Abstract
A nonlinear n-parametric eigenvalue problem called the problem P is considered. In addition to n spectral parameters, the problem P depends on n2 numerical parameters; for zero values of these parameters, the problem splits into n linear problems \( {P}_i^0,i=\overline{1,n} \). To the problem P, one can assign n nonlinear problems Pi, which, in particular, have solutions that are not related to the solutions of the problems \( {P}_i^0 \). The problems Pi are treated in this work as “nonperturbed” problems. Using the properties of eigenvalues of the problems Pi, we prove the existence of eigenvalues of the problem P; some of these eigenvalues are not related to solutions of the problems \( {P}_i^0 \).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 172, Proceedings of the Voronezh Winter Mathematical School “Modern Methods of Function Theory and Related Problems,” Voronezh, January 28 – February 2, 2019. Part 3, 2019.
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Valovik, D.V., Kurseeva, V.Y. Multiparameter Eigenvalue Problems and Their Applications in Electrodynamics. J Math Sci 267, 677–697 (2022). https://doi.org/10.1007/s10958-022-06173-4
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DOI: https://doi.org/10.1007/s10958-022-06173-4
Keywords and phrases
- nonlinear Sturm–Liouville-type problem
- multiparameter eigenvalue problem
- perturbation method
- method of integral dispersion equations