## 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:
$$\bullet$$
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 $\beta\leq\frac{-\alpha +\sqrt{\alpha^2+4\alpha}}{\alpha +\sqrt{\alpha^2+4\alpha}}e^{{(\alpha +\sqrt{\alpha^2+4\alpha}})/2}.$
$$\bullet$$
Every positive solution of the above difference except for the equilibrium solution converges to the unique period-2 cycle if $\frac{-\alpha +\sqrt{\alpha^2+4\alpha}}{\alpha +\sqrt{\alpha^2+4\alpha}}e^{{(\alpha +\sqrt{\alpha^2+4\alpha}})/2}<\beta <e^\alpha.$
$$\bullet$$
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.].

### MSC:

 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:

### References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.