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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
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