Convergence theorems for total asymptotically nonexpansive non-self mappings in spaces
© Wang et al.; licensee Springer 2013
Received: 18 December 2012
Accepted: 10 March 2013
Published: 29 March 2013
In this paper, we introduce the concept of total asymptotically nonexpansive nonself mappings and prove the demiclosed principle for this kind of mappings in spaces. As a consequence, we obtain a Δ-convergence theorem of total asymptotically nonexpansive nonself mappings in spaces. Our results extend and improve the corresponding recent results announced by many authors.
is satisfied for all and .
In 1976, Lim  introduced the concept of Δ-convergence in a general metric space. Fixed point theory in a space was first studied by Kirk [7, 8]. He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete space always has a fixed point. In 2008, Kirk and Panyanak  specialized Lim’s concept to spaces and proved that it is very similar to the weak convergence in the Banach space setting. So, the fixed point and Δ-convergence theorems for single-valued and multivalued mappings in spaces have been rapidly developed and many papers have appeared [10–25].
Let be a metric space, and let C be a nonempty and closed subset of X. Recall that C is said to be a retract of X if there exists a continuous map such that , . A map is said to be a retraction if . If P is a retraction, then for all y in the range of P.
where P is a nonexpansive retraction of X onto C.
Remark 1.2 From the definitions, it is to know that each nonexpansive nonself mapping is an asymptotically nonexpansive nonself mapping with a sequence , and each asymptotically nonexpansive mapping is a -total asymptotically nonexpansive mapping with , , and , .
Under some limit conditions, they proved that the sequence Δ-converges to a fixed point of T.
Inspired and motivated by the recent work of Chang et al. , Tang et al. , Laowang et al.  and so on, the purpose of this paper is to introduce the concept of total asymptotically nonexpansive nonself mappings and prove the demiclosed principle for this kind of mappings in spaces. As a consequence, we obtain a Δ-convergence theorem of total asymptotically nonexpansive nonself mappings in spaces. The results presented in this paper improve and extend the corresponding recent results in [19, 26, 27].
The following lemma plays an important role in our paper.
We also denote by the geodesic segment joining from x to y, that is, .
A subset C of a space is convex if for all .
Lemma 2.1 
Recall that a bounded sequence in X is said to be regular if for every subsequence of .
Proposition 2.2 
- (1)there exists a unique point such that
- (2)and both are singleton.
Let X be a space. A sequence in X is said to Δ-converge to if p is the unique asymptotic center of for each subsequence of . In this case we write and call p the Δ-limit of .
Lemma 2.4 
Every bounded sequence in a complete space always has a Δ-convergent subsequence.
Lemma 2.5 
Let X be a complete space, and let C be a closed convex subset of X. If is a bounded sequence in C, then the asymptotic center of is in C.
Now we give a connection between the ‘⇀’ convergence and Δ-convergence.
Proposition 2.7 
Let X be a space, let C be a closed convex subset of X, and let be a bounded sequence in C. Then implies that .
Lemma 2.8 Let C be a closed and convex subset of a complete space X, and let be a uniformly L-Lipschitzian and -total asymptotically nonexpansive nonself mapping. Let be a bounded sequence in C such that and . Then .
Proof By the definition, if and only if . By Lemma 2.5, we have .
In fact, it is obvious that the conclusion is true for . Suppose that the conclusion holds for , now we prove that the conclusion is also true for .
i.e., , as desired. □
The following result can be obtained from Lemma 2.8 immediately.
Lemma 2.9 Let C be a closed and convex subset of a complete space X, and let be an asymptotically nonexpansive nonself mapping with a sequence , . Let be a bounded sequence in C such that and . Then .
Lemma 2.10 
Lemma 2.11 
If and , then exists. If there exists a subsequence of which converges to zero, then .
Lemma 2.12 
Let X be a complete space, and let be a bounded sequence in X with ; is a subsequence of with , and the sequence converges, then .
3 Main results
- (1), , ;
there exist constants with such that and ;
there exists a constant such that , , .
Then the sequence defined in (3.1) Δ-converges to a common fixed point of and .
Proof We divide the proof into three steps.
Step 1. We first show that exists for each .
Since and , it follows from Lemma 2.11 that exists for each .
Step 2. We show that .
it follows from (3.17) and (3.18) that . Similarly, we also can show that .
Step 3. We show that Δ-converges to a common fixed point of and .
Let . Firstly, we show that . Let , then there exists a subsequence of such that . By Lemma 2.4 and Lemma 2.5, there exists a subsequence of such that . Since , it follows from Lemma 2.8 that . So, exists. By Lemma 2.12, we know that . This implies that . Next, let be a subsequence of with and . Since , exists. By Lemma 2.12, we know that . This implies that contains only one point. Thus, since , contains only one point and exists for each , we know that Δ-converges to a common fixed point of and . The proof is completed. □
The authors would like to express their thanks to the referees for their helpful comments and suggestions. This work was supported by the Natural Scientific Research Foundation of Yunnan Province (Grant No. 2011FB074).
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