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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.

MSC:
37C75 Stability theory for smooth dynamical systems
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
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