Asymptotic and oscillatory behavior of \(n\)th order forced functional differential equations (Q2640036)

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 \textit{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.