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

Manabu Naito

Address: Department of Mathematics, Faculty of Science, Ehime University, Matsuyama 790-8577, Japan

E-mail: jpywm078@yahoo.co.jp

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 $.

AMSclassification: primary 34C11; secondary 26D10, 34C10.

Keywords: asymptotic behavior, nonoscillatory solution, half-linear differential equation, Hardy-type inequality.

DOI: 10.5817/AM2021-1-41