ISSN 0012-2661, Differential Equations, 2017, Vol. 53, No. 7, pp. 900–907.
Pleiades Publishing, Ltd., 2017.
Original Russian Text
A.A. Abramov, L.F. Yukhno, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 7, pp. 927–934.
Nonlinear Spectral Problem
for a Self-Adjoint Vector Diﬀerential Equation
A. A. Abramov
and L. F. Yukhno
Dorodnitsyn Computing Center of the Russian Academy of Sciences,
Moscow, 119333 Russia
Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences,
Moscow, 125047 Russia
Received January 9, 2017
Abstract— We consider a spectral problem that is nonlinear in the spectral parameter for
a self-adjoint vector diﬀerential equation of order 2n. The boundary conditions depend on
the spectral parameter and are self-adjoint as well. Under some conditions of monotonicity
of the input data with respect to the spectral parameter, we present a method for counting the
eigenvalues of the problem in a given interval. If the boundary conditions are independent of
the spectral parameter, then we deﬁne the notion of number of an eigenvalue and give a method
for computing this number as well as the set of numbers of all eigenvalues in a given interval.
For an equation considered on an unbounded interval, under some additional assumptions,
we present a method for approximating the original singular problem by a problem on a ﬁnite
The importance of the problem of computing the eigenvalues of an operator, which arises in
numerous theoretical and applied studies, is well known. There is a fairly complete theory of
and eﬃcient numerical methods for ﬁnding the eigenvalues of matrices (linear operators) in linear
problems; e.g., see [1, Chaps. V and VI]. The fundamentals of the general theory for nonlinear
spectral problems were laid by Keldysh [2, 3] and further developed in quite a few papers.
The present paper develops the results in  as applied to the following problem. Consider
a nonlinear spectral problem for a self-adjoint vector ordinary diﬀerential equation of order 2n.
The coeﬃcients of the equation depend on a real spectral parameter. The boundary conditions de-
pend on the spectral parameter as well and are self-adjoint. Under some conditions of monotonicity
of the input data with respect to the spectral parameter, we prove theorems that enable one to count
the eigenvalues of this problem in a given interval without computing the eigenvalues themselves.
If the boundary conditions are independent of the spectral parameter, then we deﬁne the notion
of number of an eigenvalue. In this case, our theorems allow one to determine the number of each
eigenvalue as well as the set of numbers of all eigenvalues lying in a given interval.
We also consider coupled boundary conditions. Further, we deal with the case in which the
independent variable ranges over an unbounded interval and the input data have limit values
at inﬁnity, the boundary conditions being supplemented with the condition that the solution is
bounded at inﬁnity. We give a method for approximating this singular problem by a problem on
a ﬁnite interval, to which one can apply methods earlier described in this paper.
1. PROBLEM WITHOUT SINGULARITIES
1.1. Statement of the Problem
Consider a spectral problem for a vector diﬀerential equation of the form