Electron. J. Diff. Equ., Vol. 2010(2010), No. 146, pp. 1-7.

Existence of non-oscillatory solutions for a higher-order nonlinear neutral difference equation

Zhenyu Guo, Min Liu

Abstract:
This article concerns the solvability of the higher-order nonlinear neutral delay difference equation
$$ \Delta\Big(a_{kn}\dots\Delta\big(a_{2n} \Delta(a_{1n}\Delta(x_n+b_nx_{n-d}))\big)\Big) +\sum_{j=1}^s p_{jn}f_j(x_{n-r_{jn}})=q_n, $$
where $n\geq n_0\ge0$, $d,k,j,s$ are positive integers, $f_j:\mathbb{R}\to \mathbb{R}$ and $xf_j(x)\geq 0$ for $x\ne 0$. Sufficient conditions for the existence of non-oscillatory solutions are established by using Krasnoselskii fixed point theorem. Five theorems are stated according to the range of the sequence $\{b_n\}$.

Submitted July 30, 2010. Published October 14, 2010.
Math Subject Classifications: 34K15, 34C10.
Key Words: Nonoscillatory solution; neutral difference equation; Krasnoselskii fixed point theorem.

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Zhenyu Guo
School of Sciences, Liaoning Shihua University
Fushun, Liaoning 113001, China
email: guozy@163.com
Min Liu
School of Sciences, Liaoning Shihua University
Fushun, Liaoning 113001, China
email: min_liu@yeah.net

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