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we can write the inequality

k

v

0

(

c

)

k

6

γ <

1

. Therefore, we can prove that

the mapping

v

is compressing. We calculate

˙

W

with the help of (18):

˙

W

=

d

(

t

)

E

+

r

X

i

=2

d

(

i

1)

(

t

)

N

i

1

M

1

˙

ν

=

=

d

(

t

)

E

+

r

X

i

=2

d

(

i

1)

(

t

)

N

i

1

M

1

"

+

r

X

i

=1

A

i

+

∂p

∂z

i

d

(

i

1)

(

t

)

#

.

(20)

Taking (19) into consideration, we express

ν

(

t

)

in terms of

W

(

t

)

:

ν

(

t

) =

M

D

(

t

)

E

+

r

1

X

i

=1

d

(

i

1)

(

t

)

N

i

W

(

t

)

!

,

substitute the received relation in (20). As a result, we have the equality

˙

W

=

PW

+ [

E

M

1

A

1

PN

1

]

d

(

t

)+

+

r

1

X

i

=2

[

N

i

1

M

1

A

i

PN

i

]

d

(

i

1)

(

t

)+

+[

N

r

1

M

1

A

r

]

d

(

r

1)

(

t

)

M

1

r

X

i

=1

∂p

∂z

i

d

(

i

1)

(

t

)

PD

(

t

)

.

(21)

Now we choose matrices

N

1

, . . . , N

r

1

from the condition

E

M

1

A

1

PN

1

= 0;

N

i

1

M

1

A

i

PN

i

= 0

, i

= 2

, r

1;

N

r

1

M

1

A

r

= 0

.

(22)

With a direct substitution, we can show that the solution to system (22) of

the matrix equations is the matrices

N

1

=

M

1

(

A

2

+

KA

3

+

. . .

+

K

r

2

A

r

);

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

N

r

2

=

M

1

(

A

r

1

+

KA

r

);

N

r

1

=

M

1

A

r

.

(23)

Let us assume that the matrices

N

i

are specified by formulae (23). Then

equality (21) will take the form of

˙

W

=

PW

+

R

(

t

)

,

where

R

(

t

) =

M

1

r

X

i

=1

∂p

∂z

i

d

(

i

1)

(

t

)

PD

(

t

)

.

24

ISSN 1812-3368. Herald of the BMSTU. Series “Natural Sciences”. 2014. No. 5