Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications

TL;DR

This paper introduces new generalized projection operators in Banach spaces to extend the properties and applications of metric projections known in Hilbert spaces, addressing limitations in Banach space applications.

Contribution

It proposes a natural generalization of metric projection operators in Banach spaces, enabling solutions to variational inequalities and approximation problems.

Findings

Generalized projection operators in Banach spaces retain key properties.

New operators facilitate solving variational inequalities in Banach spaces.

Theoretical equivalence between variational inequalities and projection equations established.

Abstract

Metric projection operators can be defined in similar wayin Hilbert and Banach spaces. At the same time, they differ signifitiantly in their properties. Metric projection operator in Hilbert space is a monotone and nonexpansive operator. It provides an absolutely best approximation for arbitrary elements from Hilbert space by the elements of convex closed sets . This leads to a variety of applications of this operator for investigating theoretical questions in analysis and for approximation methods. Metric projection operators in Banach space do not have properties mentioned above and their applications are not straightforward. Two of the most important applications of the method of metric projection operators are as follows: 1. Solve a variational inequality by the iterative-projection method, 2. Find common point of convex sets by the iterative-projection method. In Banach space these problems can not be solved in the framework of metric projection operators. Therefore, in the present paper we introduce new generalized projection operators in Banach space as a natural generalization of metric projection operators in Hilbert space. In Sections 2 and 3 we introduce notations and recall some results from the theory of variational inequalities and theory of approximation. Then in Sections 4 and 5 we describe the properties of metric projection operators $P_\Omega$ in Hilbert and Banach spaces and also formulate equivalence theorems between variational inequalities and direct projection equations with these