| Title:
|
A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations (English) |
| Author:
|
Naito, Manabu |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
149 |
| Issue:
|
3 |
| Year:
|
2024 |
| Pages:
|
317-336 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The half-linear differential equation $$ (|u'|^{\alpha }{\rm sgn} u')' = \alpha (\lambda ^{\alpha + 1} + b(t))|u|^{\alpha }{\rm sgn} u, \quad t \geq t_{0}, $$ is considered, where $\alpha $ and $\lambda $ are positive constants and $b(t)$ is a real-valued continuous function on $[t_{0},\infty )$. It is proved that, under a mild integral smallness condition of $b(t)$ which is weaker than the absolutely integrable condition of $b(t)$, the above equation has a nonoscillatory solution $u_{0}(t)$ such that $u_{0}(t) \sim {\rm e}^{- \lambda t}$ and $u_{0}'(t) \sim - \lambda {\rm e}^{- \lambda t}$ ($t \to \infty $), and a nonoscillatory solution $u_{1}(t)$ such that $u_{1}(t) \sim {\rm e}^{\lambda t}$ and $u_{1}'(t) \sim \lambda {\rm e}^{\lambda t}$ ($t \to \infty $). (English) |
| Keyword:
|
half-linear differential equation |
| Keyword:
|
nonoscillatory solution |
| Keyword:
|
asymptotic form |
| MSC:
|
34C11 |
| MSC:
|
34D05 |
| MSC:
|
34D10 |
| DOI:
|
10.21136/MB.2023.0158-22 |
| . |
| Date available:
|
2024-09-11T13:45:36Z |
| Last updated:
|
2024-09-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152537 |
| . |
| Reference:
|
[1] Coppel, W. A.: Stability and Asymptotic Behavior of Differential Equations.Heath Mathematical Monographs. Heath, Boston (1965). Zbl 0154.09301, MR 0190463 |
| Reference:
|
[2] Došlý, O., Řehák, P.: Half-Linear Differential Equations.North-Holland Mathematics Studies 202. Elsevier, Amsterdam (2005). Zbl 1090.34001, MR 2158903, 10.1016/s0304-0208(05)x8001-x |
| Reference:
|
[3] Hartman, P.: Ordinary Differential Equations.John Wiley, New York (1964). Zbl 0125.32102, MR 0171038, 10.1137/1.9780898719222 |
| Reference:
|
[4] Jaroš, J., Takaŝi, K., Tanigawa, T.: Nonoscillatory half-linear differential equations and generalized Karamata functions.Nonlinear Anal., Theory Methods Appl., Ser. A 64 (2006), 762-787. Zbl 1103.34017, MR 2197094, 10.1016/j.na.2005.05.045 |
| Reference:
|
[5] Kusano, T., Manojlović, J.: Precise asymptotic behavior of regularly varying solutions of second order half-linear differential equations.Electron. J. Qual. Theory Differ. Equ. 2016 (2016), Article ID 62, 24 pages. Zbl 1389.34164, MR 3547438, 10.14232/ejqtde.2016.1.62 |
| Reference:
|
[6] Luey, S., Usami, H.: Application of generalized Riccati equations to analysis of asymptotic forms of solutions of perturbed half-linear ordinary differential equations.Int. J. Dyn. Syst. Differ. Equ. 11 (2021), 378-390. Zbl 1482.34130, MR 4318173, 10.1504/IJDSDE.2021.117360 |
| Reference:
|
[7] Luey, S., Usami, H.: Asymptotic forms of solutions of perturbed half-linear ordinary differential equations.Arch. Math., Brno 57 (2021), 27-39. Zbl 07332702, MR 4260838, 10.5817/AM2021-1-27 |
| Reference:
|
[8] Naito, M.: Asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations.Arch. Math. 116 (2021), 559-570. Zbl 1468.34076, MR 4248549, 10.1007/s00013-020-01573-x |
| Reference:
|
[9] Naito, M.: Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations. I.Opusc. Math. 41 (2021), 71-94. Zbl 1478.34064, MR 4302442, 10.7494/OpMath.2021.41.1.71 |
| Reference:
|
[10] Naito, M.: Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations. II.Arch. Math., Brno 57 (2021), 41-60. Zbl 07332703, MR 4260839, 10.5817/AM2021-1-41 |
| Reference:
|
[11] Naito, M., Usami, H.: On the existence and asymptotic behavior of solutions of half-linear ordinary differential equations.J. Differ. Equations 318 (2022), 359-383. Zbl 1497.34075, MR 4387287, 10.1016/j.jde.2022.02.025 |
| Reference:
|
[12] Řehák, P.: Asymptotic formulae for solutions of half-linear differential equations.Appl. Math. Comput. 292 (2017), 165-177. Zbl 1410.34104, MR 3542549, 10.1016/j.amc.2016.07.020 |
| Reference:
|
[13] Řehák, P.: Nonlinear Poincaré-Perron theorem.Appl. Math. Lett. 121 (2021), Article ID 107425, 7 pages. Zbl 1487.34106, MR 4268643, 10.1016/j.aml.2021.107425 |
| Reference:
|
[14] Řehák, P., Taddei, V.: Solutions of half-linear differential equations in the classes Gamma and Pi.Differ. Integral Equ. 29 (2016), 683-714. Zbl 1374.34206, MR 3498873, 10.57262/die/1462298681 |
| Reference:
|
[15] Takaŝi, K., Manojlović, J. V.: Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions.Georgian Math. J. 28 (2021), 611-636. Zbl 1476.34115, MR 4292900, 10.1515/gmj-2020-2070 |
| . |
