Extended Zabrejko theory of the Newton–Kantorovich iteration and the Pták–Potra error estimates

Abstract. A new methodology is introduced that allows to extend the application of the
Newton–Kantorovich Iteration (NKI) for solving non-linear Banach space valued operator equations containing a non-differentiable term. In particular, the Zabrejko theory is revisited. But the new majorant functions are tighter than before leading to an at least as weak semi-local convergence analysis for NKI. Moreover, the Pták–Potra error estimates are replaced with tighter ones. Furthermore, the region that determines the uniqueness of the solution is more precise. The novelty of this article is that the aforementioned benefits are obtained under the same or even weaker conditions. All these are also verified using numerical examples where the conditions are verified under the new approach but not under the earlier ones. The introduced methodology can apply analogously on other iterations containing the inverses of linear operators.