Ergodic Retractions for Families of Asymptotically Nonexpansive Mappings
© The Author(s). 2010
Received: 2 October 2009
Accepted: 14 March 2010
Published: 25 April 2010
We prove some theorems for the existence of ergodic retractions onto the set of common fixed points of a family of asymptotically nonexpansive mappings. Our results extend corresponding results of Benavides and Ramírez (2001), and Li and Sims (2002).
In 1965, Kirk  proved that if is a weakly compact convex subset of a Banach space with normal structure, then every nonexpansive mapping has a fixed point. (A nonempty convex subset of a normed linear space is said to have normal structure if each bounded convex subset of consisting of more than one point contains a nondiametral point). Goebel and Kirk  proved that if is assumed to be uniformly convex, then every asymptotically nonexpansive self-mapping of has a fixed point. This was extended to mappings of asymptotically nonexpansive type by Kirk in . However, whether normal structure implies the existence of fixed points for mappings of asymptotically nonexpansive type is a natural and still open question. Li and Sims  proved the following fixed point result in the case that has uniform normal structure (It is known that a space with uniform normal structure is reflexive and that all uniformly convex or uniformly smooth Banach spaces have uniform normal structure).
On the other hand, Bruck  initiated the study of the structure of the fixed point set in a general Banach space : if is a weakly compact convex subset of and is nonexpansive and satisfies a conditional fixed point property, then is a nonexpansive retract of . The same author  used this fact to derive the existence of fixed points for a commuting family of nonexpansive mappings. See, for example, [7, 8] for some related results.
Benavides and Ramírez  studied the structure of the set of fixed points for (weakly) asymptotically nonexpansive mappings.
Let be a Banach space and a nonempty weakly compact convex subset of . Assume that every asymptotically nonexpansive self-mapping of satisfies the -fpp. Then for any commuting family of asymptotically nonexpansive self-mappings of , the common fixed point set of , , is a nonempty nonexpansive retract of .
In this paper, we prove some theorems to guarantee the existence of nonexpansive retractions onto the common fixed points of some families of (weakly) asymptotically nonexpansive (type) mappings. The results obtained in this paper extend in some sense, for example, Theorems 1.4 and 1.5, above.
2. Nonexpansive Retractions for Families of Weakly Asymptotically Nonexpansive Mappings
We note that , ( ). Since , every closed convex -invariant subset of is also -invariant and consequently -invariant, ( ). So it is easy to see that , ( ). Therefore, for every , the set is an -invariant subset of . So, considering the fact that , we obtain . Now, we can repeat the argument used in the last paragraph to get the desired result.
Combining Theorem 1.5 [9, Theorem ] and Theorem 2.1(a), we get the following improvement of Theorem 1.5.
Let be a Banach space and a nonempty weakly compact convex subset of . Assume that every asymptotically nonexpansive self-mapping of satisfies -fpp. Then for any commuting family of asymptotically nonexpansive self-mappings of and for each , there exists a nonexpansive retraction from onto , such that , and every closed convex -invariant subset of is also -invariant.
3. Ergodic Retractions for a Semigroup of Asymptotically Nonexpansive Type
Assume that is a semigroup and is the space of all bounded real-valued functions defined on with supremum norm. For and , we define elements and in by and for each , respectively. An element of is said to be a mean on if . We often write instead of for and . A mean is said to be invariant if for each and . is said to be amenable if there is an invariant mean on . As is well known, is amenable when it is a commutative semigroup .
The following result which we need is well known (see ).
The following is our main result which is an improvement of Theorem 1.4 [4, Theorem ].
Suppose is a Banach space with uniform normal structure; is a nonempty bounded closed and convex subset of ; is a semigroup of asymptotically nonexpansive type mappings on such that is continuous on for each . Then there exists a nonexpansive retraction from onto , such that for each , and every closed convex -invariant subset of is also -invariant.
We will prove that for all . For a given , consider the set Then is a nonempty weakly compact convex subset of , because is convex and compact. Take and such as , for some . There exists , such that . Consider a subnet of such that exists for every . Now, taking , we have . Since is nonexpansive, and for every , it follows that and then . Thus satisfies the property with respect to the semigroup . Now, from Theorem 1.4, it follows that . From this and the argument used in the proof of Theorem 2.1, we obtain . Since this holds for each , .
The author would like to thank the referee for useful comments and for pointing out an oversight regarding an earlier draft of this paper. This paper is dedicated to Professor William Art Kirk. This research was in part supported by a Grant from IPM (no. 88470021).
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