Ball Convergence for a Family of Quadrature-Based Methods for Solving Equations in Banach Space

Abstract

We present a local convergence analysis for a family of quadrature-based predictor–corrector methods in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as Howk [2016] [Howk, C. L. [2016] “A classs of efficient quadrature-based predictor–corrector methods for solving nonlinear systems, Appl. Math. Comput. 276, 394–406] the [Formula: see text] order of convergence was shown on the [Formula: see text]-dimensional Euclidean space using Taylor series expansion and hypotheses reaching up to the third-order Fréchet-derivative of the operator involved although only the first-order Fréchet-derivative appears in these methods, which restrict the applicability of these methods. In this paper, we expand the applicability of these methods in a Banach space setting and using hypotheses only on the first Fréchet-derivative. Moreover, we provide computable radii of convergence as well as error bounds on the distances involved using Lipschitz constants. Numerical examples are also presented to solve equations in cases where earlier results cannot apply.