Opuscula Math. 41, no. 1 (2021), 71-94
https://doi.org/10.7494/OpMath.2021.41.1.71
Opuscula Mathematica
Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I
Abstract. We consider the half-linear differential equation of the form \[(p(t)|x'|^{\alpha}\mathrm{sgn} x')' + q(t)|x|^{\alpha}\mathrm{sgn} x = 0, \quad t\geq t_{0},\] under the assumption \(\int_{t_{0}}^{\infty}p(s)^{-1/\alpha}ds =\infty\). It is shown that if a certain condition is 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, Hardy-type inequality.
Mathematics Subject Classification: 34C11, 34C10, 26D10.
- P.R. Beesack, Hardy's inequality and its extensions, Pacific J. Math. 11 (1961), 39-61.
- O. Došlý, P. Řehák, Half-Linear Differential Equations, North-Holland Mathematics Studies, vol. 202, Elsevier, Amsterdam, 2005.
- G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1952.
- P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, vol. 38, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002 (reprinted from original John Wiley and Sons, 1964).
- J. Jaroš, T. Kusano, Self-adjoint differential equations and generalized Karamata functions, Bull. T. CXXIX de Acad. Serbe Sci. et Arts, Classe Sci. Mat. Nat. Sci. Math. 29 (2004), 25-60.
- 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.
- J. Jaroš, T. Kusano, T. Tanigawa, Nonoscillatory half-linear differential equations and generalized Karamata functions, Nonlinear Anal. 64 (2006), 762-787.
- 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.
- T. Kusano, J.V. Manojlović, Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions, Georgian Math. J., to appear.
- V. Marić, Regular Variation and Differential Equations, Lecture Notes in Mathematics, vol. 1726, Springer, Berlin, Heidelberg, New York, 2000.
- M. Naito, Asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations, Arch. Math. (Basel), to appear.
- P. Řehák, Asymptotic formulae for solutions of half-linear differential equations, Appl. Math. Comput. 292(2017), 165-177.
- P. Řehák, V. Taddei, Solutions of half-linear differential equations in the classes gamma and pi, Differential Integral Equations 29 (2016), 683-714.
- Manabu Naito
- Ehime University, Faculty of Science, Department of Mathematics, Matsuyama 790-8577, Japan
- Communicated by Petr Stehlík.
- Received: 2020-07-29.
- Revised: 2020-11-13.
- Accepted: 2020-12-11.
- Published online: 2021-02-08.
