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 Mat. Sb. (N.S.), 1969, Volume 79(121), Number 3(7), Pages 368–380 (Mi msb3593)

Compatibility of the coefficients of a generalized second order linear differential equation

I. S. Kats

Abstract: We consider a boundary value problem for the generalized second order differential equation
$$-\frac d{dM(x)}(y^+(x)-\int_{c+0}^{x+0}y(s)dQ(s))-\lambda y(x)=0,$$
where $M(x)$ is a nondecreasing function, and $Q(x)$ is the difference of two nondecreasing functions; $y^+(x)$ designates the right derivative of the function $y(x)$.
Differential equation (1) is a generalization of the differential equation
$$-y"+q(x)y-\lambda\rho(x)y=0,$$
where $\rho(x)\geqslant0$ and $q(x)$ are locally integrable real functions.
Even when equation (1) is considered on a finite interval and the functions $M(x)$ and $Q(x)$ have bounded variation there (the regular case), it may turn out that not every function in $L_M^{(2)}$ can be expanded in solutions of equation (1) (for equation (2) this is exceptional). In this paper we find a condition which is necessary and sufficient for any function $f(x)\in L_M^{(2)}$ to expand in the solutions (“eigenfunctions”) of the boundary value problem with equation of the form (1); in the case when this condition is not fulfilled, we find the class of all functions in $L_M^{(2)}$ which can be expanded in these “eigenfunctions”.
Bibliography: 5 titles.

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English version:
Mathematics of the USSR-Sbornik, 1969, 8:3, 345–356

Bibliographic databases:

UDC: 517.941.91
MSC: 34B05, 34A30, 34L05

Citation: I. S. Kats, “Compatibility of the coefficients of a generalized second order linear differential equation”, Mat. Sb. (N.S.), 79(121):3(7) (1969), 368–380; Math. USSR-Sb., 8:3 (1969), 345–356

Citation in format AMSBIB
\Bibitem{Kat69} \by I.~S.~Kats \paper Compatibility of the coefficients of a~generalized second order linear differential equation \jour Mat. Sb. (N.S.) \yr 1969 \vol 79(121) \issue 3(7) \pages 368--380 \mathnet{http://mi.mathnet.ru/msb3593} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=252746} \zmath{https://zbmath.org/?q=an:0193.04701|0196.10202} \transl \jour Math. USSR-Sb. \yr 1969 \vol 8 \issue 3 \pages 345--356 \crossref{https://doi.org/10.1070/SM1969v008n03ABEH002041} 

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This publication is cited in the following articles:
1. I. S. Kats, “Integral characteristics of the growth of spectral functions for generalized second order boundary problems with boundary conditions at a regular end”, Math. USSR-Izv., 5:1 (1971), 161–191
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