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Half-linear Euler differential equations in the critical case. (English) Zbl 1265.34126
The authors study oscillatory properties of the perturbed half-linear differential equation $$\Bigl (r(t)\Phi (x')\Bigr)^{\hskip -2pt{\prime }}+c(t)\Phi (x)=0\,,$$ where $\Phi (x)=\vert x\vert ^{p-2}x$, $p>1$, $r$, $c$ are continuous functions and $r(t)>0$. The perturbation is allowed in both the coefficients $r$ and $c$. The authors extend the results of {\it O.Došlý} and {\it S. Fišnarová} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 12, 3756--3766 (2010; Zbl 1207.34041)] and show that the perturbed half-linear Euler equation $$\left [\left (1+\frac {\lambda }{\log ^2 t}\right)\Phi (x')\right ]' +\left [\frac {\gamma _p}{t^p}+\frac {\mu }{t^p\log ^2 t}\right ]\Phi (x) =0\,, \qquad \gamma _p:=\left (\frac {p-1}{p}\right)^p,$$ considered in the limiting case $\mu -\lambda \gamma _p=\frac {1}{2}\bigl (\frac {p-1}{p}\bigr)^{p-1}$, is nonoscillatory.
Reviewer: Simona Fišnarová (Brno)

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