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Title: Solutions to conjectures on a nonlinear recursive equation (English)
Author: Öcalan, Özkan
Author: Duman, Oktay
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 70
Issue: 3
Year: 2020
Pages: 867-880
Summary lang: English
Category: math
Summary: We obtain solutions to some conjectures about the nonlinear difference equation $$ x_{n+1}=\alpha +\beta x_{n-1} {\rm e}^{-x_{n}}, \quad n=0,1,\cdots , \^^M\alpha ,\beta >0. $$ 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. (English)
Keyword: recursive equation
Keyword: nonlinear difference equation
Keyword: equilibrium point
Keyword: stability
MSC: 11B39
MSC: 39A10
MSC: 39A21
idZBL: 07250694
idMR: MR4151710
DOI: 10.21136/CMJ.2020.0572-18
Date available: 2020-09-07T09:41:22Z
Last updated: 2022-10-03
Stable URL:
Reference: [1] El-Metwally, H., Grove, E. A., Ladas, G., Levins, R., Radin, M.: On the difference equation $x_{n+1}=\alpha +\beta x_{n-1} e^{-x_{n}}$.Nonlinear Anal., Theory Methods Appl. 47 (2001), 4623-4634. Zbl 1042.39506, MR 1975856, 10.1016/S0362-546X(01)00575-2
Reference: [2] Fotiades, N., Papaschinopoulos, G.: Existence, uniqueness and attractivity of prime period two solution for a difference equation of exponential form.Appl. Math. Comput. 218 (2012), 11648-11653. Zbl 1280.39011, MR 2944008, 10.1016/j.amc.2012.05.047


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