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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 \[ (x'{}^{n*})'+\frac{\gamma_0}{t^{n+1}}[n+2(n+1)\delta(t)]x^{n*}=0, \tag{*} \] 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:
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 \(t>t_0\).
(a) If \(n>1\) and the linear equation \[ z''+\delta(e^s)z=0 \tag{**} \] is nonoscillatory then equation (*) is also nonoscillatory.
(b) If \(0<n<1\) 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 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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