In our case, where c>0 is a constant; therefore, form (3.9) we have 2 _-c 2 II Vll~,co,--~ IIFIl~,co,[I VlL, c~,+ll Vll~,~,,, or, passing to V and U, [i UII=~r IIFII~,r Uil~r II Uil,2,w,~. S 9 iO) 2. Let us consider the Weyl functions. The functions of the Weyl sequence satisfy the system of equations (3.1) with the following right-hand side: F " l+a OkS AB T "" Applying inequality (3.10) to Weyl's sequence and taking into account (3.7) we conclude that ]IUkll~,(~) <~ for an arbitrary ~ for sufficiently large k: this contradicts the definition of Weyl' s sequence.
This contradiction proves the theorem.
It remains to show that ~.=0 , when the conditions of Theorem 3 are satisfied, is not an eigenvalue of problem (1.9).
In fact, let U(~, ~) be an eigenfunction of problem (1.9), corresponding to ~=0. If we follow the argument and the estimates used in proving Theorem 3,assuming that U~(~, ~)~ U(~, ~), r and take into account that the weak convergence of U,(~, ~) was not considered, we obtain [IU(~,~)[I~:r therefore ~=0 is not an eigenvalue of problem (1.9). From this remark and Theorem 3 it follows that l=0 does not belong to the spectrum of (1.9) .
We conclude that when the consitions of Theorems 1 and 3 are satisfied, the operator corresponding to problem (1.3) is positive-definite, and an approximate solution can be found by minimizing the energy functional IV(u, v) by the mesh functions. This question is examined in detail in /3/: can estimate of the error of the approximate solution is also given. REFERENCES i. GOL'DENVEIZER A.L., Theory of thin elastic shells, (Teoriya uprugikh tonkikh obolochek) Gostekhteoretizdat, Moscow, 1953. 2. ASLANYAN L.G. and LIDSKII V.B., Distribution of the natural frequencies of thin elastic shells (Raspredelenie sobstvennykh chastot tonkikh uprugikh obolochek), NAUKA, Moscow, 1974. 3. KLABUKOVA L.S., On the differential operator of problems of the theory of momentless elastic shells and their solution by the variational-difference method. Zh. vychisl. Mat. mat. Fiz., 20, i, 208-225, 1980. Translated by.W.C.