We consider a boundary value problem for the generalized second order differential equation \begin{equation} -\frac d{dM(x)}\biggl(y^+(x)-\int_{c+0}^{x+0}y(s)dQ(s)\biggr)-\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.
@article{SM_1969_8_3_a2,
author = {I. S. Kats},
title = {Compatibility of the coefficients of a~generalized second order linear differential equation},
journal = {Sbornik. Mathematics},
pages = {345--356},
year = {1969},
volume = {8},
number = {3},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1969_8_3_a2/}
}
TY - JOUR
AU - I. S. Kats
TI - Compatibility of the coefficients of a generalized second order linear differential equation
JO - Sbornik. Mathematics
PY - 1969
SP - 345
EP - 356
VL - 8
IS - 3
UR - http://geodesic.mathdoc.fr/item/SM_1969_8_3_a2/
LA - en
ID - SM_1969_8_3_a2
ER -
%0 Journal Article
%A I. S. Kats
%T Compatibility of the coefficients of a generalized second order linear differential equation
%J Sbornik. Mathematics
%D 1969
%P 345-356
%V 8
%N 3
%U http://geodesic.mathdoc.fr/item/SM_1969_8_3_a2/
%G en
%F SM_1969_8_3_a2
I. S. Kats. Compatibility of the coefficients of a generalized second order linear differential equation. Sbornik. Mathematics, Tome 8 (1969) no. 3, pp. 345-356. http://geodesic.mathdoc.fr/item/SM_1969_8_3_a2/
[1] I. S. Kats, “Zamechaniya k state “Suschestvovanie spektralnykh funktsii obobschennykh: differentsialnykh sistem vtorogo poryadka s granichnymi usloviyami v singulyarnom kontse””, Matem. sb., 76(118) (1968), 147–152 | Zbl
[2] I. S. Kats, “Suschestvovanie spektralnykh funktsii obobschennykh differentsialnykh sistem vtorogo poryadka s granichnymi usloviyami v singulyarnom kontse”, Matem. sb., 68(110) (1965), 174–227 | Zbl
[3] M. G. Krein, “O nekotorykh zadachakh na maksimum i minimum dlya kharakteristicheskikh chisel i o lyapunovskikh zonakh ustoichivosti”, Prikl. matem. i mekhan., 15 (1951), 323–348 | MR | Zbl
[4] M. A. Naimark, Lineinye differentsialnye operatory, Gostekhizdat, M., 1954
[5] I. S. Kats, “O roste spektralnykh funktsii obobschennykh granichnykh zadach vtorogo poryadka s granichnym usloviem v regulyarnom kontse”, DAN SSSR, 181:3 (1968), 534–537 | Zbl
