Nonlinear ergodic theorems and weak convergence theorems for reversible semigroup of asymptotically nonexpansive mappings in Banach spaces
© Zhu et al.; licensee Springer. 2013
Received: 21 March 2013
Accepted: 9 August 2013
Published: 29 August 2013
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.
Baillon  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  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.  studied the existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and provided the nonlinear ergodic theorems in Banach spaces. Kim and Li  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. , Kaczor  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.
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 . 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.
for every , where A is a directed system, and is the conjugate operator of .
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 .
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 
for all .
Lemma 3.2 
for all integers , with , , and all nonexpansive mapping .
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 .
This completes the proof. □
This completes the proof. □
Thus, there are at most terms in with . Therefore, for each , there is at least one term () in satisfying .
This completes the proof. □
exists for all and .
This completes the proof. □
4 Main results
- (1)P is nonexpansive in the sense
for all and ;
for all .
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 [, Theorem 3.1] and [, Lemma 4]).
As in , we have the following ergodic theorem.
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 , Theorem 1 in . The key assumption in  that the almost orbit is almost asymptotically isometric is not necessary in our theorems.
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.
for every .
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 , Theorem 3.1 and Theorem 3.2 in . 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.
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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