On the rational recursive sequence yn=A+yn−1yn−m for small AA

This work studies the existence of positive prime periodic solutions of higher order for rational recursive equations of the form yn=A+yn−1yn−m,n=0,1,2,…, with y−m,y−m+1,…,y−1∈(0,∞)y−m,y−m+1,…,y−1∈(0,∞) and m∈{2,3,4,…}m∈{2,3,4,…}. In particular, we show that for sufficiently small A>0A>0, there exist periodic solutions with prime period 2m+Um+12m+Um+1, for almost all mm, where Um=max{i∈N:i(i+1)≤2(m−1)}Um=max{i∈N:i(i+1)≤2(m−1)}.