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

Manabu Naito
Published: 1 January 2021

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 $.
Keywords: differential equation / mathrm / behavior / prime / sgn / nonoscillatory solutions / half linear

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