Convergence theorems for total asymptotically nonexpansive non-self mappings in spaces
Journal of Inequalities and Applications volume 2013, Article number: 135 (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.
A metric space X is a space if it is geodesically connected and if every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane. That is to say, let be a metric space, and let with . A geodesic path from x to y is an isometry such that and . The image of a geodesic path is called a geodesic segment. A metric space X is a (uniquely) geodesic space if every two points of X are joined by only one geodesic segment. A geodesic triangle in a geodesic space X consists of three points , , of X and three geodesic segments joining each pair of vertices. A comparison triangle of the geodesic triangle is the triangle in the Euclidean space such that
A geodesic space X is a space if for each geodesic triangle in X and its comparison triangle in , the inequality
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. Recall that a mapping is said to be nonexpansive if
T is said to be asymptotically nonexpansive, if there is a sequence with such that
T is said to be -total asymptotically nonexpansive, if there exist nonnegative sequences , with , and a strictly increasing continuous function with such that
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.
Definition 1.1 Let X and C be the same as above. A mapping is said to be -total asymptotically nonexpansive nonself mapping if there exist nonnegative sequences , with , and a strictly increasing continuous function with such that
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 , .
Definition 1.3 A nonself mapping is said to be uniformly L-Lipschitzian if there exists a constant such that
Recently, Chang et al.  introduced the following Krasnoselskii-Mann type iteration for finding a fixed point of a total asymptotically nonexpansive mappings in spaces.
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.
In this paper, we write for the unique point z in the geodesic segment joining from x to y such that
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 
A geodesic space X is a space if and only if the following inequality holds:
for all and all . In particular, if x, y, z are points in a space and , then
Let be a bounded sequence in a space X. For , we set
The asymptotic radius of is given by
The asymptotic radius of with respect to is given by
The asymptotic center of is the set
And the asymptotic center of with respect to is the set
Recall that a bounded sequence in X is said to be regular if for every subsequence of .
Proposition 2.2 
Let X be a complete space, let be a bounded sequence in X, and let C be a closed convex subset of X. Then
there exists a unique point such that
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.
Remark 2.6 Let X be a space, and let C be a closed convex subset of X. Let be a bounded sequence in C. In what follows, we denote
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 .
Since , by induction we can prove that
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 .
Indeed, since T is uniformly L-Lipschitzian, we have
(2.8) is proved. Hence for each and , from (2.8), we have
In (2.9) taking , , we have
Letting and taking the superior limit on the both sides, we get that
Furthermore, for any , it follows from inequality (2.2) with that
Letting and taking the superior limit on the both sides of the above inequality, for any , we get
Since , for any , we have
which implies that
By (2.10) and (2.14), we have . Hence we have
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 
Let X be a space, let be a given point, and let be a sequence in with and . Let and be any sequences in X such that
for some . Then
Lemma 2.11 
Let , and be the sequences of nonnegative numbers such that
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
Theorem 3.1 Let C be a nonempty, closed and convex subset of a complete space E. Let be a uniformly L-Lipschitzian and total asymptotically nonexpansive nonself mapping with sequences and satisfying and , and strictly increasing function with , . For arbitrarily chosen , is defined as follows:
where , , , , , , and satisfy the following conditions:
, , ;
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 .
Set and , . Since , , , we know that and . For any , we have
Substituting (3.3) into (3.2), we have
Since and , it follows from Lemma 2.11 that exists for each .
Step 2. We show that .
For each , from the proof of Step 1, we know that exists. We may assume that . From (3.3), we have
Taking lim sup on both sides in (3.5), we have
In addition, since
Since , it is easy to prove that
It follows from Lemma 2.10 that
On the other hand, since
we have . Combined with (3.6), it yields that
This implies that
It is easy to show that
So, it follows from (3.12) and Lemma 2.10 that
It follows from (3.13) that
Thus, from (3.9), (3.14) and (3.16), we have
In addition, since
from (3.9), we have
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. □
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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).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Wang, L., Chang, Ss. & Ma, Z. Convergence theorems for total asymptotically nonexpansive non-self mappings in spaces. J Inequal Appl 2013, 135 (2013). https://doi.org/10.1186/1029-242X-2013-135