Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
The article investigates a second-order nonlinear difference equation of Emden-Fowler type Δ2u(k)±kαum(k)=0,{\Delta }^{2}u\left(k)\pm {k}^{\alpha }{u}^{m}\left(k)=0, where kk is the independent variable with values k=k0,k0+1,…k={k}_{0},{k}_{0}+1,\ldots \hspace{0.33em}, u:{k0,k0+1,…}→Ru:\left\{{k}_{0...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2023-10-01
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| Series: | Advances in Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/anona-2023-0105 |
| _version_ | 1797663131924692992 |
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| author | Diblík Josef Korobko Evgeniya |
| author_facet | Diblík Josef Korobko Evgeniya |
| author_sort | Diblík Josef |
| collection | DOAJ |
| description | The article investigates a second-order nonlinear difference equation of Emden-Fowler type Δ2u(k)±kαum(k)=0,{\Delta }^{2}u\left(k)\pm {k}^{\alpha }{u}^{m}\left(k)=0, where kk is the independent variable with values k=k0,k0+1,…k={k}_{0},{k}_{0}+1,\ldots \hspace{0.33em}, u:{k0,k0+1,…}→Ru:\left\{{k}_{0},{k}_{0}+1,\ldots \hspace{0.33em}\right\}\to {\mathbb{R}} is the dependent variable, k0{k}_{0} is a fixed integer, and Δ2u(k){\Delta }^{2}u\left(k) is its second-order forward difference. New conditions with respect to parameters α∈R\alpha \in {\mathbb{R}} and m∈Rm\in {\mathbb{R}}, m≠1m\ne 1, are found such that the equation admits a solution asymptotically represented by a power function that is asymptotically equivalent to the exact solution of the nonlinear second-order differential Emden-Fowler equation y″(x)±xαym(x)=0.{y}^{^{\prime\prime} }\left(x)\pm {x}^{\alpha }{y}^{m}\left(x)=0. Two-term asymptotic representations are given not only for the solution itself but also for its first- and second-order forward differences as well. Previously known results are discussed, and illustrative examples are considered. |
| first_indexed | 2024-03-11T19:10:10Z |
| format | Article |
| id | doaj.art-617a6e19b9034349bc8fe3a312ae70db |
| institution | Directory Open Access Journal |
| issn | 2191-950X |
| language | English |
| last_indexed | 2024-03-11T19:10:10Z |
| publishDate | 2023-10-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj.art-617a6e19b9034349bc8fe3a312ae70db2023-10-09T20:08:34ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2023-10-0112160361210.1515/anona-2023-0105Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler typeDiblík Josef0Korobko Evgeniya1Brno University of Technology, Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry, Faculty of Electrical Engineering and Communication, Department of Mathematics, Brno, Czech RepublicDepartment of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Brno, Czech RepublicThe article investigates a second-order nonlinear difference equation of Emden-Fowler type Δ2u(k)±kαum(k)=0,{\Delta }^{2}u\left(k)\pm {k}^{\alpha }{u}^{m}\left(k)=0, where kk is the independent variable with values k=k0,k0+1,…k={k}_{0},{k}_{0}+1,\ldots \hspace{0.33em}, u:{k0,k0+1,…}→Ru:\left\{{k}_{0},{k}_{0}+1,\ldots \hspace{0.33em}\right\}\to {\mathbb{R}} is the dependent variable, k0{k}_{0} is a fixed integer, and Δ2u(k){\Delta }^{2}u\left(k) is its second-order forward difference. New conditions with respect to parameters α∈R\alpha \in {\mathbb{R}} and m∈Rm\in {\mathbb{R}}, m≠1m\ne 1, are found such that the equation admits a solution asymptotically represented by a power function that is asymptotically equivalent to the exact solution of the nonlinear second-order differential Emden-Fowler equation y″(x)±xαym(x)=0.{y}^{^{\prime\prime} }\left(x)\pm {x}^{\alpha }{y}^{m}\left(x)=0. Two-term asymptotic representations are given not only for the solution itself but also for its first- and second-order forward differences as well. Previously known results are discussed, and illustrative examples are considered.https://doi.org/10.1515/anona-2023-0105discrete equationemden-fowler equationequation of the second-orderasymptotic behaviornonlinearity39a2239a12 |
| spellingShingle | Diblík Josef Korobko Evgeniya Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type Advances in Nonlinear Analysis discrete equation emden-fowler equation equation of the second-order asymptotic behavior nonlinearity 39a22 39a12 |
| title | Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type |
| title_full | Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type |
| title_fullStr | Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type |
| title_full_unstemmed | Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type |
| title_short | Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type |
| title_sort | asymptotic behavior of solutions of a second order nonlinear discrete equation of emden fowler type |
| topic | discrete equation emden-fowler equation equation of the second-order asymptotic behavior nonlinearity 39a22 39a12 |
| url | https://doi.org/10.1515/anona-2023-0105 |
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