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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| Veröffentlicht in: | Computational mathematics and mathematical physics Jg. 62; H. 1; S. 150 - 156 |
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| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Moscow
Pleiades Publishing
01.01.2022
Springer Nature B.V |
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| 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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| Keywords | coupled polarized electromagnetic waves multiparameter nonlinear eigenvalue problem Sturm–Liouville problem fixed point of mapping |
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| References | Vainberg (CR9) 1956 Vinokurov, Sadovnichii (CR13) 2003; 68 CR4 CR3 CR6 Angermann, Shestopalov, Smirnov, Yatsyk (CR7) 2018 Smirnov, Valovik (CR8) 2017; 294 CR5 CR18 CR17 CR16 CR15 Adams (CR21) 1975 Kupriyanova, Smirnov (CR2) 2004; 44 Atkinson, Mingarelli (CR11) 2011 CR20 Egorov, Kondrat’ev (CR12) 1996; 51 Ambrosetti, Rabinowitz (CR10) 1973; 14 Vladimirov (CR14) 2017; 208 Trenogin (CR22) 1993 Schurman, Smirnov, Shestopalov (CR1) 2005; 71 Kato (CR19) 1966 cr-split#-1612_CR17.1 cr-split#-1612_CR17.2 H. W. Schurman (1612_CR1) 2005; 71 M. M. Vainberg (1612_CR9) 1956 S. N. Kupriyanova (1612_CR2) 2004; 44 F. V. Atkinson (1612_CR11) 2011 T. Kato (1612_CR19) 1966 1612_CR20 Y. G. Smirnov (1612_CR8) 2017; 294 V. A. Vinokurov (1612_CR13) 2003; 68 1612_CR4 V. A. Trenogin (1612_CR22) 1993 1612_CR5 A. A. Vladimirov (1612_CR14) 2017; 208 1612_CR18 1612_CR6 L. Angermann (1612_CR7) 2018 1612_CR15 1612_CR16 Yu. V. Egorov (1612_CR12) 1996; 51 R. Adams (1612_CR21) 1975 1612_CR3 A. Ambrosetti (1612_CR10) 1973; 14 |
| References_xml | – volume: 208 start-page: 1298 year: 2017 end-page: 1311 ident: CR14 article-title: Majorants for eigenvalues of Sturm–Liouville problems with potentials lying in balls of weighted spaces publication-title: Sb. Math. doi: 10.1070/SM8741 – ident: CR18 – year: 2011 ident: CR11 – ident: CR3 – ident: CR4 – ident: CR15 – ident: CR16 – year: 1975 ident: CR21 – ident: CR17 – volume: 14 start-page: 349 year: 1973 end-page: 381 ident: CR10 article-title: Dual variational methods in critical point theory and applications publication-title: J. Funct. Anal. doi: 10.1016/0022-1236(73)90051-7 – volume: 294 start-page: 146 year: 2017 end-page: 156 ident: CR8 article-title: Nonlinear coupled wave propagation in an -dimensional layer publication-title: Appl. Math. Comput. – year: 1966 ident: CR19 doi: 10.1007/978-3-642-53393-8 – volume: 44 start-page: 1762 year: 2004 end-page: 1772 ident: CR2 article-title: Propagation of electromagnetic waves in cylindrical dielectric waveguides filled with a nonlinear medium publication-title: Comput. Math. Math. Phys. – ident: CR6 – year: 2018 ident: CR7 – ident: CR5 – volume: 68 start-page: 247 year: 2003 end-page: 252 ident: CR13 article-title: On the range of variation of an eigenvalue when the potential is varied publication-title: Dokl. Math. – volume: 51 start-page: 439 year: 1996 end-page: 508 ident: CR12 article-title: Estimates for the first eigenvalue in some Sturm–Liouville problems publication-title: Russ. Math. Surv. doi: 10.1070/RM1996v051n03ABEH002911 – year: 1993 ident: CR22 – volume: 71 start-page: 016614-1 year: 2005 end-page: 016614-10 ident: CR1 article-title: “Propagation of TE waves in cylindrical nonlinear dielectric waveguides,” Phys. Rev. E Stat publication-title: Nonlinear Soft Matter Phys. – year: 1956 ident: CR9 – ident: CR20 – volume: 14 start-page: 349 year: 1973 ident: 1612_CR10 publication-title: J. Funct. Anal. doi: 10.1016/0022-1236(73)90051-7 – volume: 208 start-page: 1298 year: 2017 ident: 1612_CR14 publication-title: Sb. Math. doi: 10.1070/SM8741 – volume: 294 start-page: 146 year: 2017 ident: 1612_CR8 publication-title: Appl. Math. Comput. – ident: 1612_CR20 – volume-title: Sobolev Spaces year: 1975 ident: 1612_CR21 – ident: 1612_CR5 doi: 10.1063/1.4799276 – ident: 1612_CR18 doi: 10.1007/978-3-0348-5485-6 – volume-title: Nonlinear and Inverse Problems in Electromagnetics year: 2018 ident: 1612_CR7 – volume-title: Variational Methods for Analysis of Nonlinear Operators year: 1956 ident: 1612_CR9 – volume: 68 start-page: 247 year: 2003 ident: 1612_CR13 publication-title: Dokl. Math. – ident: 1612_CR4 doi: 10.1063/1.4769885 – ident: 1612_CR16 – ident: 1612_CR6 doi: 10.14760/OWP-2014-15 – volume-title: Functional Analysis year: 1993 ident: 1612_CR22 – ident: #cr-split#-1612_CR17.1 – ident: 1612_CR15 doi: 10.1080/09500340.2019.1695004 – volume: 44 start-page: 1762 year: 2004 ident: 1612_CR2 publication-title: Comput. Math. Math. Phys. – ident: #cr-split#-1612_CR17.2 – ident: 1612_CR3 doi: 10.1063/1.4817388 – volume-title: Perturbation Theory for Linear Operators year: 1966 ident: 1612_CR19 doi: 10.1007/978-3-642-53393-8 – volume: 71 start-page: 016614-1 year: 2005 ident: 1612_CR1 publication-title: Nonlinear Soft Matter Phys. – volume-title: Multiparameter Eigenvalue Problems: Sturm–Liouville Theory year: 2011 ident: 1612_CR11 – volume: 51 start-page: 439 year: 1996 ident: 1612_CR12 publication-title: Russ. Math. Surv. doi: 10.1070/RM1996v051n03ABEH002911 |
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Y
-mappings, making it possible to prove the existence of solutions, is proposed.... 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... |
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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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