Electronic Journal of Differential Equations (Oct 2010)
Existence of non-oscillatory solutions for a higher-order nonlinear neutral difference equation
Abstract
This article concerns the solvability of the higher-order nonlinear neutral delay difference equation $$ DeltaBig(a_{kn}dotsDeltaig(a_{2n} Delta(a_{1n}Delta(x_n+b_nx_{n-d}))ig)Big) +sum_{j=1}^s p_{jn}f_j(x_{n-r_{jn}})=q_n, $$ where $ngeq n_0ge0$, $d,k,j,s$ are positive integers, $f_j:mathbb{R}o mathbb{R}$ and $xf_j(x)geq 0$ for $x e 0$. 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 ${b_n}$.
