Fixed point problems: an introduction

Resumo

This paper surveys a number of fundamental results on the existence and uniqueness of fixed points for certain classes 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 minimum norm solution of an operator equation of the form x = Ax exists and is unique, provided that is nonexpansive. This is a new result, which has interesting practical consequences in signal and image reconstruction problems and in other engineering applications.