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Keywords:
half-linear differential equation; nonoscillatory solution; asymptotic form
half-linear differential equation; nonoscillatory solution; asymptotic form
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 $).
References:
[1] Coppel, W. A.: Stability and Asymptotic Behavior of Differential Equations. Heath Mathematical Monographs. Heath, Boston (1965). MR 0190463 | Zbl 0154.09301
[2] Došlý, O., Řehák, P.: Half-Linear Differential Equations. North-Holland Mathematics Studies 202. Elsevier, Amsterdam (2005). DOI 10.1016/s0304-0208(05)x8001-x | MR 2158903 | Zbl 1090.34001
[3] Hartman, P.: Ordinary Differential Equations. John Wiley, New York (1964). DOI 10.1137/1.9780898719222 | MR 0171038 | Zbl 0125.32102
[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. DOI 10.1016/j.na.2005.05.045 | MR 2197094 | Zbl 1103.34017
[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. DOI 10.14232/ejqtde.2016.1.62 | MR 3547438 | Zbl 1389.34164
[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. DOI 10.1504/IJDSDE.2021.117360 | MR 4318173 | Zbl 1482.34130
[7] Luey, S., Usami, H.: Asymptotic forms of solutions of perturbed half-linear ordinary differential equations. Arch. Math., Brno 57 (2021), 27-39. DOI 10.5817/AM2021-1-27 | MR 4260838 | Zbl 07332702
[8] Naito, M.: Asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations. Arch. Math. 116 (2021), 559-570. DOI 10.1007/s00013-020-01573-x | MR 4248549 | Zbl 1468.34076
[9] Naito, M.: Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations. I. Opusc. Math. 41 (2021), 71-94. DOI 10.7494/OpMath.2021.41.1.71 | MR 4302442 | Zbl 1478.34064
[10] Naito, M.: Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations. II. Arch. Math., Brno 57 (2021), 41-60. DOI 10.5817/AM2021-1-41 | MR 4260839 | Zbl 07332703
[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. DOI 10.1016/j.jde.2022.02.025 | MR 4387287 | Zbl 1497.34075
[12] Řehák, P.: Asymptotic formulae for solutions of half-linear differential equations. Appl. Math. Comput. 292 (2017), 165-177. DOI 10.1016/j.amc.2016.07.020 | MR 3542549 | Zbl 1410.34104
[13] Řehák, P.: Nonlinear Poincaré-Perron theorem. Appl. Math. Lett. 121 (2021), Article ID 107425, 7 pages. DOI 10.1016/j.aml.2021.107425 | MR 4268643 | Zbl 1487.34106
[14] Řehák, P., Taddei, V.: Solutions of half-linear differential equations in the classes Gamma and Pi. Differ. Integral Equ. 29 (2016), 683-714. DOI 10.57262/die/1462298681 | MR 3498873 | Zbl 1374.34206
[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. DOI 10.1515/gmj-2020-2070 | MR 4292900 | Zbl 1476.34115
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