一类差分方程非振动解的性态

考虑下面的差分方程:A_(n+1)-A_n+pA_(n-k)-qA_(n-1)=0 (E)及其特征方程:f(λ)=λ-1+pλ~(-1)qλ~(-1)=0(＊)其中,p,q>0,K.∈Z={1,2,3,…}我们分别给出了在P≠q时差分方程(E)所有非振动解均为有界和所有非振动解均为无界的充要条件,并得到了P=q时差分方程(E)所有无界解振动和所有趋于0解振动的充要条件的代数判据。

On the Nature of Nonoscillatory Solutions of a Class of Difference Equations

Consider the difference equation A_(n+1)-A_n+pA_(n-k)-qA_(n-1)=0 (E) Where p and q are positive constants, k and l belong to Z. For p≠q, we show necessary and sufficient conditions for every nonoscillatory solution {An}of (E) to be bounded and for every nonoscillatory solution {An}of (E) to be unbounded. Moreover, for the case where p=q. We obtain nccessary and sufficient conditions for every un- bounded solution {An}of (E) to be oscillated and every solution {An}decreasing to zero of (E) to be oscillated. And these conditions will be given explicithy in terms of p. q. k and l.