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Global asymptotic stability for a higher order nonlinear rational difference equations. (English) Zbl 1110.39011

Consider the rational difference equation

x n+1 =ax n +bx n-k A+Bx n ,n=0,1,(*)

where a,b,A,B are positive real numbers, k1 is a positive integer, and the initial conditions x -k ,,x -1 ,x 0 are nonnegative real numbers. The authors solve an open problem posed by M. R. S. Kulenović and G. Ladas [Dynamics of second order rational difference equations, Chapman & Hall / CRC, Boca Raton, FL (2002; Zbl 0981.39011), p. 129]. They prove the following

Theorem: (a) If bA-a, then the zero equilibrium of Eq. (*) is globally asymptotically stable. (b) If A-a<b<A+a, then the positive equilibrium of Eq. (*) is globally asymptotically stable.

The boundedness, periodic character, invariant intervals of all nonnegative solutions of (*) are investigated.

MSC:
39A11Stability of difference equations (MSC2000)
39A20Generalized difference equations