# zbMATH — the first resource for mathematics

Global stability of a difference equation with maximum. (English) Zbl 1167.39007
The main results of the paper are the following two theorems.
Theorem 1. Every positive solution to the difference equation $x_n=\max \left \{\frac{1}{x_{n-1}^{\alpha_1}}, \frac{1}{x_{n-2}^{\alpha_2}}, \ldots, \frac{1}{x_{n-k}^{\alpha_k}}\right \}, \qquad n\in \mathbb{N}_0,$ where $$k\in \mathbb{N}$$, $$\alpha_i>0$$, $$i=1$$, $$\ldots$$, $$k$$ and $$\alpha_1\alpha_k\in (0,1)$$, converges to one.
Theorem 2. Assume that $$k\in \mathbb{N}$$, $$\alpha_i\in (0,1)$$, $$i=1$$, $$\ldots$$, $$k$$ and $$A_i>0$$, $$i=1$$, $$\ldots$$, $$k$$. Then every positive solution to $x_n=\max\left \{\frac{A_1}{x_{n-1}^{\alpha_1}},\frac{A_2}{x_{n-2}^{\alpha_2}}, \ldots, \frac{A_k}{x_{n-k}^{\alpha_k}}\right \}, \qquad n\in \mathbb{N}_0$ converges to $$\max\limits_{1\leq i\leq k}\left \{A_i^{\frac{1}{\alpha_i+1}}\right \}$$.
Moreover, Theorem 2 solves the conjecture proposed by the author of this paper [Appl. Math. Comput. 210, No. 2, 525–529 (2009; Zbl 1167.39007)] and by F. Sun [Discrete Dyn. Nat. Soc. 2008, Article ID 243291, 6 p. (2008; Zbl 1155.39008)].

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
Full Text:
##### References:
 [1] Berenhaut, K.; Foley, J.; Stević, S., Boundedness character of positive solutions of a MAX difference equation, J. differ. equations appl., 12, 12, 1193-1199, (2006) · Zbl 1116.39001 [2] Çinar, C.; Stević, S.; Yalçinkaya, I., On positive solutions of a reciprocal difference equation with minimum, J. appl. math. comput., 17, 1-2, 307-314, (2005) · Zbl 1074.39002 [3] El-Morshedy, H.A., New explicit global asymptotic stability criteria for higher order difference equations, J. math. anal. appl., 336, 1, 262-276, (2007) · Zbl 1186.39022 [4] E.M. Elsayed, S. Stević, On the max-type equation $$x_{n + 1} = \max \left\{\frac{A}{x_n}, x_{n - 2}\right\}$$, Nonlinear Anal. TMA, 2008, in press, doi:10.1016/j.na.2008.11.016. [5] Feuer, J., On the eventual periodicity of $$x_{n + 1} = \max \left\{\frac{1}{x_n}, \frac{A_n}{x_{n - 1}}\right\}$$ with a period-four parameter, J. difference equ. appl., 12, 5, 467-486, (2006) · Zbl 1095.39016 [6] Grove, E.A.; Ladas, G., Periodicities in nonlinear difference equations, (2005), Chapman & Hall, CRC Press · Zbl 1078.39009 [7] Kent, C.M.; Radin, M.A., On the boundedness nature of positive solutions of the difference equation $$x_{n + 1} = \max \left\{\frac{A_n}{x_n}, \frac{B_n}{x_{n - 1}}\right\}$$, with periodic parameters, Dyn. contin. discrete impuls. syst. ser. B appl. algorithms, suppl., 11-15, (2003) [8] Mishev, D.P.; Patula, W.T.; Voulov, H.D., A reciprocal difference equation with maximum, Comput. math. appl., 43, 8-9, 1021-1026, (2002) · Zbl 1050.39015 [9] Mishkis, A.D., On some problems of the theory of differential equations with deviating argument, Uspekhi mat. nauk, 32:2, 194, 173-202, (1977) [10] Patula, W.T.; Voulov, H.D., On a MAX type recurrence relation with periodic coefficients, J. difference equ. appl., 10, 3, 329-338, (2004) · Zbl 1050.39017 [11] Popov, E.P., Automatic regulation and control, (1966), Nauka Moscow, Russia, (in Russian) [12] Szalkai, I., On the periodicity of the sequence $$x_{n + 1} = \max \left\{\frac{A_0}{x_n}, \frac{A_1}{x_{n - 1}}, \ldots, \frac{A_k}{x_{n - k}}\right\}$$, J. differ. equations appl., 5, 25-29, (1999) · Zbl 0930.39011 [13] Stević, S., Behavior of the positive solutions of the generalized beddington – holt equation, Panamer. math. J., 10, 4, 77-85, (2000) · Zbl 1039.39005 [14] S. Stević, Some open problems and conjectures on difference equations, http://www.mi.sanu.ac.yu/colloquiums/mathcoll_programs/mathcoll.apr2004.htm. [15] S. Stević, Boundedness character of a max-type difference equation, in: Conference in Honour of Allan Peterson, Book of Abstracts, Novacella, Italy, July 26-August 02, 2007, p. 28. [16] S. Stević, On the recursive sequence $$x_{n + 1} = A + \frac{x_n^p}{x_{n - 1}^r}$$, Discrete Dyn. Nat. Soc., vol. 2007, Article ID 40963, 2007, 9 pages. [17] S. Stević, On behavior of a class of difference equations with maximum, Mathematical Models in Engineering, Biology and Medicine, in: Conference on Boundary Value Problems, Book of abstracts, Santiago de Compostela, Spain, September 16-19, 2008, p. 35. [18] Stević, S., On the recursive sequence $$x_{n + 1} = \max \left\{c, \frac{x_n^p}{x_{n - 1}^p}\right\}$$, Appl. math. lett., 21, 8, 791-796, (2008) · Zbl 1152.39012 [19] Stević, S., Boundedness character of a class of difference equations, Nonlinear anal. TMA, 70, 839-848, (2009) · Zbl 1162.39011 [20] F. Sun, On the asymptotic behavior of a difference equation with maximum, Discrete Dyn. Nat. Soc., vol. 2008, Article ID 243291, 2008, 6 pages. [21] Voulov, H.D., Periodic solutions to a difference equation with maximum, Proc. amer. math. soc., 131, 7, 2155-2160, (2003) · Zbl 1019.39005 [22] Voulov, H.D., On the periodic nature of the solutions of the reciprocal difference equation with maximum, J. math. anal. appl., 296, 1, 32-43, (2004) · Zbl 1053.39023 [23] Voulov, H.D., On a difference equation with periodic coefficients, J. difference equ. appl., 13, 5, 443-452, (2007) · Zbl 1121.39011 [24] I. Yalçinkaya, B.D. Iričanin, C. Çinar, On a max-type difference equation, Discrete Dyn. Nat. Soc., vol. 2007, Article ID 47264, 2007, 11 pages. [25] Yang, Y.; Yang, X., On the difference equation $$x_{n + 1} = \frac{\mathit{px}_{n - s} + x_{n - t}}{\mathit{qx}_{n - s} + x_{n - t}}$$, Appl. math. comput., 203, 2, 903-907, (2008) · Zbl 1162.39015 [26] Yang, X.; Liao, X., On a difference equation with maximum, Appl. math. comput., 181, 1-5, (2006) · Zbl 1148.39303
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.