一类线性脉冲时滞差分方程的振动性和渐近稳定性(英文)

The Oscillation and Asymptotic Stability of a Linear Impulsive Delay Difference Equation

研究了如下形式的作为一阶线性脉冲时滞微分方程的离散情形的脉冲时滞差分方程 : xn + 1-xn+pnxn -k=0 (n≥ 0 ,n≠nt) ,xnt+ 1-xnt=btxnt (t=1,2 ,… ) .其中 pn,k ,nt,bt 分别满足下列条件(H1) { pn}是一非负实数列 ,k为一正整数 ;(H2 )nt 为脉冲点 ,且有①nt∈ { 1,2 ,… } ,② 0 <n1<n2 <… <nj<nj+ 1<…(H3 )bt∈ (-∞ ,- 1)U(- 1,+∞ ) ,t=1,2 ,…通过方程 (1)的振动性与下列方程 (2 )的解的振动性、稳定性在一定条件下的等价性 ,我们获得了 (1)的解振动和渐近稳定的 3个充分条件 . yn + 1- yn+^pn· n -k≤nt<n (1+bt) - 1yn -k=0 . (2 )其中 ^ p =pn (n >0 ,n≠nt) ,(n =nt,t=1,2 ,… ) ,k ,pn,bt,nt 满足 (H1)～ (H3 )

The following linear impulsive delay difference equation is considered. {Xn+1 - Xn + PnXn-k = 0 (n≥0, n≠nt), Xnt+1 - Xnt = btXnt (t = 1,2, &middot&middot&middot), where Pn, k, nt bt satisfy the following conditions respectively. (H1) {Pn} is a sequence of nongative real numbers and k is a positive integer; (H2) n2 is called impulsive point and satisfies, 1 nt ∈ {1,2,&middot &middot&middot} ; 2 equation(2), yn+ 1 -t - yn + pˆ nn-k&lent ∏ (1 + bt) -1 yn-k = O, where pˆn = {pn (n ≥ 0, n ≠ nt), O (n = nt; t = 1,2,&middot &middot&middot), and k, pn, bt, nt satisfy (H1) ∼ (H3) respectively. Then three sufficient conditions for oscillation and asymptotic stability are obtained.