Characterizations of metric projections in Banach spaces and applications

Jean-Paul Penot,Robert Ratsimahalo

doi:10.1155/s1085337598000451

Abstract

This paper is devoted to the study of the metric projection onto a nonempty closed convex subset of a general Banach space. Thanks to a systematic use of semi-inner products and duality mappings, characterizations of the metric projection are given. Applications to decompositions of Banach spaces along convex cones and variational inequalities are presented.

Highlights

A number of problems can be reformulated as best approximation problems in some normed vector spaces
It is the purpose of the present paper to describe characterizations of the solutions of such problems in case the feasible set is convex. Such characterizations are known under restrictive assumptions such as Gateaux-differentiability of the norm off 0, or strict convexity of the norm. We dispense with such assumptions by making a strong use of the concepts of duality mapping and of semi-inner product
It will be shown that, when X is a vector lattice for the preorder defined by a closed convex cone C of X, this decomposition can be made more precise in the sense that it corresponds to an order decomposition w = w+ − w−, with w+ ∈ P (C, w), −w− ∈ P (−C, w), and w+⊥w−

Summary

Introduction

A number of problems can be reformulated as best approximation problems in some normed vector spaces. The choice of the norm is imposed by the nature of the problem, so that one does not always dispose of an Hilbertian structure It is the purpose of the present paper to describe characterizations of the solutions (which are not necessarily unique) of such problems in case the feasible set is convex. We dispense with such assumptions by making a strong use of the concepts of duality mapping and of semi-inner product. Metric projection, closed convex subset of a Banach space, semi-inner products, duality mappings, convex cones. Zalinescu for calling our attention on the formulation of Proposition 5.1 of the present paper

Preliminaries

Characterizations of metric projections

An application to ordered Banach spaces

An application to variational inequalities