Nonlinear Spectra for Parameter Dependent Ordinary Differential Equations
Eigenvalue problems of the form x 00 = −λf(x) + µg(x), (i), x(0) = 0, x(1) = 0 (ii) are considered. We are looking for (λ, µ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the famous Fuchik problem for piece-wise linear equations. In our considerations functions...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Vilnius University Press
2007-04-01
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| Series: | Nonlinear Analysis |
| Subjects: | |
| Online Access: | http://www.zurnalai.vu.lt/nonlinear-analysis/article/view/14715 |
| _version_ | 1818384699047804928 |
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| author | F. Sadyrbaev A. Gritsans |
| author_facet | F. Sadyrbaev A. Gritsans |
| author_sort | F. Sadyrbaev |
| collection | DOAJ |
| description | Eigenvalue problems of the form x 00 = −λf(x) + µg(x), (i), x(0) = 0, x(1) = 0 (ii) are considered. We are looking for (λ, µ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the famous Fuchik problem for piece-wise linear equations. In our considerations functions f and g may be super-, sub- and quasi-linear in various combinations. The spectra obtained under the normalization condition (otherwise problems may have continuous spectra) structurally are similar to usual Fuchik spectrum for the Dirichlet problem. We provide explicit formulas for Fuchik spectra for super and super, super and sub, sub and super, sub and sub cases, where superlinear and sublinear parts of equations are of the form |x| 2α x and |x| 1 2β+1 respectively (α > 0, β > 0.) |
| first_indexed | 2024-12-14T03:26:24Z |
| format | Article |
| id | doaj.art-ccf28276a8c647978d6efcbda173f301 |
| institution | Directory Open Access Journal |
| issn | 1392-5113 2335-8963 |
| language | English |
| last_indexed | 2024-12-14T03:26:24Z |
| publishDate | 2007-04-01 |
| publisher | Vilnius University Press |
| record_format | Article |
| series | Nonlinear Analysis |
| spelling | doaj.art-ccf28276a8c647978d6efcbda173f3012022-12-21T23:18:53ZengVilnius University PressNonlinear Analysis1392-51132335-89632007-04-0112210.15388/NA.2007.12.2.14715Nonlinear Spectra for Parameter Dependent Ordinary Differential EquationsF. Sadyrbaev0A. Gritsans1Daugavpils University, LatviaDaugavpils University, LatviaEigenvalue problems of the form x 00 = −λf(x) + µg(x), (i), x(0) = 0, x(1) = 0 (ii) are considered. We are looking for (λ, µ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the famous Fuchik problem for piece-wise linear equations. In our considerations functions f and g may be super-, sub- and quasi-linear in various combinations. The spectra obtained under the normalization condition (otherwise problems may have continuous spectra) structurally are similar to usual Fuchik spectrum for the Dirichlet problem. We provide explicit formulas for Fuchik spectra for super and super, super and sub, sub and super, sub and sub cases, where superlinear and sublinear parts of equations are of the form |x| 2α x and |x| 1 2β+1 respectively (α > 0, β > 0.)http://www.zurnalai.vu.lt/nonlinear-analysis/article/view/14715nonlinear spectrajumping nonlinearityasymptotically asymmetric nonlinearitiesFuchik spectrum |
| spellingShingle | F. Sadyrbaev A. Gritsans Nonlinear Spectra for Parameter Dependent Ordinary Differential Equations Nonlinear Analysis nonlinear spectra jumping nonlinearity asymptotically asymmetric nonlinearities Fuchik spectrum |
| title | Nonlinear Spectra for Parameter Dependent Ordinary Differential Equations |
| title_full | Nonlinear Spectra for Parameter Dependent Ordinary Differential Equations |
| title_fullStr | Nonlinear Spectra for Parameter Dependent Ordinary Differential Equations |
| title_full_unstemmed | Nonlinear Spectra for Parameter Dependent Ordinary Differential Equations |
| title_short | Nonlinear Spectra for Parameter Dependent Ordinary Differential Equations |
| title_sort | nonlinear spectra for parameter dependent ordinary differential equations |
| topic | nonlinear spectra jumping nonlinearity asymptotically asymmetric nonlinearities Fuchik spectrum |
| url | http://www.zurnalai.vu.lt/nonlinear-analysis/article/view/14715 |
| work_keys_str_mv | AT fsadyrbaev nonlinearspectraforparameterdependentordinarydifferentialequations AT agritsans nonlinearspectraforparameterdependentordinarydifferentialequations |