Extending the Kantorovich’s theorem on Newton’s method for solving strongly regular generalized equation

Abstract

The aim of this paper, is to extend the applicability of Newton’s method for solving a generalized equation of the type $f(x)+F(x)\ni 0$ in Banach spaces, where f is a Fréchet differentiable function and F is a set-valued mapping. The novelty of the paper is the introduction of a restricted convergence domain. Using the idea of a weaker majorant, the convergence of the method, the optimal convergence radius, and results of the convergence rate are established. That is we find a more precise location where the Newton iterates lie than in earlier studies. Consequently, the Lipschitz constants are at least as small as the ones used before. This way and under the same computational cost, we extend the semilocal convergence of the Newton iteration for solving $f(x)+F(x)\ni 0$. The strong regularity concept plays an important role in our analysis. We finally present numerical examples, where we can solve equations in cases not possible before without using additional hypotheses.