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Global asymptotic stability in some discrete dynamical systems. (English) Zbl 0933.37016
The paper deals with the stability of the equilibrium of a discrete dynamical system $x_{n+1}=Tx_n$ in a metric space $(M,d)$. Under some natural conditions the authors show that the unique equilibrium is globally asymptotically stable. Taking for $d$ the part-metric, the authors obtain the strong negative feedback property as a special case. Finally, they apply these results to the Putnam equation, showing that the equilibrium $x^*=1$ of $x_{n+1}= {x_n+x_{n-1}+ x_{n-2} \cdot c_{n-3} \over x_nx_{n-1}+ x_{n-2}+ x_{n-3}}$ with positive initial conditions $x_0, \dots, x_{-3}$ is globally asymptotically stable.

37C75Stability theory
37C50Approximate trajectories (pseudotrajectories, shadowing, etc.)
Full Text: DOI
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