Convergence of Newton's method under Vertgeim conditions: new extensions using restricted convergence domains

Abstract

We present new sufficient convergence conditions for the semilocal convergence of Newton’s method to a locally unique solution of a nonlinear equation in a Banach space. We use Hölder and center Hölder conditions, instead of just Hölder conditions, for the first derivative of the operator involved in combination with our new idea of restricted convergence domains. This way, we find a more precise location where the iterates lie, leading to at least as small Hölder constants as in earlier studies. The new convergence conditions are weaker, the error bounds are tighter and the information on the solution at least as precise as before. These advantages are obtained under the same computational cost. Numerical examples show that our results can be used to solve equations where older results cannot.