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Zbl 0958.34029
Elbert, Á.; Schneider, A.
Perturbations of the half-linear Euler differential equation.
(English)
[J] Result. Math. 37, No.1-2, 56-83 (2000). ISSN 1422-6383; ISSN 0378-6218/e

The authors investigate oscillation/nonoscillation properties of the perturbed half-linear Euler differential equation $$(x'{}^{n*})'+\frac{\gamma_0}{t^{n+1}}[n+2(n+1)\delta(t)]x^{n*}=0, \tag{*}$$ where the function $\delta(t)$ is piecewise continuous on $(t_0,\infty)$, $t_0\geq 0$, $n>0$ is a fixed real number and $u^{n*}=|u|^n \text{sgn} u$. The number $\gamma_0=\frac{n^n}{(n+1)^{n+1}}$ is a critical constant, which means that in case $\delta(t)\equiv 0$ the half-linear Euler differential equation is for $\gamma\leq \gamma_0$ 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: \par Suppose that there exists the finite limit $$\lim_{T\to\infty} \int_{t_0}^T \delta(t)\frac{dt}{t}$$ such that $\int_{t}^\infty \delta(s)\frac{ds}{s}\geq 0$ for $t>t_0$.\par (a) If $n>1$ and the linear equation $$z''+\delta(e^s)z=0 \tag{**}$$ is nonoscillatory then equation (*) is also nonoscillatory.\par (b) If $0<n<1$ and equation (*) is nonoscillatory then the linear equation (**) is also nonoscillatory.\par In addition, the authors establish an asymptotic form of the solution to (*) provided that the solutions to (**) satisfy two integral inequalities.
[Zuzana Došlá (Brno)]
MSC 2000:
*34C10 Qualitative theory of oscillations of ODE: Zeros, etc.

Keywords: half-linear differential equations; principal solutions; Riccati equations

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