Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mce.biophys.msu.ru/archive/doc15398/doc.pdf Дата изменения: Fri Sep 21 19:48:14 2007 Дата индексирования: Mon Oct 1 20:01:53 2012 Кодировка:

. . (, ) T - . , .

= µ ( ) + (t , , x,

)

(1)

: ) x R m , R n - m , R n - m , R m , = ( , ) , R s - s ; ) µ ( ), (t , , x, ) -- n - m - -; ) (1) , ­ t

0 (r ) = { : - r}, -- , r > 0 -- ; ) :
+ c 3 (r ) - ,

= [ 0, T ] в R

n-m

в (r ) в 0 (r ) ,

( r ) = { x R m : x r} ,
-

(t , , x , ) - (t , , x , ) c1 (r ) - + c 2 (r ) x - x +

21


5. Part 5. Mathematical Theories

c1 (r ) 0, c2 (r ) 0, c3 (r ) 0 r 0 ,

µ ( ) - µ ( ) p(r ) - , p(r ) 0 r 0 ,
c, c1 , c2 , c3 , p -- , (t , ,0, ) = 0 , lim ( t , , x, ) = 0 t , , R ) (
x 0
n-m

в

0

(r )в[

0, ] ;

) µ ( ) = 0 ;
0

)

µ + = ( ) = + 1 ( ) ,
1 ( ) - 1 ( ) ( ) - , ( ) 0 0, (0, 0 ] .

(

)

= { R

n-m

:

0

}

- t F ( , t ) , F (t , ) d , F (t , ) - F (t , ) q1 t - t + q2 - , q1 0, q2 0 d 0 C (d ) . , - t F (t , ) C (d ) x (1), , [0,T]. (1) ­ , (1) -.

= µ ( ) + (t , , F (t , ), ) ,
F

(0 ) = 0 , : t = R
F

. t = R (t , 0 , ) , t = R F (t , 0 , ) . t [0, ]

(t

, 0 , ) .

t - t ( c3 ( r ) + p ( r ) ) e

( c1 ( r ) + c2 ( r ) q2 )

- .

(1.1)

22


. . -- -- 2006, . 2, .21­27 Baeva O. V. -- MCE -- 2006, v. 2, p. 21­27

[1]. , (1) ) ­ ).

µ ( ) + ( t , t , F ( t , t ) , ) dt = 0 .
0



(2)

-- . -- , (2) . = + ),


-

-1 ( t , t , F ( t , t ) , ) + 1 ( ) dt , =- 0

(3)

= , + . (2) ( )

(

)



F

( )

=-

-1 ( t , t , F ( t , t ) , ) + 1 ( ) dt . 0



. -- , d1 > 0, 1 > 0 , F (t , ) C (d1 ) (3)

{

R

n-m

: 1} .

. , F ( R

{

n-m

: 1} .

)



23


5. Part 5. Mathematical Theories

R

{

, , , n-m

: 1} ,

F ( ) - F ( ) - , 0 < < 1 .
, , ,

F

( )

-

F

( )





-1



(
0



t , t, F ( t , t ) , ) + 1 ( ) -

-1

- ( t , t, F ( t , t) , ) - 1 ( ) dt




0



( ( t , t, F ( t , t ) , ) -

- ( t , t, F ( t , t) , ) + 1 ( ) - 1 ( ) )dt



-1



(c ( r )
1 0





t - t + c2 ( r ) F ( t , t ) - F ( t , t) + c3 ( r ) - +

+ ( ) - )dt =




-1



[ ( c3 ( r ) + (

) ) -

+

+ (c1 ( r ) + c2 ( r ) q2 ) t - t dt .
0

(1.1)

F

( )

-

F

( )





-1



[(c3 ( r ) + ( )) - +
( c1 ( r ) + c2 ( r ) q2 )

+ (c1 ( r ) + c2 ( r ) q2 ) ( c3 ( r ) + p ( r ) ) e
0



- dt ] =

= e

-1

[c3 ( r ) + ( ) + (c1 ( r ) + c2 ( r ) q2 ) ( c3 ( r ) + p ( r ) ) ] - = - ,

( c1 ( r ) + c2 ( r ) q2 )

24


. . -- -- 2006, . 2, .21­27 Baeva O. V. -- MCE -- 2006, v. 2, p. 21­27



-1

[c3 ( r ) + ( ) + (c1 ( r ) + c2 ( r ) q2 ) ( c3 ( r ) + p ( r ) )

e

( c1 ( r ) + c2 ( r ) q2 )

] 0 r 0 , 0 d 0 .

,

F ( ) - F ( ) - .
R

, F ( ,

{

,
n-m

:

1

}

F (

)

-

.

)

1 .



F

( )
d 0





-1






0



( (

t , t , F ( t , t ) ,

)

+ 1 (

)

)

dt .

lim (t , , F (t , ), ) = 0 t , , , d1 F (t , ) C (d1 ) , -

( t , t , F ( t , t ) ,

)





2B

1 -1

.

(1.2)

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

) 1 (

) ( )

-

1 ( )



2B

1 -1

.

(1.3)

(1.2) (1.3),

25


5. Part 5. Mathematical Theories




F

( )

1 -1

+



-1






0



( (

t , t , F ( t , t ) ,

)

+ 1 (

)

)
)

dt



-1





(
0



2B

2B

1 -1

)dt = 1.

, F (



{

R

n-m

:

1

}

. , d1 > 0, 1 > 0 , F ( , t ) C (d1 ) R F F ( ) , (3). .
1. .. // : . . . / . .. . ­ , 1996. ­ . 76­86.

{

n-m

:

1

}

-

26


. . -- -- 2006, . 2, .21­27 Baeva O. V. -- MCE -- 2006, v. 2, p. 21­27

THE EXISTENCE OF NONTRIVIAL PERIODICAL SOLUTIONS OF NONLINEAR SYSTEM OF DIFFERENTIAL EQUATIONS WITH A PARAMETR. Baeva O. V.

(Russia, Rjazan) The article studies a nonlinear non-autonomous finite-dimensional T periodical system of differential equations with a vector parameter .

27