Fixed point problems: an introduction

• Author(s): Paulo J. S. G. Ferreira
• Date: 2012
• Origin: Electrónica e Telecomunicações
• Subject(s): fixed points; nonlinear operators; signal reconstruction; image reconstruction; fixed points; nonlinear operators; signal reconstruction; image reconstruction
• Document Type: Article
• Language: English

Description

This paper surveys a number of fundamental results on the existence and uniqueness of fixed points for certainclasses of possibly nonlinear operators. I do not try to be exhaustive, but merely to present the results that are more useful in the context of signal and image reconstruction. Some specific aspects pertaining to linear operators, and linear operators in finite dimensional spaces, are also discussed. It is shown that the set of fixed points of a nonexpansive operator is either empty or convex. Under rather general conditions this shows that the minimnum norm sulution of an operator equation of the form x = Ax exists and is unique, provided thatA is nniiexpansive. This is a new result, which lias interestingpractical consequences in signal and image reconstruction problems and in other engineering applications. This paper surveys a number of fundamental results on the existence and uniqueness of fixed points for certainclasses of possibly nonlinear operators. I do not try to be exhaustive, but merely to present the results that are more useful in the context of signal and image reconstruction. Some specific aspects pertaining to linear operators, and linear operators in finite dimensional spaces, are also discussed. It is shown that the set of fixed points of a nonexpansive operator is either empty or convex. Under rather general conditions this shows that the minimnum norm sulution of an operator equation of the form x = Ax exists and is unique, provided thatA is nniiexpansive. This is a new result, which lias interestingpractical consequences in signal and image reconstruction problems and in other engineering applications.