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

This article is cited in 1 scientific paper (total in 1 paper)

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
\begin{equation} -\frac d{dM(x)}(y^+(x)-\int_{c+0}^{x+0}y(s)dQ(s))-\lambda y(x)=0, \end{equation}
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
\begin{equation} -y"+q(x)y-\lambda\rho(x)y=0, \end{equation}
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
Received: 23.09.1968

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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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  crossref  mathscinet  zmath
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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