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On global stability in a nonlinear discrete model. (English) Zbl 0842.39005

For difference equations of the form \[ \mu \Delta x_n= -x_{n+ 1}+ f(x_{n- m}), \qquad n\in \mathbb{N}\cup \{0\} \] with continuous \(f: \mathbb{R}\to \mathbb{R}\), parameter \(\mu>0\) and delay \(m\in \mathbb{N}\), the global attractivity of a stationary solution is studied by means of the interval maps described by the functions \(f(x)\) and \(F(x):= {1\over {\mu+1}} f(x)\), respectively. Some sufficient and some necessary conditions are derived and applied to particular model equations related to population dynamics.

MSC:

39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations
92D25 Population dynamics (general)
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