%0 Journal Article
%T Global Asymptotic Stability for Linear Fractional Difference Equation
%A A. Brett
%A E. J. Janowski
%A M. R. S. Kulenovi£¿
%J Journal of Difference Equations
%D 2014
%R 10.1155/2014/275312
%X Consider the difference equation ,£¿£¿ , where all parameters ,£¿£¿ , and the initial conditions ,£¿£¿ are nonnegative real numbers. We investigate the asymptotic behavior of the solutions of the considered equation. We give easy-to-check conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation. 1. Introduction Consider the difference equation where , the parameters ,£¿£¿ , and the initial conditions , are nonnegative real numbers. The important special cases of (1) are the well-known Riccati equation the second order linear fractional difference equation and the third order linear fractional difference equation that we get from (1) for . The global behavior and the exact solutions of (2) even for real parameters have been found in [1]. The global behavior of solutions of (3), in many subcases when one or more parameters are zero, was established in [1]. There are still some conjectures left whose answers will complete the global picture of the asymptotic behavior for the solutions of (3). As far as the third order linear fractional difference equation is concerned, there are a large number of sporadic results that are systemized in a book [2]. The characterization of the global asymptotic behavior of the solutions of (1) for seems to be much harder than for the second order equation (3). Consequently an attempt at giving the characterization of the global asymptotic behavior for the solutions of (1) seems to be a formidable task at this time. However using some known global attractivity results we can describe the global asymptotic behavior for the solutions of (1) in some subspaces of the parametric space and the space of initial conditions. See [2¨C6] for a complete description of the behavior of some special cases of (1), in particular for the cases known as periodic trichotomies. See [7] where the difference in global behavior between the second and third order linear fractional difference equation is emphasized. The results on the global periodicity, that is, the results which describe all special cases of (1) where all solutions are periodic of the same period, were obtained in [8, 9]. Most results in [2¨C6, 10, 11] are based on known global attractivity or global asymptotic stability results obtained in [1, 2, 12¨C17]. This paper is an attempt at establishing some global stability results for the equilibrium solution(s) of (1). Our results give effective conditions for global asymptotic stability of the equilibrium solution(s) of (1) expressed in terms of the inequalities on the coefficients.
%U http://www.hindawi.com/journals/jde/2014/275312/