Estimating the radius of an attraction ball

Given a nonlinear mapping G differentiable at a fixed point x ∗ , the Ostrowski theorem offers the sharp sufficient condition ρ ( G ′ ( x ∗ ) ) < 1 for x ∗ to be an attraction point, where ρ denotes the spectral radius. However, no estimate for the size of an attraction ball is known. w in this note that such an estimate may be readily obtained in terms of ‖ G ′ ( x ∗ ) ‖ < 1 (with ‖ ⋅ ‖ an arbitrary given norm) and of the Hölder (in particular Lipschitz) continuity constant of G ′ . An elementary example shows that this estimate may be sharp. sumptions do not necessarily require G to be of contractive type on the whole estimated ball.