Electron. J. Diff. Eqns., Vol. 2007(2007), No. 09, pp. 1-7.

Existence of non-oscillatory solutions to higher-order mixed difference equations

Qiaoluan Li, Haiyan Liang, Wenlei Dong, Zhenguo Zhang

Abstract:
In this paper, we consider the higher order neutral nonlinear difference equation
$$\displaylines{ \Delta^{m}(x(n)+p(n)x(\tau(n)))+f_1(n,x(\sigma_{1}(n))) -f_2(n,x(\sigma_{2}(n)))=0, \cr \Delta^{m}(x(n)+p(n)x(\tau(n)))+f_1(n,x(\sigma_{1}(n))) -f_2(n,x(\sigma_{2}(n)))=g(n), \cr \Delta^{m}(x(n)+p(n)x(\tau(n)))+\sum_{i=1}^{l}b_i(n)x(\sigma_i(n))=0. }$$
We obtain sufficient conditions for the existence of non-oscillatory solutions.

Submitted April 30, 2006. Published January 2, 2007.
Math Subject Classifications: 39A05, 39A10.
Key Words: Nonoscillatory; existence; neutral equation.

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Qiaoluan Li
College of Mathematics and Information Science
Hebei Normal University
Shijiazhuang, 050016, China
mail: qll71125@163.com
Haiyan Liang
College of Mathematics and Information Science
Hebei Normal University
Shijiazhuang, 050016, China
  Wenlei Dong
College of Mathematics and Information Science
Hebei Normal University
Shijiazhuang, 050016, China
Zhenguo Zhang
College of Mathematics and Information Science
Hebei Normal University, Shijiazhuang, 050016, China
Information College, Zhejiang Ocean University
Zhoushan, Zhejiang, 316000, China
email: Zhangzhg@mail.hebtu.edu.cn

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