关于差分方程x_n=(A+x_(n-1)~p)/(B+x_(n-k)~p)正解的有界性(英文)

On the Boundedness Nature of Positive Solutions of the Difference Equation x_n=(A+x_(n-1)~p)/(B+x_(n-k)~p)

证明差分方程x_n=(A+x_(n-1)~p)/(B+x_(n-k)~p),n=0,1,2,…,(其中k≥2,A,B,p∈(0,+∞))在p^(k-1)≥k^k/(k-1)^(k-1)时,有无界的解,并且当p^(k-1)<k^k/(k-1)^(k-1)时,每个正解都有界.

Abstract：The difference equation Xn=A＋xn-1^p/B＋xn-k^p, n = 0,1,2, …, are considered in this paper, where k≥2 and A,B,p∈（0,＋∞））. We show that if p^k-1≥k^k/（k-1）^k-1,then this equation has positive unbounded solutions,and if p^k-1≥k^k/（k-1）^k-1,then every positive solution of this equation is bounded