 ( x ) )



,

«» . A C [ 0,1] , A -
C [ 0,1] = = (1 , 2 , 3 ,

{

4

) :

i

C [ 0, 1] , i = 1, 4

}



C [ 0 ,1]

= max
1i 4

i C [ 0 ,1]

.

4.4.
A : C [ 0, 1] C [ 0, 1]


A2 : C [ 0, 1] C [ 0, 1]

= 1 ­
0 = ( 0 , 0 , 0 ,
0

)



,

0

( x)

1

.

27


.. , .. , ..

. A 0 = 0 . - A 0 :

(
=

A
1

01

)

= ( SK
d
2

0

) ( x ) + ( K 00 ) ( x ) + ( K0 ) ( x )
2

=

1- x



(1 - x )
0

+
1

+

1



1+ (
0

1

d x -

)

2

+

x




0

1

d = x +2
2

1 = arctg 1- x

=0

+ arctg ( - x

)

1

=0

+ arctg


x

1

=0

=

=

1 1 1 + arctg (1 - x ) + arctgx + arctg = arctg 1- x x =

1 + = 1 = 0 ( x ) . 2 2 , -

- A 0 0 ( x ) . A2 0 = 0 . ,
A2 = (4.9)
0

- ).

(

. - 1 (4.9) ­ 0 .

µ + = sup µ : 0 - µ 1 0 , µ - = inf µ : 0 - µ 1 0 .
x [ 0,1] . , µ + µ - . µ0 µ + µ - , ( , µ0 ­ ).
= 0 - µ0 1 .
28

{

}

{

}

(4.10)




­ (4.9). = A ,
= A , = A. (4.11)

- ­ . , - = (1 , 2 , 3 ,
4

)





[ 0,1] . , - , x0 [ 0,1] , . , 1 ( x0 ) = 0 .
(4.11) x0

1 ( x0 ) =

1 - x0



(1
0

1

2 ( ) d
- x0

)

2

+

2

+

1 + ( x0 - 0

1

1

3 ( ) d

)

2

1 x0 4 ( ) d +2 = 0 . (4.12) 0 x0 + 2

, 2 , 3 4 ,
2 ( ) 0, 3 ( ) 0, 4 ( ) 0, 0 < x0 < 1, x0 = 1, 2 ( 0 ) = 0, 3 ( ) 0, 4 ( ) 0, ( ) 0, ( ) 0, ( 0 ) = 0, x0 = 0. 3 4 2 3 ( ) 0 . (4.11)

3 ( x) =

1+ ( x - 0

1

1

1 ( ) d

)

2

+

1- x



(1 - x ) + (1 - )
2 0

1

2 ( ) d

2

1 x 4 ( ) d +2 0. 0 x +2

, 1 , 2 4 ,

1 ( ) 0, 2 ( ) 0, 4 ( ) 0 .

(4.13)

1 ( ) 0 , (4.12) x0 = 1 2 . , 2 ( ) 0 . (4.12)

29


.. , .. , ..

2 ( x) =

x

0 x 2 + (1 -



1

1 ( ) d

)

2

+

1- x



(1 - x ) + (1 - )
2 0

1

3 ( ) d

2

+

1



1+ (
0

1

4 ( ) d
x -

)

2

0.

, 1 . 3 4 ,

1 ( ) 0, 3 ( ) 0, 4 ( ) 0 .

(4.14)

(4.13) (4.14) , - ( x ) , .. 0 ( x ) - µ0 1 ( x ) 0 . - 0 ( x ) 1 ( x ) . , - ( x ) . µ0 . , - 0 ( x

)

, -

= 1 . , A -
0 ( x ) ( -

), = 1 . , , = 1 , A2 . A2 ( ), = 1 . , A ( ), = 1 . 4.4 . 4.1.

(

I + A) = 0

. . - 1 ( x

)



( I + A ) = 0 . A1 = -1 , , A2 1 = 1 .

30




A2 ( ), = 1 ,
1 ( x ) 0 ( x ) ,

- . A 0 = - 0 . 0 ­ A : A 0 = 0 , 0 ( x ) 0 , 0 . 4.1 . 4.5.
A:C C ,

C = C [ 0, 1] , Ak , k = 1, 2, ... ,

( A) = A


k C C

= 1.

. - C [ 0,1]
- C 0 ( x ) ( x ) C 0 ( x ) ,

(4.15)

0 ( x ) - A , = 1 . A (4.15), A ,
- C 0 ( x ) A ( x ) C 0 ( x ) .

()
C

,
A .
C


A
C C

1.

(4.16)

, = 1 A , (4.16)
A
C C

= 1.

(4.17)

31


.. , .. , ..

(4.17) k = 2, 3, ... A
k C C

= 1, , ,
k k

( A ) = lim
4.5 .

A

k C C

= 1.

5. x0 [ 0, 1] ­ . {hk = hk ( x

)}

-

x0 , > 0 > 0 , x [ 0,1] , x - x0 < , hk ( x ) - hk ( x0 ) < k = 1, 2, ... . 5.1. [ 0, 1] :
hk ( x ) C , x [ 0, 1] , k = 1, 2, ... ,

{

h = hk ( x

)}

­

x = 0 . { gk = Khk [ 0, 1] . . :
1 x hk ( ) d hk ( 0 ) Khk ( x ) - Khk ( 0 ) = 2 - = 0 x +2 2 1 1 h ( 0 ) d hk ( 0 ) 1 h ( 0) x hk ( ) d x +2 - - k2 + arctg - k 2 2 0 x + 0 x + x 2

}



hk ( 0



)

1 h (0) x arctg - k + x 2
2C


0

1

hk ( ) - hk ( 0 x2 +
2

)

d I1 + I 2 .

(5.1)

> 0 - . x0 > 0 ,



x0 <


3

.

(5.2)

32




0 < x < x0
I1 =

hk ( 0 ) 1 arctg - x2

1 hk ( 0 ) - arctg = 2 x C



C



arctg <



x<

C



x0 <


3

.

(5.3)

I 2 :
I2 =

x


1


0
1 x0

1

hk ( ) - hk ( 0 x2 +
2

)

d =

= xu 1 hk ( xu ) - hk ( 0 ) = du = d = xdu 0 1 + u2
1

1 x

=




0

hk ( xu ) - hk ( 0 1+ u
22 2

)

du +




1 x0

1 x

hk ( xu ) - hk ( 0 1+ u
2

)

d u I 21 + I 22 .

I
1



I 22 =




1 x0

1 x

hk ( xu ) - hk ( 0 1+ u
2

)

du

1




1 x0

1 x

2Cdu 2C = arctgu 1 + u2 2C

1 x

=
1 x0

2C 1 1 arctg - arctg = x x0 2Cx0

2C



[

arctgx0 - arctgx ]



arctgx0 <



<. 3

(5.4)

(5.4) (5.2). x0 , (5.2). I 21 . {h = hk

}



x = 0 , > 0 > 0 , t : 0 < t <
hk ( t ) - hk ( 0 ) <


3

.

(5.5) 1 x0

x x < x0 . t = xu 0 < u < (5.5) hk ( xu ) - hk ( 0 ) <


3

. :

33


.. , .. , ..

I 21 =

1

1 x0




0

hk ( xu ) - hk ( 0 1+ u
2

)

du <

du 1 = arctg < . (5.6) 2 3 1 + u 3 x0 3 2 3 0

+

1 x0

(5.1), (5.3), (5.4) (5.6),
= . 333 , + Khk ( x ) - Khk ( 0 ) <





{ g k = Khk } x = 0 . [ 0,1] . > 0 - . > 0 , , x - y < , g ( x ) - g ( y ) < . { g k = Khk } x = 0 0 > 0 , , 0 < x g ( x ) - g ( 0 ) <
0




3

.

0 { g k = Khk

}

[ 0 ,1] .

3.1 { g k = Khk } C1 [ 0 ,1] . C [ 0 ,1] , 1 > 0 , ,
x, y [ 0 ,1] , x - y < 1 , -

g ( x ) - g ( y ) <


3

.

= min { 0 , 1} . x, y [ 0,1] x - y < . :
1 ) 0 x, y 0 ; 2 ) 0 x , y 1 ; 3 ) 0 x < 0 < y 1 .

1)
g ( x ) - g ( y ) g ( x ) - g ( 0) + g ( 0) - g ( y ) <


3

+


3

< .

2) g ( x ) - g ( y ) <


3

.

34




3)
g ( x ) - g ( y ) < g ( x ) - g ( 0 ) + g ( 0 ) - g (
+ g (
0 0

)

+

)

- g ( y) < .

5.1 . 5.2. {hk = hk ( x
2) x = 0 .

)}

C [ 0, 1] :
C

1) [ 0, 1] , hk

C;

{

k

= I-K

(

2

)

-1

hk

}

x = 0 . . K
C C

1 = , [13] 2

= (I - K

2 -1

)

h = h + K 2 h + K 4 h + ...

[ 0,1] .
K 2nh
C

K

2n

C C

h

C



C C = n, 22 n 4


K 2nh + K
=
2 ( n +1)

1 1 h + ... C n + n+1 + ... = C 4 4

C 11 . (5.7) 1 + + 2 + ... = n -1 44 3 4 C n , n-1 < . (5.7) : 4 C 4n K 2nh + K
2 ( n +1)

h + ... <
C


3

.

(5.8)



( x ) - ( 0) = ( I - K

2 -1

) h( x) - (

I -K

2 -1

) h ( 0)

=

= h ( x ) - h ( 0 ) + K 2 h ( x ) - K 2 h ( 0 ) + ... .

35


.. , .. , ..

{hk

= hk ( x )} ) 0 > 0 ,
h ( x ) - h ( 0) <

> 0 (

x < 0


3n

.

(5.9)

2k

(

k = 1, 2, ..., n - 1) 5.1, k > 0 , K 2k h ( x ) - K 2k h ( 0) <

x < k


3n

(

k = 1, 2, ..., n - 1) .

(5.10)

= min { 0 , 1 , ..., ,

n -1

}

. (5.9) (5.10) x <

h ( x ) - h ( 0 ) + K 2 h( x) - K 2 h(0) + ... + + K
2 ( n -1)

h( x ) - K

2 ( n -1)

h(0) < n = . 3n 3 h( x ) - K
2 ( n +1)





(5.11)

(5.8)
K 2 n h( x) - K 2 n h(0) + K
2 ( n +1)

h(0) + ... 2 . 3

K 2nh ( x ) + K + K 2n h ( 0 ) + K
2 ( n +1)

2 ( n +1)

h ( x ) + ... +

33 (5.11) (5.12)

h ( 0 ) + ... <



+



=

(5.12)

( x ) - ( 0) < .
5.2 . 5.3. {hk = hk ( x

)}

C [ 0, 1] :
C

1) [ 0, 1] , hk

C;

2) [ 0, ] , 0 < 1 .



{

k

= I-K

(

2

)

-1

hk

}

[ 0, ] .

36




. > 0 ­ . n , (5.8).

{hk

= hk ( x

)}



[

0, ] ,

{ {

K 2 h , K 4 h , ..., K

}{

}{

2 ( n -1)

h

}
2 ( n -1)

[ 0,1] 5.1 , ,
( x) = h + K 2 h + K 4 h + ... + K h

}

[ 0, ] . > 0 ,

( x ) - ( y ) < ,
3

(5.13)

x, y [ 0, ] ,

x - y < .

( x ) - ( y ) ( x ) - ( y ) + Rn h ( x ) + Rn h ( y ) ,
Rn h ( t ) = ( t ) - ( t ) , (5.8) (5.13) x- y < .

( x) - ( y ) < x, y [ 0, ],
5.3 .

37


.. , .. , ..

6.
A : C C C = C [ 0,1] , I A (4.7),

(

)

(4.8) C [ 0,1] . 6.1.

(

I + A ) : C [ 0, 1] C [ 0, 1]

, d > 0 ,

(

I + A)

C

d

C

(6.1)

­ C [ 0, 1] . . . (6.1). ­ k ,

kC

= 1, k = (1k , 2 k , 3k , k + A 0

4k

)



, k = 1, 2, ... ,

(6.2)


kC

k .

(6.3)

(6.3)


1k + KS1k K 01k K1k

SK 2 k + K 03k + K 4 k = 1k 0, + 2 k + SKS3k + K 0 4 k = + SKS 2 k + 3k + KS 4 k = + K 0 2 k + SK3k + 4 k =
2k 3k

0, 0,

(6.4)

4k

0,

x [ 0,1] . (6.2) ,



ik C

1, i = 1, 4, k = 1, 2, ... .

38




{
ik

}

, i = 1, 4,

k = 1, 2, ... . -

, , { . K 0 - , :
( ( ( (
ik

}

-

) K 0 4 k ) K 01k ) K 0 2 k )
K 0
3k

( ( ( (

x ) 13 ( x ) , x ) 24 ( x ) , k x ) 31 ( x ) , x ) 42 ( x ) ,

(6.5)

x [ 0,1] ;
( SK 2 k ) ( x ) 12 ( x ) , ( SKS3k ) ( x ) 23 ( x ) , k ( KS4 k ) ( x ) 34 ( x ) , ( SK3k ) ( x ) 43 ( x ) ,

(6.6)



x [ 0,1] ,

[

0, ] , 0 < < 1 ; ( K 4 k ) ( x ) 14 ( x ) , ( KS1k ) ( x ) 21 ( x ) , k ( SKS2 k ) ( x ) 32 ( x ) , ( K1k ) ( x ) 41 ( x ) ,

(6.7)



x [ 0,1] ,

[ ,1]

, 0 < < 1;

(6.4), (6.5)-(6.7),
1k 2 k 3 k 4 k

( ( ( (

13 23 32 x ) - 41 ( x ) - 42

x ) -12 ( x ) - x ) - 21 ( x ) - x ) -31 ( x ) -

( ( ( (

x ) - 14 ( x ) 1 ( x ) , x ) - 24 ( x ) 2 ( x ) , k , x ) - 34 ( x ) 3 ( x ) , x ) - 43 ( x ) 4 ( x ) ,

(6.8)

39


.. , .. , ..

x [ 0,1] , [ , ] :
0 < < < 1. , (6.8)



x

x k( ) [0,1] ,

( x ) , k , ( x) = (1 , 2 , 3 ,

4

)

T

(6.9)

-

[ , ] : 0 < < < 1 . , - ( x ) [ 0,1] ( 0,1) . (6.9) ,
A
k

( x)

A ( x ) , k ,



x [ 0,1]



[ , ]

:

0 < < < 1. , K (. 3.3) ,

KS1k KS1 SK SK , 2k 2 SKS3k SKS3 K 4 k K 4 ,

, K1k K1 , SKS 2 k SKS2 , , SK3k SK3 , KS4 k KS4 ,

k x [ 0,1] ,

[ , ]

: 0 < < < 1.

K 0 (. 3.5 4.1)
K 0
mk

( x)

K 0

x , k ( x

{

)}

( x ) [0,1] . ( x )
m

k ; m = 1, 2, 3, 4 ;

-

[ 0,1] . , (6.9)











[ , ]

, 0< < < 1 , -

k ( x

{

)}

x = 0 x = 1 .

{1k ( x

. -

)} {4 k ( x )}
40

x = 0 .





1k ( x ) + ( K4 ( K1k ) ( x ) +
k 4

) ( x ) = 1k ( x ) k ( x ) = 4k ( x )

1k 1k

- SK 2 k - K 03k , - K 02 k - SK3k ,

(6.10)

(6.4). (6.10) K , , :

- K 2 = h


( = 1

k

,

h = h1k = 1k - K 4 k ) ,

(6.11) (6.12)

(
2 -1

I - K2 = h.

)

I - K 2 , (6.12)

{3k ( x )} , 3.1, {( SK 2 k ) ( x )} {( SK3 { 1k ( x )} , { 4k ( x )} , {( K02k ) ( [0,1] . { 1k
{h = h1k ( x

= (I - K

)

h .

{2k ( x )}



k

[0,1]



)}

) ( x )} [0, ] . x )} {( K 03k ) ( x )} ( x )} { 4 k ( x )} [ 0, ] .

5.1. 5.3 -

= 1k = I - K [ 0, ] .

{

(

2 -1

)

h1k

}



= 4 k ( x) , h = h4 k ( x) = 4 k ( x) - K 1k ( x) (6.11)

,

[ 0, ] .

{

= 4 k = I - K

(

2 -1

)

h4

k

}

{1k ( x

)}

{

4k

( x )}

-

x = 0 .

41


.. , .. , ..



{1k ( x

{3k ( x )}

{2k ( x )}
)}



x = 1 . x = 0 .
2 k ( x ) + ( SKS3k ) ( x ) = 2 k ( x ) , ( SKS 2 k ) ( x ) + 3k ( x ) = 3k ( x ) ,

{4k ( x )}


(6.13)


2 k ( x ) 2 k - KS1k - K 04 k , 3k ( x ) 3k - K 01k - KS4 k , (6.4). (6.13) SKS .




2k

( x ) - ((

SKS )
2

2k

)

( x ) = 2k ( x ) - (
2

SKS 3

k

)( x)

,

- ( SKS ) = h ,
=
2k

(6.14) . (6.15) (6.16)

( x)

,

h = h2 k = 2 k ( x ) - ( SKS 3

k

)( x)

(6.14)

- SK 2 S = h .
S ,

(

I-K
2

2

)

S = Sh .

I - K
S = I - K

, (6.16) : .

=S I -K

{4k ( x )} [0, ] {( KS1k ) ( x )} {( KS 4 k ) ( x )} [1 - ,1] . { 2 k ( x )} , { 3k ( x )} , {( K 01k ) ( x )} {( K 04 k ) ( x )}
42

( (

2 -1 2 -1

) )

Sh . S

Sh . {1k ( x

)}






[ 0,1] . { 2 k ( x

[ 0,1] [1 - ,1] . , 5.3,

)}

{ 3k ( x

)}

-

[1 - ,1] . (6.15)

{

= 2 k = S ( I - K

2 -1

)

Sh2

k

}

= 3k ( x) , h = h3k ( x ) = 3k ( x ) - ( SKS 2

k

)( x)

,

[1 - ,1] .

{

= 3 k ( x ) = S ( I - K

2 -1

)

Sh3

k

}

{

2k

( x )}

{

3k

( x )} )}

-

x = 1 .

x = 1 {

{1k ( x
2k

( x )}
k 2

x = 0 .
, ,

1k ( x ) + ( SK 2 ( KS1k ) ( x ) +

) ( x ) = 1k ( x ) k ( x ) = 2k ( x )

(6.17)


1k ( x ) 1k - K 03k - K4 k , 2k ( x ) 2 k - SKS 2 k - K 0 4 k , (6.4). (6.17) SK , ,

1k ( x ) - ( SK 2 S1k ) ( x ) = 1k ( x ) - ( SK 2k ) ( x ) ,



((

I -K

2

)

S1k

)

( x ) = ( S 1k ) ( x ) - ( K 2k ) ( x )

.

43


.. , .. , ..



( S1k ) ( x )
. ,

= I-K

(

2 -1

) ((

S 1k

) ( x ) - ( K 2k ) ( x ) )

K 0 , { K 0 { {SKS

1k ( x ) = S ( I - K

2 -1

) ((

S 1k

) ( x ) - ( K 2k ) ( x ) )

.
3k

{K04 k } {4k }

­ [ 0,1] .
2k

}



}

x = 1

x = 0 ( 5.1) 2k

}

{ K

4k

}

[ 0,1] .
2k

{ 1k = 1k - K 03k - K4k } { 2k = [ 0,1] {1k [ 0,1] .

- SKS 2 k - K 0

4k

}

}



(6.17) KS , 2 k ( x ) - K 2 2 k ( x ) = 2k ( x ) - ( KS 1k ) ( x ) ,

(

)


{ 1k } { 2k }
2k

2 k = ( I - K

2 -1

) ( ( x ) - (
2k

KS 1k

)( x ))

.

[ 0,1] , {
x = 1 {

{1k
2k

}

[ 0,1] .

}

}



x = 0 .

, {1k } { 2 k } [ 0,1] . [ 0,1] . {
3k ( x ) + ( KS4 k ) ( x ) = 3k ( x ) 3k - K 01k - SKS 2 k , ( SK3k ) ( x ) + 4 k ( x ) = 4k ( x ) 4 k - K1k - K 02 k , (6.4).
44
3k

}

{

4k

}










3k

( x)

= I -K

(

2 -1

4k

( x)

=S I -K

)} { 4k ( x )} [ 0,1] , {3k ( x )} {4 k ( x )} , (6.18), [ 0,1] . , , {1k ( x )} , {2 k ( x )} , {3k ( x )} {4k ( x )} [0,1] . ( -

{ 3k ( x

(

2 -1

) ( ( x ) - ) ( S ( x )
3k 3k

KS 4k ( x ) ) , - K 4k ( x ) ) .

(6.18)

)
k ( x ) = (1k ( x ) ,
2k

( x ) , 3 k ( x ) , 4 k ( x ) )




( x ) = (1 ( x ) , 2 ( x ) , 3 ( x ) , 4 ( x

))



,

C. (6.4) k ,


1 + KS1 K 01 K1

SK2 + K 03 + K 4 = 0, + 2 + SKS3 + K 0 4 = 0, + SKS 2 + 3 + KS 4 = 0, + K 02 + SK3 + 4 = 0,
C

(6.19)



kC

= 1 k -

0 ,

C

= 1. -

4.1, (6.19) . 6.1 . 6.2. (4.8) C [ 0, 1] . . 6.1 , I + A C [ 0,1] . C [ 0,1]
­ ­ . ,
+ A =

(6.20)

45


.. , .. , ..

C [ 0,1] . 0 < < 1 ­ . ­ C [ 0,1] ,
+ A = , (6.21)


= ( I + A

)

-1

.


A
C C

(

I + A

)

-1



= 1 0 < < 1 .

(6.1) > 1 -

C

d : 2
C

1 1 + A = + A + (1 - ) A C d d 1 1- + A + A C C d d 1 1- 1 1 + + . C C C C d d d 2 , 2 -1 ( I + A) C C C d d 1 - < < 1 - C [ 0,1] . 2 d 1 - < , < 1. 2



- = ( I + A - -

)

-1

- ( I + A

)

-1

= (I + A
I + A

-1 ) ( - ) (

I + A

)

-1

.

, :
C

(

I + A

)

-1

C C

(

)

-1

C C



C

-

4 . C d2

1 -

d < k < 1, k 1, k . , 2

= k , = m , , . ,
k

{}

k

-

k . (6.21)

46




= k , k , (6.20). , I + A C [ 0,1] . (6.1) ( I + A

)

-1



(

I+A

)

-1 C C



1 . d

6.2 . . , -
( x ) = ( 1 ( x ) ,
2

( x), 3 ( x ), 4 ( x ))



, 1 ( 0 ) = 4 ( 0 ) , 1 (1) = 2 ( 0 ) , 3 (1) = 2 (1) , 3 ( 0 ) = 4 (1) . (6.22) 6.3. -
( x ) = (
1

( x)

,

2

( x) ( x)

, ,

3

( x) ( x)

,

4

( x ))



(6.22).
( x ) = (1 ( x ) ,
2 3

,

4

( x ))



(2.23) . . x = 1 x = 0 (2.23)
1 (1) + SK2 (1) + K 03 (1) + K 4 (1) = 1 (1) , KS1 ( 0 ) + 2 ( 0 ) + SKS3 ( 0 ) + K 0 4 ( 0 ) = 2 ( 0 ) .

(6.23)


SK (1) = KS ( 0 ) =

(0
2

)

, K 0 (1) = KS (1) ,

, SKS ( 0 ) = KS (1) , K 0 ( 0 ) = K (1) , 2 (6.23), :
1 (1) 1 (1) 2 (6.24), +

(1)

2 ( 0
2

)

+ KS3 (1) + K4 (1) = 1 (1) , (6.24)

+ 2 ( 0 ) + KS3 (1) + K4 (1) = 2 ( 0 ) .

(2.23), : 1 (1) = 2 ( 0 ) .

47


.. , .. , ..

(2.23) 2 (1) = 3 (1) , x = 0
x = 1 (2.23), 3 ( 0 ) = 4 (1) , x = 0 (2.23)

x = 1 -

1 ( 0 ) = 4 ( 0 ) .

6.3 . 7. K , [ 0,1] . 7.1. { :
k

( x )}

­



k L [ 0 ,1]

N , k = 1, 2, ... .

0 L [ 0, 1]

{

k

j

( x )}

{
lim
j

k

( x )}

,


0

x

(



k

j

( )

- 0 ( ) d = 0, x [ 0, 1] .

)

.


{
k

k

( x ) = k ( ) d
0

x

.

(7.1)

}

:


x

k

( x)



k L [ 0;1]

x

k L [ 0;1]

N

,

k ( x ) - k ( x ) = k ( ) d
x

k L [ 0;1]

x - x K x - x , K = const ,

(7.2)

48




.. k ( x ) . , , , ,
0

( x)

,
k 0 x 1

lim max

k

( x ) - 0 ( x )
0

= 0.

(7.3)

,

0

( x)

, -

(7.2). ,

( x)

-

. 0 ( x ) M , , 0 ( x ) = 0 ( x ) L [ 0;1] . -



0

( x)


x

0 ( x ) = 0 ( ) d ,
0

(7.4)

0 ( 0 ) = 0 . , (7.3), (7.1) (7.4), :
lim max
k 0 x 1

( ( )
k 0

x

- 0 ( ) ) d = 0.

7.1 . 7.2. {
k

( x )}

-





k L [ 0 ,1]

N , k = 1, 2, ... ,


lim
k

( ( )
k 0 k

x

- 0 (

))

d = 0 .


lim (
k

)(

x , y ) = (

0

)(

x, y ) ,



(



)(

x, y ) =

x




0

1

x2 + ( y -

( ) d

)

2

,

0 < x 1, 0 y 1 .

49


.. , .. , ..

. , :

( k ) (


x, y ) =
1

x

0 x2 + ( y -
1



1

k ( ) d


)

2

=

x

( s )
k

ds

x + ( - y
x
2 1

0 2

)

2

+
=0

2x




0

(

-y
2

) k ( s )
0

ds

(

x + ( - y

)

22

)

d =

=

x + (1 - y

(

( s ) ))
2 k 0

ds +

2x





0

1

(

x

2

( - y ) + ( - y )


22

)

( s )
0 0

dsd +

1 2x ( - y ) + 0 x 2 + ( - y

(

)

22

)

(
0

k ( s ) - 0 ( s ) ) ds d.


k ( ) =

(

x

2

( - y ) + ( - y )

22

)

( ( s )
k
0

- 0 ( s ) ) ds

[ 0;1] . , , :
lim (
k k

)(

x, y ) =

22

x2 + ( y - 0
0

x

1

0 ( ) d

)

2

+

+

2x




0

1

(

x

2

( - y ) + ( - y )

)

( s )
0

dsd = (

0

)(

x, y ) .

7.2 . 7.3.

L [ 0;1] .


(

x , y ) G = ( 0, 1) â ( 0, 1)

(

)(

x, y )

L [ 0;1]

.

50




.

( ) (

x, y ) =

x2 + ( y - 0
1

x

1

( ) d
d

)

2



x2 + ( y - 0

x

1

( ) d

)

2





x2 + ( y - 0

x

)

2

(

)

L [ 0;1]

=

1 1- x x + arct g ( arct g y y 7.3 .

)

L [ 0;1]

(

)

L [ 0 ;1]

.

8. ( x ) ­ .
1 ( x ) = lim ( + x ) d , 0 < h h0 < h 0 + 2 h -h
h

(8.1)

x , (8.1). D . ([12], 5, . 275), , D ( x ) = ( x

()

()

D ( x ) ( x ) = 0 .

()

)

.


u ( x, y ) = y




-



y + (x -
2

( ) d

)

2



y

-





y + (x -
2

( ) d

)

2

,

y > 0,

( x, 0 ) y > 0 :
u ( x, y ) , u ( x, y ) = ( x ) ,

y > 0, - < x < , y = 0, x D .

()

(8.2)

51


.. , .. , ..

8.1. ( x , 0 ) , x D , u ( x , y )
y 0+

()

lim u ( x , y ) = ( x ) ,

x D .

()

(8.3)

.
y
-






y + (x -
2

d

)

2

=1



y > 0, - < x < ,

(8.4)

u ( x, y ) - ( x ) =
= y


y





-



( ) - ( x
y2 + ( x -

) d = 2 )


- x =
I1 + I 2 .



-



( x + ) - ( x) y d = 2 2 y +




+

y



>



(8.5)

I1 :
I1 = y


2

-



( x + ) - ( x) d = y2 + 2
dv = ( x + ) - ( x ) d v=

u=

y y +

(

2

)
22

(

)

du = -

(y +
2
2

2 y d

)

( (
0



x + s ) - ( x ) ds
2y


)

=

y = y2 +

(

)

( v ( ) - v ( -

))

+



-



(

v (
y2 +

)

22

)

d .

(8.6)

:
v ( ) - v ( - ) = ( x + s ) - ( x ) ds -
0 -




0

( x + s ) - ( x ) ds =


-

(

x + s ) - ( x ) ds ,



-



(

v (
y2 +

)

22

)

d = + =
0 -



0

= -

52





=


0

(

v (
y +
2

)

22

)

d -




0

(

v ( -
y +
2 2

) d = v ( ) 2 2

)

0

(

- v ( - ) d = 22 y +



)


0

(

2 2 w ( y +
2 2

)

) d 2

,


1 w ( , x ) = 2

-





1 ( x + s ) - ( x ) ds = v ( ) - v ( - ) . 2

, w ( , x ) > 0 , ( x ) ,
x D w ( , x ) 0, 0 .

()

> 0 - . x D

()

>0
0< <





,







sup w ( , x ) <


3

.

(8.6),
I1
2 y 4y 2 w ( , x ) + 2 y +



0

(

2 d
y +
2

22

)

sup w ( , x ) .
0< <

(8.7)



(y +
2

2 y

22

)

+

4y




0

(

2 d
y +
2

22

)

=

2



arctg


y

< 1 , w ( , x ) sup w ( , x ) ,
0< <

(8.7) :
I1 sup w ( , x ) <
0< <


2

.

(8.8)

I 2 . :
I
2



y




>



( x + ) - ( x) y d 2 2 y +
d y2 + 2
L (

>



(x + ) + (x
y2 +
2

)

d

4y







)

=

4 - arctg y 2

L (

)

.

y0 > 0 , 0 < y < y0

53


.. , .. , ..

4 - arctg 2 y

L (

)

<


2

.


I
2




2

.

(8.9)

(8.8) (8.9)
= . (8.10) 22 > 0 ­ , (8.10) (8.3). 8.1 .
1

u ( x, y ) - ( x ) I

+I

2

<



+



( y ) : ( y ) > 0 y > 0

( y ) 0 y 0 + . x0 D ( y )
G ( x0 , ) =

()

{(

x, y ) : x0 - x y ( y ) , y > 0} .
u ( x, y ) = ( x0 ) .

u ( x, y ) G ( x0 , ) ­ ( x0 , 0 ) ,
( x , y )( x0 ,0 ) ( x , y )G( x0 )

lim

u ( x, y )

( x , y )( x0 ,0 ) ( x , y )G( x0 )

lim

(8.11)

8.1. ( x ) ­ x0 D . u ( x , y ) , (8.2), G ( x0 , ) ­ ( x0 , 0 ) . . x0 D (8.1), > 0 h0 > 0 ,
1 ( x0 ) - h ( x0 + ) d < 2 2h -
h

()

()

0 < h < h0 .

(8.12)

(8.4), (8.5),
u ( x, y ) - ( x0 ) = I1 + I 2 ,

54





I1 = = y


y





( x + ) - ( x0 ) d , y2 + 2 ( x + ) - ( x0 ) d . y2 + 2
(8.13)

I

2



>



:
I1 =
u=

y





-



( x + ) - ( x0 ) d = y2 + 2
dv = ( x + ) - ( x0


y y2 +

=
du = -

(

2

)
22

)
=

(y +
2

2 y d

)

v = ( x + s ) - ( x0 ) ds
0



y = y2 +

(

2

)


v (

)
= -

+

2y





-



(

v (
y2 +

)
2

)

2

d .


v ( ) - v ( - ) = - = + = ( x + s ) - ( x0 ) ds ,
0 0 0
0

-

0



-

-



-



(

v (
y2 + =


)

22

)

d = + =
0 -



= -
0




0

(

v (
y2 +

)

22

)

d +





(

- v ( - y2 +

22

)

)

( -d )

=

2 v ( ) - v ( - ) d = 2 w ( , x, x0 ) d , = 2 y2 + 2 2 0 0( ( y2 + 2 ) )


1 w ( , x, x0 ) = 2

-

(



x + s ) - ( x0 ) ds ,

55


.. , .. , ..

(8.13)
2 2 y 1 4 y w ( , x0 , x0 ) I1 = 2 w ( , x, x0 ) + d + y + 2 2 y2 + 2 2 0

(

)

+

4y






0

(


2

2 22

y +

)

w ( , x, x0 ) - w ( , x0 , x0 ) d .

(8.14)

w ( , x, x0 ) x x = x0 : 0 < . ,
1 w ( , x, x0 ) = 2


-



1 ( x + s ) ds - ( x0 ) = x + s = u = 2

x + x - x+ x-

x + x-

( u )

du - ( x0 ) .

x < x , :
1 w ( , x, x0 ) - w ( , x, x0 ) = 2
x



( u ) du -



( u ) du =

(8.15)



1 = ( s - ) - ( s + ) ds . 2 x

(8.15), : 0 < x x , : w ( , x, x0 ) w ( , x, x0
w ( , x, x0 )

(8.12)

) w ( w (


x x , ,
, x0 , x0 , x, x0

) )

x x0 .

(8.16)

0 < h < h0 . (8.17) 3 : 0 < < h0 ­ . (8.16) = ( ) > 0 ,
w ( , x, x0 ) - w ( , x0 , x0 ) <

w ( h, x0 , x0 ) <


3



x - x0 < .
2 . 3

(8.18)

(8.17) (8.18)
w ( , x, x0 ) w ( , x, x0 ) - w ( , x0 , x0 ) + w ( , x0 , x0 ) <

56




, (8.14) 2 y 1 w ( , x, x0 ) < , (8.19) 2 2 y + 2 3 0 < < h0 x - x0 < . (8.14)
2 4 y w ( , x0 , x0




0

(

y2 +

2

)

2

) d <

3

,

(8.20)

. . :
4y





0

(



2 22

y2 +
4
y

)

= ys w ( , x, x0 ) - w ( , x0 , x0 ) d = d = yds =
s
2 22

=




0

(
0

1+ s

)

w ( ys, x, x0 ) - w ( ys, x0 , x0 ) ds =

2 = y

y



(1

s +s

22

)

x ( t - ys ) - ( t + ys ) dt ds . x
0

, :
4y



2 y


0

(


2

2 2

y + s +s

)

2

w ( , x, x0 ) - w ( , x0 , x0 ) d
0

y


0

(1
y

22

)

x ( t - ys ) + ( t + ys ) dt ds x

2 y


0

(1

sds +s

22

)

2

L (

)



2 x0 - x y

L ( )



2 ( y



)

L ( )

.

(8.21)

57


.. , .. , ..

( x, y ) G ( x0 , ) , y0 > 0 , 0 < y < y0
, ( x, y ) G ( x0 , ) , 0 < y < y0 . 3 (8.19), (8.20), (8.21) (8.22), : 2 ( y



)

L (( ))

<



(8.22)

I1 < .

(8.23)

I 2 . :
I
2



y




>



( x + ) - ( x0 ) y d 2 2 y +


>



( x + ) + ( x0
y2 +
2

)

d

4y







d y2 + 2

L ( ) ( )

=

4 - arctg y 2

L ( )) (

.

y0 > 0 ,
(8.22) 0 < y < y0 4 - arctg 2 y
L ( )) (

< .


I
2

. < + = 2 .

(8.24)

(8.23) (8.24) :
u ( x, y ) - ( x0 ) I1 + I
2

(8.25)

> 0 ­ , (8.25) (8.11). 8.1 .

58




9. 1.1 ­ 1.3 1.1. ) ) , u ( x, y ) ( , . [11]). ) ). , u ( x, y ) G . ( x, y ) = u ( x, y

)



. -

1 ( x ) = ( x, 0 ) , 2 ( x ) = (1, x ) , 3 ( x ) = ( x,1) , 3 ( x ) = ( 0, x ) .
, 6.2,
( x ) = (1 ( x ) , 2 ( x ) , 3 ( x ) , 4 ( x v ( x, y ) =

))



(2.23). G

v (
i =1 i i

4

x, y ) ,


vi ( x, y ) = ( i

)(

x, y ) , i = 1, 4 ,

i , i = 1, 4 (1.4). -
( x ) = ( 1 ( x ) ,
2

( x ), 3 ( x ), 4 ( x ))



, , 6.3, - ( x ) . , - ( x, y

)

-

, , 2.1,
v ( x, y ) G . v ( x, y

( x, y ) : v ( x, y [1] G : u ( x, y ) v ( x, y ) .

)



= ( x, y

) ).

(1.2) ) ) . ) ) 2.1. 1.1 . 59


.. , .. , ..

1.2. ) ). u ( x, y G ,
u ( x, y ) M ,

)

­

(

x, y ) G.


1- 1- u ( x, y ) = u x + , y + , 0 < < 1. 2 2

u ( x, y ) G , u ( x, y )
-1 + G = , 2 2

1

-1 + â , 2 2

1 ,

G = G . u ( x, y ) G : 0 < < 1 . u ( x, y )



(

x, y ) :

(

x, y ) = u ( x, y

)



.


-


( x ) = ( 1 ( x ) , 2 ( x ) , 3 ( x ) , 4 ( x ) )
2

, ( 0, t ) .





1

(t )

= (t, 0) ,

(t )

= (1, t ) ,

3

(t )

= ( t ,1) ,


4

, : 0 < < 1 -

(t ) = ( x )



. 6.1 6.2 -



( x ) = (1 ( x ) , 2 ( x ) , 3 ( x ) , 4 ( x ) )
(
I + A ) =




,



( x)
,


C

1 d

C

d ­ .

60




u ( x, y

)



G = G , , 1.1 ),
u =


i =1

4

ui , ui = ( i

i

)(

x, y ) ,
k

(
k

x, y ) G, i = 1, 4 .

(9.1)

{ {

}:

1 - 0 k . -

, 7.1, , -

} 1,1 ( x ) , 2,1 ( x )
k


,



3,1

( x)

,



4 ,1

( x)

L [ 0;1] ,

:
k

lim

(
0

x



i

k

( ) - i ,1 ( ) ) d
x, y ) ,

= 0,

i = 1, 4 .

8.2
ui ( x, y ) ui
k

,1

(

(

x, y ) G, i = 1, 4 k .

(9.2)

(9.1) = k , k ,
(9.2), lim u ( x, y ) =
k
k

u (
i =1 i1

4

x, y ) .

(9.3)

, u :
lim u ( x, y ) = u ( x, y ) .
k
k

(9.4)

(9.3) (9.4) , u ( x, y

)



, (1.4). ) ) . ) ). 1 = T , 2 = S , 3 = ST , 4 = ,

( ) (

x, y ) =

x2 + ( y - 0

x

1

( ) d

)

2

, (Tu ) ( x, y ) = u ( y, x ) , ( Su ) ( x, y ) = u (1 - x, y ) ,

, 7.1, ) ). 1.2 .

61


.. , .. , ..

1.3. . . ( x, y ) , -
( x ) = ( 1 ( x ) ,
2

( x ), 3 ( x ), 4 ( x ))



,

i , i = 1, 4 ­ , C [ 0,1] . , 6.1, (2.23) C [ 0,1] , , 6.3, . , 1.1, , u ( x, y ) = ( x, y 1.3 .

(

)

)

-

G - ( x, y ) .

10. , ­ L1,loc ( ) ­ , . , . 10.1. x ( ) ­ , [12,

. 275] L1,loc ( ) ,

( x ) ±
1 lim ( x + t ) - ( x ) dt = 0 . h 0 h 0
h

L1,loc ( ) , [12, . 275] x ( ) .

62




, L1,loc ( ) . 10.2. x ( ) L1,loc ( ) ,
1 lim ( x + t ) dt = a+ ( x, ) , h0 + h 0 1 lim ( x + t ) dt = a- ( x, ) , h0 + h -h
0 h

(10.1) (10.2)

a+ ( x, ) a- ( x, ) ­ . , , . . x0 L1,loc ( )

, a+ ( x0 , ) = a- ( x0 , ) = ( x0 ) . : . ,

( x) =

x0 = 0

0, x 0, 1, x > 0,

,

a+ ( x0 , ) = 1 , a- ( x0 , ) = 0 .

,
1 1 lim ( x + t ) - ( 0 ) dt = lim ( x + t ) dt h 0 h h 0 h 0 0
h h

. : x 0 . , , ­ . ,

63


.. , .. , ..

( x) =

0, x - , 1, x - ,

( ) . 10.3. x ( ) L1,loc ( ) , a+ ( x, ) = a- ( x, ) , a ( x, ) a+ ( x, ) = a- ( x, ) . 10.4. x ( )

L1,loc ( ) , : v ( ) = ( s ) ds
0



[12, . 323],
v ( x + h) - v ( x - h) . h 0 + 2h , . , x s ( x ) = lim

1 ( a+ ( x ) + a- ( x ) ) . , 2 .

, s ( x ) =

5 [12, . 275] x ( )
s ( x ) = ( x ) . . ,
1 1 sgn x, 2 n-1 x 2 n- 2 , 2 2 1 1 ( x ) = 0, < x < 2 n-2 , 22 n 2 x > 1. 0, x0 = 0 ( x ) ,
v ( x0 + h ) - v ( x0 - h ) = ( s ) ds 0 ,
-h h

h( ) ,

64





h0 +

lim

v ( x0 + h ) - v ( x0 - h ) = 0. 2h
h

x0 = 0 ( x ) . 1 , : h1n
h
1n


0

2 1 ( s ) ds = , h = h1n = 2 n 3 2
h

-1

1 h2

2n


n0

( s ) ds = ,

4 3

h = h

2n

1 1 = 2 n . , lim ( s ) ds . h0 + h 2 0
h

1 ( x ) , lim ( s ) ds h0 - h 0

. 10.5. ( x ) , ,
. 0, . , . L ( ) .


( x) =

s ( x ) , x - ,

u ( x, y; ) u ( x, y ) =
y 0 + .

y



-



y2 + ( x -

( ) d

)

2

(10.3)

10.1. L ( ) x0 ( ) .
y 0+

lim u ( x0 , y ) = s ( x0 ) .

65


.. , .. , ..


. v ( ) = ( s ) ds
x0

u ( ) = 2 y +( -2 ( du ( ) = y2 + (

1 x0 - x0 -

)

2

(

- x0 ) d

)

22

)

dv ( ) = ( ) d , v ( ) = ( s ) ds x
0




u ( x0 , y ) =

y



-



y + ( x0 -
2

( ) d

)

2

=

y

(s)
x0





ds

y 2 + ( x0 -
2y


)

2

+
= -

+

2y





-



(

(
2

- x0 ) v (

)

y + ( x0 -
=

)

22

)

d = = x0 + t =



-


)

(

tv ( x0 + t y2 + t

22

)

)

dt =

2 y tv ( x0 + t




0

(

y2 + t

22

)

)

dt +

2y



-



0

(

tv ( x0 + t y2 + t

22

)

dt =

=

2 y tv ( x0 + t




0

(

y2 + t

22

)

)

dt -

2 y tv ( x0 - t




0

(

y2 + t

22

)

)

dt =

2 y t ( v ( x0 + t ) - v ( x0 - t




0

(

y2 + t


22

)

))

dt =

1 ( v ( x0 + t ) - v ( x0 - t 4 y 2t = 2 y2 + t 2 0


))

(

)

- s ( x0 ) t 2 dt + 4 ys ( x0



)


0

(
2

t

2 22

y2 + t

)

dt I1 + I 2 .

. > 0 ­ .
I1 4y






0

1 ( v ( x0 + t ) - v ( x0 - t ) ) - s ( x0 ) t 2t

(

y +t
2

22

)

dt =

=

4y




0

1 ( v ( x0 + t ) - v ( x0 - t 2t

))

- s ( x0 ) t

2

(

y +t
2

22

)

dt +

66




+

4y







1 ( v ( x0 + t ) - v ( x0 - t ) ) - s ( x0 ) t 2t

2

(

y +t
2

22

)

dt I11 + I12 .

: 0 < <


2

,

1 ( v ( x0 + t ) - v ( x0 - t 2t t 2 dt y2 + t

))

- s ( x0 ) <



0




I11 < 4 y

(


22

)

=

1 t t arctg - 2y y 2 y2 + t

(

2

)

,

(10.4)


=


0

(

t
2

2 22

y +t

)

4 y 1 t t arctg - dt = 2 2y y 2 y +t

(

2

)





=
t =0

2 4 y 1 arctg - < arctg < . 2 2 2y y 2 y + y 1 . ( v ( x0 + t ) - v ( x0 - t ) ) - s ( x0 ) 2t , c ,

(

)

1 ( v ( x0 + t ) - v ( x0 - t ) ) - s ( x0 ) < c 2t



0
I12 . y0 =
I12 < 4cy



4c

0 < y < y0 ,









(

t

2 22

y2 + t

)

4cy 1 t t arctg - dt = 2y y 2 y2 + t

(

2

)

=
t =

4cy 1 1 = - arctg + 2 2y 2 2y y 2 y +

(

2

)

=

2c y y = arctg + 2 y2 +

(

2

)

2c y y 4cy < + = < .

67


.. , .. , ..

,
I1 < 2



0 < y < y0 .

(10.5)

(10.4), I 2 :
4 ys ( x0 ) 1 t t arctg - I2 = 2 2y y 2 y +t

(

2

)





=
t =0

4 ys ( x0 ) = s ( x0 ) . 4y

(10.5)
u ( x0 , y ) - s ( x0 ) = I1 < 2



0 < y < y0 .

10.1 . , L ( ) , (10.3) u ( x, y; ) u ( x, y; ) . , , , L ( ) . 10.1 . 10.1. L ( ) x0 ( ) .
y 0+

lim u ( x0 , y; ) = ( x0 ) .

10.6. , u ( x, y ) ,
(10.3), ( x0 , 0 ) , x0 ( ) , -

: 0 < < +
y 0 + x - x0 < y

lim u ( x, y ) .

, ( x0 , 0 ) , x0 ( ) , . L ( ) x0 ( ) .
a+ = a+ ( x0 ) , a- = a- ( x0 ) , a+ ( x0 (10.1) (10.2). :

)

a- ( x0

)



68





I + ( h, y ) = I - ( h, y ) =
x

2y


2y


0 0

( (

(

t - h)t

y2 + (t - h

)

22

)

1 v ( x0 + t ) - a+ dt , t 1 v ( x0 + t ) - a- dt , t



-



(

t - h)t

y2 + (t - h

)

22

)

v ( x ) = ( s ) ds ­ ,
0

I 0 ( h, y ) =

2a+ y






0

(

(
2

t - h)t

y + (t - h

)

22

)

dt +

2a- y



-



0

(

(
2

t - h)t

y + (t - h

)

22

)

dt .

u ( x, y )
u ( x, y ) = I 0 ( h, y ) + I + ( h, y ) + I - ( h, y ) , (10.6)

x h h = x - x0 . , ,
u ( x, y ) = + y




-



y2 + ( x -

( ) d

)

2

=

y


y2 + ( x -

v (

)



)

2

+
= -

2y





-



(

(
2

- x ) v (

)

y + (x -

)

22

)

d =

2y



-



(

(
2

- x ) v (

)

y + (x -

)

22

)

d .

, = x0 + t
h = x - x0 , u ( x, y ) =

2y





-



(

(

t - h ) v ( x0 + t y2 + (t - h

)

22

)

)

dt =

2y





-



(

(

t - h)t

y2 + (t - h

)

22

)

1 v ( x0 + t ) dt = t

=

2y



-



0

(

(
2

t - h)t

y + (t - h

)

22

)

2a- y 1 v ( x0 + t ) - a- dt + t

-



0

(

(
2

t - h)t

y + (t - h

)

22

)

dt +

+

2a+ y






0

(

(

t - h)t

y2 + (t - h

)

22

)

dt +

2y






0

(

(

t - h)t

y2 + (t - h

)

22

)

1 v ( x0 + t ) - a+ dt = t

= I - ( h, y ) + I 0 ( h, y ) + I + ( h, y ) .

69


.. , .. , ..

10.2. L ( ) x0 ( ) .
h y y 0+

lim I

+

(

h, y ) = 0 ,

h y y 0+

lim I

-

(

h, y ) = 0 .

( 0, ]
1 v ( x0 + t ) - a+ < t

. > 0 ­ . > 0 ,



0
t ( ,

)



1 v ( x0 + t ) - a+ ­ , , t

1 v ( x0 + t ) - a+ < c+ t ( , ) , c+ > 0 ­ . t

,
( + 4 y0 = min , 12c+ 2 + 4 I + ( h, y ) , h < y,

)

,



0< y< y

0

(10.7)

, y0 ,
h < y < y0


2

.


I + ( h, y ) 2 y


2y






0

(

(

t - h)t

y2 + (t - h

)

2

)

2

1 v ( x0 + t ) - a+ dt < t


<




0

(

(
2

t - h)t

y + (t - h

)

2

)

2

dt +

2 yc+





(

(
2

t - h)t

y + (t - h


)

2

)

2

dt .


I
+1

(

h, y ) =

2y






0

(

(
2

t - h)t

y + (t - h

)

2

)

2

dt ,

I

+2

(

h, y ) =

2y





(

(
2

t - h)t

y + (t - h

)

2

)

2

dt ,

70





I + ( h, y ) < I
+1

(

h, y ) + c+ I

+2

(

h, y ) .

(10.8)

I

+1

(

h, y ) . : ) h < 0 , ) h > 0 .


) h < 0 . ( t - h ) t = ( t - h ) t , ,
I
+1

(

2y 1 t-h 1 t h, y ) = -2 arctg 2y y 2 y + ( t - h )2

=
t =0

=

1 -h h y arctg + arctg - 2 y y y + ( - h 1 + 2 2 y - y 2 + ( - h

)

2

- 0

(

)

2

)

<1.

) h > 0 . h <
I
+1


2

.

(

h, y ) =

2y




0

h

(

(
2

t - h)t

y + (t - h

)

2

)

2

dt +

2y






h

(

(
2

t - h)t

y + (t - h

)

2

)

2

dt


2y




0

h

(

(
2

t - h)t

y + (t - h

)

2

)

2

dt = -

2y




0

h

(

(
2

t - h)t

y + (t - h

)

2

)

2

dt =

1 0 h y h = - arctg + arctg + 2 - 0 = y y y

= 2y


h1 hh h - arctg =, y y y h


=


h

(

(
2

t - h)t

y + (t - h

)

2

)

2

dt =

2y






h

(

(
2

t - h)t

y + (t - h

)

2

)

2

dt =

1 -h 0 y h arctg + arctg - 2 - 2 y y y + ( - h )2 y 1 -h y h 1 h arctg - + <+ = 2 y 2 h y y 2 + ( - h )


1 +, 2

(

)

71


.. , .. , ..

I

+1

(

h, y ) <

1 2 + . 2

, h <
I


2
1

, I

+2

(

h, y ) .


+2

(

t-h h, y ) = arctg y



-
t =

y

t

y2 + (t - h

)

2 t =

=
= y

1 -h y - arctg - 0 - 2 2 y 2 y + ( - h ) 1 y y 1 y = arctg + < + - h y 2 + ( - h )2 - h y 2 =

(

)

(

+ ( - h

)

2

)

<

2 2 0 + 2 (10.8)

<



1y +

y



=

1 2y 4y 6y + . =

1 2 I + ( h, y ) < + 2

6y 1 2 < + + c+ 2

6 y0 + 4 + c+



(

h, y ) : h < y, 0 < y < y0 ,

(10.7). , > 0 ,
h y y 0 +

lim I + ( h, y ) = 0 .

lim I - ( h, y ) = 0
h y y 0 +

. 10.2 . . 10.1. L ( ) x0 ( ) . : 0 < < +
x

x - x0 a + + a - a - a- lim u ( x , y ) - + arctg . = - x y 2 y y 0+
0

(10.9)

72




. I 0 ( h, y ) , (10.6). ( t - h ) tdt = 1 arctg t - h - t , 2 2 22 2y y 2 y2 + (t - h) y + (t - h)

(

)

(

)


2y 1 t -h t I 0 ( h, y ) = a+ arctg - 2y y 2 y2 + (t - h

(

)

2

)





+
t =0

2y 1 t -h t + a- arctg - y 2y 2 y2 + (t - h

(


)

2

)



0

=
t = -

1 t-h = a+ arctg y 1 t-h + a- arctg y
0





-
t =0

y

t

y2 + (t - h
y

)

2
t =0

+ =

0

-
t = -

y2 + (t - h



t

)

2
t = -

= a+ + a-

-h y 1 arctg ( + ) - arctg - ( 0 - 0 ) + y -h g y y - arctg ( - ) - - ( 0 - 0 ) =
h y 1 1 h + a- - arctg y 2 1 h + ( a+ - a- ) arctg . y =

1 arct

1 1 = a+ + arctg 2 1 = ( a+ + a- ) 2


1 1 ( a+ + a- ) + ( a+ - a- ) arctg 2 , (10.6) 10 . 10.1 . I0 = h . y .2, -

73


.. , .. , ..

u ( x, y .

)

10.2. L ( ) x0 ( ) . (10.3) ( x0 , 0 ) , , x0 . u ( x , y ) , -

. . u ( x, y ) -

( x0 , 0 ) , : 0 < < +
x - x0 < y y 0 +

lim u ( x, y ) = A .

, = 1 . (10.9), : a+ - a- x - x0 a + a- lim arctg = A- + . x x-0+y y 2 y
0

(10.10)

( x, y ) ( x0 , 0 ) , ,
x - x0 y y 0 +

lim arctg lim arctg

x - x0 1 = arctg 2 ( x - x0 ) 2 x - x0 1 = arctg 3 ( x - x0 ) 3

y = 2 ( x - x0 ) , y = 3 ( x - x0 ) ,

x - x0 y y 0 +

, (10.10) a+ - a- = 0 . , x0 . . . x0 ­

L ( ) , a+ = a- . (10.9)
x - x0 < y y 0 +

lim u ( x, y ) =

a+ + a- . 2

. 10.2 .
74




11. u ( x, y ) ,
G = ( 0, 1) â ( 0, 1) , .

11.1. ( x, y ) , ( x, y ) , G u ( x, y ) , ) lim u ( x, y ) = ( x, 0 ) 1 ( x ) x : 0 < x < 1;
y 0 +
x1-

) lim u ( x, y ) = (1, y ) ) lim u ( x, y ) = ( x,1)
y 1- x 0 +

2

( y)

y : 0 < y < 1; x : 0 < x < 1; y : 0 < y < 1.

3

( x)
( y)

) lim u ( x, y ) = ( 0, y ) ( x, y

4

)



u ( x, y ) , u = .

L [ 0,1] .
u ( x, y ) = ( 1 ) ( x, y ) =

y 2 + ( - x 0

y

1

( ) d

)

2

,

y > 0.

(11.1)

, u ( x, y ) , (11.1), , ( ) . , ( ) . (11.1) . , L [ 0, 1] . L ( ) , [ 0, 1] . « » « » x = 0 x = 1 , :

75


.. , .. , ..

) x = 0 , ,
a- ( 0 ) = 0, a+ ( 0 ) = 2 s ( 0 ) ,

) x = 1 , ,
a+ (1) = 0, a- (1) = 2 s (1) .

, u ( x, y ) , (10.3), ( 1 ) ( x, y ) , (11.1) ( x ) , [ 0, 1] . 10.1 . 11.1. L [ 0, 1] x0 .
y 0+

lim ( 1

)(

x , y ) = s ( x0 ) .
4

- = (1 , 2 , 3 , ( x, y ) .

)

T



11.2. ( x, y ) : ) ( x, 0 ) , x ( 0, 1) ­ 1 ; ) (1, y ) , y ( 0, 1) ­ 2 ; ) ( x, 1) , x ( 0, 1) ­ 3 ; ) ( 0, y ) , y ( 0, 1) ­ 4 . 11.3. ( x, y ) : ) ( 0, 0 ) , :
1 a+ ( 0; 1 ) lim 1 ( s ) ds h 0+ h 0
h h



a+ ( 0;

4

)

1 lim 4 ( s ) ds ; h 0+ h 0

) (1, 0 ) , :

76




1 a- (1; 1 ) lim 1 (1 + s ) ds h0 + h -h
0

0



a+ ( 0;

2

)

1 lim 2 ( s ) ds ; h 0+ h 0
0

h

) (1, 1) , :
a- (1;
2

)

1 lim 2 (1 + s ) ds h 0 + h -h
h



1 a- (1; 3 ) lim 3 (1 + s ) ds ; h0 + h -h
0

) ( 0, 1) , :
1 a+ ( 0; 3 ) lim 3 ( s ) ds h0 + h 0



a- (1;

4

)

1 lim 4 (1 + s ) ds . h 0 + h -h

11.4. ( x, y ) : ) ( x, 0 ) , x ( 0, 1) ­ 1 ; ) (1, y ) , y ( 0, 1) ­ 2 ; ) ( x, 1) , x ( 0, 1) ­ 3 ; ) ( 0, y ) , y ( 0, 1) ­ 4 . 11.5. ( x, y ) : ) ( 0, 0 ) , :
h h 1 lim 1 ( s ) ds + 4 ( s ) ds ; h0 + 2 h 0 0

) (1, 0 ) , :
0 h 1 lim 1 (1 + s ) ds + 2 ( s ) ds ; h0 + 2 h 0 -h

) (1, 1) , :
0 0 1 2 (1 + s ) ds + 3 (1 + s ) ds ; lim h0 + 2 h -h -h

) ( 0, 1) , :
h 0 1 lim 3 ( s ) ds + 4 (1 + s ) ds . h0 + 2 h -h 0

77


.. , .. , ..

i L [ 0, 1] , i = 1, 4 , [ 0, 1] .
u1 ( x, y; 1 ) = u 2 ( x, y; y




0

1

y2 + ( x -
2

1 ( ) d

)

2

,

2

)

=

1- x


1- y

(1 - x ) + (
0 1 2 0

1

2 ( ) d
y - x -

)

2

,

u3 ( x, y; 3 ) = u 4 ( x, y;


4

(1 - y ) + (
1

3 ( ) d

)

2

,

)

=

x2 + ( y - 0

x
4

4 ( ) d

)

2

, (11.2)

u ( x, y ) =

u (
i =1 i

x , y; i ) .

11.1. i L [ 0, 1] x0 ( 0, 1) ­ 1 . : 0 < < +
x

a lim u ( x , y ) - - x < y y 0
0

+

(

x0 , 1 ) - a



-

(

x0 ,

1

)

arctg

x - x0 = y

=

x0 , 1 ) + a - ( x0 , 1 ) + u2 ( x0 , 0; 2 , a
+

(

2

)

+ u3 ( x0 , 0;

3

)

+ u4 ( x0 , 0; 4 ) .

u-

a+ ( x0 ,1 ) - a- ( x0 ,1



)

arctg

x - x0 = y
4

a ( x , ) - a- ( x0 ,1 ) x - x0 = u1 - + 0 1 arctg + ( u2 + u3 + u y

)

u2 , u3 , u4 ( x0 , 0 ) ,
10.1 . :

11.2. i L [ 0, 1] x0 ( 0, 1) ­ 1 . u ( x , y ) ( x0 , 0 ) , x0 78




1 .
x - x0 < y y 0

lim u ( x , y ) = a ( x0 ,1 ) + u2 ( x0 , 0;

2

)

+ u3 ( x0 , 0;

3

)

+ u4 ( x0 , 0; 4 ) .

1 2 ­ , : 0 < 1 < 2 < + . :

) = {( x, y ) G : 1x y 2 x} , G (1, 0 ) = G1,0 (1 , 2 ) = {( x, y ) G : 1 (1 - x ) y 2 (1 - x )} , G (1,1) = G1,1 (1 , 2 ) = {( x, y ) G : 1 (1 - x ) 1 - y 2 (1 - x )} G ( 0,1) = G0,1 (1 , 2 ) = {( x, y ) G : 1 x 1 - y 2 x} .
G ( 0, 0 ) = G0,0 (1 ,
2

,

(

0, 1)

(1, 1)
y = 2x

G ( 0, 0

)

y = 1 x

(

0, 0

)

(1, 0 )

11.6. ,
u ( x, y ) ( x0 , y
0

)



G , 1 , 2 : 0 < 1 < 2 < +
( x , y )( x0 ( x , y )G( x0

lim

, y0 , y0

) )

u ( x, y ) .

79


.. , .. , ..

11.3. 1 , 4 L [ 0, 1] t = 0 ­ .
( x , y )( 0 , 0 ) ( x , y )G ( 0 , 0 )

lim

u1 ( x , y; 1 ) + u4 ( x , y;

4

)

-

-

1 x y a ( 0,1 ) + a+ ( 0, 4 ) a+ ( 0, 1 ) arctg + a+ ( 0, 4 ) arctg = + . (11.3) y x 2 . t = 0 ­ 1 ( t ) ,
1 1 a+ ( 0,1 ) = lim 1 ( t ) dt , a- ( 0,1 ) = lim 1 ( t ) dt = 0 . h 0 + h h 0 + h -h 0
h 0



4 ( t ) :
a+ ( 0,
4

)

1 = lim 4 ( t ) dt , a- ( 0, h 0 + h 0

h

4

)

1 = lim 4 ( t ) dt = 0 . h 0+ h -h

0

10.1 u1 ( x, y; 1 ) x0 = 0 ,

=

1



,

1

a ( 0,1 ) x a ( 0,1 ) lim u1 ( x, y; 1 ) - + arctg = + , 1 y 2 x y 1
y 0 +

(11.4)

u4 ( x, y;
a lim u4 ( x, y; 4 ) - + y x x 0 +
2

) ( 0,4 )
4

x0 = 0 , = 2 ,
y a ( 0, 4 ) arctg = + . x 2 (11.5)

G0,0 (1 ,

2

)

, 2

(11.4) (11.5). G0,0 (1 ,

)



(11.4) (11.5) . , (11.3). 11.3 .

80




, u2 ( x, y; :

2

)

u3 ( x, y;

3

)

-

( 0, 0 ) . 11.3 11.4. i L [ 0, 1] , i = 1, 4 t = 0 ­ 1 4 .
( x , y )( ( x , y )G (

lim

x0 , y0 x 0 , y0

1 x y u ( x , y ) - a+ ( 0,1 ) arctg y + a+ ( 0, 4 ) arctg x = ) )

a+ ( 0,1 ) + a+ ( 0, 4 ) + u2 ( 0, 0; 2 ) + u3 ( 0, 0; 3 ) . 2 11.7. ( x0 , y0 ) G

( x, y ) , : 1) a+ ( 0, 4 ) = a+ ( 0,1 ) a41 , ( x0 , y0 ) = ( 0, 0 ) ;
2) a- (1,1 ) = a+ ( 0, 3) a- (1, 4) a+

) ( 0,3 )
2

= a- = a-

) (1,3 ) (1,4 )
2

a12 , ( x0 , y0 ) = (1, 0 ) ; a23 , ( x0 , y0 ) = (1, 1) ; a34 , ( x0 , y0 ) = ( 0, 1) .

11.5. i L [ 0, 1] , i = 1, 4

(

0, 0

)

1 4 . u ( x , y ) , . 3a lim u ( x , y ) = 41 + u2 ( 0, 0; 2 ) + u3 ( 0, 0; 3 ) . (11.6) ( x , y )( x , y ) 2
(
x , y )G ( x0 , y
0 0 0

)

. ( 0, 0 ) , a+ ( 0,
4

)

= a+ ( 0,1 ) = a41
1 x y a+ ( 0,1 ) arctg + a+ ( 0, 4 ) arctg = y x = x y 1 1 a41 arctg + arctg = a41 = a41 . y x 22 1

81


.. , .. , ..

u2 ( x, y;

2

)

u3 ( x, y;

3

)



(

0, 0 ) (11.6).

. 11.6. ( x , y ) L ( ) . G (11.2) u , ( x , y ) , , :
u u u u
1
2

3

4

( ( ( (

x ) = 1 ( x ) + SK y x y

( x ) + K 4 ( x ) , ) = KS1 ( y ) + 2 + SKS 3 ( y ) + K 0 4 ( y ) ) = K 01 ( x ) + SKS 2 ( x ) + 3 ( x ) + KS 4 ( x ) ) = K1 ( y ) + K 0 2 ( y ) + SK 3 ( y ) + 4 ( y ) ,
2

( x) ( y)

+ K 0

3

, ,

(11.7)



(

K

)(t )

=

( ) d , t2 + 2 0
t
1

(

K 0

)(t )
S

=

1

0 1 + (t -
= (1 - t ) .



1

( ) d

)

2

,

(

)(t )

. 1 ­ G ( x, y ) 1 ­ ( x, y ) L ( ) , , y = 0 0 < x < 1 . x 1 : = (1 , 2 , 3 ,
y 0 +
4

)

T

. 11.1,

lim u1 ( x, y ) = lim

y

y 0 +




0

1

y2 + ( x -

1 ( ) d

)

2

= 1 ( x

)

x ( 0, 1) . u2 ( x, y ) , u3 ( x, y ) , u4 ( x, y ) ( 0, 1) ,
y 0 +

lim u2 ( x, y ) = lim

1- x

y 0 +



(1 - x ) + (
2 0

1

2 ( ) d
y -

)

2

= ( SK

2

)( x)

,

82




y 0 +

lim u3 ( x, y ) = lim

1- y

y 0+


y 0 +

(1 - y ) + (
2 0

1

3 ( ) d
x -

)

2

= ( K 0

3

)( x)
.

,

y 0 +

lim u4 ( x, y ) = lim

x

0 x2 + ( y -



1

4 ( ) d

)

2

= ( K

4

)( x)

,
u
1

= lim u ( x, y ) =
y 0 +


i =1

4

y 0 +

lim ui ( x, y ) = 1 ( x ) + SK 2 ( x ) + K 03 ( x ) + K4 ( x

)

x ( 0, 1) . . 11.6 . , (11.7) x y . y x . (11.7) x y :
u u u u
SK 0 KS 0 A= K 0 SKS K0 K
1

= 1 + SK2 + K 03 + K 4 , = KS1 + 2 + SKS3 + K 0 4 , = K 01 + SKS2 + 3 + KS 4 , = K1 + K 02 + SK3 + 4 .
K J K0 , I = 0 0 KS 0 0 0 J 0 0 0 0 0 , J i = i , i = 1, 4 , J 0 0 J : 0

2

3

4

:
K
0

SKS 0 SK
11.6



u = ( I + A) .

, , § 2.

83


.. , .. , ..

12. (. 1.1),
u = 0



G;

u =

(12.1)

, . u ( x, y ) G = G ­ [1] 1 1 u ( M ) = (Q ) dsQ . ln nQ RMQ

(

I + A) = ,

(12.2)

­ -, . -
A : C [ 0, 1] C [ 0, 1]

,
= (I + A

)

-1



(12.2) . (12.1). (12.2) -, . . L ( ) ­ . + ,
+ + + + + = 1 1 + 22 + 33 + 44 ,

(12.3)



(



+ 11

)(

x, y ) =

y2 + ( x + t 0
84

y

1 2

1 ( t ) dt

)

2

+

y




1 2

1

y + (2 - x - t
2

1 ( t ) dt

)

2

,




( (


+ 2

2

)( )(

x, y ) =

1- x


1- y

(1 - x ) + (
2 0 1 2 2

1 2

2 ( t ) dt
y+t

) ) )

2

+

1- x


1- y

(1 - x ) + (
2 1 2 1 2

1

2 ( t ) dt
2- y -t

)

2

,



+ 33

x, y ) =



(1 - y ) + (
0

3 ( t ) dt
x+t

2

+


x
1

(1 - y ) + (
1 2

3 ( t ) dt
2- x-t

)

2

,

(
(


+ 4

4

)(

x, y ) =

x

0 x2 + ( y + t



1 2

4 ( t ) dt

2

+

1 x2 + ( 2 - y - t
2



4 ( t ) dt

)

2

.

+ + , 1 , + 3 + : 2 4
+ 1 1 ( x, y ) = +1 4 + 3

)

( (

3

) ( y, x ) , ( ) ( x, y ) = ( ) (1 ) ( x, y ) = ( ) (1 - y, x ) .
+ 2 2 + 4 2 + 43

- x, y ) ,

:
0 =
4


i =1

i ,0 ,

2 = 2 \ 0 , 0

1 3 1,0 = ( x, 0 ) : - x 0 ( x, 0 ) : 1 x , 2 2

2 ,0

1 3 = (1, y ) : - y 0 (1, y ) : 1 y , 2 2 1 3 = ( x, 1) : - x 0 ( x, 1) : 1 x , 2 2



3 ,0

1 3 = ( 0, y ) : - y 0 ( 0, y ) : 1 y . 2 2 , ,
4 ,0

G . , , 2 . 0

85


.. , .. , ..

12.1. L ( ) . + 2 . 0 .
t t +s
2 2

(

)(

x , y ) -



, , ( s, t ) = ( 0, 0 ) .
1 L1 0, 2
t

0 t2 + (s +
( s, 0 ) : -



1 2

( ) d

)

2

(12.4)

,
1 s 0 , ­ . 2 (12.4) t 2 2 . t +s
+ , 1 1 ,

(12.4) : s = x, t = y, = , = 1 . -







,







1 ( x, 0 ) : - x 0 . 2
+ , 1 1 ,

(12.4) : s = 1 - x, t = y, = 1 - , = 1 . -

,
3 ( x, 0 ) : 1 x . 2
+ , 1 1 ,


1 3 1,0 = ( x, 0 ) : - x 0 ( x, 0 ) : 1 x . 2 2

86




, , i+i ­ , i ,0 , i = 2, 3, 4 . , , , , 0 =
4


i =1

i ,0 , 2 = 2 \ 0 . 0

12.1 . 12.1. C ( ) . + 2 . 0 G , , 2 . + ( x, y ) 0 G , , C ( ) , L ( ) . 12.2. x = 0 1 1 L 0, . 2
x , y )( 0 , 0 ) x 0 , y 0, x + y 0

(

)(

x , y ) -

(

)

(

lim

a ( 0, 1 ) x a ( 0, 1 ) v1 ( x , y ) - + arctg = + , y 2
1 2

(12.5)


v1 ( x , y ) = y




0

y2 + ( x + t

1 ( t ) dt

)

2

.

.
v1 ( x, y ) = ya+ ( 0, 1 y




0

1 2

(1 ( t )
dt

y + (x + t
2

- a+ ( 0, 1 ) ) dt

)

2

+

+



)


0

1 2

y2 + ( x + t

)

2

I1 ( x, y ) + I 2 ( x, y ) .

. , : 87


.. , .. , ..

I1 ( x, y ) =

y




0

1 2

(1 ( t )
+ y

y + (x + t
2

- a+ ( 0, 1 ) ) dt

)

2

=

y

1

y2 + ( x + t

s 2 ( 1( ) )0

t

- a+ ( 0, 1 ) ) ds

t s =0

+




0

1 2

(

2( x + t y2 + ( x + y 1

) t)

2

)

2

t (1 ( s ) - a+ ( 0, 1 ) ) ds dt = 0

=



1 y2 + x + 2

2

( ( s )
1 0

1 2

- a+ ( 0, 1 ) ) ds +

+

2y




2y


0 1 2

( (

( (

x + t )t

y2 + ( x + t x + t )t

) )

2

) )

2

1 t (1 ( s ) - a+ ( 0, 1 ) ) ds dt + t 0 1 t (1 ( s ) - a+ ( 0, 1 ) ) ds dt t 0

+







y2 + ( x + t

2

2

I11 ( x, y ) + I12 ( x, y ) + I13 ( x, y ) ,

0 < <

1 ­ . 2 1 1 L 0, , 2

I11 ( x, y ) 0 y 0 . > 0 . x = 0 1 1 , 0 < < , 2 1 I12 ( x, y ) < . , 0 < y0 < 2 , 0 < y < y0 I13 ( x, y ) < . , , I1 ( x, y ) 0 y 0 . , 1 x+ a ( 0, 1 ) 2 - a+ ( 0, 1 ) arctg x . I 2 ( x, y ) = + arctg y y
88




( x, y ) ( 0, 0 ) , x 0, y 0, x + y 0
v1 ( x, y ) + a+ ( 0, 1



)

arctg

a ( 0, 1 ) x = I1 ( x, y ) + + arctg y

x+ y

1 2,

(12.5). 12.2 . u = + , (12.3) G . ( 0, 0 ) . u = +
u ( x, y ) = v00 ( x, y ) + w00 ( x, y ) ,


v00 ( x, y ) = y
1

y




0

1 2

y2 + ( x + t +

1 ( t ) dt

)

2

+

x
1 2

0 x2 + ( y + t



1 2

4 ( t ) dt

)

2

,

w00 ( x, y ) =




1 2

y2 + ( x + t

1 ( t ) dt

)

1- x

2



(1 - x ) + (
2 0 1 2 2

2 ( t ) dt
y+t

)

2

+

+

1- x


+

(1 - x ) + (
2 1 2 1

1

2 ( t ) dt
2- y -t

)

2

+

1- y


+

(1 - y ) + (
0

3 ( t ) dt
x+t .

)

2

+

1- y



(1 - y ) + (
2 1 2

3 ( t ) dt
2- x -t

x

)

2

1 x2 + ( y + t
2



1

4 ( t ) dt

)

2

, , w00 , ( 0, 0 )
w00 w0
1 2 1 2

0

(

1 1 2 ( t ) dt 1 2 ( t ) dt 1 3 ( t ) dt 1 3 ( t ) dt 0, 0 ) = + + + . 0 1+ t2 1 1 + ( 2 - t )2 1 + t 2 1 1 + ( 2 - t )2 0 1 2 2

x = 0 1 4 , 12.2 v00 ( x, y ) ,

89


.. , .. , ..

( x , y ) ( 0 , 0 ) ( x , y )G \( 0 , 0 )

lim

a+ ( 0,1 ) x a ( 0, 4 ) y arctg + + arctg = v00 + y x

1 ( a+ ( 0,1 ) + a+ ( 0,4 ) ) . 2 , , =

u = + ( 0, 0 ) :
( x , y )( 0 , 0 ) ( x , y )G \( 0 , 0 )

lim

a ( 0,1 ) x a ( 0, 4 ) y u ( x, y ) + + arctg + + arctg - w00 = y x

1 (12.6) ( a+ ( 0,1 ) + a+ ( 0,4 ) ) . 2 , =
( x , y )(1, 0 ) ( x , y )G \(1, 0 )

lim

a- (1,1 ) 1 - x a+ ( 0, 2 ) y + - w10 = arctg arctg u ( x, y ) + y 1- x

= lim

1 ( a- (1,1 ) + a+ ( 0, 2

2

))

,

(12.7)

( x , y )(1, 1) ( x , y )G \(1, 1)

a- (1, 2 ) 1 - y a- (1,3 ) 1- x arctg + arctg - w11 = u ( x, y ) + 1- x 1- y

= lim

1 ( a1 (1,2 ) + a- ( 0, 2

3

))

,

(12.8)

( x , y )( 0, 1) ( x , y )G \( 0 , 1)

a+ ( 0,3 ) a ( 0, 4 ) x 1- y arctg +- arctg - w01 = u ( x, y ) + 1- y x

=

1 ( a+ ( 0,3 ) + a- (1, 2

4

))
1 2

,

(12.9)


1 1 3 ( t ) dt 1 3 ( t ) dt 1 4 ( t ) dt 1 4 ( t ) dt w10 w10 (1, 0 ) = + + + , 0 1+ t2 1 1 + ( 2 - t )2 1 + t 2 1 1 + ( 2 - t )2 0 1 2 1 2 1 2 2 1 1 1 1 ( t ) dt 1 1 ( t ) dt 1 4 ( t ) dt 1 4 ( t ) dt w11 w11 (1,1) = + + + , 0 1+ t2 1 1 + ( 2 - t )2 1 + t 2 1 1 + ( 2 - t )2 0 2 2 1 2

90




w01 w0

1

(

1 1 1 ( t ) dt 1 1 ( t ) dt 1 2 ( t ) dt 1 2 ( t ) dt 0,1) = + + + . 0 1+ t2 1 1 + ( 2 - t )2 1 + t 2 1 1 + ( 2 - t )2 0 1 2 2

1 2

1 2

: 12.1.

(

x0 , y

0

)



L ( ) , (12.6) ­ 12.9) ( ­ ). u = +

(

x0 , y0 ) .

12.2. ( x0 , y

0

L

) () .



u = + , , . . ( x0 , y0 ) = ( 0, 0 ) . .
( x , y )( 0 , 0 ) ( x , y )G \( 0 , 0 )

lim

u ( x , y ) = u 00 .

(*1)
( x , y ) ( 0 , 0 ) ( x , y )G \( 0 , 0 )

lim

a+ ( 0,1 ) x a ( 0, 4 ) y arctg + + arctg - w00 = a14 . y x

, , : ) x = 0 y 0 + ,
( x , y ) ( 0 , 0 ) ( x , y )G \( 0 , 0 )

lim

a+ ( 0,1 ) a ( 0, x a ( 0, 4 ) y arctg + + arctg - w00 = + 2 y x

4

)

= a14 ;

) y = 0 x 0 + ,

91


.. , .. , ..

( x , y ) ( 0 , 0 ) ( x , y )G \( 0 , 0 )

lim

a+ ( 0,1 ) a ( 0,1 ) x a ( 0, 4 ) y = a14 . arctg + + arctg - w00 = + y x 2

: a+ ( 0,1 ) = a+ ( 0, 4 ) ,

(

0, 0 ) .

. a+ ( 0,1 ) = a+ ( 0, 4 ) . x 0 , y 0 ,
x + y > 0

arctg

x y + arctg = , y x2

(12.6) u = +

(

x, y ) ( 0, 0 ) , ( x, y ) G \ ( 0, 0 ) .

. 12.2 . 12.2. C ( ) . +

(

)(

x, y

)

-

G . , ,
C ( ) , G -

,
a+ ( 0, a-

) ( 1,2 )
4

= a- (1,3 ) = ( 1, 1) ,

= a+ ( 0,1 ) = ( 0, 0 ) , a- ( 1,1 ) = a+ ( 0,

a+ ( 0,3 ) = a-

) (1,4 )
2

= ( 1, 0 ) , = ( 0, 1) .

.

L [ 0, 1] F F0 :

+ . 1 2

(

1 ( s ) ds t ( s ) ds t F ) ( t ) = 2 2 + 2 0 t +s 1 t + (2 - s 2 1 2

)

2

,

( F0 ) ( t )

=

1 + (t + s 0

1

( s ) ds

)

2

+

1+ (2 - t - s 1
2

1

1

( s ) ds

)

2

.

, ( F ) ( t ) ( F0 ) ( t ) ( 0, 1] [ 0, 1] , .
92




L [ 0, 1] t = 0 . , ( F ) ( t ) t = 0 ( F ) ( 0 ) = [ 0, 1] .
a+ ( 0, ) , 2

C ( 0 , 1] ( 0 , 1] . F : L [ 0, 1] C ( 0 , 1] ­ , F0 : L [ 0, 1] C [ 0, 1] ­ . L ( ) . + 12.1 , :
+ =
1 + + = 1 1 + + 2 + 33 + + 2 4 4 y =0 1 2

=

1- x



(1 - x )
0

1 2

2 ( ) d
2

+

2

+

1- x



(1 - x ) + (
2 1 2 1 2

1

2 ( ) d
2 -

)

2

+

1+ ( x + 0
=

1

3 ( ) d

)

2

+

+

1+ (2 - x - 1
2

1

1

3 ( ) d

)

2

1 ( ) d x 4 ( ) d x +2 + 24 0 x + 2 1 x + (2 - 2

)

2

= ( SF

2

) ( x ) + ( F03 ) ( x ) + ( F4 ) ( x )

2

, 0 < x < 1,
4 x =1

+ =

+ + = 1 1 + + 2 + 33 + + 2 4

= +

y




0

1 2

y + (1 +
2 1

1 ( ) d

)

2

+

y




1 2

1

y + (1 -
2

1 ( ) d

)

2

+

1- y



(1 - y ) + (1 + )
2 0

1 2

3 ( ) d

2

1- y



(1 - y ) + (1 - )
2 1 2

3 ( ) d

2

+

1



1+ (
0 3

1 2

4 ( ) d
y +

)

2

+

1



1+ (
1 2 4 y =1

1

4 ( ) d
2 - y -

)

2

=

= ( FS1

)( y) + (

3

SFS

) ( y ) + ( F04 ) ( y )

, 0 < y < 1,
=

+

+ + + = 1 1 + 2 2 + 33 + + 4

93


.. , .. , ..

=

1+ ( x + 0
1- x
1

1

1 2

1 ( ) d

)

2

+

1 + (2 - x - 1
2

1

1

1 ( ) d

)

2

+

1- x



(1 - x ) + (1 + )
2 0

1 2

2 ( ) d

2

+



(1 - x ) + (1 - )
2 1 2

2 ( ) d

2

+

x 2 + (1 + 0
2

x

1 2

4 ( ) d

)

2

+

x 2 + (1 - 1
2

x

1

4 ( ) d

)

2

=

= ( F01 ) ( x ) + ( SFS
+ =
1 2
4

) ( x ) + ( FS4 ) ( x )
1 2

, 0 < x < 1,
4 x =0

+ + = 1 1 + + 2 + 33 + + 2 4

= +

( ) d y 1 ( ) d y + 21 2 y + 1 y + (2 - 0
1 2

)

2

+

1+ ( y + 0
1

1

2 ( ) d

)

2

1



1+ (
1 2

1

2 ( ) d
2 - y -

)

2

+

1- y



(1 - y )
0

1 2

3 ( ) d
2

+

2

+

1- y



(1 - y ) + (
2 1 2

3 ( ) d
2 -

)

2

=

= ( F1

) ( y ) + ( F02 ) ( y ) + ( SF3 ) ( y )

, 0 < y < 1.

, + :
+ + + +
1

= + ( x, 0 ) = ( SF

2

) ( x ) + ( F03 ) ( x ) + ( F4 ) ( x )
3

, , (12.10)

1

= + ( 1, y ) = ( FS1 ) ( x ) + ( SFS = + ( x, 1) = ( F01 ) ( x ) + ( SFS = + ( 0, y ) = ( F1
2

) ( x ) + ( F04 ) ( x )
x ) + ( FS
4

3

)(

)(

x),

4

) ( y ) + ( F02 ) ( y ) + ( SF3 ) ( y )

.

:
+ = A+ ,


(12.11)
F F0 . FS 0


+ A = 0 FS F0 F SF 0 SFS F0
94

F0 SFS 0 SF

(12.12)




, : 12.3. L ( ) .

(

+

)(

x, y

)

+ ,

(12.10).
13.

- , + - , (12.3). , , - + , . , G , ­ G . L ( ) , , , , . . 13.1. L ( ) . :
1) -

(

+

)

G

-
2) -

(

+

)

;


(

+

)

-


+ =

+

(

x , y ) ;

95


.. , .. , ..

3) + ­ , , , ; 4) = B . . 1) G , +
2 0

+

G 2 - + G , 0 2.1 ­ . 11.6 ( 11.7) :

1
3

(

)

= 1 + SK 2 + K 03 + K 4 ,



2

= KS1 + 2 + SKS3 + K 04 ,

4

= K 01 + SKS 2 + 3 + KS4 ,

= K1 + K 02 + SK3 + 4 ,



( K ) (

1 x ( ) d x) = 2 , 0 x +2

( K 0 ) ( x )

=

1+ ( x - 0

1

1

( ) d

)

2

,

( S ) ( x ) = (1

- x) .

:
= ( I + A ) .

12.3 + , (12.3) (12.12):
+ = A+ . - +


(

)

- + ,


(

)

(

- + = ( I + B) ,


)

­ B :
0 B12 B13 B 0 B23 B = 21 B31 B32 0 B41 B42 B43 B12 = SK - SF , B13 = K 0 - F0 B14 B24 , B34 0
, B14 = K - F ,

(13.1)

B21 = KS - FS , B23 = SKS - SFS , B24 = K 0 - F0 ,

96




B31 = K 0 - F0 , B32 = SKS - SFS , B34 = KS - FS , B41 = K - F , B42 = K 0 - F0 , B43 = SK - SF ,
1 2

(

1 ( s ) ds t ( s ) ds t F ) ( t ) = 2 2 + 2 0 t +s 1 t + (2 - s 2 1 2

)

2

,

( F0 ) ( t )
,

=

1 + (t + s 0
+

1

( s ) ds

)

2

+

1+ (2 - t - s 1
2

1

1

( s ) ds

)

2

.

( (


- + - +

)

1

= 1 ( x ) +

( (

x, 0 ) , x, 1) ,

( (

- + - +

)



=
2

2

( y) ( y)

+ + ( 0, y ) , + + ( 0, y ) ,

)



= 3 ( x ) +
3

+

)



=
4

4

( x, 0 ) + (1, y ) + ( x,1) + ( 0, y )

+

= ( B12 = = =

) ( x ) + ( B133 ) ( x ) + ( B144 ) ( x ) , ( B211 ) ( y ) + ( B233 ) ( y ) + ( B244 ) ( y ) , ( B311 ) ( x ) + ( B322 ) ( x ) + ( B344 ) ( x ) , ( B411 ) ( y ) + ( B422 ) ( y ) + ( B433 ) ( y ) .
2 +

1) . 2).

(

x, y ) ( 0, 0 ) , (1, 0 ) , (1, 1) ,

(

0, 1) . , ( 0, 0 ) :

lim
x 0

+

(

x, 0 ) = ( B12

2

) ( 0 ) + ( B133 ) ( 0 ) + ( B144 ) ( 0 )

=

1 11 1 1 = - d + 2 2( ) 2 1 1 + 1 + ( 2 - ) 2
y 0


1 2

1

1 1 - 2 1+ (2 - 1 +
3

)

2

3 ( ) d ,

lim + ( 0, y ) = ( B211 ) ( 0 ) + ( B23 = 1

) ( 0 ) + ( B244 ) ( 0 )

=




1 2

1

1 1 - 2 1 + 1 + ( 2 -

)

2

1 2 ( ) d +


1 2

1

1 1 - 2 1+ (2 - 1 +

)

2

3 ( ) d .

97


.. , .. , ..

lim
x 0

+

(

x, 0 ) = lim + ( 0, y ) , y 0



+

(

x, y ) ( 0, 0 ) .
+



(

x, y ) -

. 2) . 3). ,

+ +

(

x, 0 ) , + (1, y ) ,

( x,1)

, + ( 0, y ) , 0 < x < 1 0 < y < 1 ,
+



(

x, y ) -

. 3) . 4) 1) 2). 13.1 . , . 2) . 3) , : 13.1. B ­ L [ 0, 1] C [ 0, 1] ,
L [ 0, 1] B ,

, - B . 13.1 : 13.2. C ( ) . :
1) -

(

+

)

­ G
+

G ;
2) + = -

(

)

- ­ ;


3) + ­ , , , ; 4) = B .
+

98




. 1). C ( ) L ( ) , - + G 1) 13.1. C ( ) , , , . 13.1 . 1) . 2) - + . 2) - + G . 3) 4) , , 3) 4) 13.1 13.2 .

(

)

(

)

(

)

14. , (13.1) B L [ 0,1] , , C [ 0,1] C [ 0,1] . B , . 14.1. - B :
B
L [ 0 ,1] L [ 0 ,1]

= m ,

(14.1) (14.2)
.

B C [0 ,1]C [ 0,1] = m .


m = 1



arctg

3 4

1 3 0, 204 < arctg 4 < 0, 205

99


.. , .. , ..

. (14.1). L [ 0,1] . B

()

1

- B :


( B )

1

( x ) ( B122 ) ( x ) + ( B133 ) ( x ) + ( B144 ) ( x )

1 1- x 1- x 1 - 2 2 2 (1 - x ) + ( 2 - 1 (1 - x ) + 2
1 1 + 0 1+ ( x -
1 2

)

2

d +
1 1+ (2 - x - d +

)

2

-

1 1+ ( x +

)

2

1 d +


1 2

1

1 1+ ( x -

)

2

-

)

2

d 2 ) 2 .
1 1 x x + 2 -2 2 1 x + x + (2 -

L [ 0 , 1]

.

1 1- x 1- x - 2 2 (1 - x ) + 2 (1 - x ) + ( 2 - 1
1 2

)
1

2

1 d 1 d = 1 1+ 2 1

1- x + 2 1- x
1



d 1 + 1 1+ 2
1

-2
1- - 1-

x 2 x

= 1 arctg 2 1- x

=

1 2

2 - - arctg 1- x

=

1 2

=

1 3 - 1 1 1 -1 = arctg - arctg 2 - arctg - arctg 2 = 1- x 1- x 1- x 1- x 1 3 1 1 = 2arctg - arctg 2 - arctg 2 . 1- x 1- x 1- x

100




:
1 1 1+ ( x - 0
1 2

)

2

-

1 1+ ( x +

)

2

d =

=

1 2arctg x - arct
1 1 1 1+ (x - 1 2

1 1 g + x + arctg - x , 2 2 - 1 1+ (2 - x -

)

2

)

2

d =

=

1 1 3 2arctg (1 - x ) - arctg 2 - x - arctg 2 - x ,

1 1 x x -2 2 2 x + x + (2 - 1 2

)

2

1 d =

1 1 3 2arctg - arctg - arctg . x 2x 2x

,

( B )

1

( x)



1

1 1 2 arctg + arctg (1 - x ) + 2 arctg x + arctg - x 1- x
3 g2- x .

1 3 1 -arctg 2 - arctg 2 - arctg 2 - arct 1- x 1- x x

1 -arctg + x - arct 2

3 g - x 2

L [ 0 , 1]


1 arctg A + arctg = A


2

, A > 0, (14.3)

- , A < 0, 2

( B )

1

( x)



2 (1 - x ) 1 arctg 2 (1 - x ) + arctg ( 2 x ) + arctg + 3 1 + 2x g - arct 2 3 - 2 x g 2
L [ 0 , 1]

2x +arctg - arct 3

.

101


.. , .. , ..


A+ B arctg , 1 - AB arctg A + arctg B = sgn A, 2 A+ B arctg 1 - AB + sgn A, AB < 1, AB = 1, AB > 1. (14.4)

1 2 (1 - x ) 2 x 1 2 (1 - x ) 2 x = 1 x = , 2

2 (1 - x ) 2 x 1 1 = 4 x (1 - x ) < 1 , 3 39 9 1 + 2x 3 - 2x 1 1 = ( 3 + 4 x (1 - x ) ) < 1 2 2 4 4


2 2 2 b ( z ) = arctg + arctg 3 - arctg , z 1 1- z 1- 1 - (3 + z ) 9 4 :

( B )

1

(

2 1 2 3 x ) arctg + arctg - 1 1 - 4 x (1 - x ) 1 - 4 x (1 - x ) 9 1 b ( 4 x (1 - x

2 - arctg 1 1 - ( 3 + 4 x (1 - x ) ) 4
z = 4 x (1 - x

L [ 0 , 1]



))



L [ 0 , 1]

. (14.5)

)

, , 0 x 1 ,

0 z 1. (14.3) 1 arctg A = - arctg A > 0 , 2 A 1- z 9- z 1- z b ( z ) = - arctg - arctg - arctg . 2 2 6 8
102

(14.6)






1- z 9 - z 1 9 < 1, , 2 6 26 24 - 8 z 1 b ( z ) = - arctg + arctg 2 2 3 + 10 z - z 24 - 8 z 0 z 1 > 3 + 10 z - z 2 (14.6),

(14.4), -z . 8 0 , ,

3 + 10 z - z 2 1- z b ( z ) = arctg + arctg . 24 - 8 z 8


3 + 10 z - z 2 1 - z 3 + 10 1 < < 1, 24 - 8 z 8 24 - 8 8 , (14.4), , 48 (1 + z ) . b ( z ) = arctg 189 - 71z + 11z 2 - z 3

(1

89 - 71z + 11z 2 - z

3

)

= -71 + 22 z - 3 z 2 < 0 0 z 1 ,

0 z 1, , , b ( z ) .

3 max b ( z ) = b (1) = arctg . 0 z 1 4 (14.5) 1 3 B arctg . L [ 0, 1] 1 L [ 0, 1] 4

()





. , 1 3 B = max B arctg . (14.7) L [ 0 , 1] i L [ 0, 1] 1i 4 4

()

1 3 B 0 = B 0 = arctg , (14.8) L [ 0, 1] C [ 0 , 1] 4

103


.. , .. , ..


0 ( x ) = (0 ( x ) , 0 ( x ) , 0 ( x ) , 0 ( x

))

T

, 0 ( x ) 1, x [ 0, 1] .

, (14.7)
B
L [ 0 ,1] L [ 0 ,1]

= m .

(14.1) . (14.2) (14.8)
B C [0 ,1]C [ 0 ,1] B
L [ 0 ,1] L [ 0 ,1]

0 C [ 0, 1] . 14.1 .

15. , . , B , , . , , G . . 15.1. A, B , C , D ­ .
f ( x, y ) = C (1 - y ) Ay By + + 2 2 2 x +y (1 - x ) + y 2 (1 - x ) + (1 - y
2

)

2

+

x 2 + (1 -

D (1 - y

) y)

2

G A = 0, B = 0, C = 0, D = 0 .
104




. . f ( x, y ) , ( x, y ) ( 0, 0

)

y , x2 + y2

. , , A = 0 . ( x, y ) . 15.1 . G : G = G00 G10 G11 G01 , = 00 10 11 01 ,
3 G00 = 0, 4 3 1 3 â 0, , G10 = , 1 â 0, , 4 4 4 1 â , 1 , 4

1 1 3 G11 = , 1 â , 1 , G01 = 0, 4 4 4

00 = G 00 ,

10 = G10 ,

11 = G11 ,

01 = G 01 .

u = - + :
u ( x, y ) =

u (
i =1 i

4

x, y ) 1

(
4
i =1

ii - i+

i

)(

x, y ) ,



( 11 ) (

x, y ) =
1 2

0 y2 + ( x -



1

y1 ( ) d

)

2

, y1 ( ) d

(



+ 11

)(

x, y ) =

1




0

y2 + ( x + 1
1

y1 ( ) d

)

2

+

1




1 2

1

y2 + ( 2 - x -

)

2

,

( 22 ) (

x, y ) =
1 2



(1 - x ) + (
2 0

1- x y - 1
1

)

2

2 ( ) d ,

(


+ 2

2

)(

x, y ) =

1



(1 - x ) + (
2 0

(1

- x )2 ( ) d y +

)

2

+



(1 - x ) + (
2 1 2

(1

- x ) 2 ( ) d 2 - y -

)

2

,

105


.. , .. , ..

( 33 ) (

x, y ) =

1



(1 - y ) + (
2 0

1

(1

- y ) 3 ( ) d x -

)

2

,

(



+ 33

)(

x, y ) =

1



(1 - y ) + (
2 0

1 2

(1

- y ) 3 ( ) d x +

)

2

+

1



(1 - y ) + (
2 1 2

1

(1

- y ) 3 ( ) d

2 - x -

)

2

,

( 44 ) (

x, y ) =

1

0 x2 + ( y -



1

x 4 ( ) d

)

2

,

(


+ 4

4

)(

x, y ) =

x2 + ( y + 0

1

1 2

x 4 ( ) d

)

2

+

x2 + ( 2 - y - 1
2

1

1

x 4 ( ) d

)

2

.

u = u ( x, y

)

,

:

u u = w00+ w 00 + u00 , u00 = u2 + u3 = w00 + w + 00 u u =w100+ w10 +u1100,, u10 = u3 + u4 = w1 + w + u u == w11 +w11 +u1111,, u11 = u4 + u1 u w11 + w + u = = w0+ w 01 + u01,, u01 = u1 + u2 u w01 1 + w + 1

y1 ( ) d

G00 , G10 , G11 , G01 ,
x4 ( ) d

(15.1) (15.2) (15.3) (15.4)

w00 = 11 - 1,01 + 44 - 4,0 4 ,

(

1,01 ) ( x, y ) =
w 00 w00

1

0 y2 + ( x +



1

)

2

,

(
2

4,0

4

)(

x, y ) =

1

0 x2 + ( y +
1 ( ) d +



1

)

2

,

(

1 1 y x, y ) = 2 1 y +(x + 2

)

-

y y2 + ( 2 - x -

)

2

1 1 x + 2 1 x + ( y + 2

)

2

-

d , 2 4( ) x2 + ( 2 - y - ) x

w10 = 11 - 1,11 + 2 2 - 2,02 ,

106




(

1,11 ) ( x, y ) =

1

0 y2 + ( 2 - x -
1



1

y1 ( ) d
1 2

)

2

,

(
)
2

2,0

2

)(

x, y ) =
y

1



(1 - x ) + (
2 0
2

1

(1

- x )2 ( ) d y +

)

2

,

w1100 ( x, y ) = - w




0

y 2 y + (x +

-

y2 + (2 - x -

)

1 ( ) d +

1 1 1- x + 2 1 (1 - x ) + ( y + 2

)

2

-

(1
1

-x

)

2

d , 2 2( ) + (2 - y - ) 1- x

w11 = 22 - 2,1 2 + 33 - 3,13 ,

(

2,1

2

)(
3

x, y ) =

1



(1 - x ) + (
2 0

(1

- x ) 2 ( ) d 2 - y -

)
2

2

,

(
w1111 ( x, y ) = - w
1 2

3,1

)(

x, y ) =

1



(1 - y ) + (
2 0 2

1

(1

- y ) 3 ( ) d x -

)

,
2 ( ) d -

1



0

1 2

1- x 2 (1 - x ) + ( y +

)

-

(1

-x

)

2

+ (2 - y -

1- x

)

2

-

1 1- y 1- y (1 - y )2 + ( x + )2 - (1 - y )2 + ( 2 - x - )2 0 w01 = 33 - 3,03 + 44 - 4,1 4 , 1

3 ( ) d ,
1

(

3,0

3

)(

x, y ) =


1

(1 - y ) + (
2 0

1

(1

- y )1 ( ) d x +

)

2

,
-

(

4,1

4

)(
2

x, y ) =

x2 + ( 2 - y - 0

1

x4 ( ) d

)

2

,

w 0011 ( x, y ) = w

1



1
2 1 2

1- y 2 (1 - y ) + ( x +

)

2

(1

-y

)

+ (2 - x -

1- y

)

2

3 ( ) d -

1 x x -2 2 ( ) d . x + ( y + )2 x + ( 2 - y - )2 4 0 K i j , -

Gi j R =

1 . , G i j K i j . 2

107


.. , .. , ..

15.2. w

ij

ui

j



K i j , i , j = 0, 1 . . w
ij

ui j

K i j . , , w ij ui j , K i j . 15.2 . 1 , 1,0 , 4 :
1 ( ) d , 1 ( ) d , 4 ( ) d , 4 ( ) d .

4 ,0

( (



* 11

)(

x, y ) = x, y ) =

1


1


0 1

1

y2 + ( x - x +

x -

)

2



* 1,0 1

)(

y2 + ( x + 0 x2 + ( y - 0
1 1
1

)

2

( (

* 44

)(

x, y ) = x, y ) =

y -

)

2


,0

* 4 ,0 4

)(

0 x + ( y +
2



1

y +

)

2

v0
vk
,0

(

x, y ) 0 k 1 :

vk -1,0 y1( k -1) (1) = - x y 2 + (1 - x

(

)

2

+

x 2 + (1 + y
vk
,0

(

x

(
4

k -1)

(1)

)

2

)

+

2 y 1(

(

)(

-

y1(
(
4

k -1)

y 2 + (1 + x
k -1)

(1)

)

2

k -1)

( x2 + y

( 0) -

2

)

( 0)

)

)(
k -1)

+

x

(
4

k -1)

x 2 + (1 - y

(1)

)

2

)

+

, k = 4k1 + 1 ,
y1(

(15.5)

vk -1,0 y1( k -1) (1) = - x y 2 + (1 - x

(

)

2

)(
)
2

+

y 2 + (1 + x

(1)

)

2

)

+

+

x + (1 - y
2

(

x

(
4

k -1)

(1)

)

2

) (

-

x
2

(
4

k -1)

x + (1 + y

(1)

)

, k = 4k1 + 2

(15.6)

108




vk

,0

vk -1,0 y1( k -1) (1) = - x y 2 + (1 - x

(

)

2

-

x + (1 + y
2

(

x

(
4

k -1)

(1)

)

2

)

+

2 y 1(

(

) (

-

y1(
2

k -1)

y + (1 + x
(
4 k -1)

(1)

)

2

k -1)

( x2 + y

( 0) +

2

)
+

( 0)

)

) (
k -1)

+

x
2

(
4

k -1)

x + (1 - y

(1)

)

2

)

-

, k = 4k1 + 3 ,

(15.7)

vk

,0

vk -1,0 y1( k -1) (1) = - x y 2 + (1 - x

(

)

2

)(
)
2

y1(

y 2 + (1 + x

(1)

)

2

)

-

-

x 2 + (1 - y

(

x

(
4

k -1)

(1)

)

2

)(

+

x

(
4

k -1)

x 2 + (1 + y

(1)

)

, k = 4k1 + 4 .

(15.8)

15.3. - = ( 1 , 2 , 3 ,

4

)

T



[

0, 1] n . -

k w 00 , k = 1, n x k
( ( ( vk ,0 + 11( k ) + 1,01 k ) - * 4 k ) - * ,0 4 k ) , k = 4k1 + 1, 4 4 (k) (k) (k) (k) k w00 vk ,0 + 11 - 1,01 - 4 4 + 4 ,0 4 , k = 4k1 + 2, (15.9) = * (k) * (k) (k) (k) x k vk ,0 + 11 + 1,01 + 4 4 + 4,0 4 , k = 4k1 + 3, (k) (k) (k) (k) vk ,0 + 11 - 1,01 + 4 4 - 4 ,0 4 , k = 4k1 + 4. , :

y 2 y + (x ± x 2 x + ( y ± y - 2 x + ( y ±

) ) )

2

y = ± 2 y + (x ± x y ± = 2 x + ( y ± x x = ± 2 x + ( y ± x

) ) )

2

, , .

2

2

2

2

109


.. , .. , ..

k = 0, n
vk ,0 (1 - x )1( k ) (1) + (1 + x )1( k ) (1) - x4( k ) (1) - vk ,1 = 2 y y 2 + (1 - x )2 y 2 + (1 + x ) x 2 + (1 - y

(

)(
(
k

-

x + (1 + y
2

(

x

(k )
4

(1)

)

2

)

-
k

2 x 1(
)

)

( x2 + y
-

( 0 ) - 4( k ) ( 0 )
2

)

)

)(
,

)

2

)

- (15.10)



k = 4k1 ,

vk ,1 =

vk

,0

y -

-

y 2 + (1 -
(k )
4

(1

(

- x )1(

(1) 2 x)

)(

y 2 + (1 +
(k )
4

(1

+ x )1(

k

)

(1) 2 x)

)

- (15.11) k = 4k1 + 1,
+

x 2 + (1 -

(1

(

- y )

(1) 2 y)

)(
)(
(
k

-

x 2 + (1 +

(1

+ y )

(1) 2 y)

)

,



vk ,0 (1 - x )1( k ) (1) + (1 + x )1( k ) (1) + x4( k ) (1) vk ,1 = - 2 y y 2 + (1 - x ) 2 y 2 + (1 + x ) x 2 + (1 - y

(

+

x + (1 + y
2

(

x

(k )
4

(1)

)

2

)
2

-

2 x 1(

)

( x2 + y

( 0 ) + 4( k ) ( 0 )
2

)
2

)

)(
,

)

2

)



k = 4k1 + 2, (15.12)

vk ,1 =

vk

,0

y +

-

y + (1 -
(k )
4

(1

(

- x )1(

k

)

(1) 2 x)

) (

-

(1

y + (1 +
(k )
4

+ x )1(

k

)

(1) 2 x)

)

+ (15.13) k = 4k1 + 3 .

x 2 + (1 -

(1

(

- y )

(1) 2 y)

)(

+

x 2 + (1 +

(1

+ y )

(1) 2 y)

)

,

15.4 - = ( 1 , 2 , 3 ,

4

)

T



[

0, 1] n + 1 .


k +1 w00 = x k y v v v v
k ,1 k ,1 k ,1 k ,1

k + 1 w 00 , k = 0, n x k y
) ) ) )

- * 1 - * 1 - * 1 - * 1

(
1

k +1 k +1 k +1 k +1

- *,0 1 - *,0 1

(
1

k +1

) )

+ 4 - * 4 - 4 + * 4

(
4

k +1 k +1 k +1

) )

+ 4 ,0 + * ,0 4 - 4 ,0 - * ,0 4

(
4

k +1 k +1 k +1

) )

, k = 4k1 , , k = 4k1 + 1, , k = 4k1 + 2, , k = 4k1 + 3.

(
1

+ *,0 1 + *,0 1

(
1

k +1 k +1

(
4

(
4

(
1

(
1

) )

(
4

) )

(
4

) )

(15.14)

(
1

(
1

k +1

(
4

k +1

(
4

k +1

110




, :
y 2 y + (x ± x 2 x + ( y ± y ± 2 x + ( y ±

) ) )

2

x ± = 2 y + (x ± y x = ± 2 x + ( y ± y y ± = ± 2 x + ( y ± y

) ) )

2

, , .

2

2

2

2

1 ( x ) 4 ( x ) C [ 0, 1] .
z1 ( x, y; 1 ) = 1


1


0

1

y y + (x -
2

)

2

1 ( x ) - 1 ( ) d , 1 ( x ) - 1 ( ) d ,

z1,0 ( x, y; 1 ) = z 4 ( x, y; z


1


0 1

1

y y2 + ( x + x

)

2

4

)

=

x2 + ( y - 0
1
1

)

2

4

( y ) - 4 ( ) d

,

4 ,0

(

x, y;

4

)

=

x2 + ( y + 0
1

x

)

2



4

( y ) - 4 ( ) d

,

z ( x, y; 1 ) =
* 1


1


0

1

y + (x -
2

x -

)

2

1 ( x ) - 1 ( ) d , 1 ( x ) - 1 ( ) d ,

z

* 1,0

(

x, y; 1 ) =


1


0 1

1

y2 + ( x + y -

x +

)

2

z ( x, y;
* 4

4

)

=

x2 + ( y - 0
1
1

)

2

4

( y ) - 4 ( ) d

,

z

* 4 ,0

(

x, y;

4

)

=

x2 + ( y + 0

y +

)

2



4

( y ) - 4 ( ) d

.

111


.. , .. , ..

15.5. 1 ( x )

4

( x)



[ 0, 1] . zi , zi ,0 , zi*
zi*,0 , i = 1; 4 G .
* . z1 ( x, y; 1 ) z1 ( x, y; 1 ) .













(

x, y ) 2 \ 1 ,



1 = [ 0, 1] â {0} . z1 ( x, y; 1

)



* z1 ( x, y; 1 ) ( x, y ) ( x0 , 0 ) , ( x, y ) G , x0 [ 0, 1] . -

,
z1 ( xn , yn ; 1

)

* z1 ( xn , yn ; 1

)



(

xn , yn ) G ,

(

xn , yn ) ( x0 , 0 ) n .

. [12, . 139]. [12, . 139]. E

{ f ( )}
n

,

1) F ( ) .

K , , n 2)
fn ( ) < K ,
lim f n ( ) d = F ( ) d .
n E E



1. 1)
F ( ) E .

2. 2) E . z1 ( xn , y
z1 ( xn , y
n

) 1 1 1 ( xn ) - 1 ( ) yn ( xn - ) d . )= 2 2 0 xn - yn + ( xn - )
n




1 ( xn ) - 1 ( ) y (x - ) 2n n , 2 xn - f n ( ) = yn + ( xn - ) 0,

xn , = xn

112




n x0
F ( ) 0 .

1 ( x ) ,

1 ( xn ) - 1 ( ) k1 xn - ,
k1 ­ 1 ( x ) .
12 k1 xn - 2 yn + ( xn - 2 xn - yn + ( xn - )

1 ( xn ) - 1 ( ) y x - f n ( ) 2n n xn - yn + ( xn -
y

)

(

)
2

2

)

2

=

k1 . 2

. ,
lim
n



n


0

1

1 ( xn ) - 1 (
2 yn + ( xn -

)

) d 2

= lim

1

n



f ( )
n
0

1

d =

1



F ( )
0

1

d = 0 .

,
z1 ( x, y; 1 ) ( x, y ) ( x0 , 0 ) , ( x, y ) G , 0 x0 1 . ,

z1 ( x, y; 1 ) G .
* z1 ( xn , yn ; 1 )

1 1 1 ( xn ) - 1 ( ) ( xn - ) z ( xn , yn ; 1 ) = 0 xn - y 2 + ( xn - 2 * 1

)

2

d


2 1 ( xn ) - 1 ( ) ( xn - ) 2 f n ( ) = xn - yn + ( xn - 0,

)

2

,

xn , = xn

n x0 F ( ) =

1 ( x0 ) - 1 ( ) . x0 -

1 ( x ) ,
2 1 ( xn ) - 1 ( ) ( xn - ) f n ( ) 2 xn - yn + ( xn -

)

2



k1 xn - = k1 xn -

. ,

113


.. , .. , ..
1 1 1 ( xn ) - 1 ( ) ( xn - ) lim n xn - y 2 + ( xn - 0 2 1 1 1 ( x0 ) - 1 ( ) d = d . 0 x0 -

)

2

1 ( x ) . ,
* z1 ( x, y; 1 ) ( x, y ) ( x0 , 0 ) , ( x, y ) G , 0 x0 1 . ,
* z1 ( x, y; 1 ) G .

G . 15.5 . . 15.5 , 1 4 :
1) , z1 , z1,0 , z
4

z 4 ,0 ;
2) : 0 < < 1 ,
* * * * z1 , z1,0 , z4 z4,0 .

1). 1 ( x ) [ 0, 1] ( x, y ) ( x0 , 0 ) , ( x, y ) G , 0 x0 1 . z1 ( x, y; 1 ) . 0 < x0 < 1 , 2.1
(
x , y )( x0 , 0

lim

y
)




0

1

y2 + ( x -

1 ( ) d
d

)

2

= 1 ( x0 ) , = 1 ( x0 ) .

(

x , y )( x0 , 0

lim

y1 ( x
)

)


0

1

y + (x -
2

)

2


(
x , y )( x0 , 0

lim

)

z1 ( x, y; 1 ) = 0 ,

0 < x0 1 .

114




x0 = 0 , ,
z1 ( x, y; 1 ) =



y 1 ( x ) - 1 (


0

y + (x -
2

)

) d 2

+



1 y 1 ( x ) - 1 (



y + (x -
2

)

2

) d

,

.



y 1 ( x ) - 1 (


0

y + (x -
2

)

) d 2

max 1 ( x ) - 1 (
0
0

)
0

1



y + (x -
2

yd

)

2



max 1 ( x ) - 1 ( ) .

> 0 ­ . > 0 , 0 x


2


max 1 ( x ) - 1 ( ) <
0

. (15.15) 3 . 1



0 x


2

,
y y2 + ( x -

y 0 +
[ , 1] x 0, , 0 < y y

)

2

(1 ( x ) - 1 (

))



. y0 > 0 , 2 0
y y2 + ( x -



)

2

(1 ( x ) - 1 (

))

<


2

.




x 0, 2

1 y 1 ( x ) - 1 (



y + (x -
2

)

) d < 2

2

.

(15.16)
z1 ( x, y; 1 ) < ,

(15.15) (15.16)

y ( 0, y0 ) . , > 0 ,
(
x , y ) ( 0 , 0

lim

)

z1 ( x, y; 1 ) = 0 .

115


.. , .. , ..

x0 = 1 . ,
z1 ( x, y; 1 ) ( x, y ) ( x0 , 0 ) , ( x, y ) G , 0 x0 1 . ,

z1 ( x, y; 1 ) G . z1,0 ( x, y; 1 ) , z4 ( x, y; : 0 < < 1 ,
4

)

z

4 ,0

(

x, y;

4

)

G .

2). 1 ( x )

1 ( x ) - 1 ( x ) k1 x - x .


( x, y ) ( x0 , 0 ) , ( x, y ) G , 0 x0 1 . ,
(
x , y )( x0 , 0

lim

)

* * z1 ( x, y; 1 ) = z1 ( x0 , 0; 1 ) ,

(15.17)



1 1 1 ( x0 ) - 1 ( ) z ( x0 , 0; 1 ) = d . 0 x0 - * 1

, 1 ( x ) . > 0 ­ .
e ,0 = [ 0, 1] ( x0 - , x0 + ) , e ,1 = [ 0, 1] \ e ,0 ,
* * z1 ( x, y; 1 ) - z1 ( x0 , 0; 1 ) = I1 + I 2 + I 3 ,

* * z1 ( x, y; 1 ) - z1 ( x0 , 0; 1 )



I1 =

1




e
,0

(

x - ) (1 ( x ) - 1 ( y2 + ( x -

)

))

2

d ,

I2 =

1




e
,0

1 ( x0 ) - 1 ( ) d , x0 -

( x - ) (1 ( x ) - 1 ( ) ) 1 ( x0 ) - 1 ( ) - d 2 e x0 - y2 + ( x - ) > 0 ­ . I3 = 1
,1

, x - x0 <


2

. 1

I1




e
,0

y + (x -
2

(

x -

)

2

)

1 ( x ) - 1 (
2

x -



)

d x -

1-



116









k1


e
,0

d x -

1-

k x - x0 + = 1

+


x0 - x +



2k 3 < 1 . 2

2k 3 , I 2 < 1 . > 0 , 2 2k 3 1 < . 2 3 I1 <



3

,





.

. 3

I2 <



(15.18)



e

,1



(

x - ) (1 ( x ) - 1 ( y2 + ( x -

)

))

2



(

x, y ) ( x0 , 0

)



1 ( x0 ) - 1 ( x0 -

)

. 1 : 0 < 1 <


2



y0 > 0 , x - x0 < 1 0 < y < y0

(


x - ) (1 ( x ) - 1 ( y2 + ( x -

)

2

) ) 1 ( x0 ) - 1 ( ) - <
x0 - 3

.

I3

1




e
,1


3

d <


3


0

1

d =


3

.

(15.19)

(15.18) (15.19) (15.17).
* * z1,0 ( x, y; 1 ) , z4 ( x, y;
4

)

z

* 4 ,0

(

x, y;

4

)

G .

.
= (1 , 2 ,3 ,
4


n

,



-

)

T

C [ 0, 1] -

n , k 0 , 2k n ,

117


.. , .. , ..
( 4 1( ( 2 ( 3 ) ) ) )

2k 2k 2k 2k

(0) (1) (1) (0)

= ( -1) 1(
k

= = =

( 0) , k ( -1) 2( 2 k ) ( 0 ) , k ( -1) 3( 2 k ) (1) , k ( -1) 4( 2k ) (1) .
2k

)

(15.20)

. 15.1. - = ( 1 , 2 , 3 ,
4

)

T



n 1 , n - . - n , n +n u , n1 + n2 n , x n y n
1 2 1 2

G . 15.2. - = ( 1 , 2 , 3 ,
4

)

T



n 1 , n - .
n +n u , n1 + n2 n , G , x n y n
1 2 1 2

= ( 1 , 2 , 3 ,

4

)

T



n . 15.1 15.2 : 15.3. - = ( 1 , 2 , 3 ,
4

)

T



n 1 , n - . , n +n u , n1 + n2 n , x n y n
1 2 1 2

G , , -

118




- = ( 1 , 2 , 3 ,

4

)

T

-

n . 15.1. n1 n2 ­
n +n u . . x n y n
1 2 1 2

u = u ( x, y )
2k 2k u k u = ( -1) , y 2 k x 2 k

k 0,


n +n u k , n2 = 2k2 , ( -1) n +n n +n u x = , x n y n n +n u k ( -1) n +n -1 , n2 = 2k2 + 1. x y
1 2 2 1 2 1 2 1 2 1 2 2 1 2

G :

ku ) k x



k 1;

k +1u ) k x y



k 0.

). -

ku G . x k G i j , i, j = 0, 1.

ku x k
G 00 . (15.1). 15.2 w
00

u

00



K

00

G 00 . ,
00

k w w x

00 k

k u 00 x k

G

k 1 .

119


.. , .. , ..

,
ku G x k
00





k w00 G 00 . x k

k w00 15.3 . x k vk ,0 , k 1 , (15.5)-(15.9)

K

00

G 00 .
,0

k1 = 0 . vk

k = 1 . v0
y1(
2

,0

(

x, y ) 0

4 ( 0 ) = 1 ( 0 ) , (15.5)
v1,0 = -

y + (1 - x
2

(

y1(

k -1)

(1)

)

2

) (
)
2

-

k -1)

y + (1 + x x
2

(1)

)

2

)
)

+

+

x + (1 - y
2

(

x

(
4

k -1)

(1)

) (

64

+

(
4

k -1)

x + (1 + y

(1)

2

)

.

(15.21)

v1,0 K
00

( x, y ) K 00 . -

G 00 .

k = 2 . (15.6)
v1,0 y1( k -1) (1) = - x y 2 + (1 - x y1(
2

v2

,0

(

)
2

2

) (
-

+

k -1)

y + (1 + x
(
4

(1)

)
2

2

)
.

+

+

x + (1 - y
2

(

x

(
4

k -1)

(1)

)

) (

x
2

k -1)

x + (1 + y

(1)

)

)

(15.22)

v1,0 K 00 G 00 (15.21), x ,






64

( x, y ) K 00 .

120




k = 3 . (15.7) 4 ( 0 ) = -1( 0 ) v3
,0

v2,0 y1( k -1) (1) = - x y 2 + (1 - x

(

)
2

2

) (
-

-

y1(
2

k -1)

y + (1 + x
(
4 k -1)

(1)

)
2

2

)
.

+

+

x 2 + (1 - y

(

x

(
4

k -1)

(1)

)

)(

x

x 2 + (1 + y

(1)

)

)

(15.23)

v2,0 K 00 G 00 (15.22), x ,







64 k = 4 . (15.8)
v4
,0

( x, y ) K 00 .
y1(
(
4

v3,0 y1( k -1) (1) = - x y 2 + (1 - x

(

)
2

2

)(
+

+

k -1)

y 2 + (1 + x
x
2

(1)

)
2

2

)
.

-

-

x + (1 - y
2

(

x

(
4

k -1)

(1)

k -1)

)

) (

x + (1 + y

(1)

)

)

(15.24)

v3,0 K 00 G 00 (15.23), x ,






64

( x, y ) K 00 .
00

, (15.5)-(15.8) K
k1 = 0 .

G

00



vk ,0 k = 4k1 + i, i = 1, 4 k1 1 , . G (15.9). k = 1 .
00



( 11 ) (

x, y ) + ( 1,01 ) ( x, y ) - *4 4
1 = z1(4) ( x, y ) + z

(

)(

x, y ) - * ,0 4 4

(

)(

x, y ) =

(1) 14

(

x, y ) ,

121


.. , .. , ..
* z1(1) ( x, y4 ) = - z1 ( x, y; 1 ) - z1,0 ( x, y; 1 ) + z4 ( x, y; 4 ) + z 4



* 4 ,0

(

x, y; 4 ) ,

z

(1)
14

(

2 1 ( x ) 1- x 1 + x 4 ( y ) x + (1 - y x, y ) = arctg + arctg + 2 ln 2 y y x + (1 + y

) )

2 2

z1(1) G 4

00



15.5, , z1(1) , 4 G , , G 00 . ( z 11) G 00 . 4 ( x, y ) ( x0 , 0 ) , ( x, y ) G00 , 0 < x0
(
x , y )( x0 , 0

3 . 4 (15.25)

lim

)

z

(1) 14

(

x, y ) = 1 ( x0 ) .

3 ( x, y ) ( 0, y0 ) , ( x, y ) G00 , 0 < y0 . 4 2 1 ( y ) 1 - y0 ( lim z 11)( x, y ) = 1 ( 0 ) arctg + 4 0 ln . (15.26) 4 ( x , y )( 0, y ) y0 1 + y0
0

, ( x, y ) ( 0, 0 ) , ( x, y ) G00
(
x , y ) ( 0 , 0

lim

)

z

(1) 14

(

x, y ) = 1 ( 0 ) .

(15.27)

, , (15.27), (15.25) x0 0 (15.26) y0 0 . ( z 11) G 00 . 4 , G 00 . k = 2 .
ku k = 1 x k

( 11) (

x, y ) - ( 1,01) ( x, y ) - ( 4 4 ) ( x, y ) + ( 4,0 4 ) ( x, y ) =
2 = z1(4 ) ( x, y ) + z

(2 14

)

(

x, y ) ,

122






z1(

2 4

)

(

x, y ) = - z1 ( x, y; 1) + z1,0 ( x, y; 1) + z4 ( x, y; 4 ) - z1 ( x, y; 4 ) ,
z
(1)
14

(

x, y ) =

1( x ) 1- x 1+ x - arctg arctg + y y

+

4 ( y ) 1+ y 1- y 2 x - arctg arctg + (1( x ) + 4 ( y ) ) arctg - 4 ( y ) . x x y
2 4

z1(

)

G

00



2 15.5, , z1(4 ) , -

G , , G 00 .
z
(2 14 )

G 00 . ( x, y ) ( x0 , 0 ) , ( x, y ) G00 , 0 < x0
3 . 4 (15.28)


(
x , y )( x0 , 0

lim

)

z

(2 14

)

(

x, y ) = 1( x0 ) .

3 ( x, y ) ( 0, y0 ) , ( x, y ) G00 , 0 < y0 . 4 ( lim z 12)( x, y ) = -4 ( y0 ) . (15.29) 4
(
x , y )( 0, y0

)

, ,

(

x, y ) ( 0, 0 ) ,

(

x, y ) G00 ,
(
x , y )( 0 , 0

lim

)

z

(1) 14

(

x, y ) = - 4 ( 0 ) .

(15.30)

, , (15.30), (15.28) x0 0 4 ( 0 ) = -1( 0 ) , (15.29) ­ y0 0 .
z
(2 14 )

G

00

.

ku , k = 2 x k

G 00 .

123


.. , .. , ..

k = 3 .

( 11) (

z1(
z
3 4

x, y ) + ( 1,01) ( x, y ) + * 4 ( x, y ) + * ,0 4 ( x, y ) = 4 4

(

)

(

)

= z1(
)

3 4

)

(

x, y ) + z

( 3) 14

(

x, y4 ) = - z1 ( x, y; 1) - z1,0 ( x, y;

( x, y ) , * 1) - z4 (

x, y; 4) - z

* 4 ,0

(

x, y; 4) ,

( 3)
14

(

2 1( x ) 1- x 1 + x 4( y ) x + (1 - y x, y ) = arctg + arctg - 2 ln 2 y y x + (1 + y
3 4

) )

2 2

z1(

)

G

00



3 15.5, , z1(4 ) , -

G , , G 00 .
z
( 3) 14

G 00 . ( x, y ) ( x0 , 0 ) , ( x, y ) G00 , 0 < x0
3 . 4 (15.31)


(
x , y )( x0 , 0

lim

)

z

( 3) 14

(

x, y ) = 1( x0 ) .

3 ( x, y ) ( 0, y0 ) , ( x, y ) G00 , 0 < y0 . 4 2 1 ( y ) 1 - y0 ( lim z 13)( x, y ) = 1( 0 ) arctg - 4 0 ln . (15.32) ( x , y )( 0, y ) 4 y0 1 + y0
0

, ( x, y ) ( 0, 0 ) , ( x, y ) G00
(
x , y )( 0 , 0

lim

)

z

(1) 14

(

x, y ) = 1( 0 ) .

(15.33)

, , (15.33), (15.31) x0 0 (15.32) ­ y0 0 .
z
( 3) 14

G

00

.

124




ku , k = 3 x k

G 00 . k = 4 .

(

11IV

)(

x, y ) - 1,01IV

(

)(

x, y ) + 4
(4 14

(

IV 4

)(

x, y ) - 4,0

(

IV 4

)( (

x, y ) =

4 = z1(4 ) ( x, y ) + z

)

(

x, y

)
IV 4


z1(
4 4

)

(

x, y ) = - z1 x, y; 1IV + z1,0 x, y; 1IV - z4 x, y;
z
(4)
14

(

)

(

)

(

)

+z

4 ,0

x, y;

IV 4

)

,

(

1IV ( x ) 1- x 1+ x x, y ) = - arctg arctg - y y

-



IV 4



( y)

1+ y 1 - y 2 IV - arctg arctg + 1 ( x ) - x x

(

IV 4

( y)

)

arctg
4 4

x + y

IV 4

( y)

.

z1(

)

G

00



4 15.5, , z1(4 ) , -

G , , G 00 .
z
(4 14 )

G 00 . ( x, y ) ( x0 , 0 ) , ( x, y ) G00 , 0 < x0
3 . 4 (15.34)


(
x , y )( x0 , 0

lim

)

z

(4 14

)

(

x, y ) = 1IV ( x0 ) .

3 ( x, y ) ( 0, y0 ) , ( x, y ) G00 , 0 < y0 . 4 I ( lim z 14)( x, y ) = 4V ( y0 ) . (15.35) 4
(
x , y )( 0, y0

)

, ,

(

x, y ) ( 0, 0 ) ,

(

x, y ) G00

125


.. , .. , ..

(

x , y )( 0 , 0

lim

)

z

(4 14

)

(

x, y ) =

IV 4

(0)

.

(15.36)

, (15.36), (15.34)
x0 0 1IV ( 0 ) =
IV 4

( 0)

, (15.35) y0 0 .
(4 14 )

z

G

00

.

, G 00 .

ku k = 4 x k

ku k 5 x k

. ) . ). ), k +1u , k 0 x k y

G 00 . (15.1). 15.2
w
00

u

00

K

k +1

00

G 00 . -

, G
00

w

00

x k y

k +1u00 x k y

k 0 .

,
k +1u G x k y k +1w00 G x k y
00

k 0 .

00

k +1w00 15.4 k 0 . x k y

vk ,1 , (15.10)-(15.13), K
00

G

00

k 0 .

126






k1 = 0 . vk

,1



k = 0 . v0,0 0



4 ( 0 ) = 1 ( 0 ) , (15.10),
v0,1 = - -

y 2 + (1 - x

x + (1 - y
2

(

x 4 (1)

(

(1

- x )1 (1)

)

2

)(

+

y 2 + (1 + x

(1

+ x )1 (1)

)

2

)

2

) (

-

x 2 + (1 + y

x 4 (1)

)

- (15.37)

)

2

)

64

(15.37),



(

x, y ) K 00 . v0,1 K

00

G 00 . -
(15.38)

k = 1 . (15.11) v (1 - x )1 (1) - (1 + x )1 (1) v1,1 = 1,0 - y y 2 + (1 - x )2 y 2 + (1 + x )
-

x 2 + (1 - y

(

(1

- y ) 4 (1)

(

)

2

) (

-

(1

x 2 + (1 + y

+ y ) 4 (1)

)(

2

)

)

2

)

K
00

v1,0 K y

00

(15.21),

,

(15.38)

( x, y ) K 00 . 64 k = 2 . (15.12), 4 ( 0 ) = -1( 0 ) ,
v2,1 = + v2,0 (1 - x )1(1) - y y 2 + (1 - x ) x4 (1)



(

2

x + (1 - y
2

(

)

2

) (

+

x 2 + (1 + y

x 4 (1)

) (

+

(1

y + (1 + x
2 2

+ x )1(1)

)

2

)

+
(15.39)

)

)

K
00

v2,0 K y

00

(15.22),

, ,

(15.39)


64

( x, y ) K 00 .

127


.. , .. , ..

k = 3 . (15.13) v (1 - x )1(1) v3,1 = 3,0 - y y 2 + (1 - x )
+

x 2 + (1 - y

(

(1 - y )4(1)

(

2

)

2

) (


+

(1 + y )4(1) x 2 + (1 + y

) (

-

(1

y + (1 + x
2 2

+ x )1(1)

)

2

)

+ (15.40)

)

)

K
00

v3,0 K y

00

(15.23),

,

(15.40)

( x, y ) K 00 . 64 , (15.10)-(15.13) K

00



k1 = 0 .

vk ,1 k = 4k1 + i , i = 0, 3
k1 1 , .

G (15.14). k = 0 .

00



1 * * (1) -11 - 1,01 + 44 + 4,0 4 = 1(4) + 14 , 4 ,0

1 * * 1(4) = z1 ( x, y; 1 ) + z1,0 ( x, y; 1 ) - z4 ( x, y; 4 ) - z

(

x, y; 4 ) ,



(1)
14

1 1(4) G

1 ( x ) y 2 + (1 - x = ln 2 2 y + (1 + x

) )

2 2

+

4 ( y ) 1- y 1+ y + arctg arctg . x x
00

-

1 15.5, , 1(4) , G , , G 00 .

G 00 .

(1)

14

( x, y ) ( x0 , 0 ) , ( x, y ) G00 , 0 < x0
(
x , y )( x0 , 0

3 . 4 (15.41)

lim

)



(1)
14

(

x, y ) =

1 ( x0 ) 1 - x0 24 ( 0 ) 1 ln arctg . + 1 + x0 x0

128




3 ( x, y ) ( 0, y0 ) , ( x, y ) G00 , 0 < y0 . 4 (1) lim 14 ( x, y ) = 4 ( y0 ) . (15.42)

, ( x, y ) ( 0, 0 ) , ( x, y ) G00 ,
(
x , y ) ( 0 , 0

(

x , y )( 0, y0

)

lim

)



(1)
14

(

x, y ) = 4 ( 0 ) .

(15.43)

, , (15.43), (15.41) x0 0 (15.42) y0 0 .

(1)
14

G

00

.

k = 1 .
* * -11 + 1,01 - * 4 + * ,01 = 1(42) + 4 4

( 2)
14

,


* * * 1(42) = z1 ( x, y; 1) - z1,0 ( x, y; 1) + z4 ( x, y; 4 ) - z * 4 ,0

+

( 2)
14

=

1(42) G G , , G 00 .

4 ( y ) 2 ln x + (1 - y 2

(

1( x ) 2 ln y + (1 - x 2

(

x, y; 4 ) ,

(

)
2

2

)

2

)(

x 2 + (1 + y

)

)

+ + 1( x ) + 4 ( y ) ln x 2 + y 2 .
2

)(

y 2 + (1 + x

)

)

(

)

00

-

15.5, , 1(42) ,
( 2)
14

G 00 . ( x, y ) ( x0 , 0 ) , ( x, y ) G00 , 0 < x0
(
x , y )( x0 , 0

3 . 4

lim

)


0

( 2)
14

(

x, y ) =

+

2



( ( x )
1

+ 4 ( 0 ) ) ln x0

1( x0 ) ( 0 ) 2 2 ln (1 - x0 ) + 4 ln (1 + x0 ) + . (15.44)

129


.. , .. , ..

3 ( x, y ) ( 0, y0 ) , ( x, y ) G00 , 0 < y0 . 4 ( y ) ( 0 ) 2 2 (2) lim 14 ( x, y ) = 1 ln 1 + y0 + 4 0 ln 1 - y0 + ( x , y )( 0, y ) (15.45) 2 + (1( 0 ) + 4 ( y0 ) ) ln y0 .
0

(

)

(

)

1( 0 ) 4 ( 0 ) , 1( x ) 4 ( y ) x = 0 y = 0 -



:
1( x ) - 1( 0 x

)

k1( 2) ,

4 ( y ) - 4 ( 0 y

)

k

( 2)
4

(

k1( 2) > 0, k

( 2)
4

>0 .

)

1( 0 ) + 4 ( 0 ) = 0 , ( x, y ) ( 0, 0 ) ,

(

x, y ) G00 ,
(
x , y )( 0 , 0

lim

)



( 2)
14

(

x, y ) =

(

x , y )( 0 , 0

lim

1
)

+

1


1

(

x , y ) ( 0 , 0

lim

1( x ) - 1( 0
)

) )



1( x ) + 4 ( y ) ln ( x 2 + y
2

2

)




k1(

(

x , y )( 0 , 0

lim

x 4 ( y ) - 4 ( 0 y
)

x ln x 2 + y

(

) )

+ (15.46) y ln x 2 + y

)

y ln x 2 + y + k
( 2)
4

(

2

2

) (
x , y )( 0 , 0



lim

x ln x + y
2

(

2

)



(

x , y ) ( 0 , 0

lim

)

(

2

)

=0

, , (15.46), (15.44) x0 0 (15.45) ­ y0 0 , 1( 0 ) + 4 ( 0 ) = 0 .

( 2)
14

G

00

.

k = 2 .
3 * * ( 3) -11- 1,01- 44 - 4,0 4 = 1(4 ) + 14 , * * 1(43) = z1 ( x, y; 1) + z1,0 ( x, y; 1) - z4 ( x, y; 4) - z



4 ,0

(

x, y; 4) ,



( 3)
14

1( x ) y 2 + (1 - x = ln 2 2 y + (1 + x

) )

2 2

-

4 ( y ) 1- y 1+ y arctg + arctg . x x

130



3 1(4 ) G

00

-

3 15.5, , 1(4 ) , -

G , , G 00 .
3 1(4

)

G 00 . ( x, y ) ( x0 , 0 ) , ( x, y ) G00 , 0 < x0
(
x , y )( x0 , 0

3 . 4

lim

)



( 3)
14

(

x, y ) =

1( x0 ) 1 - x0 24 ( 0 ) 1 ln - arctg . 1 + x0 x0

(15.47)

3 ( x, y ) ( 0, y0 ) , ( x, y ) G00 , 0 < y0 . 4
(
x , y )( 0, y

lim

0

)



( 3)
14

(

x, y ) = - 4( y0 ) .

(15.48)

, ( x, y ) ( 0, 0 ) , ( x, y ) G00 ,
(
x , y ) ( 0 , 0

lim

)



( 3)
14

(

x, y ) = - 4 ( 0 ) .

(15.49)

, , (15.49), (15.47) x0 0 (15.48) ­ y0 0 .

( 3)
14

G

00

.

k = 3 .
* * -11IV + 1,01IV - * 4

IV 4

+ * ,0 4

IV 4

(4) = 1(44) + 14 ,
IV 4



* * * 1(44) = z1 ( x, y; 1IV ) - z1,0 ( x, y; 1IV ) - z4 ( x, y;

)

+z

* 4 ,0

(

x, y;

IV 4

)

,


-

( 4)
14

1IV ( x ) 2 = ln y + (1 - x 2

(

)

2

)(

y 2 + (1 + x

)

2

)

-
ln x 2 + y 2 .



IV 4

2

( y)

ln x 2 + (1 - y

(

)

2

)(

x + (1 + y
2

)

2

)

IV + 1 ( x ) -

IV 4

( y)

(

)

131


.. , .. , ..

1(44) G

00

-

15.5, , 1(44) , G , , G 00 .

( 4)
14

G 00 . ( x, y ) ( x0 , 0 ) , ( x, y ) G00 , 0 < x0
(
x , y )( x0 , 0

3 . 4

lim

)



+

2



( (
IV 1

1IV ( x0 ) 4IV ( 0 ) 2 2 ln (1 - x0 ) - ln (1 + x0 ) + 14 ( x , y ) = . IV x0 ) - 4 ( 0 ) ) ln x0
(4)

(15.50)

3 ( x, y ) ( 0, y0 ) , ( x, y ) G00 , 0 < y0 . 4
(
x , y )( 0 , y0

lim

)



+

2



( (
IV 1

4IV ( y0 ) 1IV ( 0 ) 2 2 ln (1 + y0 ) - ln (1 - y0 ) + 14 ( x , y ) = . (15.51) I 0 ) - 4V ( y0 ) ) ln y0
(4)

V 1 ( 0 )

, 1IV ( x ) x = 0 y = 0 :

(0) 4IV ( y )
V 4

-

1IV ( x ) - 1IV ( 0
x

)

k1 x ,

( 4)



IV 4

( y ) - 4IV ( 0 )
y

k

(4)
4

y

(

k1( 4) > 0, k
IV 4

( 4)
4

>0 .

)











1IV ( 0 ) -

( 0)

= 0,



(

x, y ) ( 0, 0 ) , ( x, y ) G00 ,

132




(

x , y )( 0 , 0

lim

)



( 4)
14

(

x, y ) =

(

x , y )( 0 , 0

lim

1
)

+

1


1

(

x , y ) ( 0 , 0

lim

1IV ( x ) - 1IV ( 0
)

)



1IV ( x ) -

IV 4

( y)

ln x 2 + y

(

2

)



x

x ln x 2 + y y ln x 2 + y k
(4)
4

(

2

) )
)

+ (15.52) y ln x 2 + y


k1(

(

x , y )( 0 , 0

lim


)

IV 4

( y ) - 4IV ( 0 )
y

(
(

2

4

) (
x , y ) ( 0 , 0



lim

)

x ln x + y
2

(

2

)

+



x , y )( 0 , 0

lim

(

2

)

=0

, , (15.52), (15.50) I x0 0 (15.51) ­ y0 0 , 1IV ( 0 ) - 4V ( 0 ) = 0 . z ( 4 ) G .
14

00

, G 00 .

u k = 3 x k y

k +1

k +1u k 4 x k y

. ) . 15.1 . 15.2. 15.2. n (15.20). (15.20). (15.1). w 00 u00 K
15.2), G 00 .
00

G

00

(

ku ) k , 1 k n . x k u k w00 k . x x k ( ( ( ( 11( k ) , 1,01( k ) , 4 4k ) , * 4k ) , 4,0 4k ) , * ,0 4k ) 4 4 k w0 (15.9) , x k vk ,0 , (15.5)-(15.8).
0



, vk ,0 , 1 k n . 133


.. , .. , ..

k = 1 . v1,0 , 15.1 (15.5) ( x, y ) ( 0, 0 ) , ( x, y ) G00 1 ( 0 ) - 4 ( 0 ) = 0 . k = 3 . v3,0 , 15.1 (15.7) ( x, y ) ( 0, 0 ) , ( x, y ) G00 1( 0 ) + 4 ( 0 ) = 0 .

1(

2k

)

k ( 0 ) = ( -1)

( 2k )
4

(0)

-

k .
k +1u ) k , 0 k n - 1 . x y k +1u k +1w00 . x k y x k y
* * 11( k ) , 1,01( k ) , 4

(k )
4

, * 4

(k )
4

, 4,0

(k )
4

, * ,0 4

(k )
4



k +1w00 (15.14) , x k y vk ,1 , (15.10)-(15.13).

, vk ,1 , 0 k n - 1 . k = 0 . v0,1 , 15.1 (15.10) ( x, y ) ( 0, 0 ) , ( x, y ) G00 1 ( 0 ) - 4 ( 0 ) = 0 . k = 2 . v2,1 , 15.1 (15.12) ( x, y ) ( 0, 0 ) , ( x, y ) G00 1( 0 ) + 4 ( 0 ) = 0 .

1(

2k

)

k ( 0 ) = ( -1)

( 2k )
4

(0)

-

k . ) ) ). , u = u ( x, y )
G
2k 2k u k u = ( -1) . y 2 k x 2 k

134




,
n n +n u k = ( -1) x n y n x
1 2 2 1 2 1 2 1 2 2 1

+ n2

u

n1 + n2

,



n2 = 2k

2

n +n u n +n u k , n2 = 2k2 + 1 . = ( -1) x n y n x n + 2 k y 15.2 . 15.3. . 15.1.
1 2 1 2

.

n +n u x n y n
1 2 1 2

G . G . n 15.2. 15.3 .

16. 1.3 , G
= 0 G, u =, (16.1) , ( x, y ) . u = , ( x, y ) ­ , . § 6.

(

I + A) = ,

(16.2)

­ -, . (16.2) . , ,

135


.. , .. , ..

A : C [ 0, 1] C [ 0, 1] ,

, , , , -
I + A : C [ 0, 1] C [ 0, 1]

. . +
, - + (16.1), -

u = - + ,
u = . -

(

)

(16.2) (. 13.1)

(

I + B) = ,

(16.3)

-, . § 14, B r ( B

)

B . (16.3) . . , , , u , y > 0 , u = . , . :
u = - + - u+ ,

(

)

u + G , :
136




u+ = - + + ,

(

)

+ ­ , - , ( I + B ) = B . B § 13, § 14. B . 16.1. L [ 0, 1] ­ -. - = B n 0 n . .

( 2k )
4

+

( 2k )
1

( 2k )
2

( 2k )
3

( 0) (1) (1) ( 0)

= ( -1)
k

= = =

( 0) , k ( -1) 2( 2 k ) ( 0 ) , k ( -1) 3( 2 k ) (1) , k ( -1) 4( 2 k ) (1)
(2k )
1

(16.4)

k 0 . (16.4).
( n)
1

( 0 ) 4( n) ( 0 )

. = B -

:
1 ( x ) = B12 2 2 ( x ) = B211 3 ( x ) = B311 ( x ) = B 41 1 4

( ( ( (

x ) + B133 ( x ) + B14 4 ( x ) ,

x ) + B322 ( x ) + B34 4 ( x ) , x ) + B42 2 ( x ) + B433 ( x
4

x ) + B233 ( x ) + B244 ( x ) ,

(16.5)

)

1 ( x )

( x)

:
1

1 ( x) =
+ 1
1

1




1 2

1

b1 ( x, t ) 2 ( t ) dt +



b ( x, t ) ( t )
2 3 0

1 2

dt +



b ( x, t ) ( t )
3 3 1 2

dt +

1



b ( x, t ) ( t )
4 4 1 2

1

dt ,

(16.6)

137


.. , .. , ..

4 ( x) =
+ 1

1


1


1 2

1

b4 ( x, t )1 ( t ) dt +

1


1

b ( x, t ) ( t )
2 2 0

1 2

dt +



b ( x, t ) ( t )
3 2 1 2

dt +

1



b ( x, t ) ( t )
1 3 1 2

dt ,

(16.7)


b1 =

(1 - x

1- x

)

2

+t 1

2

-

(1 - x
-

)

1- x
2

+ (2 - t 1

)

2

,

b2 = b4 =

1 1+ (x - t
2

)

2

-

1 1+ ( x + t

)

2

,

b3 =

1+ ( x - t

)

2

1+ (2 - x - t

)

2

,

x x -2 2 x +t x + (2 - t

)

2

.

,



( n)
1

(

n 1 b1 ( x, t ) 1 b2 ( x, t ) x) = 2 ( t ) dt + 3 ( t ) dt + 1 x n 0 x n

1

n

1 2

2 1n 1n 1 b3 ( x, t ) 1 b4 ( x, t ) + 3 ( t ) dt + 4 ( t ) dt , 1 x n 1 x n 2 1 n 2 1 2

(16.8)



4

(

n 1 b4 ( x, t ) 1 b2 ( x, t ) x) = 1 ( t ) dt + 2 ( t ) dt + 1 x n 0 x n 2 1n 1n 1 b3 ( x, t ) 1 b1 ( x, t ) + 2 ( t ) dt + 3 ( t ) dt . 1 x n 1 x n 2 2

(16.9)



( n)
1

( x ) 4( n) ( x )

-

n , bi , i = 1, 4 :
1 1 1 1 1 b1 = - - + + , 2 x - 1 - it x - 1 + it x - 1 - i ( 2 - t ) x - 1 + i ( 2 - t )
b2 = 1 1 1 1 1 - - + , i2 x - t - i x - t + i x + t - i x + t + i

138




b3 =

1 i2

1 1 1 1 - - + , x -t -i x -t +i x +t -2-i x +t -2+i

1 1 1 1 1 b4 = + - - . 2 x - it x + it x - i ( 2 - t ) x + i ( 2 - t )

n - bi ( x, t ) , i = 1, 4 x , n - 1 g ( x ) = : x+a
g
( n)

( x)

=

(

( -1)

n

n!
n +1

x+a

)

,

n a .
n +1 nb1 ( x, t ) ( -1) n! 1 = n x 2 ( x - 1 - it

)

n +1

+

1

(

x - 1 + it

)

n +1

-

-

1

(

x -1- i(2 - t

))
-

n +1

-

1

(

x -1 + i(2 - t 1

))

n +1

,

n nb2 ( x, t ) ( -1) n! 1 = n x i2 ( x - t - i

)

n +1

(

x-t +i

) )

n +1

- -

1

( (

x+t -i 1 x-t +i

) )

n +1

+ -

1

(

x+t +i

)

n +1

,

n nb3 ( x, t ) ( -1) n! 1 = n x i2 ( x - t - i

n +1

n +1

-

1

(

x+t -2-i

)

n +1

+

1

(

x+t -2+i

)

n +1

,

n nb4 ( x, t ) ( -1) n! 1 = x n 2 ( x - it

)

n +1

+

1

(

x + it

)

n +1

-

-

1

(

x - i(2 - t

))

n +1

-

1

(

x + i(2 - t

))

n +1

.

139


.. , .. , ..

x = 0 :
nb1 ( 0, t ) n! 1 = n x 2 (1 + it

)

n +1

+

1

(1

- it

)

n +1

-

1

(1 + i (
n +1 +1

2-t

))

n +1

-

1

(1 - i (
,

2-t

))

n +1

,

nb2 ( 0, t ) n! 1 + ( -1) = - x n i 2 ( t + i )n nb3 ( 0, t ) n! 1 = - n i2 (t + i x

+

1 + ( -1)

n +1

(

t -i

)

n +1

)

n +1

+

1

(

t -i -

)

n +1

+

1

(

-t + 2 + i
n n +1

)

n +1

-

1

(

-t + 2 - i i
n

)

n +1

=

n n ! ( -i ) in = + 2 (1 - it ) n+1 (1 + it

)

n +1

(1 - i
n +1

( -i ) ( 2 - t ))
1

-

(1 + i (
+

2-t 1

))

n +1

, =

nb4 ( 0, t ) n! 1 1 = - - n +1 n x 2 ( it ) ( -i t )

+

(i (

2-t

))

n +1

( -i (

2-t

))

n +1

n +1 n! ( -i ) + i = - 2 t n+1

n +1

+

( -i ) + n (2 - t )
n +1

i

n +1

+1

.

n : ) n = 4k , k 0 , ) n = 4k + 2, k 0 . ). n = 4k .
4 k b1 ( 0, t ) ( 4k = x 4 k 2 - 1

)

! 1 (1 + it -

)

4 k +1

+ 1

1

(1 - it

)

4 k +1

-

(1 + i ( 2 - t

))

4 k +1

(1 - i (
4 k +1 k +1

2-t +

))

4 k +1

,
4 k +1

4 k b2 ( 0, t ) ( 4k )! 1 + ( -1) = - x 4 k i 2 ( t + i )4 4 k b3 ( 0, t ) ( 4k = x 4 k 2 -

1 + ( -1)

(

t -i

)

4 k +1

0, - 1

)

4k ! ( -i ) i 4k + 4 k +1 (1 + it ) (1 - it )

4 k +1

(1

-i

( -i ) ( 2 - t ))
4k

4 k +1

-

i

4k

(1 + i (

2-t

))

4 k +1

( 4k = 2

)

! 1 (1 - it

)

4 k +1

+

(1

+ it

)

4 k +1

-

140




-

1

(1 - i (

2-t

))

4 k +1

-

1

(1 + i (
)

2-t

))

4 k +1

4 k b ( 0, t ) 1 = , x 4 k

4 k b4 ( 0, t ) ( 4k = x 4 k 2

! -i + i -i + i 0, - 4 k +1 + 4 k +1 t (2 - t)

(16.8) (16.9),



( 4k )
1

(

1 4k 1 4k 1 b1 ( 0, t ) 1 b3 ( 0, t ) 0) = 2 ( t ) dt + 3 ( t ) dt = 1 x 4 k 1 x 4 k 2 2 1 4k 1 4k 1 b1 ( 0, t ) 1 b1 ( 0, t ) = 2 ( t ) dt + 3 ( t ) dt 1 x 4 k 1 x 4 k 2 2 1 4k 1 4k 1 b3 ( 0, t ) 1 b1 ( 0, t ) 0) = 2 ( t ) dt + 3 ( t ) dt = 1 x 4 k 1 x 4 k 2 1 4k 1 2 4k 1 b1 ( 0, t ) 1 b1 ( 0, t ) = 3 ( t ) dt + 2 ( t ) dt 1 x 4 k 1 x 4 k 2 2

(16.10)



( 4k )
4

(

(16.11)

(16.10) (16.11) :
( 4k )
4

(0) =

( 4k )
1

( 0)

,

k 0.

) . ). n = 4k + 2 .

4k +2

b1 ( 0, t ) ( 4k + 2 )! 1 = 4 k +2 2 x (1 + it - 1

)

4 k +3

+

1

(1 - it
4 k +3

)

4 k +3

-

(1 + i ( 2 - t

))

4 k +3

-

1

(1 - i (

2-t +

))

,
4 k +3



4k +2

b2 ( 0, t ) ( 4k + 2 )! 1 + ( -1) = - 4 x 4 k + 2 i2 (t + i )
4k +2

4 k +3 k +3

1 + ( -1)

(

t -i

)

4 k +3

0,

4k +2 b3 ( 0, t ) ( 4k + 2 )! ( -i ) i 4 k +2 = + 4 k +3 4 x 4 k + 2 2 (1 - it ) (1 + it )

k +3

-

141


.. , .. , ..

-

( -i ) (1 - i ( 2 - t ) )
4k +2

4 k +3

-

i

4k +2

(1 + i (
1

2-t

))

4 k +3

( 4 k + 2 )! 1 = - 2 (1 - it

)

4 k +3

-

1

(1 + it

)

4 k +3

+

+

(1 - i (

4k +2

2-t

))

4 k +3

+

1

(1 + i (

2-t

))

4 k +3

4 k + 2b1 ( 0, t ) =- , x 4 k + 2
+3

b4 ( 0, t ) ( 4k + 2 )! -i + i -i + i = - 4 k +3 + 4k 4k +2 x 2 (2 - t ) t (16.8) (16.9),

0,



(
1

4k +2

)

(0)
=

=

1 1




1 2

4k +2

1 4k +2 b1 ( 0, t ) 1 b3 ( 0, t ) 2 ( t ) dt + 3 ( t ) dt = x 4 k + 2 1 x 4 k + 2 2

1 1




1 2

4k +2

1 4 k +2 b1 ( 0, t ) 1 b1 ( 0, t ) 2 ( t ) dt + 3 ( t ) dt , 1 x 4 k + 2 x 4 k + 2 2 4k +2 1 4 k +2 b3 ( 0, t ) 1 b1 ( 0, t ) 2 ( t ) dt + 3 ( t ) dt = x 4 k + 2 1 x 4 k + 2 2 4k +2 1 4k +2 b1 ( 0, t ) 1 b1 ( 0, t ) 3 ( t ) dt - 2 ( t ) dt . 1 x 4 k + 2 x 4 k + 2 2

(16.12)



(
4

4k +2

)

(0)

=

1




1 2

1

=-

1 1




1 2

(16.13)

(16.12) (16.13) , , :



(
4

4k +2

)

(0)

= -

(
1

4 k +2

)

( 0)

,

k 0.

) . (16.4) . 16.1 . C ( ) -.

(
+

I + B ) = B ,

+

(16.14)

­ -. 14.1 B - C [ 0,1] B C[0,1]C[0,1]

142




r ( B ) ,
3 ( 0, 2048 < r < 0, 2049 ) . 4 (16.14) r ( B) = r = arctg 1

C [ 0, 1]
= (I + B
+

)

-1

B = B ( I + B
+

)

-1

.

(16.15)

13.2 - [ 0, 1] . . 16.1. C ( ) .
+

u+ = -

(

+

)

+ ,

(16.16)

(16.15), G . . + , - u + = - + + . (16.15)
16.1 , - [ 0, 1] . +

+

(

)

, 15.1,
u = - +

(

)

+

-

G . 16.1 . 16.2. C ( ) . (16.1) u+ (16.16).
u= -

(

+

)

- u+ ,

(16.17)

. 15.2 - + G . ,
u = - + - u

(

)

(

)

+

143


.. , .. , ..

G , - u , u , :
u = ( I + B) - ( I + B

) ((

I + B ) B .
1

)

, u = . 16.2 . 16.3. - n , n - . - n , n (16.17) (16.1) G , u+ (16.16). . 15.1 n - + G . 16.1 , u + G . , n
u = - + - u

(

)

(

)

+

G . 16.3 . B : C [ 0,1] C [ 0,1] ,

( (
I+B

I+B
n

)

-1

= I - B + B 2 - B 3 + ... r n+1 , n . 1- r [0 ,1]C[0,1]

(16.18)



)

-1

-

( -1)
k =0

k

B

k C

(16.19)

u = u ( x, y )
u n ( x, y ) = -

(

*

) ( -1)
n k =0

k

B k , n = 1, 2, ... .

(16.20)

144




, u = u ( x, y

)

un = un ( x, y )

n , , . u - un . . 16.2. L ( ) (16.1) :
u= -

(

*

) ( -1 )
k =0



k



k

(

x, y ) ,

(16.21)



k

(

x , y ) = B
k

u ( x , y ) - un ( x , y

) C(G )

M r n , n = 1, 2, ... ,

1 B . - * C [ 0 , 1] C ( G ) C [ 0 , 1] 1- r . , (16.18)-(16.20) r < 1 (16.21).

M=

u - un = -

(

+

) ( -1)
k = n +1



k

Bk =

= ( -1)

n +1

(

-
+

) ( -1)
k =1



k

Bk Bn B ,

()

(16.22)

n = 1, 2, ... . 10.3 - B [ 0,1] . (16.22) u - un . (16.22),
u -u -
+ C [ 0 ,1]C nC

- (G )

(

*

) ( -1)
k =0



k

Bk Bn B

()

CG

()



(G )


k =0



n k B C[0,1]C[0,1] B C[0,1]C[0,1] B = C [ 0 ,1]
+ C [ 0 ,1]C G

rn - 1- r 16.2 . =

()

B

C [ 0 ,1]

.

145


.. , .. , ..

1. , .. / .., .. . -.:, 1966. ­ 724 . 2. , .. / .. .-.:, 1961. ­ 400 . 3. , . / ., ..-.:, 1979. ­ 587 . 4. , .. / ... - .: ,1967. ­ 436 . 5. , . / ., .. -.:, 1974. ­ 333 . 6. , . / .. - .:, 1974. ­ 160 . 7. , .. / .., .. . -.: , 1979. ­ 222 . 8. , .. / .., ...// . 1981. ­ . 35. . 3-35. 9. , .. / ... ­ .: , 1974. ­ 187 . 10. , .. / .., ... -.:, 1967. ­ 304 . 11. , .. / ... .2. .:, 2004. ­ 720 . 12. , .. / .. . -.:-, 1999. ­ 560 . 13. , .. / .., ... -.:, 1999. -496 .

146




...................................................................................................... 3 . ...................................................................................... 4 . ........................................... 8 . K .............. 16 . , , ................................................................ 24 . .................................. 32 . .......................................................... 38 . K ........................................................... 48 . ............................... ..... 51 . 1.1 ­ 1.3 ................................................................... 59 0. ................................................................................................ 62 1. ............... 75 2. .................................................. 84 3. ......... 95 4. , .................................................................. 99 5. ................................................................................... 104 6. ................................................. 135 ..................................... ................................ 146

1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1

147