Extended Newton-type Method for Generalized Equations with Hölderian Assumptions

Abstract

In the present paper, we consider the generalized equation $0\in f(x)+g(x)+\mathcal F(x)$, where $f:\mathcal X\to \mathcal Y$ is Fr\'{e}chet differentiable on a neighborhood $\Omega$ of a point $\bar{x}$ in $\mathcal X$, $g:\mathcal X\to \mathcal Y$ is differentiable at point $\bar{x}$ and linear as well as $\mathcal F$ is a set-valued mapping with closed graph acting between two Banach spaces $\mathcal X$ and $\mathcal Y$. We study the above generalized equation with the help of extended Newton-type method, introduced in [ M. Z. Khaton, M. H. Rashid, M. I. Hossain, Convergence Properties of extended Newton-type Iteration Method for Generalized Equations, Journal of Mathematics Research, 10 (4) (2018), 1--18, DOI:10.5539/jmr.v10n4p1, under the weaker conditions than that are used in Khaton et al. (2018). Indeed, semilocal and local convergence analysis are provided for this method under the conditions that the Frechet derivative of $f$ and the first-order divided difference of $g$ are Hölder continuous on $\Omega$. In particular, we show this method converges superlinearly and these results extend and improve the corresponding results in Argyros (2008) and Khaton $et$ $al.$ (2018).