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].
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.
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.
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
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
Then we have the following theorem.
Theorem 2.1 (see ).
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].
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
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.
which is contradiction.
3. The Methods of Hilbert Spaces
It is consequence of the above definitions.
4. Main Result
We are now in position to prove our main result.
Thus, . This implies that see  for details. Thus for every and is a retraction of onto .
then we have the following corollary.
- Bruck RE Jr.: Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Transactions of the American Mathematical Society 1973, 179: 251–262.MathSciNetView ArticleMATHGoogle Scholar
- Domínguez-Benavides T, Lorenzo Ramírez P: Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 2001,129(12):3549–3557. 10.1090/S0002-9939-01-06141-XMathSciNetView ArticleMATHGoogle Scholar
- Goebel K, Kirk WA: A fixed point theorem for transformations whose iterates have uniform Lipschitz constant. Studia Mathematica 1973, 47: 135–140.MathSciNetMATHGoogle Scholar
- Goebel K, Kirk WA: Classical theory of nonexpansive mappings. In Handbook of Metric Fixed Point Theory. Edited by: Kirk WA, Sims B. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:49–91.View ArticleGoogle Scholar
- Bruck RE: Asymptotic behavior of nonexpansive mappings. In Fixed Points and Nonexpansive Mappings, Contemp. Math.. Volume 18. Edited by: Sine RC. American Mathematical Society, Providence, RI, USA; 1983:1–47.View ArticleGoogle Scholar
- Sędłak E, Wiśnicki A: On the structure of fixed-point sets of uniformly Lipschitzian mappings. Topological Methods in Nonlinear Analysis 2007,30(2):345–350.MathSciNetMATHGoogle Scholar
- Górnicki J: Remarks on the structure of the fixed-point sets of uniformly Lipschitzian mappings in uniformly convex Banach spaces. Journal of Mathematical Analysis and Applications 2009,355(1):303–310. 10.1016/j.jmaa.2009.02.003MathSciNetView ArticleMATHGoogle Scholar
- Górnicki J: A remark on fixed point theorems for Lipschitzian mappings. Journal of Mathematical Analysis and Applications 1994,183(3):495–508. 10.1006/jmaa.1994.1156MathSciNetView ArticleMATHGoogle Scholar
- Lifshitz EA: A fixed point theorem for operators in strongly convex spaces. Voronezhskĭ Gosudarstvennyĭ Universitet imeni Leninskogo Komsomola. Trudy Matematicheskogo Fakul'teta 1975, 16: 23–28.MathSciNetGoogle Scholar
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