Opuscula Mathematica
Opuscula Math. 43, no. 2 (2023), 221-246

Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations

Manabu Naito
Abstract. We consider the half-linear differential equation \[(|x'|^{\alpha}\mathrm{sgn}\,x')' + q(t)|x|^{\alpha}\mathrm{sgn}\,x = 0, \quad t \geq t_{0},\] under the condition \[\lim_{t\to\infty}t^{\alpha}\int_{t}^{\infty}q(s)ds = \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}.\] It is shown that if certain additional conditions are satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as \(t\to\infty\).
Keywords: asymptotic behavior, nonoscillatory solution, half-linear differential equation.
Mathematics Subject Classification: 34C11, 34C10.
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References ▼
  1. [1] O. Došlý, Half-linear Euler differential equation and its perturbations, Electron. J. Qual. Theory Differ. Equ. 2016, Paper no. 10, 14 pp. https://doi.org/10.14232/ejqtde.2016.8.10
  2. [2] O. Došlý, P. Řehák, Half-Linear Differential Equations, North-Holland Mathematics Studies, vol. 202, Elsevier, Amsterdam, 2005.
  3. [3] Á. Elbert, Studia Sci. Math. Hungar. Asymptotic behaviour of autonomous half-linear differential systems on the plane, 19 (1984), 447-464.
  4. [4] Á. Elbert, A. Schneider, Perturbations of the half-linear Euler differential equation, Results Math. 37 (2000), 56-83. https://doi.org/10.1007/BF03322512
  5. [5] J. Jaroš, T. Kusano, T. Tanigawa, Nonoscillation theory for second order half-linear differential equations in the framework of regular variation, Results Math. 43 (2003), 129-149. https://doi.org/10.1007/BF03322729
  6. [6] J. Jaroš, T. Kusano, T. Tanigawa, Nonoscillatory half-linear differential equations and generalized Karamata functions, Nonlinear Anal. 64 (2006), 762-787. https://doi.org/10.1016/j.na.2005.05.045
  7. [7] T. Kusano, J.V. Manojlović, Precise asymptotic behavior of regularly varying solutions of second order half-linear differential equations, Electron. J. Qual. Theory Differ. Equ. 2016, Paper no. 62, 24 pp. https://doi.org/10.14232/ejqtde.2016.1.62
  8. [8] T. Kusano, J.V. Manojlović, Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions, Georgian Math. J. 28 (2021), 611-636. https://doi.org/10.1515/gmj-2020-2070
  9. [9] J.V. Manojlović, Asymptotic analysis of regularly varying solutions of second-order half-linear differential equations, Kyoto University, RIMS Kokyuroku 2080 (2018), 4-17.
  10. [10] M. Naito, Asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations, Arch. Math. (Basel) 116 (2021), 559-570. https://doi.org/10.1007/s00013-020-01573-x
  11. [11] M. Naito, Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I, Opuscula Math. 41 (2021), 71-94. https://doi.org/10.7494/OpMath.2021.41.1.71
  12. [12] M. Naito, Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, II, Arch. Math. (Brno) 57 (2021), 41-60. https://doi.org/10.5817/AM2021-1-41
  13. [13] P. Řehák, Asymptotic formulae for solutions of half-linear differential equations, Appl. Math. Comput. 292 (2017), 165-177. https://doi.org/10.1016/j.amc.2016.07.020
  14. [14] P. Řehák, V. Taddei, Solutions of half-linear differential equations in the classes gamma and pi, Differential Integral Equations 29 (2016), 683-714. https://doi.org/10.57262/die/1462298681
About the authors ▼
Manabu Naito
Ehime University, Faculty of Science, Department of Mathematics, Matsuyama 790-8577, Japan
About the article ▼
Communicated by Josef Diblík.
Received: 2022-12-15.
Revised: 2023-02-14.
Accepted: 2023-02-21.
Published online: 2023-03-27.
Cite this article as ▼
Manabu Naito, Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations, Opuscula Math. 43, no. 2 (2023), 221-246, https://doi.org/10.7494/OpMath.2023.43.2.221.
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