Lipschitz-like mapping and its application to convergence analysis of a variant of Newton's method

Let $X$ and $Y$ be Banach spaces. Let $f: \Omega\to Y$ be a Fréchet differentiable function on an open subset $\Omega$ of $X$ and $F$ be a set-valued mapping with closed graph. Consider the following generalized equation problem: $0 \in f(x)+F(x)$. In the present paper, we study a variant of Newton's method for solving generalized equation (1) and analyze semilocal and local convergence of this method under weaker conditions than those considered by Jean-Alexis and Piétrus [13]. In fact, we show that the variant of Newton's method is superlinearly convergent when the Frechet derivative of f is $(L,p)$-Hölder continuous and $(f+F)^{-1}$ is Lipzchitz-like at a reference point. Moreover, applications of this method to a nonlinear programming problem and a variational inequality are given. Numerical experiments are provided which illustrate the theoretical results.