This paper aims to investigate the global stability of negative solutions of the difference equation , , where the initial conditions is a positive integer, and the parameters . By utilizing the invariant interval and periodic character of solutions, it is found that the unique negative equilibrium is globally asymptotically stable under some parameter conditions. Additionally, two examples are given to illustrate the main results in the end. 1. Introduction The study of nonlinear difference equations has always attracted a considerable attention (see, e.g., [1–30] and the references cited therein). In particular, some references investigated the dynamical behavior of positive solutions of difference equations (see, e.g., [3–5, 11]), and some references examined the dynamical behavior of negative solutions of some difference equations (see, e.g., [6, 7]). Gibbons et al. [4] studied the behavior of nonnegative solutions to the recursive sequence with , and also presented an open problem, which had been solved by in [8]. and Ladas, in addition, considered (1) in their book [9]. [10] considered the behavior of nonnegative solutions of the following second-order difference equation where is a nonnegative increasing mapping. Douraki et al. [11] studied the qualitative behavior of positive solutions of the difference equation where the initial values , is a positive integer, and . Moreover, (3) is a special case of the following open problem (see also in [11]), which was proposed by and Ladas in [9]. Open Problem (equation in [9]). Assume that . Investigate the global behavior of positive solutions of (3). [12] considered the boundedness, oscillatory behavior, and global stability of nonnegative solutions of the difference equation where , and is a continuous function nondecreasing in each variable such that . It is worthwhile to note that the above mentioned references ([4, 10–12]), especially [4, 11], only discussed the dynamical behavior of positive solutions of difference equation. Furthermore, inspired by the above work and [6, 7], the main goal of this paper is to study the global behavior of negative solutions of the difference equation where is a positive integer, , and the initial conditions . In fact, it is easy to see that (5) is an extension of an open problem introduced by and Ladas in [9] and also is a special case of (4) by a simple change. However, here we establish some results regarding the global stability, invariant interval, and periodic character of negative solutions of (5). 2. Linearized Stability and Period 2 Solutions The aim of this
References
[1]
M. T. Aboutaleb, M. A. El-Sayed, and A. E. Hamza, “Stability of the recursive sequence ,” Journal of Mathematical Analysis and Applications, vol. 261, no. 1, pp. 126–133, 2001.
[2]
X. Yang, Y. Yang, and J. Luo, “On the difference equation ,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 918–926, 2007.
[3]
R. DeVault, W. Kosmala, G. Ladas, and S. W. Schultz, “Global behavior of ,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 7, pp. 4743–4751, 2001.
[4]
C. H. Gibbons, M. R. S. Kulenovi？, and G. Ladas, “On the recursive sequence ,” Mathematical Sciences Research Hot-Line, vol. 4, no. 2, pp. 1–11, 2000.
[5]
F. Sun, X. Yang, and C. Zhang, “On the recursive sequence ,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 608976, 8 pages, 2009.
[6]
X.-M. Jia and L.-X. Hu, “Global attractivity of a higher-order nonlinear difference equation,” International Journal of Difference Equations, vol. 5, no. 1, pp. 95–101, 2010.
[7]
M. Saleh and M. Aloqeili, “On the difference equation with ,” Applied Mathematics and Computation, vol. 176, no. 1, pp. 359–363, 2006.
[8]
S. Stevi？, “On the recursive sequence ,” Taiwanese Journal of Mathematics, vol. 6, no. 3, pp. 405–414, 2002.
[9]
M. R. S. Kulenovi？ and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjecture, Chapman and Hall/CRC, Boca Raton, Fla, USA, 1st edition, 2001.
[10]
S. Stevi？, “On the recursive sequence ,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 12, pp. 1767–1774, 2002.
[11]
M. J. Douraki, M. Dehghan, and M. Razzaghi, “The qualitative behavior of solutions of a nonlinear difference equation,” Applied Mathematics and Computation, vol. 170, no. 1, pp. 485–502, 2005.
[12]
S. Stevi？, “On the recursive sequence ,” Taiwanese Journal of Mathematics, vol. 9, no. 4, pp. 583–593, 2005.
[13]
K. C. Cunningham, M. R. S. Kulenovi？, G. Ladas, and S. V. Valicenti, “On the recursive sequence ,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 7, pp. 4603–4614, 2001.
[14]
L.-X. Hu, W.-T. Li, and S. Stevi？, “Global asymptotic stability of a second order rational difference equation,” Journal of Difference Equations and Applications, vol. 14, no. 8, pp. 779–797, 2008.
[15]
V. L. Koci？ and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, vol. 256 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.
[16]
M. R. S. Kulenovi？ and O. Merino, “Global attractivity of the equilibrium of for ,” Journal of Difference Equations and Applications, vol. 12, no. 1, pp. 101–108, 2006.
[17]
W.-T. Li and H.-R. Sun, “Global attractivity in a rational recursive sequence,” Dynamic Systems and Applications, vol. 11, no. 3, pp. 339–345, 2002.
[18]
W. Li, Y. Zhang, and Y. Su, “Global attractivity in a class of higher-order nonlinear difference equation,” Acta Mathematica Scientia: Series B, vol. 25, no. 1, pp. 59–66, 2005.
[19]
W. Liu and X. Yang, “Quantitative bounds for positive solutions of a Stevi？ difference equation,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 235808, 14 pages, 2010.
[20]
W. Liu, X. Yang, and J. Cao, “On global attractivity of a class of nonautonomous difference equations,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 364083, 13 pages, 2010.
[21]
S. Stevi？, “On the difference equation ,” Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1159–1171, 2008.
[22]
S. Stevi？, “Global stability of a max-type difference equation,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 354–356, 2010.
[23]
S. Stevi？, “On a generalized max-type difference equation from automatic control theory,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1841–1849, 2010.
[24]
Y.-H. Su, W.-T. Li, and S. Stevi？, “Dynamics of a higher order nonlinear rational difference equation,” Journal of Difference Equations and Applications, vol. 11, no. 2, pp. 133–150, 2005.
[25]
Y.-H. Su and W.-T. Li, “Global attractivity of a higher order nonlinear difference equation,” Journal of Difference Equations and Applications, vol. 11, no. 10, pp. 947–958, 2005.
[26]
X.-X. Yan, W.-T. Li, and H.-R. Sun, “Global attractivity in a higher order nonlinear difference equation,” Applied Mathematics E-Notes, vol. 2, pp. 51–58, 2002.
[27]
X.-X. Yan and W.-T. Li, “Global attractivity in the recursive sequence ,” Applied Mathematics and Computation, vol. 138, no. 2-3, pp. 415–423, 2003.
[28]
X. Yang, W. Liu, and J. Liu, “Global attractivity of a family of max-type difference equations,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 506373, 12 pages, 2011.
[29]
X. Yang, Y. Y. Tang, and J. Cao, “Global asymptotic stability of a family of difference equations,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2643–2649, 2008.
[30]
Y. Yang and X. Yang, “On the difference equation ,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 903–907, 2008.
