一类非线性差分方程解的稳定性及振动性

Solution Stability and Oscillation for a Class of Nonlinear Difference Equations

研究非线性差分方程xn+1=(xn xn-1+a)/(xn+xn-1+b),(n≥0;a,b∈[0,∞);x0,x-1∈(0,∞))解的稳定性及振动性,得到该差分方程存在唯一非负平衡解x,且x为全局渐近稳定的,同时根据a和b是否为0,分别研究了解关于的振动性,得到该差分方程任意解,下述结论之一成立:(1)当n>0时,xn单调减收敛于;(2)当n>0时,xn≡;(3)解关于严格振动,可能除第1个半环外,每个负半环的长为2,且每个正半环的长为1。

In this paper, the stability and oscillation are studied for the given nonlinear difference equation xn＋1=（xnxn-1＋a）/（xn＋xn-1＋b）,（n≥0;a,b∈[0,∞）;x0,x-1∈（0,∞）） . The only non-negative equilibrium solution ~ is obtained for the difference equation, where x^- is globally asymptotically stable. For the cases where a and b are 0 and non-zero, the oscillation of any solutionx^- of the difference equation is explored. In conclusion, one of the following statements is held true. （1） If n 〉 0, then x monotonously decreases and converges to x^-. （2） If n 〉 0, then x ≡x^-. （3） The solution is oscillated by x^-. Except the first one, the length of the negative semicircle is 2, and the length of the positive semicircle is 1.