## Oscillation of higher-order neutral nonlinear difference equations.(English)Zbl 0946.39002

The paper deals with the higher-order neutral nonlinear difference equation $\Delta(r_n(\Delta^d (x_n - p_n x_{n-r}))^{\delta}) + f(n, x_{n-\sigma}) = 0, \tag{1}$ where $$n \in \{ n_0+1, n_0+2,...\}, n_0$$ and $$\sigma$$ are nonnegative integers, $$\tau$$ and $$d$$ are positive integers, $$\Delta x_n = x_{n+1} - x_n,$$ $$\delta$$ is a quotient of odd positive integers, $$0 \leq p_n < 1, r_n > 0,$$ and $$f$$ is continuous in $$x$$ for $$x \in \mathbb{R}.$$ A solution of (1) is called nonoscillatory if it is eventually positive or eventually negative. Some necessary and sufficient conditions for the existence of nonoscillatory solutions to (1) are obtained. The author gives necessary and sufficient conditions for all bounded solutions to (1) to be oscillatory or to tend to zero.

### MSC:

 39A11 Stability of difference equations (MSC2000)
Full Text:

### References:

  Zhou, X. L.; Yan, J. R., Oscillatory property of higher order nonlinear difference equations, Computers Math. Applic., 31, 12, 61-68 (1996) · Zbl 0855.39016  Zhou, X. L.; Yan, J. R., Oscillation of higher order difference equations, Chinese Ann. Math., 15A, 6, 692-700 (1994), (In Chinese) · Zbl 0815.39004  Li, W. T.; Cheng, S. S., Classifications and existence of positive solutions of second order neutral nonlinear difference equations, Funkcialaj Ekvacioj, 40, 371-393 (1997) · Zbl 0894.39002  Zafer, A.; Dahiya, R. S., Oscillation of a neutral difference equations, Appl. Math. Lett., 6, 2, 71-74 (1993) · Zbl 0772.39001  He, X. Z., Oscillatory and asymptotic behavior of second order nonlinear difference equations, J. Math. Anal. Appl., 175, 482-498 (1993) · Zbl 0780.39001  Hooker, J. W.; Patula, W. T., Riccati type transformations for second order linear difference equations, J. Math. Anal. Appl., 82, 451-462 (1981) · Zbl 0471.39007  Cheng, S. S.; Li, H. J.; Patula, W. T., Bounded and zero convergent solutions of second order difference equations, J. Math. Anal. Appl., 141, 463-483 (1989) · Zbl 0698.39002  Cheng, S. S.; Patula, W. T., An existence for a nonlinear difference equations, Nonlinear Analysis TMA, 20, 193-203 (1993) · Zbl 0774.39001  Hooker, J. W.; Patula, W. T., A second order nonlinear difference equation: Oscillation and asymptotic behavior, J. Math. Anal. Appl., 91, 9-29 (1983) · Zbl 0508.39005  Szmanda, B., Characterisations of oscillation of second order nonlinear difference equations, Bull. Polish. Acad. Sci. Math., 34, 3/4, 133-141 (1986) · Zbl 0598.39004  Yan, J. R.; Liu, B., Asymptotic behavior of a nonlinear delay difference equations, Appl. Math. Lett., 8, 6, 1-5 (1995) · Zbl 0840.39007  Li, W. T., Classifications and existence of nonoscillatory solutions of second order nonlinear neutral differential equations, Annales Polonic Mathematici, LXV.3, 283-302 (1997) · Zbl 0873.34065
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.