Czechoslovak Mathematical Journal, Vol. 70, No. 3, pp. 867-880, 2020

Solutions to conjectures on a nonlinear recursive equation

Özkan Öcalan, Oktay Duman

Received December 25, 2018.   Published online February 24, 2020.

Abstract:  We obtain solutions to some conjectures about the nonlinear difference equation $x_{n+1}=\alpha+\beta x_{n-1} {\rm e}^{-x_n}$, $n=0,1,\cdots$, $\alpha,\beta > 0$. 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.
Keywords:  recursive equation; nonlinear difference equation; equilibrium point; stability
Classification MSC:  39A10, 39A21, 11B39

[1] H. El-Metwally, E. A. Grove, G. Ladas, R. Levins, M. Radin: On the difference equation $x_{n+1}=\alpha +\beta x_{n-1} e^{-x_n}$. Nonlinear Anal., Theory Methods Appl. 47 (2001), 4623-4634. DOI 10.1016/S0362-546X(01)00575-2 | MR 1975856 | Zbl 1042.39506
[2] N. Fotiades, G. Papaschinopoulos: Existence, uniqueness and attractivity of prime period two solution for a difference equation of exponential form. Appl. Math. Comput. 218 (2012), 11648-11653. DOI 10.1016/j.amc.2012.05.047 | MR 2944008 | Zbl 1280.39011

Affiliations:  Özkan Öcalan, Akdeniz University, Faculty of Science, Department of Mathematics, Dumlupinar Boulevard, 07058 Campus, Antalya, Turkey, e-mail:, Oktay Duman (corresponding author), TOBB University of Economics and Technology, Faculty of Arts and Sciences, Department of Mathematics, Sögütözü Avenue 43, Sögütözü, Ankara, 06560, Turkey, e-mail:,

PDF available at: