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.


37C75 Stability theory for smooth dynamical systems
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
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[1] Amer. Math. Monthly, 734-736 (1965)
[3] Bauer, H.; Bear, H. S., The part metric in convex sets, Pacific J. Math., 30, 15-33 (1969) · Zbl 0176.42801
[4] Elaydi, S., An Introduction to Difference Equations (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0855.39003
[5] Krause, U.; Nussbaum, R. D., A limit set trichotomy for self-mappings of normal cones in Banach spaces, Nonlinear Anal., 20, 855-870 (1993) · Zbl 0833.47047
[6] Ladas, G., Open problems and conjectures, J. Differ. Equations Appl., 4, 497-499 (1998)
[7] Thompson, A. C., On certain contraction mappings in a partially ordered vector space, Proc. Amer. Math. Soc., 14, 438-443 (1963) · Zbl 0147.34903
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