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].