The existence and uniqueness of the minimum norm solution to certain linear and nonlinear problems

It is shown that the set of fixed points of a nonexpansive operator is either empty or closed and convex. Under rather general conditions this shows that the minimum norm solution of an operator equation of the form x = Tx exists and is unique, provided that T is nonexpansive. This holds in any strictly convex Banach space, a class of spaces that includes Hilbert spaces as particular case, and has consequences in signal and image reconstruction, as well as in other engineering applications.