An extension of the characterization of the domain of attraction of an asymptotically stable fixed point in the case of a nonlinear discrete dynamical system

The fixed point x of (1) is ”stable” provided that given any ball B(x, ε) = {x ∈ Ω/‖x− x‖ < ε}, there is a ball B(x, δ) = {x ∈ Ω/‖x− x‖ < δ} such that if x ∈ B(x, δ) then g(x) ∈ B(x, ε), for k = 0, 1, 2, ... [4]. If in addition there is a ball B(x, r) such that g(x) → x as k → ∞ for all x ∈ B(x, r) then the fixed point x is ”asymptotically stable”.[4]. The domain of attraction DA(x) of the asymptotically stable fixed point x is the set of initial states x ∈ Ω from which the system converges to the fixed point itself i.e. DA(x) = {x ∈ Ω|g(x) k→∞ −→ x} (3) It is known that x is a fixed point for system (1) if and only if 0 ∈ R is a fixed point for the system