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Ergodic Retractions for Families of Asymptotically Nonexpansive Mappings
Fixed Point Theory and Applications volume 2010, Article number: 281362 (2010)
Abstract
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).
1. Introduction
Let be a Banach space and a nonempty closed and convex subset of . We recall some definitions.
Definition 1.1.
A mapping is said to be
(i)nonexpansive if
(ii)asymptotically nonexpansive if there exists a sequence of positive numbers satisfying the property and
(iii)of asymptotically nonexpansive type if for each in , we have
(iv)weakly asymptotically nonexpansive if it satisfies the condition
(v)retraction if . A subset of is called a nonexpansive retract of if either or there exists a retraction of onto which is a nonexpansive mapping.
Definition 1.2.
We say that a nonempty closed convex subset of satisfies property with respect to
(i)a mapping if for every where
(ii)a semigroup of mappings if for every where
Obviously, itself verifies .
Definition 1.3.

(i)
A mapping is said to satisfy the fixed point property (fpp) if has a fixed point in every nonempty closed convex subset of which satisfies with respect to .

(ii)
A semigroup is said to satisfy the fpp if has a common fixed point in every nonempty closed convex subset of which satisfies with respect to the semigroup .

(iii)
A family is said to satisfy the fpp if has a common fixed point in every nonempty closed convex subset of which satisfies with respect to each .
In 1965, Kirk [1] 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 [2] proved that if is assumed to be uniformly convex, then every asymptotically nonexpansive selfmapping of has a fixed point. This was extended to mappings of asymptotically nonexpansive type by Kirk in [3]. 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 [4] 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).
Theorem 1.4.
Suppose is a Banach space with uniform normal structure; is a nonempty bounded subset of . Then
(i)every continuous and asymptotically nonexpansive type mapping satisfies fpp;
(ii)every semigroup of asymptotically nonexpansive type mappings on such that is continuous on for each satisfies fpp.
On the other hand, Bruck [5] 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 [6] 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 [9] studied the structure of the set of fixed points for (weakly) asymptotically nonexpansive mappings.
Theorem 1.5.
Let be a Banach space and a nonempty weakly compact convex subset of . Assume that every asymptotically nonexpansive selfmapping of satisfies the fpp. Then for any commuting family of asymptotically nonexpansive selfmappings 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
Theorem 2.1.
Let be a nonempty weakly compact convex subset of a Banach space , and a family of weakly asymptotically nonexpansive mappings on such that . Assume one of the following assumptions is satisfied:
(a) satisfies the fpp;
(b) is a nonexpansive retract of .
Then for each , there exists a nonexpansive retraction from onto , the common fixed points of , such that , and every closed convex invariant subset of is also invariant.
Proof.
Consider with the product topology induced by the weak topology on . Now, consider an and define
By applying an argument similar to that in the proof of [9, Theorem ], it follows that is compact (the topology on is that of weak pointwise convergence) and there is a minimal element in the following sense:
First, we assume the case (a). We shall prove that for all . For a given , consider the set Then is a nonempty weakly compact convex subset of , because is convex and compact. We will show that for all , satisfies property with respect to . Fix and 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 , it follows that and then . Thus satisfies the property with respect to . Since, satisfies the fpp (by (a)), it follows that . So, there exists with . Let . Then , and by using the minimality of , we have
So, we get . Since this is so for each and belongs to , it follows that and .
Now, we assume the case (b). From (b), there is a nonexpansive retraction from onto . Put . Since , we can replace by in the above assertions to obtain a minimal element in the sense (*), where ia defined here as
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.
A nonexpansive retraction satisfying the thesis of Theorem 2.1 is usually called an ergodic retraction (see e.g., [10, 11]).
Combining Theorem 1.5 [9, Theorem ] and Theorem 2.1(a), we get the following improvement of Theorem 1.5.
Corollary 2.2.
Let be a Banach space and a nonempty weakly compact convex subset of . Assume that every asymptotically nonexpansive selfmapping of satisfies fpp. Then for any commuting family of asymptotically nonexpansive selfmappings 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 realvalued 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 [12].
The following result which we need is well known (see [13]).
Lemma 3.1.
Let be a function of a semigroup into such that the weak closure of is weakly compact. Then, for any , there exists a unique element in such that for all . Moreover, if is a mean, then
We can write by . As a direct consequence of Lemma 3.1, we have the following lemma.
Lemma 3.2.
Let be a nonempty closed convex subset of a Banach space , a semigroup of weakly asymptotically nonexpansive mappings on such that weak closure of is weakly compact for each , and a mean on .
If we write instead of , then the following hold.
(i) for each .
(ii) for each .
(iii)If is invariant, then for each and is a nonexpansive mapping from into itself.
Proof.
We only need to prove that is nonexpansive: consider and . Then for each , we have
Therefore, , for every . Consequently, we get
The following is our main result which is an improvement of Theorem 1.4 [4, Theorem ].
Theorem 3.3.
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.
Proof.
Consider with the product topology induced by the weak topology on . Now, define
We note that , because the mapping in Lemma 3.2 belongs to . By applying an argument similar to that in the proof of [9, Theorem ] (see also the proof of [7, Lemma ]), it follows that is compact and there is a minimal element in the following sense:
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 , .
References
 1.
Kirk WA: A fixed point theorem for mappings which do not increase distances. Proceedings of the American Mathematical Society 1965, 72: 1004–1006.
 2.
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1972, 35: 171–174. 10.1090/S00029939197202985003
 3.
Kirk WA: Fixed point theorems for nonlipschitzian mappings of asymptotically nonexpansive type. Israel Journal of Mathematics 1974, 17: 339–346. 10.1007/BF02757136
 4.
Li G, Sims B: Fixed point theorems for mappings of asymptotically nonexpansive type. Nonlinear Analysis: Theory, Methods & Applications 2002, 50: 1085–1091. 10.1016/S0362546X(01)007441
 5.
Bruck RE: Properties of fixedpoint sets of nonexpansive mappings. Transactions of the American Mathematical Society 1973, 179: 251–262.
 6.
Bruck RE: A common fixed point theorem for a commuting family of nonexpansive mappings. Pacific Journal of Mathematics 1974, 53: 59–71.
 7.
Saeidi S: Ergodic retractions for amenable semigroups in Banach spaces with normal structure. Nonlinear Analysis: Theory, Methods & Applications 2009,71(7–8):2558–2563. 10.1016/j.na.2009.01.091
 8.
Saeidi S: The retractions onto the common fixed points of some families and semigroups of mappings. Nonlinear Analysis: Theory, Methods & Applications 2009,71(3–4):1171–1179. 10.1016/j.na.2008.11.049
 9.
Benavides Td, Ramírez PL: 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/S000299390106141X
 10.
Kirk WA, Sims B: Handbook of Metric Fixed Point Theory. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:xiv+703.
 11.
Lau AT, Shioji N, Takahashi W: Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces. Journal of Functional Analysis 1999,161(1):62–75. 10.1006/jfan.1998.3352
 12.
Day MM: Amenable Semigroups. Illinois Journal of Mathematics 1957, 1: 509–544.
 13.
Takahashi W: A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space. Proceedings of the American Mathematical Society 1981,81(2):253–256. 10.1090/S0002993919810593468X
Acknowledgments
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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Saeidi, S. Ergodic Retractions for Families of Asymptotically Nonexpansive Mappings. Fixed Point Theory Appl 2010, 281362 (2010). https://doi.org/10.1155/2010/281362
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Keywords
 Nonexpansive Mapping
 Minimal Element
 Weak Topology
 Common Fixed Point
 Pointwise Convergence