LITERATURE CITED V. I. Khrabustovskii, "Spectral analysis of periodic systems with a degenerate weight on the axis and the semiaxis," Teor. Funktsii Funktsional. Anal. i Prilozhen. (Kharkov), No. 44, 122-133 (1985). V. I. Khrabustovskii, "Expansions in the eigenfunctions of periodic systems with weight," Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 5, 26-29 (1984). V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients, Vols. 1 and 2, Wiley, New York (1975). F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York
i.
.
3. 4.
(t964).
5.
V. I. Khrabustovskii, "The spectral matrix of a periodic symmetric system with a degenerate weight on the axis," Teor. Funktsii Funktsional. Anal. i Prilozhen. (Kharkov), No. 35, 111-119 (1981). Yu. L. Daletskii (Ju. L. Daleckii) and M. G. Krein, Stability of Solutions of Differen ~ tial Equations in Banach Space, Am. Math. Soc., Providence (1974).
6.
OUTER-INNER FACTORIZATION OF j-EXPANDING INVERTIBLE MATRIX-FUNCTIONS P. M. Yuditskii
UDC 517.5
This paper is a continuation of [2]. matrix-function.
Let B(~)
be a j-expanding invertible analytic
We define F (t)= lira{]-- B -v~ (~)]B -x (~)},A
A we associate an interpolation problem [2].
=
F x/2(! + F)-I/2,~ t J = l .
With the given
Any function of the form
[w~)] = B (.~)[o~ ~)] (b~ (~)~ (~) + b~2 (~))_~ ' where
~(~)
is an analytic contractive function, B = IlbH!i, is a solution of it.
by virtue of Theorem 3, there exists an analytic function
]A (~) = lira{]
Therefore,
A(~), defined by
+ P+ [A (DF~A)] (~)}R
E~O
and possessing the properties / - - A-~* (:)
iA-'
(~) = lira ( ( D + S) - 1 (~ - - T ) - I A ,
($ - - T ) - a A ) ,
j - - A-~* (t) ]A -~ (t) = A (I - - A 2)-, A = F (t), J t : = I.
Since the boundary values of the j-forms of the matrices follows that the matrix
Bi ($) = A -1(~)B (~)
is j-expanding in the unit circle.
A(~) and B ( ~ )
has j-unitary boundary values.
coincide, it
We prove that B~ ($)
To this end we derive the splitting-off inequality, the
exact analogue of the inequality used in [i] for the separation of the Blaschke-Potapov factors
]J-5 Fg*g/ I
i --
~$
(s0i)
j
x being an arbitrary vector in L~(C2), I$I< I. For the proof of the inequality we apply the following Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 46, pp. 132-135, 1986. Original articl e submitted December 5, 1984.
0090-4104/90/4805-0607512.50
9 1990 Plenum Publishing Corporation
607
LEMMA.
The block operator (in orthogonal expansion.)
is a contraction if and only if
['-/"*, jr,r] Proof of the Splitting-Off Inequality (SO1).
>.
o.
We represent the given function B-X(~)
by a linear fractional transformation of a contraction: B-I(~) = [pS(~) + p =-I/2(I + ]), q =
q]lqS(r + p]-1
where
I/2(1--]). Moreover, the j-form of the matrix B-I(~) has the form ] - - B -a* (~) jB -~ (~.) = [qS (~) q-- p]-'* [I--S* (~) S ($)1 [qS (~)q-p]-X.
(2)
Being restricted to the unit circumference, the identity (2) yields the equality
1 - - S ' S - - (qS* + p)*A ( I - - A 2 ) - * A (qS + p) = O, a consequence of which is the inequality
'5
w = I A (qS -l- P)
oI
is contracting. In accordance with this, the operator Pw
in the space H = H 2(C~ ) ~ L ~(C 2)
represents
also a contraction,
We decompose the space H into the orthogonal sum
~= The c o r r e s p o n d i n g
. @bL
block decomposition
~ C H , ~ 6 C ~,
of the operator
b(t)--~_t.~. Pw~ h a s t h e f o r m
I-~'~.~< 1 l
0
J
@ bPw*~. As we c a n s e e ,
the decomposition
is triangular
and,
by v i r t u e
o f t h e lemma, we h a v e t h e
inequality
IXL
0 J
We define the vector =
608
, x 6 L~ (C)~"
(3)
and we compute tile obtained blocks of (3).
Since
n[--S*P+qAx + (p + S*q)Ax I = [ 0P+ O][S*P-q~x+ pAx] = P+pAx]
Pw*~ = ,- [
Ax
Ax
l'
we have
(~, ~> - - ( Pw*~, Pw*~> = (P +qAx, Ax> + ( x, x> -- ( P +pAx, Ax) -- (Ax, Ax) ~ ((1 - - A ~"- - AP+/A) x, x) = ( Dx, x). Another block of the inequality (3) can be reduced to the form
=\
/A. p -[- qS ($) X~ T--;
~' / "
Introducing these expressions into (3), we obtain the inequality
(Dx, x> I,<(~--T)-aA, x> (p + qS (~)) ~z ]
I
~,l--S*(t) S(Oc~
] ~0.
Making use of relation (2), we can rewrite the last inequality into the form of the SO1. Now it is easy to show that Bi(~) = A-I(~)B(~)
is a j-contracting matrix function.
Indeed, from the SOI there follows the inequality i _ B-I.(O/R-~(~ ) 1--~
- - ( ( D + e ) - ~ ( ~ - - T ) - l A , ( ~ - - T ) - I A > > O, V e i > O .
Taking in this inequality the limit as
e-~0, by property (i) we obtain
] - - B - ' * C) ]B-L (t)
From h e r e
we h a v e
- - A -1. ( t ) / A -1 (t) >~ O,
B*(.~)A -x*($)/A - 1 ( ~ ) B ( ~ ) - ] > I 0 .
Thus, we have established that i.
The resolvent matrix
A(~)
of the interpolation problem of [2] has the property that
each j-expanding analytic matrix function j-form, admits the representation
B($),
having the same boundary values of the
B(~)----A(OB~($)
function in the class of j-contracting ones~
(o.-i.f.), where
Bi(r
is a j-inner matrix
The latter means that
,, - - / > O, 1 - - ;~ > 0, B~* (~) jB~ (2)
Bi(t)/B~(t)--]=O,
It]=
1.
2. An arbitrary invertible j-expanding analytic matrix function B (~) position (o.-i.f.), where
A(~)
admits a decom-
is the resolvent matrix of the corresponding problem of [2]. LITERATURE CITED
i.
A. V. Efimov and V. P. Potapov, "J-expanding matrix-valued functions, and their role in the analytic theory of electrical circuits," Usp. Mat. Nauk, 28, No. i (169), 65-130
(1973).
609
.
P. M. Yuditskii, "On the restoration of a j-contracting analytic matrix function from the boundary values of its j-form and the related "interpolation" problem," Teor. Funktsii Funktsional. Anal. i Prilozhen. (Kharkov), No. 44, 141-143 (1984).
AVERAGING OF DENSELY PERFORATED CYLINDRICAL SHELLS
I. Yu. Chudinovich
UDC 517.946
It is well known that the computation of the deformation of thin elastic shells, containing various kinds of micrononhomogeneities, is a very complicated problem.
When the
number of the micrononhomogeneities is large, the straightforward application of the standard numerical methods is not possible.
It is natural to replace shells of this type by bodies
with specially selected efficient elastic characteristics. In this note we present a method of averaging boundary value problems for circular cylindrical shells, weakened by a large number of fine openings.
The proofs of the averaging
theorems are based on the variational methods developed in [i]. A similar problem on the averaging of a perforated plate has been solved in [2].
In
[3, 4] one has carried out the averaging theorems are based on the variational methods developed in [i]. A similar problem on the averaging of a perforated plate has been solved in [2].
In
[3, 4] one has carried Out the averaging of boundary value problems for elliptic systems of differential equations in domains with holes.
Since the system of equations of the theory of
thin shells is elliptic [5], this note has to be considered as a continuation of [3, 4]. Moreover, we shall use the notations introduced in those papers [3, 4]. Thus, we consider a circular cylindrical shell of length a and radius R.
We parametrize
the points of its median surface with the aid of the variables x = (x z, x2), where x I is the length of an arc of a meridian and x 2 is the length of an arc of a parallel. ness of the shell will be denoted by 2o.
Let U(x) = (utl~(x), ut2~(x), uCm(x))
vector of the points of the median surface, where u(~
The small thick-
be the displacement
(i= 1,2) are tangential displacements
and u(S)(x) is the normal displacement. The relation between the strains of the shell and the displacements are given by the formulas OuO)
~(U) =-~,
au (~)
%(U)-~ Ox~
u (3)
au tl)
e ' ,o(U) = ~
Ou (2)
+ a,,, '
O'~vt3) a~-u (3) aZu (3) I au (2) ~, ( U ) = ox---~-' xa ( U ) = ax--~- ' ~ ( U ) - - ax~ax~ + ~ ax~ 9
(i)
We introduce the domain of the variation of the parameters ~ = { ( x 1,x~):0
L~(U, V) = ~W(U, V)dx,
Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 46, pp. 136-139, 1986. Original article submitted December 5, 1984.
610
0090-4104/90/4805-0610 $12.50
9 1990 Plenum Publishing Corporation
