Extended Newton-type Method for Generalized Equations with Hölderian Assumptions

Authors : Most Zamilla KHATON, Mohammed Harunor RASHİD

Pages : 1-13

Doi:10.33434/cams.738324

View : 47 | Download : 8

Publication Date : 2021-03-29

Article Type : Research Paper

Abstract :In the present paper, we consider the generalized equation $0\in finsert ignore into journalissuearticles values(x);+ginsert ignore into journalissuearticles values(x);+\mathcal Finsert ignore into journalissuearticles values(x);$, 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 insert ignore into journalissuearticles values(4); insert ignore into journalissuearticles values(2018);, 1--18, DOI:10.5539/jmr.v10n4p1, under the weaker conditions than that are used in Khaton et al. insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(2008); and Khaton $et$ $al.$ insert ignore into journalissuearticles values(2018);.
Keywords : Divided difference, Extended Newton type method, Generalized equations, Lipschitz like mappings, Semilocal convergence

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