Estimating the radius of an attraction ball (Q1021805)

Given a nonlinear mapping \(G\) differentiable at a fixed point \(x^*\), the Ostrowski theorem offers the sharp sufficient condition \(\rho(G'(x^*))<1\) for \(x^*\) to be an attraction point, where \(\rho\) denotes the spectral radius. However, no estimate for the size of an attraction ball is known. The spectral radius gives some global information regarding the convergence rate of all sequences of successive approximations converging towards \(x^*\), while the spectral elements of \(G'(x^*)\) characterize the convergence rate of each individual such sequense. In this paper, the author shows that the estimate may be readily obtained in terms of \(\|G'(x^*)\|<1\) and of the Hölder (in particular, Lipschitz) continuity constant of \(G'\). In the hypothesis, \(G\) is not necessarily required to be of contractive type on the whole estimate ball. For the justification, an example is also given.