Results on Newton methods. Part II: Perturbed Newton-like methods in generalized Banach spaces

Abstract

In this paper, we provide convergence results and error estimates for perturbed Newton-like methods in generalized Banach spaces. We use the idea of a generalized norm which is defined to be a map from a linear space into a partially ordered Banach space. We find out that this way the metric properties of the examined problem can be analyzed more precisely. The convergence results are improved compared with the real norm theory. Since the iterates can rarely be computed exactly, we have considered perturbed Newton-like methods which converge to a solution of a nonlinear operator equation. We have achieved that by managing to control the “size” of the allowable error. Special cases of our results reduce to ones already in the literature, but even then our results are simpler and easier to apply. Finally applications to nonlinear integral and differential equations are suggested.