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nonlinear difference equtions; oscillation; eventually positive solutions; characteristic equation
In this paper we consider the nonlinear difference equation with several delays \[ (ax_{n+1}+bx_{n})^k-(cx_{n})^k+\sum \limits _{i=1}^{m} p_{i}(n)x^k_{n-\sigma _{i}}=0 \] where $a,b,c\in (0,\infty )$, $k=q/r, q, r$ are positive odd integers, $m$, $\sigma _{i}$ are positive integers, $\lbrace p_{i}(n)\rbrace $, $i=1,2,\dots ,m, $ is a real sequence with $p_{i}(n)\ge 0$ for all large $n$, and $\liminf _{n\rightarrow \infty }p_{i}(n)=p_{i}<\infty $, $i=1,2,\dots ,m$. Some sufficient conditions for the oscillation of all solutions of the above equation are obtained.
[1] R. P. Agarwal, P. J. Y. Wong: Advanced Topics in Difference Equations. Kluwer Academic Publishers, 1997. MR 1447437
[2] I. Györi, G. Ladas: Oscillation Theory of Delay Differential Equations with Applications. Oxford Science Publications, 1991. MR 1168471
[3] G. H. Hardy, J. E. Littlewood, G. Polya: Inequalities. Second edition, Cambridge University Press, 1952. MR 0046395
[4] B. Li: Discrete oscillations. J. Differ. Equations Appl. 2 (1996), 389–399. DOI 10.1080/10236199608808073 | MR 1444268 | Zbl 0881.39007
[5] S. T. Liu, S. S. Cheng: Oscillation criteria for a nonlinear difference equation. Far East J. Math. Sci. 5 (1997), 387–392. MR 1620642
[6] J. Yang, X. P. Guan et al.: Nonexistence of positive solution of certain nonlinear difference equations. Acta Math. Appl. Sinica 23 (2000), 562–567. (Chinese) MR 1814400
[7] J. S. Yu, B. G. Zhang, X. Z. Qian: Oscillations of delay differential equations with oscillatory coefficients. J. Math. Anal. Appl. 177 (1993), 432–444. DOI 10.1006/jmaa.1993.1267 | MR 1231491
[8] Y. Zhou, B. G. Zhang: Oscillations of delay differential equations in a critical state. Comput. Math. Appl. 39 (2000), 71–89. DOI 10.1016/S0898-1221(00)00066-3 | MR 1746156
[9] B. G. Zhang, Yong Zhou: Oscillation of delay differential equations with several delays. (to appear).
[10] B. G. Zhang, Y. J. Sun: Existence of nonoscillatory solutions of a class of nonlinear difference equations with a forced term. Math. Bohem. 126 (2001), 639–647. MR 1970266
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