Smoothness and approximative properties of solutions of the singular nonlinear Sturm-Liouville equation
It is known that the eigenvalues λn(n = 1, 2, ...) numbered in decreasing order and taking the multiplicity of the self-adjoint Sturm-Liouville operator with a completely continuous inverse operator L−1 have the following property (∗) λn → 0, when n → ∞, moreover, than the faster convergence to zer...
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| Format: | Article |
| Language: | English |
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Academician Ye.A. Buketov Karaganda University
2020-12-01
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| Series: | Қарағанды университетінің хабаршысы. Математика сериясы |
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| Online Access: | http://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/391 |
| _version_ | 1797372640331038720 |
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| author | M.B. Muratbekov M.M. Muratbekov |
| author_facet | M.B. Muratbekov M.M. Muratbekov |
| author_sort | M.B. Muratbekov |
| collection | DOAJ |
| description |
It is known that the eigenvalues λn(n = 1, 2, ...) numbered in decreasing order and taking the multiplicity of the self-adjoint Sturm-Liouville operator with a completely continuous inverse operator L−1 have the following property (∗) λn → 0, when n → ∞, moreover, than the faster convergence to zero so the operator L−1 is best approximated by finite rank operators.
The following question:
- Is it possible for a given nonlinear operator to indicate a decreasing numerical sequence characterized by the property (∗)?
naturally arises for nonlinear operators. In this paper, we study the above question for the nonlinear Sturm-Liouville operator. To solve the above problem the theorem on the maximum regularity of the solutions of the nonlinear Sturm-Liouville equation with greatly growing and rapidly oscillating potential in the space L2(R) (R = (−∞, ∞)) is proved. Twosided estimates of the Kolmogorov widths of the sets associated with solutions of the nonlinear SturmLiouville equation are also obtained. As is known, the obtained estimates of Kolmogorov widths give the opportunity to choose approximation apparatus that guarantees the minimum possible error.
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| first_indexed | 2024-03-08T18:38:45Z |
| format | Article |
| id | doaj.art-2d435f29c1f245c2a48cf4b60f26d6d4 |
| institution | Directory Open Access Journal |
| issn | 2518-7929 2663-5011 |
| language | English |
| last_indexed | 2024-03-08T18:38:45Z |
| publishDate | 2020-12-01 |
| publisher | Academician Ye.A. Buketov Karaganda University |
| record_format | Article |
| series | Қарағанды университетінің хабаршысы. Математика сериясы |
| spelling | doaj.art-2d435f29c1f245c2a48cf4b60f26d6d42023-12-29T10:20:08ZengAcademician Ye.A. Buketov Karaganda UniversityҚарағанды университетінің хабаршысы. Математика сериясы2518-79292663-50112020-12-01100410.31489/2020m4/113-124Smoothness and approximative properties of solutions of the singular nonlinear Sturm-Liouville equationM.B. MuratbekovM.M. Muratbekov It is known that the eigenvalues λn(n = 1, 2, ...) numbered in decreasing order and taking the multiplicity of the self-adjoint Sturm-Liouville operator with a completely continuous inverse operator L−1 have the following property (∗) λn → 0, when n → ∞, moreover, than the faster convergence to zero so the operator L−1 is best approximated by finite rank operators. The following question: - Is it possible for a given nonlinear operator to indicate a decreasing numerical sequence characterized by the property (∗)? naturally arises for nonlinear operators. In this paper, we study the above question for the nonlinear Sturm-Liouville operator. To solve the above problem the theorem on the maximum regularity of the solutions of the nonlinear Sturm-Liouville equation with greatly growing and rapidly oscillating potential in the space L2(R) (R = (−∞, ∞)) is proved. Twosided estimates of the Kolmogorov widths of the sets associated with solutions of the nonlinear SturmLiouville equation are also obtained. As is known, the obtained estimates of Kolmogorov widths give the opportunity to choose approximation apparatus that guarantees the minimum possible error. http://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/391maximum regularitysingular nonlinear equationSturm-Liouville equationsmoothness of solutionsapproximative propertiesapproximate numbers |
| spellingShingle | M.B. Muratbekov M.M. Muratbekov Smoothness and approximative properties of solutions of the singular nonlinear Sturm-Liouville equation Қарағанды университетінің хабаршысы. Математика сериясы maximum regularity singular nonlinear equation Sturm-Liouville equation smoothness of solutions approximative properties approximate numbers |
| title | Smoothness and approximative properties of solutions of the singular nonlinear Sturm-Liouville equation |
| title_full | Smoothness and approximative properties of solutions of the singular nonlinear Sturm-Liouville equation |
| title_fullStr | Smoothness and approximative properties of solutions of the singular nonlinear Sturm-Liouville equation |
| title_full_unstemmed | Smoothness and approximative properties of solutions of the singular nonlinear Sturm-Liouville equation |
| title_short | Smoothness and approximative properties of solutions of the singular nonlinear Sturm-Liouville equation |
| title_sort | smoothness and approximative properties of solutions of the singular nonlinear sturm liouville equation |
| topic | maximum regularity singular nonlinear equation Sturm-Liouville equation smoothness of solutions approximative properties approximate numbers |
| url | http://mathematics-vestnik.ksu.kz/index.php/mathematics-vestnik/article/view/391 |
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