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recursive equation; nonlinear difference equation; equilibrium point; stability
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.
[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. DOI 10.1016/S0362-546X(01)00575-2 | MR 1975856 | Zbl 1042.39506
[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. DOI 10.1016/j.amc.2012.05.047 | MR 2944008 | Zbl 1280.39011
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