## 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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### References:

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