Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, II

Manabu Naito

Archivum Mathematicum (2021)

  • Volume: 057, Issue: 1, page 41-60
  • ISSN: 0044-8753

Abstract

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We consider the half-linear differential equation of the form ( p ( t ) | x ' | α sgn x ' ) ' + q ( t ) | x | α sgn x = 0 , t ≥ t 0 , under the assumption that p ( t ) - 1 / α is integrable on [ t 0 , ∞ ) . 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 → ∞ .

How to cite

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Naito, Manabu. "Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, II." Archivum Mathematicum 057.1 (2021): 41-60. <http://eudml.org/doc/297252>.

@article{Naito2021,
abstract = {We consider the half-linear differential equation of the form \[ (p(t)|x^\{\prime \}|^\{\alpha \}\mathrm \{sgn\}\,x^\{\prime \})^\{\prime \} + q(t)|x|^\{\alpha \}\mathrm \{sgn\}\,x = 0\,, \quad t \ge t\_\{0\} \,, \] under the assumption that $p(t)^\{-1/\alpha \}$ is integrable on $[t_\{0\}, \infty )$. 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 \rightarrow \infty $.},
author = {Naito, Manabu},
journal = {Archivum Mathematicum},
keywords = {asymptotic behavior; nonoscillatory solution; half-linear differential equation; Hardy-type inequality},
language = {eng},
number = {1},
pages = {41-60},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, II},
url = {http://eudml.org/doc/297252},
volume = {057},
year = {2021},
}

TY - JOUR
AU - Naito, Manabu
TI - Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, II
JO - Archivum Mathematicum
PY - 2021
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 057
IS - 1
SP - 41
EP - 60
AB - We consider the half-linear differential equation of the form \[ (p(t)|x^{\prime }|^{\alpha }\mathrm {sgn}\,x^{\prime })^{\prime } + q(t)|x|^{\alpha }\mathrm {sgn}\,x = 0\,, \quad t \ge t_{0} \,, \] under the assumption that $p(t)^{-1/\alpha }$ is integrable on $[t_{0}, \infty )$. 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 \rightarrow \infty $.
LA - eng
KW - asymptotic behavior; nonoscillatory solution; half-linear differential equation; Hardy-type inequality
UR - http://eudml.org/doc/297252
ER -

References

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  10. Manojlović, J.V., Asymptotic analysis of regularly varying solutions of second-order half-linear differential equations, Kyoto University, RIMS Kokyuroku 2080 (2018), 4–17. (2018) 
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  12. Naito, M., Asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations, to appear in Arch. Math. (Basel). 
  13. Naito, M., 10.7494/OpMath.2021.41.1.71, Opuscula Math. 41 (2021), 71–94. (2021) DOI10.7494/OpMath.2021.41.1.71
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  15. Řehák, P., Taddei, V., Solutions of half-linear differential equations in the classes gamma and pi, Differ. Integral Equ. 29 (2016), 683–714. (2016) MR3498873

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