Method of Y-Mappings for Study of Multiparameter Nonlinear Eigenvalue Problems

For the study of nonlinear multiparameter eigenvalue problems, a method of Y -mappings, making it possible to prove the existence of solutions, is proposed. The problem of propagation of coupled polarized electromagnetic waves in a nonlinear layer with saturating nonlinearity is studied. The concept...

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Published in:	Computational mathematics and mathematical physics Vol. 62; no. 1; pp. 150 - 156
Main Author:	Smirnov, Yu. G.
Format:	Journal Article
Language:	English
Published:	Moscow Pleiades Publishing 01.01.2022 Springer Nature B.V
Subjects:	Computational Mathematics and Numerical Analysis Eigenvalues Eigenvectors Electromagnetic radiation Existence theorems Fixed points (mathematics) Mathematics Mathematics and Statistics Nonlinearity Partial Differential Equations Wave propagation coupled polarized electromagnetic waves multiparameter nonlinear eigenvalue problem Sturm–Liouville problem fixed point of mapping
ISSN:	0965-5425, 1555-6662
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Abstract	For the study of nonlinear multiparameter eigenvalue problems, a method of Y -mappings, making it possible to prove the existence of solutions, is proposed. The problem of propagation of coupled polarized electromagnetic waves in a nonlinear layer with saturating nonlinearity is studied. The concept of a Y -mapping, which puts into correspondence to the potential a special nonlinear function of several arguments: eigenfunctions of a linear problem, is defined. The multiparameter nonlinear eigenvalue problem is reduced to the problem of finding fixed points of Y -mappings. Using the Schauder theorem, the existence of an infinite set of fixed points of Y -mappings and, accordingly, solutions in a nonlinear multiparameter eigenvalue problem for sufficiently small values of the nonlinearity coefficient is proved.
AbstractList	For the study of nonlinear multiparameter eigenvalue problems, a method of Y -mappings, making it possible to prove the existence of solutions, is proposed. The problem of propagation of coupled polarized electromagnetic waves in a nonlinear layer with saturating nonlinearity is studied. The concept of a Y -mapping, which puts into correspondence to the potential a special nonlinear function of several arguments: eigenfunctions of a linear problem, is defined. The multiparameter nonlinear eigenvalue problem is reduced to the problem of finding fixed points of Y -mappings. Using the Schauder theorem, the existence of an infinite set of fixed points of Y -mappings and, accordingly, solutions in a nonlinear multiparameter eigenvalue problem for sufficiently small values of the nonlinearity coefficient is proved. For the study of nonlinear multiparameter eigenvalue problems, a method of Y-mappings, making it possible to prove the existence of solutions, is proposed. The problem of propagation of coupled polarized electromagnetic waves in a nonlinear layer with saturating nonlinearity is studied. The concept of a Y-mapping, which puts into correspondence to the potential a special nonlinear function of several arguments: eigenfunctions of a linear problem, is defined. The multiparameter nonlinear eigenvalue problem is reduced to the problem of finding fixed points of Y-mappings. Using the Schauder theorem, the existence of an infinite set of fixed points of Y-mappings and, accordingly, solutions in a nonlinear multiparameter eigenvalue problem for sufficiently small values of the nonlinearity coefficient is proved.
Author	Smirnov, Yu. G.
Author_xml	– sequence: 1 givenname: Yu. G. surname: Smirnov fullname: Smirnov, Yu. G. email: smirnovyug@mail.ru organization: Penza State University
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Cites_doi	10.1070/SM8741 10.1016/0022-1236(73)90051-7 10.1007/978-3-642-53393-8 10.1070/RM1996v051n03ABEH002911 10.1063/1.4799276 10.1007/978-3-0348-5485-6 10.1063/1.4769885 10.14760/OWP-2014-15 10.1080/09500340.2019.1695004 10.1063/1.4817388
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Copyright	Pleiades Publishing, Ltd. 2022. ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2022, Vol. 62, No. 1, pp. 150–156. © Pleiades Publishing, Ltd., 2022. Russian Text © The Author(s), 2022, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2022, Vol. 62, No. 1, pp. 159–165.
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SubjectTerms	Computational Mathematics and Numerical Analysis Eigenvalues Eigenvectors Electromagnetic radiation Existence theorems Fixed points (mathematics) Mathematics Mathematics and Statistics Nonlinearity Partial Differential Equations Wave propagation
Title	Method of Y-Mappings for Study of Multiparameter Nonlinear Eigenvalue Problems
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