# Nonlinear spectral problem for a self-adjoint vector differential equation

Nonlinear spectral problem for a self-adjoint vector differential equation 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 interval. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

# Nonlinear spectral problem for a self-adjoint vector differential equation

, Volume 53 (7) – Aug 23, 2017
8 pages

Publisher
Subject
Mathematics; Ordinary Differential Equations; Partial Differential Equations; Difference and Functional Equations
ISSN
0012-2661
eISSN
1608-3083
D.O.I.
10.1134/S0012266117070060
Publisher site
See Article on Publisher Site

### 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 interval.

### Journal

Differential EquationsSpringer Journals

Published: Aug 23, 2017

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