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Perturbations of the half-linear Euler differential equation. (English) Zbl 0958.34029

The authors investigate oscillation/nonoscillation properties of the perturbed half-linear Euler differential equation

${\left({x}^{\text{'}}{}^{n*}\right)}^{\text{'}}+\frac{{\gamma }_{0}}{{t}^{n+1}}\left[n+2\left(n+1\right)\delta \left(t\right)\right]{x}^{n*}=0,\phantom{\rule{2.em}{0ex}}\left(*\right)$

where the function $\delta \left(t\right)$ is piecewise continuous on $\left({t}_{0},\infty \right)$, ${t}_{0}\ge 0$, $n>0$ is a fixed real number and ${u}^{n*}={|u|}^{n}\text{sgn}u$. The number ${\gamma }_{0}=\frac{{n}^{n}}{{\left(n+1\right)}^{n+1}}$ is a critical constant, which means that in case $\delta \left(t\right)\equiv 0$ the half-linear Euler differential equation is for $\gamma \le {\gamma }_{0}$ nonoscillatory while for $\gamma >{\gamma }_{0}$ is oscillatory. Using a Riccati technique and a transformation of the independent variable, there are proved the following main results:

Suppose that there exists the finite limit

$\underset{T\to \infty }{lim}{\int }_{{t}_{0}}^{T}\delta \left(t\right)\frac{dt}{t}$

such that ${\int }_{t}^{\infty }\delta \left(s\right)\frac{ds}{s}\ge 0$ for $t>{t}_{0}$.

(a) If $n>1$ and the linear equation

${z}^{\text{'}\text{'}}+\delta \left({e}^{s}\right)z=0\phantom{\rule{2.em}{0ex}}\left(**\right)$

is nonoscillatory then equation (*) is also nonoscillatory.

(b) If $0 and equation (*) is nonoscillatory then the linear equation (**) is also nonoscillatory.

In addition, the authors establish an asymptotic form of the solution to (*) provided that the solutions to (**) satisfy two integral inequalities.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
##### References:
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