Convergence Properties of Extended Newton-type Iteration Method for Generalized Equations

In this paper, we introduce and study the extended Newton-type method for solving generalized equation $0\in f(x)+g(x)+\mathcal F(x)$, where $f:\Omega\subseteq\mathcal X\to \mathcal Y$ is Fr\'{e}chet differentiable in a neighborhood $\Omega$ of a point $\bar{x}$ in $\mathcal X$, $g:\Omega\subseteq \mathcal X\to \mathcal Y$ is linear and differentiable at a point $\bar{x}$, and $\mathcal F$ is a set-valued mapping with closed graph acting in Banach spaces $\mathcal X$ and $\mathcal Y$. Semilocal and local convergence of the extended Newton-type method are analyzed.