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Asymptotic and oscillatory behavior of \(n\)th order forced functional differential equations. (English) Zbl 0719.34119
The authors study the forced differential equation \(x^{(n)}(t)+q(t)| x(g(t))^{\alpha}| sgn x(g(t))=\eta^{(n)}(t)\), where \(\alpha\) is a quotient of positive odd integers, g(t)\(\leq t\) and g(t)\(\to \infty\) as \(t\to \infty\). They prove that if \(\eta (t)t^{1-n}\to 0\) as \(t\to \infty\) and \(\limsup_{t\to \infty}\int^{t}_{g(t)}[g(s)]^{\alpha (n-1)}q(s)ds>M>0\), then either every solution is oscillatory or \([x^{(n-1)}(t)-\eta^{(n-1)}(t)]\to 0\) monotonically as \(t\to \infty\). The result improves those of A. G. Kartsatos [J. Math. Anal. Appl. 76, 98-106 (1980; Zbl 0443.34032)] in the sense that the forcing term need not be small.

MSC:
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K25 Asymptotic theory of functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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