A sequential implicit function theorem for the chords iteration

Mathematica Slovaca

Diana Nedelcheva¹ 

关键词: Newton-type method;    generalized equations;    variational inequalities;    strong regularity;    implicit function theorem;    set-valued mapping;    linear convergence;    chords method;   

DOI  :  10.2478/s12175-013-0157-7

学科分类：数学（综合）

来源: Slovenska Akademia Vied * Matematicky Ustav / Slovak Academy of Sciences, Mathematical Institute

【 摘 要 】

In this paper we study the local convergence of the method $$0 in fleft( {p,x_k } ight) + Aleft( {x_{k + 1} - x_k } ight) + Fleft( {x_{k + 1} } ight),$$ in order to find the solution of the generalized equation $$find x in X such that 0 in fleft( {p,x} ight) + Fleft( x ight).$$ We first show that under the strong metric regularity of the linearization of the associated mapping and some additional assumptions regarding dependence on the parameter and the relation between the operator A and the Jacobian $$abla _x fleft( {ar p,ar x} ight)$$, we prove linear convergence of the method which is uniform in the parameter p. Then we go a step further and obtain a sequential implicit function theorem describing the dependence of the set of sequences of iterates of the parameter.

【 授权许可】

Unknown