Solutions to conjectures on a nonlinear recursive equation

We obtain solutions to some conjectures about the onlinear difference equation xn+1=α+βxn−1e−xn,n=0,1,…,α,β'0. More precisely, we get not only a condition under which the equilibrium point of the above equation is globally asymptotically stable but also a condition under which the above equation has a unique positive cycle of prime period two. We also prove some further results. © 2020, Mathematical Institute, Academy of Sciences of Cz