## Asymptotic behavior of solutions to third-order delay differential equations.(English)Zbl 0916.34059

The authors study asymptotic and oscillatory properties of solutions to third-order differential equations $\left (\frac {1}{p(t)}\left (\frac {1}{r(t)} x'(t)\right)'\right)' +q(t)f(x(h(t)))=0 \tag{$$N,h$$}$ and $\left (\frac {1}{r(t)}\left (\frac {1}{p(t)} z'(t)\right)'\right)' -q(t)f(z(h(t)))=0,\tag{$$N^A,h$$}$ with $$r,p,q,h\in C^{0}([a,\infty),\mathbb{R}),\;r(t)>0,\;p(t)>0,\^^Mq(t)> 0$$, $$f\in C^0(\mathbb{R}, \mathbb{R}),$$ $$f(u)u> 0$$ for $$u\neq 0$$, \hbox {$$h(t)\leq t$$}, $$\lim _{t\to \infty }h(t)=\infty$$ and $\int _a^\infty r(t) dt=\int _a^\infty p(t) dt=\infty .$ The equation $$(N,h)$$ is said to have property A if any proper solution $$x$$ to $$(N,h)$$ is either oscillatory or satisfies $| x^{[i]}(t)| \downarrow 0\;\;\text{as }t\to \infty ,\;i=0,1,2,$ and $$(N^{A},h)$$ is said to have property B if any proper solution $$z$$ to $$(N^{A},h)$$ is either oscillatory or satisfies $| z^{[i]}(t)| \uparrow \infty \quad\text{as }t\to \infty ,\;i=0,1,2.$ Here, $$x^{}=x$$, $$x^{}={x'\over r}$$, $$x^{}={1\over p}\left ( {x'\over r}\right)$$, the quasiderivatives $$z^{[i]}$$ are defined in a similar way.
In the first part of the paper, two comparison theorems on property A and property B in case $$f(x)\equiv x$$ (i.e., for linear equations) are given, and in the second part, the authors study relations (concerning properties A, B) between equations ($$N,h$$), ($$N^A,h$$) and corresponding linear equations.
Reviewer: O.Došlý (Brno)

### MSC:

 34K11 Oscillation theory of functional-differential equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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