Local convergence of two Newton-like methods under Hölder continuity condition in Banach spaces (Q2073488)

Summary: The main objective of this paper is to study the local convergence analysis of two cubically convergent Newton-like methods, harmonic mean Newton's method (HNM) and midpoint Newton's method (MNM), for approximating a unique solution of a nonlinear operator equation in Banach spaces. Unlike the earlier works using hypotheses up to the third-order Fréchet-derivative, we provide the convergence analysis using the only assumption that the first-order Fréchet derivative is Hölder continuous. Therefore, this study not only boosts the applicability of these schemes but also offers the convergence radii of these methods. Furthermore, the uniqueness of the solution and the error estimates are discussed. Finally, various numerical examples are provided to show that our study is applicable to solve such problems where earlier studies fail.