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### Newton's method for solving inclusions using set-valued approximations

 Citace: CIBULKA, R. Newton's method for solving inclusions using set-valued approximations. Caparica - Lisbon, Portugal, 2013. PŘEDNÁŠKA, POSTER eng Newton's method for solving inclusions using set-valued approximations 2013 Ing. Radek Cibulka Ph.D. Given Banach spaces $X$ and $Y$, a single-valued mapping $f: X \to Y$ and a multivalued mapping $F:X\rightrightarrows Y$, we investigate the convergence properties of Newton-type iterative process for solving the generalized equation. The problem is to $$\label{Eqn1} \mbox{find}\quad x\in X \quad \mbox{such that}\quad 0\in f(x)+F(x).$$ This model has been used to describe in a unified way various problems such as equations, inequalities, variational inequalities, and in particular, optimality conditions. We study the following iterative process: {\it Choose a sequence of set-valued mappings $A_k: X\times X\rightrightarrows Y$ approximating the function $f$ and a starting point $x_0 \in X$, and generate a sequence $(x_k)$ in $X$ iteratively by taking $x_{k+1}$ to be a solution to the auxiliary generalized equation $$\label{Newton-Seq} 0\in A_k(x_{k+1},x_k)+F(x_{k+1}) \quad \mbox{for each} \quad \quad k \in \{0,1,2, \dots\}.$$} In the first part, we present a result concerning the stability of metric regularity under set-valued perturbations. In the latter, this statement is applied in the study of (super-)linear convergence of the iterative process (\ref{Newton-Seq}). Also some particular cases are discussed in detail. This work is based on a forthcoming joint paper with Samir Adly and Huynh Van Ngai.

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