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Let us consider the differential equation d~y/dt2--k2Q(t)=F(t) (t~R) (1) in the Banach space B. It is assumed that q is a sufficiently smooth operator function 9 Q(t) are !bounded and uniformly invertible, and, moreover 9 there exists a continuous branch of the root Ql/2(t), such that for all t e R, the operators -Q1/2(t) generate contraction semigroups.It is proved that if the spectra of the limit operators Q(• do not contain negative numbers, ithen, for sufficiently large k, the equation will have a unique bounded solution for any ~ounded F(t). This result turns out to be stronger than those corollaries that may be obtained for this problem from the results on solvability given in [1][2][3].We also consider a boundary-value problem with explicitly stipulated conditions at infinitely!distant points:It is prqved that problem (i), (2) [regardless of whether the condition imposed on Q(• holds] fqr F E LI(R , B) is solvable, and a formula which constitutes ageneralization of the following estimate is established: G(t,s)=-(2k)-tQ-'~'(t)Q-'~'(s) exp(,sign(t-s)kI[ O'~(s)ds) (i+O(k-') )for the Green's function of the scalar problem of the form (1) 9 (2) (formula (3) is an obvious consequence of the VKB estimates for the solution of a homogeneous scalar equation; cf. [4 9 5]).In the operator case, however, in place of exponential functions in the expression for the principal term of the asymptote evolutionary operators of the following Cauchy problem are encoUntered:Asymptotic decomposition of abstract differential equations has been investigated previously in [2,6,7] for the Cauchy problem. Questions related to the construction of the asymptotes of particular solutions determined by a point %(t) of the discrete spectrum of the coefficient Q(t) both in the case of systems of scalar equations and in the case of abstract equations (cf. [5,8] and the literature cited therein) have been subjected to extensive study. The essential part of the results presented in here involves the approximate reduction of a boundary-value problem to an evolutionary-type problems; these results are not relaged to a decomposition of a solution with respect to the spectrum Q(t).Note that problems such as (i), (2) arise under natural conditions in the description i. of harmonic fields in waveguides [though Q(t) may then be unbounded] 9 and the reduction referred tO above asserts, speaking in terms of physics 9 that it is possible to take into ac-! --P Ros~ov State University.

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