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Global asymptotic stability of a system of two nonlinear difference equations. (English) Zbl 1207.39024

The authors consider the system of rational difference equations

z n+1 =t n +z n-1 t n z n-1 +a,t n+1 =z n +t n-1 z n t n-1 +a,n=0,1,2,,

where the parameter a(0,) and the initial values are positive, i.e., z k ,t k (0,) for k=-1,0. The change of variables (z n ,t n )=(ax n ,ay n ) reduces this system to

x n+1 =y n +x n-1 y n x n-1 +1,y n+1 =x n +y n-1 x n y n-1 +1,n=0,1,2,,

with initial values x k ,y k (0,), k=-1,0. As their main result the authors show that the positive equilibrium point (x ¯,y ¯)=(1,1) of the reduced system is globally asymptotically stable.

MSC:
39A30Stability theory (difference equations)
39A20Generalized difference equations
39A22Growth, boundedness, comparison of solutions (difference equations)