The Methods of Hilbert Spaces and Structure of the Fixed-Point Set of Lipschitzian Mapping
© Jarosław Górnicki. 2009
Received: 16 May 2009
Accepted: 25 August 2009
Published: 28 September 2009
The purpose of this paper is to prove, by asymptotic center techniques and the methods of Hilbert spaces, the following theorem. Let be a Hilbert space, let be a nonempty bounded closed convex subset of and let be a strongly ergodic matrix. If is a lipschitzian mapping such that , then the set of fixed points is a retract of . This result extends and improves the corresponding results of [7, Corollary 9] and [8, Corollary 1].
Let be a Banach space and let be a nonempty bounded closed convex subset of . We say that a mapping is nonexpansive if
The result of Bruck  asserts that if a nonexpansive mapping has a fixed point in every nonempty closed convex subset of which is invariant under and if is convex and weakly compact, then , the set of fixed points, is nonexpansive retract of (i.e., there exists a nonexpansive mapping such that ). A few years ago, the Bruck results were extended by Domínguez Benavides and Lorenzo Ramírez  to the case of asymptotically nonexpansive mappings if the space was sufficiently regular.
On the other hand it is known that, the set of fixed points of -lipschitzian mapping can be very irregular for any .
It is not difficult to see that and the Lipschitz constant of tends to if .
In 1973, Goebel and Kirk  introduced the class of uniformly -lipschitzian mappings, recall that a mapping is uniformly -lipschitzian, , if
and proved the following theorem.
Then has a fixed point in . Note that in a Hilbert space, .
Let be a Hilbert space, a nonempty bounded closed convex subset of and a uniformly - lipschitzian mapping with . Then has a fixed point in and is a retract of .
In this paper we shall continue this work. Precisely, by means of techniques of asymptotic centers and the methods of Hilbert spaces, we establish some result on the structure of fixed point sets for mappings with lipschitzian iterates in a Hilbert space. The class of mappings with lipschitzian iterates is importantly greater than the class of uniformly lipschitzian mappings; see [8, Example 1].
2. Asymptotic Center
Denote by the Lipschitz norm of :
Lifshitz  significantly extended Goebel and Kirk's result and found an example of a fixed point free uniformly lipschitzian mapping which leaves invariant a bounded closed convex subset of . The validity of Lifshitz's Theorem in a Hilbert space for remains open.
A more general approach was proposed by the present author using the methods of Hilbert spaces, asymptotic techniques, and strongly ergodic matrix. We recall that a matrix is called strongly ergodic if
(i)for all ,
(ii)for all ,
(iii)for all ,
Then we have the following theorem.
Theorem 2.1 (see ).
then has a fixed point in .
This result generalizes Lifshitz's Theorem (in case of a Hilbert space) and shows that the theorem admits certain perturbations in the behavior of the norm of successive iterations in infinite sets; see [8, Example 1].
Let be a Banach space. Recall that the modulus of convexity is the function defined by
and uniform convexity means for . A Hilbert space is uniformly convex. This fact is a direct consequence of parallelogram identity.
Now we prove some version of Sęd ak and Wiśnicki's result [6, Lemma 2.1]. Let be a nonempty bounded closed convex subset of a real Hilbert space , let be a strongly ergodix matrix, and let be a mapping such that for all , and
Let we use
to denote the asymptotic radius of at and the asymptotic radius of in , respectively. It is well known in a Hilbert space  that the asymptotic center of in :
is a singleton.
Let denote a mapping which associates with a given a unique , that is, . The following Lemma is a crucial tool to prove Theorem 4.1.
Let be a Hilbert space and let be a nonempty bounded closed convex subset of . Then the mapping is continuous.
On the contrary, suppose that there exists and such that for all there exists such that and , where .
for all but finitely many .
which is contradiction.
This contradiction proves the continuity of mapping .
3. The Methods of Hilbert Spaces
Let , be as above. We define functionals
where . Let in be an asymptotic center of with respect to and , which minimizes the functional over in (for fix ).
One has .
It is consequence of the above definitions.
One has for all .
as , we get thesis.
One has for every .
Taking , we get, .
4. Main Result
We are now in position to prove our main result.
then is a retract of .
over in .
Thus, . This implies that see  for details. Thus for every and is a retraction of onto .
If is the Cesaro matrix, that is, for
then we have the following corollary.
then is a retract of .
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