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Báo cáo hóa học: "TOWARDS VISCOSITY APPROXIMATIONS OF HIERARCHICAL FIXED-POINT PROBLEMS"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: TOWARDS VISCOSITY APPROXIMATIONS OF HIERARCHICAL FIXED-POINT PROBLEMS | TOWARDS VISCOSITY APPROXIMATIONS OF HIERARCHICAL FIXED-POINT PROBLEMS A. MOUDAFI AND P.-E. MAINGÉ Received 10 February 2006 Revised 14 September 2006 Accepted 18 September 2006 We introduce methods which seem to be a new and promising tool in hierarchical fixed-point problems. The goal of this note is to analyze the convergence properties of these new types of approximating methods for fixed-point problems. The limit attained by these curves is the solution of the general variational inequality 0 e I - Q xx NFixP xM where NFixP denotes the normal cone to the set of fixed point of the original nonexpan-sive mapping p and Q a suitable nonexpansive mapping criterion. The link with other approximation schemes in this field is also made. Copyright 2006 A. Moudafi and P.-E. Mainge. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction In nonlinear analysis a common approach to solving a problem with multiple solutions is to replace it by a family of perturbed problems admitting a unique solution and to obtain a particular solution as the limit of these perturbed solutions when the perturbation vanishes. Here we will introduce a more general approach which consists in finding a particular part of the solution set of a given fixed-point problem that is fixed points which solve a variational inequality criterion. More precisely the main purpose of this note consists in building methods which hierarchically lead to fixed points of a nonex-pansive mapping p with the aid of a nonexpansive mapping Q in the following sense find x e Fix P such that x Q X x - x 0 Vx e Fix P 1.1 where Fix P x e C x P x is the set of fixed points of p and C is a closed convex subset of a real Hilbert space GX. It is not hard to check that solving 1.1 is equivalent to the fixed-point problem find x e C such that x projFix P oQ x .