二阶不稳定中立型非线性差分方程有界解的振动性

Oscillation for the Bounded Solution of Unstable Type Neural Second-order Nonlinear Difference Equations

研究了二阶不稳定中立型非线性差分方程Δ2 ( x( n) -p( n) x( n-τ) ) =f ( n,x( g( n) ) ) , n≥ n0有界解的振动性 .其中 Δ为前差分算子 ,即 Δx( n) =x( n+ 1 ) -x( n) ;p( n)为实数序列 ;τ为一非负整数 ;g( n)为非减整数序列 ,满足 limn→∞ g( n) =∞ ,且当 n>N0 时 ,g( n)≤n成立 .f:S+×R→ R,并对任意 u≠ 0 ,有 f( n,u) /u≥q( n)≥ 0 ,且 q( n) 0成立 .给出了该差分方程有界解振动的一些充分条件 ,并给出了示例 .

The problem of oscillation for unstable type second-order neutral difference equation is considered: Δ 2(x(n)-p(n)x(n-τ))=f(n,x(g(n))), n≥n 0. Where Δ is the forward difference operator,defined by Δx(n)=x(n+1)-x(n),p(n) is a sequence of real,τ is nonnegative integer and g(n) is a sequence of nondecreasing integers with lim n→∞g(n)=∞,and g(n) ≤n for n>N 0.f:S +×R→R with f(n,u)/u≥q(n)≥0 for any u≠0,where q(n)0. Some sufficient conditions for the bounded solutions of the equation to be oscillatory are obtained,at the same time,some examples for these theorems are given.