Extended Newton-type method and its convergence analysis for nonsmooth generalized equations

Abstract

Let X and Y be Banach spaces and \(\Omega \) be an open subset of X. Suppose that \(f:{\Omega \subseteq X}\rightarrow {Y}\) is a single-valued function which is nonsmooth and it has point based approximations on \(\Omega \) and \(F:X\rightrightarrows 2^Y\) is a set-valued mapping with closed graph. An extended Newton-type method is introduced in the present paper for solving the nonsmooth generalized equation \(0\in {f(x)+F(x)}\). Semilocal and local convergence of this method are analyzed based on the concept of point-based approximation.