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Mat. Sb., 2017, Volume 208, Number 9, Pages 26–41 (Mi msb8666)  

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

The spectral properties of some nonlinear operators of Sturm-Liouville type

D. V. Valovik

Penza State University

Abstract: For certain quasilinear second-order ordinary differential equations, we study the eigenvalue problem of Sturm-Liouville type on a closed interval with conditions of the first kind. To determine the discrete eigenvalues we use an additional (local) condition on one of the boundaries of the interval. The problem is (equivalently) reduced to a transcendental equation with respect to the spectral parameter. The analysis of this equation makes it possible to prove the existence of infinitely many (isolated) eigenvalues, indicate their asymptotics, find conditions under which the eigenfunctions are periodic, calculate the period, and give an explicit formula for the zeros of the eigenfunction. Several comparison theorems are obtained. We also study a problem to which perturbation theory cannot be applied.
Bibliography: 27 titles.

Keywords: nonlinear problem of Sturm-Liouville type, nonlinear differential equation, asymptotics of eigenvalues, comparison theorem.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-00206-а
Ministry of Education and Science of the Russian Federation 1.894.2017/4.6
MK-4684.2016.1
This research was supported by the Russian Foundation for Basic Research (grant no. 15-01-00206-a) in the framework of the state commission of the Ministry for Higher Education and Science of the Russian Federation (contract no. 1.894.2017/4.6), as well as in the framework of the Programme of the President of the Russian Federation for state support for young Russian scientists (grant no. MK-4684.2016.1).


DOI: https://doi.org/10.4213/sm8666

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English version:
Sbornik: Mathematics, 2017, 208:9, 1282–1297

Bibliographic databases:

UDC: 517.927.4+517.988.57
MSC: Primary 34L15; Secondary 34B09, 34B24, 34L20
Received: 03.02.2016 and 14.03.2017

Citation: D. V. Valovik, “The spectral properties of some nonlinear operators of Sturm-Liouville type”, Mat. Sb., 208:9 (2017), 26–41; Sb. Math., 208:9 (2017), 1282–1297

Citation in format AMSBIB
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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. Kurseeva V.Yu., Tikhov S.V., Valovik D.V., “Nonlinear Multiparameter Eigenvalue Problems: Linearised and Nonlinearised Solutions”, J. Differ. Equ., 267:4 (2019), 2357–2384  crossref  isi
  • Математический сборник Sbornik: Mathematics (from 1967)
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