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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

Citations:

Zbl 1042.39506
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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
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