Let be a metric space with . Recall that a mapping is non-expansive if satisfies
for all .
Theorem. Let be a nonempty, closed, convex subset of a normed linear space with nonexpansive and a subset of a compact set of . Then has a fixed point.
Proof. Let . For define
Since is convex and , we see that and it is clear that is a contradiction. Therefore each has a unique fixed point . That is
In addition, since lies in a compact subset of , there exist a subsequence of integers and a with as in .
Thus as in . By continuity, as in which claims that .
The main theorem of this topic is a result proved independently by Browder, Gohde and Kirk in 1965. We state it as follows.
Theorem. Let be a nonempty, closed, bounded, convex set in a (real) Hilbert space . Then each nonexpansive map has at least one fixed point.
Remark. Notice that uniqueness need not hold as the example, , shows.
Remark. In fact the above Theorem, it is enough to assume that is a uniformly convex Banach space.
In the proof of the above Theorem, we will need the following two technical results.
Claim 1. Let be a Hilbert space with , and let be constants with . If there exists an with
Proof. This comes from the so-called parallelogram law, that is,
Claim 2. Let be a Hilbert space, a bounded set and a nonexpansive map. Suppose and . Let denote the diameter of and let with and . Then
Proof of main theorem. Assume that . (A modified argument from the one given below holds for any , therefore for simplicity we let .) Also assume that (otherwise we are finished). For each , notice that
is a contradiction. Now the first theorem guarantees that there exists a unique with
For each , let
is a decreasing sequence of nonempty closed sets. Let
and since the s are decreasing we have
with for each . Consequently, with .
Now is a decreasing sequence of closed, nonemplty sets. We now show that
To see this, let . Then
Also since we have
It is easy to check that is non-expansive. As a result
is a non-expansive map. This guarantees that there exists with . If , then
and has a fixed point. If does not belong to then
Source: Ravi P. Agarwal, Maria Meehan, Donal O’Regan, Fixed point theory and applications, Cambridge University Press, 2001.