## Forced oscillation of $$n$$th-order functional differential equations.(English)Zbl 0997.34059

Here, the authors consider the oscillation of forced functional-differential equations $x^{(n)}(t)+a(t)f(x(q(t)))=e(t),\tag{1}$ when the forcing term is not required to be the $$n$$th derivative of an oscillatory function. Several new oscillation criteria and explicit oscillation results are given. For example, if $$a(t)\geq 0$$, $$\beta>0$$ and $\lim_{t\rightarrow\infty}\inf \frac{1}{(t-t_0)^{\beta}} \int_{t_0}^t (t-s)^{\beta}e(s) ds =-\infty,\;\lim_{t\rightarrow\infty}\sup \frac{1}{(t-t_0)^{\beta}} \int_{t_0}^t (t-s)^{\beta}e(s) ds =+\infty,$ then equation (1) is oscillatory. Forced neutral differential equations are also considered.

### MSC:

 34K11 Oscillation theory of functional-differential equations 34K40 Neutral functional-differential equations

### Keywords:

oscillation; forced term; delay; neutral equation
Full Text:

### References:

 [1] Agarwal, R. P.; Grace, S. R., Forced oscillation of $$n$$ th order nonlinear differential equations, Appl. Math. Lett., 13, 53-57 (2000) · Zbl 0958.34050 [2] Erbe, L. H.; Kong, Qingkai; Zhang, B. G., Oscillation Theory for Functional Differential Equations. Oscillation Theory for Functional Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics (1995), Dekker: Dekker New York, p. 190 [3] Grace, S. R.; Lalli, B. S., Asymptotic and oscillatory behavior of $$n$$ th order forced functional differential equations, J. Math. Anal. Appl., 140, 10-25 (1989) · Zbl 0719.34119 [4] Hamedani, G. G., Oscillatory behavior of $$n$$ th order forced functional differential equations, J. Math. Anal. Appl., 195, 123-134 (1995) · Zbl 0844.34068 [5] Kartsatos, A. G., On the maintenance of oscillation under the effect of a small forcing term, J. Differential Equations, 10, 355-363 (1971) · Zbl 0211.11902 [6] Kartsatos, A. G., Recent results on oscillation of solutions of forced and perturbed nonlinear differential equations of even order, Stability of Dynamic Systems: Theory and Application. Stability of Dynamic Systems: Theory and Application, Lecture Notes in Pure and Applied Mathematics, 28 (1977), Dekker: Dekker New York, p. 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] Wong, J. S.W., Second nonlinear forced oscillations, SIAM J. Math. Anal., 19, 667-675 (1988) · Zbl 0655.34023 [9] Wong, J. S.W., Oscillation criteria for a forced second-order linear differential equation, J. Math. Anal. Appl., 231, 235-240 (1999) · Zbl 0922.34029 [10] Wong, J. S.W., Oscillation criteria for second order nonlinear differential equations involving general means, J. Math. Anal. Appl., 247, 489-505 (2000) · Zbl 0964.34028
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.