Czechoslovak Mathematical Journal, first online, pp. 1-14

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
DOI:  10.21136/CMJ.2020.0572-18

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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:,

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