Abstract
In this paper, we provide the nonlinear ergodic theorems and weak convergence theorems for almost orbits of a reversible semigroup of asymptotically nonexpansive mappings in a uniformly convex Banach space without assuming that X has a Fréchet differentiable norm. Since almost orbits in this paper are not almost asymptotically isometric, new methods have to be introduced and used for the proofs. Our main results include many well-known results as special cases and are new even for reversible semigroup of nonexpansive mappings.
MSC:47H20.
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1 Introduction
Baillon [1] proved the first nonlinear ergodic theorem for nonexpansive mappings in the framework of Hilbert space. Baillon′s theorem was extended to various semigroups in Hilbert spaces [2–4] or Banach spaces [5–13]. For instance, Takahashi [2] proved the ergodic theorem for right reversible semigroups of nonexpansive mappings in a Hilbert space by using the methods of invariant means. Lau et al. [5] studied the existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and provided the nonlinear ergodic theorems in Banach spaces. Kim and Li [6] proved the ergodic theorem for the almost asymptotically isometric almost orbits of right reversible semigroups of asymptotically nonexpansive mappings in a uniformly convex Banach space with a Fréchet differentiable norm. Many papers about weak convergence of asymptotically nonexpansive semigroups in a uniformly convex Banach space with a Fréchet differentiable norm have appeared [6, 10, 11, 14–16]. In 2001, Falset, et al. [14], Kaczor [15] proved the weak convergence theorems of almost orbits of commutative semigroups of asymptotically nonexpansive mappings under the assumptions that the Banach space is uniformly convex, and its dual space has the Kadec-Klee property.
This paper is devoted to the study of the nonlinear ergodic theorem and weak convergence for almost orbits of reversible semigroups of asymptotically nonexpansive mappings. Using the technique of product net, we first obtain the nonlinear ergodic theorems without assuming that the uniformly convex Banach space has a Fréchet differentiable norm, which extend and unify many previously known results in [2, 6, 10, 11, 16]. Next, we establish the convergence theorem in the case of reversible semigroup and the uniformly convex Banach space whose dual space has the Kadec-Klee property, which improves the known ones (see [2, 11, 14–16]) for commutative semigroups of asymptotically nonexpansive mappings in a uniformly convex Banach space. It is safe to say that the many general and key assumptions in the situation of reversible semigroup, such as the almost orbit is almost asymptotically isometric, and the subspace D has a left invariant mean (see [2, 6, 10]), are not necessary in this paper. Our main results are new even for the reversible semigroup of nonexpansive mappings.
2 Preliminaries
Let C be a nonempty bounded closed convex subset of a Banach space X. Let be the dual of X, then the value of at will be denoted by , and we associate the set
It is clear from the Hahn-Banach theorem that for all . Then the multi-valued operator is called the normalized duality mapping of X. We say that X has a Fréchet differentiable norm, i.e., for each , exists uniformly in , . We say that X has the Kadec-Klee property if for every sequence in X, whenever with , it follows that . Recall that X has the Kadec property if for every net in X, whenever with , it follows that , where I is a directed system. It is well known that within the class of reflexive spaces, the Kadec-Klee property is equivalent to the Kadec property [17]. We also would like to remark that a uniformly convex Banach space with a Fréchet differentiable norm implies that its dual has Kadec-Klee property, while the converse implication fails [14, 15].
Let G be a semitopological semigroup, i.e., G is a semigroup with a Hausdorff topology such that for each , the mappings and from G to G are continuous. G is called right reversible if any two closed left ideals of G have nonvoid intersection. In this case, is a directed system when the binary relation ≤ on G is defined by if and only if , . Right reversible semitopological semigroups include all commutative semigroups and all semitopological semigroups, which are right amenable as discrete semigroups.
Let be the Banach space of all bounded real valued functions on G with the supremum norm. Then for each and , we can define in by for all . Let D be a subspace of containing constant functions and invariant under for every . Let be the dual space of D, then the value of at will be denoted by . A linear function μ on D is called a mean on D if . Further, a mean μ on D is left invariant if for all and , . For each , we define a point evaluation on D by for every . A convex combination of point evaluation is called a finite mean on G.
By Day [18], if D has a left invariant mean, then there exists a net of finite means on G such that
for every , where A is a directed system, and is the conjugate operator of .
Let be a semigroup acting on C, i.e., for all and . Recall that ℑ is said to be asymptotically nonexpansive [19–21] if there exists a function with such that for all and ,
If for every , then ℑ is said to be nonexpansive. It should be pointed out that there is a notion of asymptotically nonexpansive mappings defined dependent on right ideals in a semigroup in [22, 23].
A function is said to be an almost orbit of ℑ [16] if
Suppose that is an almost orbit of ℑ such that for each , the function is in D. For each , since X is reflexive, there exists a unique in X such that for all . We denote by . If λ is a finite mean on G, say , where , , , and , then
Throughout this paper, let C be a nonempty bounded closed convex subset of uniformly convex Banach space X, and let be a reversible semigroup of asymptotically nonexpansive mappings acting on C. Let denote the set of all fixed points of ℑ, i.e., . For each and , we set
It should be noted that if for any , there exists such that for all , , then , and thus by the continuity of .
We denote by the set of all almost orbits of ℑ and by the set . Denote by the set of all weak limit points of subnets of the net .
3 Lemmas
In this section, we prove some lemmas, which play a crucial role in the proof of our main theorems in next section.
Lemma 3.1 [24]
Let X be a Banach space and J be the normalized duality mapping. Then for given , the following inequality holds:
for all .
Lemma 3.2 [25]
Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space X. Then there exists a strictly increasing continuous convex function with such that
for all integers , with , , and all nonexpansive mapping .
From Lemma 3.2, we can get for all with , ,
where .
To simplify, in the following, for each , we define
and
where is as in Lemma 3.2. Then is nonempty for each , and if , then for all , . And it should also be noted for all .
Lemma 3.3 For all ,
Proof Since is closed, we only need to prove that for all ,
Let , , , , and . Then
This completes the proof. □
Lemma 3.4 For all ,
Proof Let and , where and , then
This completes the proof. □
Lemma 3.5 Let and , then there exists an such that for all and ,
Proof Let and , there is an satisfying
For any given and , we can take a number
For each and , we set
Noting and
we get
Suppose that there are k elements in such that , then
Hence
Thus, there are at most terms in with . Therefore, for each , there is at least one term () in satisfying .
Putting
. It is easy to see that there are at most N elements in such that . Since
we can conclude that for all ,
By Lemma 3.3, we get for all ,
Using Lemma 3.4 and
we obtain
This completes the proof. □
Lemma 3.6 Let be an almost orbits of ℑ. Then
exists for all and .
Proof We only need to show that
In fact, for any , there are and such that for any , and , where . Then for all ,
Hence . Thus, there exists such that
So, for any , we get
Hence we have
Since is arbitrary, we can conclude
This completes the proof. □
4 Main results
Theorem 4.1 Let X be a uniformly convex Banach space, and let C be a nonempty bounded closed convex subset of X. Let be a reversible semigroup of asymptotically nonexpansive mappings on C. If D has a left invariant mean, then there exists a retraction P from onto satisfying the following properties:
-
(1)
P is nonexpansive in the sense
-
(2)
for all and ;
-
(3)
for all .
Proof Since D has a left invariant mean, there exists a net of finite means on G such that for every , where A is a directed system. Put . For , , we define if and only if , . In this case, I is also a directed system. For each , define , and . Then for every ,
Let . Taking any , since is bounded, without loss of generality, suppose that is weakly∗ convergent. Then for all , exists. We define
It is easy to see that for all , . Next, we shall show that . In fact, for any given , there exists such that for any , . Also, we can suppose that for all , then , . By Lemma 3.5, for any , there exists such that for all and ,
Noting for all ,
we have for any ,
By (4.1), we obtain
Combining it with the definition of Pu, we get for all ,
Thus, by the Lemma 3.3, we can conclude that for all , . The continuity of then implies that . Obviously, for any ,
and for any and ,
Thus,
This completes the proof. □
Remark 4.1 It should be noted that in Theorem 4.1, we do not assume . In fact, we can find a fixed point . It also should be pointed out that in the case of reversible semigroup, if D has a left invariant mean, then (see [[6], Theorem 3.1] and [[10], Lemma 4]).
As in [6], we have the following ergodic theorem.
Theorem 4.2 Let X be a uniformly convex Banach space and C a nonempty bounded closed convex subset of X. Let of a reversible semigroup of asymptotically nonexpansive mappings on C. If D has a left invariant mean and there exists a unique retraction P from onto , which satisfies the properties (1)-(3) in Theorem 4.1, then for every strongly regular net on D and ,
where .
Remark 4.2 By Theorem 4.1 and Theorem 4.2, we can get many known theorems in [2, 6, 10, 11, 16], such as Theorem 3.1 and Theorem 3.2 in [6], Theorem 1 in [11]. The key assumption in [6] that the almost orbit is almost asymptotically isometric is not necessary in our theorems.
Theorem 4.3 Let X be a uniformly convex Banach space, and let C be a nonempty bounded closed convex subset of X. Let of a reversible semigroup of asymptotically nonexpansive mappings on C, and let be an almost orbit of ℑ. If
for every , then
Proof For any given , there exists such that for any , . Let , then there exists a subnet in with such that for all , , where A is a directed system. By Lemma 3.5, for any , there exists a such that for all ,
Noting for each ,
we get
Since for every , we have , . Then for all ,
Consequently, from Lemma 3.3, we can conclude that for all , , which implies . This completes the proof. □
Remark 4.3 In Theorem 4.1, Theorem 4.2 and Theorem 4.3, we do not assume that X has a Fréchet differentiable norm.
Theorem 4.4 Let X be a uniformly convex Banach space whose dual has the Kadec-Klee property, and let C be a nonempty bounded closed convex subset of X. Let of a reversible semigroup of asymptotically nonexpansive mappings on C and be an almost orbit of ℑ. Then the following statements are equivalent:
-
(1)
.
-
(2)
-
(3)
for every .
Proof (1) ⇒ (2). It suffices to show that is a singleton. Since X is reflexive, it is nonempty. Let f and g be two elements in , then by (1), we can obtain . For any , by Lemma 3.6, exists. Setting
then for a given , there exists such that for all ,
Hence for all ,
where . Let us note , then
which means . Since ε is arbitrary, we get
It then follows from that there exists a subnet in such that , where A is a directed system. Putting
then for , , we define if and only if , . In this case, I is also a directed system. For arbitrary , define , , , , then and . By Lemma 3.1, we have
Applying Lemma 3.6 and noting the inequality , we obtain
Then for each , there exists such that and
Obviously, is also a subnet of I, then . Putting
In as much as X is reflexive, is also reflexive, we can conclude that the set of all weak limit points of is nonempty. Hence, without loss of generality, we may assume that . Therefore, . Since
passing the limit for , we get , which implies . Hence we can get
i.e., . Therefore, and . Since is reflexive and has Kadec-Klee property, it has the Kadec property, and this implies that . Taking the limit for in (4.2), we obtain , i.e., , which implies .
(2) ⇒ (3). Obviously.
(3) ⇒ (1). See Theorem 4.3. This completes the proof. □
Remark 4.4 By Theorem 4.4, we can get many known theorems in [2, 6, 10, 11, 14–16], such as Theorem 4.3 and Theorem 8.1 in [14], Theorem 3.1 and Theorem 3.2 in [15]. And in [6, 10], it is assumed that the almost orbit is almost asymptotically isometric and the subspace D has a left invariant mean. Those key conditions are not necessary by the theorem above.
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Acknowledgements
This research is supported by the National Science Foundation of China (11201410, 11271316 and 11101353), the Natural Science Foundation of Jiangsu Province (BK2012260) and the Natural Science Foundation of Jiangsu Education Committee (10KJB110012 and 11KJB110018).
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Zhu, L., Huang, Q. & Li, G. Nonlinear ergodic theorems and weak convergence theorems for reversible semigroup of asymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 2013, 231 (2013). https://doi.org/10.1186/1687-1812-2013-231
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DOI: https://doi.org/10.1186/1687-1812-2013-231