Stability problem of some nonlinear difference equations

Abstract

In this paper, we investigate the asymptotic stability of the recursive sequence xn+1=α+βxn21+γxn−1, n=0,1,… and the existence of certain monotonic solutions of the equation xn+1=xnpf(xn,xn−1,…,xn−k), n=0,1,… which includes as a special case the rational recursive sequence xn+1=βxnp1+∑i=1kγixn−1p−r, where α≥0,β>0,γ>0,γi≥0, i=1,2,…,k,∑i=1kγi>0, p∈{2,3,…} and r∈{1,2,…,p−1}. The case when r=0 has been investigated by Camouzis et. al. [1], and for r=0 and p=2 by Camouzis et. al. [2].

Keywords

• Difference Equations
• Monotonic solutions
• stability.