# zbMATH — the first resource for mathematics

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
##### Keywords:
forced differential equation; oscillatory
Full Text:
##### References:
 [1] Dahiya, R.S.; Akinyele, O., Oscillation theorems of nth order functional differential equations with forcing terms, J. math. anal. appl., 109, 325-332, (1985) · Zbl 0587.34029 [2] Foster, K.E.; Grimmer, R.C., Nonoscillatory solutions of higher order delay equations, J. math. anal. appl., 77, 150-164, (1980) · Zbl 0455.34053 [3] Grace, S.R.; Lalli, B.S., Oscillation theorems for nth order delay differential equations, J. math. anal. appl., 91, 352-366, (1983) · Zbl 0546.34055 [4] Grace, S.R.; Lalli, B.S., On oscillation of solutions of higher order forced functional differential inequalities, J. math. anal. appl., 91, 525-535, (1983) · Zbl 0543.34055 [5] {\scS. R. Grace and B. S. Lalli}, A comparison theorem for general nonlinear ordinary differential equations, J. Math. Anal. Appl., in press. · Zbl 0628.34037 [6] Kartsatos, A.G., Recent results on oscillation of solutions of forced and perturbed nonlinear differential equations of even order, (), 17-72 [7] Kartsatos, A.G., The oscillation of a forced equation implies the oscillation of the unforced equation—small forcing, J. math. anal. appl., 76, 98-106, (1980) · Zbl 0443.34032 [8] Kartsatos, A.G., On the maintenance of oscillations under the effect of a small forcing term, J. differential equations, 10, 355-363, (1970) · Zbl 0211.11902 [9] Mahfoud, W.E., Oscillation and asymptotic behavior of solutions of nth order nonlinear delay differential equations, J. differential equations, 24, 75-98, (1977) · Zbl 0341.34065 [10] Z̆ilina, R.O., Oscillation of linear retarded differential equation, Czech. math. J., 34, 371-377, (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.