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On the oscillatory and asymptotic behavior of damped functional differential equations. (English) Zbl 0732.34056
The author analyzes the equations $$[a(t)x'(t)]'\pm p(t)x'(\sigma (t))+q(t)f(x(g(t)))=0,$$ where $$a>0$$, p,q$$\geq 0$$; g, $$\sigma$$ are real- valued satisfying $$\lim_{t\to \infty}g(t)=\lim_{t\to \infty}\sigma (t)=\infty$$ and $$xf(x)>0$$, $$f'(x)\geq 0$$, $$x\neq 0$$. Conditions on a,p,$$\sigma$$,q,g are given under which any solution x(t) is either oscillatory or satisfies x(t)$$\to 0$$ monotonically as $$t\to \infty$$. Certain sufficient hypotheses for the equations to be oscillatory are stated. Examples are included.

##### MSC:
 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 34K25 Asymptotic theory of functional-differential equations