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Global stability of a rational difference equation
Authors:Lin-Xia Hu  Wan-Tong Li  
Institution:

aDepartment of Mathematics, Tianshui Normal University, Tianshui, Gansu 741001, People’s Republic of China

bSchool of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China

Abstract:In this paper, we study the global stability of the difference equation View the MathML source, where the parameters a,aiset membership, variant(0,) for i=0,…,k, x-k,…, x-1set membership, variant0,) and x0set membership, variant(0,). We prove that the unique positive equilibrium is globally asymptotically stable if and only if it is locally asymptotically. Also we provide sufficient condition for it to be globally asymptotically stable and our results solve the open problem proposed by Kulenović and Ladas (Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002).
Keywords:Difference equation  Global attractor  Globally asymptotically stable
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