Solutions to conjectures on a nonlinear recursive equation. (English) Zbl 1513.39044

The authors investigate the boundedness and long-term behavior of all solutions of the difference equation \[x_{n+1}=\alpha +\beta x_{n-1}e^{-x_n},\quad n=0,1,2,\dots,\ \alpha,\beta>0.\] The main results can be summarized as follows:
The equilibrium solution \(\bar{x}\) of the above equation is globally asymptotically stable if the following condition holds: \[\frac{-\alpha +\sqrt{\alpha^2+4}}{2}e^\alpha <\beta\leq\frac{-\alpha +\sqrt{\alpha^2+4\alpha}}{\alpha +\sqrt{\alpha^2+4\alpha}}e^{{(\alpha +\sqrt{\alpha^2 +4\alpha}})/2}.\] With this in mind, since the equilibrium solution \(\bar{x}\) is known to be globally asymptotically stable if \(\beta\leq\frac{-\alpha +\sqrt{\alpha^2+4}}{2}e^\alpha\) [H. El-Metwally et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 47, No. 7, 4623–4634 (2001; Zbl 1042.39506)], one can conclude that the equilibrium solution \(\bar{x}\) is globally asymptotically stable if \[\beta\leq\frac{-\alpha +\sqrt{\alpha^2+4\alpha}}{\alpha +\sqrt{\alpha^2+4\alpha}}e^{{(\alpha +\sqrt{\alpha^2+4\alpha}})/2}.\]
Every positive solution of the above difference except for the equilibrium solution converges to the unique period-2 cycle if \[\frac{-\alpha +\sqrt{\alpha^2+4\alpha}}{\alpha +\sqrt{\alpha^2+4\alpha}}e^{{(\alpha +\sqrt{\alpha^2+4\alpha}})/2}<\beta <e^\alpha.\]
If \(\beta\geq e^\alpha\), then the above difference equation has no (bounded) periodic solution.
The above results solve conjectures and open problems presented in [loc. cit.].


39A30 Stability theory for difference equations
39A23 Periodic solutions of difference equations
39A20 Multiplicative and other generalized difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations


Zbl 1042.39506
Full Text: DOI


[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 · doi:10.1016/S0362-546X(01)00575-2
[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 · doi:10.1016/j.amc.2012.05.047
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