Abstract
The purpose of this paper is to study the existence theorems of fixed points, △-convergence and strong convergence theorems for total asymptotically nonexpansive nonself mappings in the framework of spaces. The convexity and closedness of a fixed point set of such mappings are also studied. Our results generalize, unify and extend several comparable results in the existing literature.
MSC:47J05, 47H09, 49J25.
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1 Introduction
In recent years, spaces have attracted the attention of many authors as they have played a very important role in different aspects of geometry [1]. Kirk [2, 3] showed that a nonexpansive mapping defined on a bounded closed convex subset of a complete space has a fixed point. Since then, the fixed point theory in spaces has been rapidly developed and many papers have appeared (see, e.g., [4–8]).
In 1976, the concept of △-convergence in general metric spaces was coined by Lim [9]. In 2008, Kirk et al. [8] specialized this concept to spaces and proved that it is very similar to the weak convergence in the Banach space setting. Dhompongsa et al. [6] and Abbas et al. [10] obtained △-convergence theorems for the Mann and Ishikawa iterations in the space setting.
Motivated by the work going on in this direction, the purpose of this paper is twofold. We investigate the existence theorems of fixed points, the convexity and closedness of a fixed point set in spaces for total asymptotically nonexpansive nonself mappings which is essentially wider than that of the asymptotically nonexpansive nonself mappings and the asymptotically nonexpansive mapping in the intermediate sense. We also study sufficient conditions for △-convergence and strong convergence of a sequence generated by finite or an infinite family of total asymptotically nonexpansive nonself mappings in spaces.
2 Preliminaries
Let be a metric space and 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 a 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 .
A thorough discussion of these spaces and their important role in various branches of mathematics are given in [11–15].
Let C be a nonempty subset of a metric space . Recall that a mapping is said to be nonexpansive if
T is said to be an asymptotically nonexpansive nonself mapping if there exists a sequence with such that
Let be a metric space, and 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 2.1 [16]
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 2.2 From the definitions, it is to know that each nonexpansive mapping is an asymptotically nonexpansive nonself mapping with a sequence , and each asymptotically nonexpansive nonself mapping is a -total asymptotically nonexpansive nonself mapping with , , and , .
Definition 2.3 [16]
A nonself mapping is said to be uniformly L-Lipschitzian if there exists a constant such that
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 2.4 [17]
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 2.5 [18]
Let X be a complete space, be a bounded sequence in X and C be a closed convex subset of X. Then
-
(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 △-converge to if q is the unique asymptotic center of for each subsequence of . In this case we write and call q the △-limit of .
Lemma 2.7
-
(1)
Every bounded sequence in a complete space always has a △-convergent subsequence [8].
-
(2)
Let X be a complete space, C be a closed convex subset of X. If is a bounded sequence in C, then the asymptotic center of is in C [20].
Remark 2.8
-
(1)
Let X be a space and C be a closed convex subset of X. Let be a bounded sequence in C. In what follows, we define
(2.13)
where .
-
(2)
It is easy to know that if and only if .
Nanjaras et al. [21] established the following relation between △-convergence and weak convergence in a space.
Lemma 2.9 [21]
Let be a bounded sequence in a space X, and let C be a closed convex subset of X which contains . Then
-
(i)
implies that ;
-
(ii)
the converse of (i) is true if is regular.
Lemma 2.10 [16]
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 .
Lemma 2.11 [16]
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 2.12 [22]
Let X be a space, be a given point and be a sequence in with and . Let and be any sequences in X such that
and
for some . Then
Lemma 2.13 [22]
Let , and be the sequences of nonnegative numbers such that
If and , then the limit exists. If there exists a subsequence of which converges to 0, then .
Lemma 2.14 [6]
Let X be a complete space, be a bounded sequence in X with , and let be a subsequence of with and let the sequence converge, then .
3 Main results
Theorem 3.1 Let X be a complete space, C be a nonempty bounded closed and convex subset of X. If is a uniformly Lipschitzian and total asymptotically nonexpansive nonself mapping, then T has a fixed point in C.
Proof For any given point , define
where P is a nonexpansive retraction of X onto C.
Since T is a total asymptotically nonexpansive nonself mapping, ζ is a strictly increasing continuous function, one gets
for any . Letting and taking superior limit on the both sides of the above inequality, we have
It is easy to know that the function is lower semi-continuous, and C is bounded closed and convex, there exists a point such that . Letting in (3.1), for each , we have
By using (2.7) with , for any positive integers , we obtain
Let and take superior limit in (3.3). It follows from (3.2) that
This implies that
As T is a total asymptotically nonexpansive nonself mapping, so
which implies that is a Cauchy sequence in C. Since C is complete, let . In view of the continuity of TP, we have
Since , , this shows that , i.e., v is a fixed point of T in C.
The proof is completed. □
Remark 3.2 Theorem 3.1 is a generalization of Kirk [2, 3] and Abbas et al. [10] from nonexpansive mappings and asymptotically nonexpansive mappings in the intermediate sense to total asymptotically nonexpansive nonself mappings.
Theorem 3.3 Let X be a complete space, C be a nonempty bounded closed and convex subset of X. If is a uniformly Lipschitzian and total asymptotically nonexpansive nonself mapping, then the fixed point set of T, , is closed and convex.
Proof As T is continuous, so is closed. In order to prove that is convex, it is enough to prove that whenever . Setting , by using (2.7) with , for any , we have
Since T is a total asymptotically nonexpansive nonself mapping, using (2.8), we obtain
Similarly,
Substituting (3.5) and (3.6) into (3.4) and simplifying, we have
for any . Hence , in view of the continuity of TP, we have
Since C is convex, this shows that . Therefore , which implies that , i.e., .
The proof is completed. □
Now we prove a △-convergence theorem for the following implicit iterative scheme:
where C is a nonempty closed and convex subset of a complete space X for each , is a uniformly -Lipschitzian and -total asymptotically nonexpansive nonself mapping defined by (2.4), and for each positive integer n, and are the solutions to the positive integer equation . It is easy to see that (as ).
Remark 3.4 Letting , , and , then is a finite family of uniformly L-Lipschitzian and -total asymptotically nonexpansive nonself mappings defined by (2.4).
Theorem 3.5 Let X be a complete space, C be a nonempty bounded closed and convex subset of X. If is a finite family of uniformly L-Lipschitzian and -total asymptotically nonexpansive nonself mappings satisfying the following conditions:
-
(i)
, ;
-
(ii)
there exists a constant such that , ;
-
(iii)
there exist constants with such that .
If , then the sequence defined by (3.7) △-converges to some point .
Proof Since for each , is a -total asymptotically nonexpansive nonself mapping, by condition (ii), for each and any , we have
where P is a nonexpansive retraction of X onto C.
(I) We first prove that the following limits exist:
In fact, since and , , is a total asymptotically nonexpansive nonself mapping, is nonexpansive, it follows from Lemma 2.4 and (3.8) that
Simplifying it and using condition (iii), we have
Since (as ), there exists a positive integer such that for all . Therefore one has
and so
where and . By condition (i), and . Therefore it follows from Lemma 2.13 that the limits and exist for each .
(II) Next we prove that for each ,
For each , from the proof of (I), we know that exists, we may assume that
From (3.11) we get
which implies that
In addition, since
From (3.14), we have
It follows from (3.14)-(3.16) and Lemma 2.12 that
Now, by Lemma 2.4, we obtain
Hence, from (3.17) and (3.18), one gets
and for each ,
Since , , is uniformly L-Lipschitzian and for each and P is a nonexpansive retraction of X onto C, we have , where , and . Hence it follows from (3.19) and (3.20) that
where . Consequently, for any , from (3.20) and (3.21) it follows that
This implies that the sequence
Since for each , is a subsequence of , therefore we have
Conclusion (3.13) is proved.
(III) Now we show that △-converges to a point in ℱ.
Let . We first prove that .
In fact, let , then there exists a subsequence of such that . By Lemma 2.7, there exists a subsequence of such that . In view of (3.13), . It follows from Lemma 2.10 that ; so, by (3.9), the limit exists. By Lemma 2.14, . This implies that .
Next let be a subsequence of with , and let . Since , from (3.9) the limit exists. In view of Lemma 2.14, . This implies that consists of exactly one point. We know that △-converges to some point .
The conclusion of Theorem 3.5 is proved. □
Now we prove the strong convergence results for the following iterative scheme:
where C is a nonempty closed and convex subset of a complete space X for each , is a uniformly -Lipschitzian and -total asymptotically nonexpansive nonself mapping defined by (2.4), and for each positive integer , and are the unique solutions to the following positive integer equation:
Lemma 3.6 [23]
-
(1)
The unique solutions to the positive integer equation (3.23) are
where denotes the maximal integer that is not larger than x.
-
(2)
For each , denote
then , .
Theorem 3.7 Let X be a complete space, C be a nonempty bounded closed and convex subset of X, and for each , let be a uniformly -Lipschitzian and -total asymptotically nonexpansive nonself mapping defined by (2.4), satisfying the following conditions:
-
(i)
, ;
-
(ii)
there exists a constant such that , , ;
-
(iii)
there exist constants with such that .
If and there exist a mapping and a nondecreasing function with and , , such that
then the sequence defined by (3.22) converges strongly (i.e., in metric topology) to some point .
Proof We observe that for each , is a -total asymptotically nonexpansive nonself mapping. By condition (ii), for each and any , we have
(I) We first prove that the following limits exist:
In fact, since and is nonexpansive, it follows from Lemma 2.4, (3.25) that
and
Substituting (3.27) into (3.28) and simplifying it, we have
and so
where , . By condition (i), and . By Lemma 2.13, the limits and exist for each .
(II) Next we prove that for each , there exists a subsequence such that
In fact, for each given , from the proof of (I), we know that exists. Without loss of generality, we may assume that
From (3.27) one gets
Since
we have
In addition, it follows from (3.29) that
which implies that
From (3.32)-(3.35) and Lemma 2.12, one has
Since
Taking lim inf as on both sides in the inequality above, from (3.36) we have
which combined with (3.33) implies that
Using (3.27) we have
This implies that
Similarly, we have that
This together with (3.32) and (3.39) and Lemma 2.12 yields that
Therefore we obtain
Furthermore, it follows from (3.36) that
Now combined with (3.41) this shows that
From Lemma 3.6, (3.37), (3.40), (3.42) and (3.43), we have that for each given positive integer , there exist subsequences , and such that
Conclusion (3.31) is proved.
(III) Now we prove that converges strongly (i.e., in metric topology) to some point .
In fact, it follows from (3.31) and (3.24) that for a given mapping , there exists a subsequence of such that
and
Taking lim sup on both sides of the above inequality, one has
By the property of f, this implies that
Next we prove that is a Cauchy sequence in C.
In fact, it follows from (3.29) that for any ,
where and . Hence, for any positive integers m, k, we have
Since for each , , one gets
and
where . By (3.44) we have
This shows that the subsequence is a Cauchy sequence in C. Since C is a closed subset in a space X, it is complete. Without loss of generality, we can assume that the subsequence converges strongly (i.e., in metric topology in X) to some point . By Theorem 3.3, we know that ℱ is a closed subset in C. Since , . By using (3.26), it yields that the whole sequence converges in the metric topology to some point .
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 Science Foundation of Sichuan Province (No. 08ZA008).
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Yang, L., Zhao, F.H. Strong and △-convergence theorems for total asymptotically nonexpansive nonself mappings in spaces. J Inequal Appl 2013, 557 (2013). https://doi.org/10.1186/1029-242X-2013-557
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DOI: https://doi.org/10.1186/1029-242X-2013-557