As previous examples illustrate, neither iterative sequences nor approximate fixed point sequences for nonexpansive mappings typically converge, at least in the strong sense. However, in certain instances, these sequences may converge in the weak (or weak*) topology to fixed points, or in some other way determine invariant sets which contain fixed points.

Throughout this chapter we shall, as usual, assume that K is a nonempty, closed and convex subset of a Banach space X but, in general, we shall not assume K is bounded.

Suppose T: K→K is nonexpansive and fix x0 ∈ K. We begin by considering the iterative sequence {xn} = {Tnx0}. The set of points O(x0) = {xn: n = 0, 1, …} is called the orbit of x0 under T, and its closure is called the closed orbit. Since T is nonexpansive, if O(x0) is bounded for at least one x0 ∈ K then all other orbits O(x), x ∈ K, are bounded. Thus the nonexpansive self-mappings of K fall into two categories: those with bounded orbits and those with unbounded orbits. Obviously all nonexpansive mappings having fixed points have bounded orbits. (Less obvious is the fact that in a finite dimensional Banach space O(x0) is bounded for a nonexpansive mapping T whenever {xn} has a convergent subsequence (see Roehrig and Sine, 1981).)

In our previous examples of fixed point free, nonexpansive mappings none of the interative sequences contain convergent subsequences. However Edelstein has shown (1964) that even if such convergent subsequences exist, a nonexpansive mapping may remain fixed point free.