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Study of a Nonlinear Eigenvalue Problem by the Integral Characteristic Equation Method

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Abstract

We consider an eigenvalue problem for a quasilinear nonautonomous second-order differential equation with a cubic nonlinearity. The problem is posed on an interval with boundary conditions of the first kind and with an auxiliary (local) condition at one of the endpoints of the interval. We prove that the problem in question has infinitely many negative and infinitely many positive eigenvalues. The corresponding linear problem has infinitely many negative and finitely many (or none) positive eigenvalues. Moreover, the first terms of the asymptotics of the negative eigenvalues of the nonlinear and linear problems coincide, while the asymptotics of the positive eigenvalues of the nonlinear problem is expressed in terms of a transcendental function of the eigenvalue number. The results are derived with the use of a nonclassical approach.

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Funding

This work was supported by the Russian Science Foundation, project no. 18-71-10015.

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Authors and Affiliations

  1. Penza State University, Penza, 440026, Russia

    D. V. Valovik

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  1. D. V. Valovik
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Correspondence to D. V. Valovik.

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Translated by V. Potapchouck

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Valovik, D.V. Study of a Nonlinear Eigenvalue Problem by the Integral Characteristic Equation Method. Diff Equat 56, 171–184 (2020). https://doi.org/10.1134/S0012266120020032

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