Nonlinear spectral problem for a self-adjoint vector differential equation


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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.

About the authors

A. A. Abramov

Dorodnitsyn Computing Center of the Russian Academy of Sciences; Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences

Author for correspondence.
Email: alalabr@ccas.ru
Russian Federation, Moscow, 119333; Moscow, 125047

L. F. Yukhno

Dorodnitsyn Computing Center of the Russian Academy of Sciences; Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences

Email: alalabr@ccas.ru
Russian Federation, Moscow, 119333; Moscow, 125047

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