In this paper, we study the strong and △-convergence theorems of SP-iteration for nonexpansive mappings on a space. Our results extend and improve many results in the literature.
MSC:47H09, 47H10.
Hinweise
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.
Dedication
Dedicated to Professor Hari M Srivastava
1 Introduction
A space plays a fundamental role in various areas of mathematics (see Bridson and Haefliger [1], Burago et al. [2], Gromov [3]). Moreover, there are applications in biology and computer science as well [4, 5]. 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. It is well known that any complete, simply connected Riemannian manifold having non-positive sectional curvature is a space. The complex Hilbert ball with a hyperbolic metric is a space (see [6]). Other examples include pre-Hilbert spaces, R-trees (see [1]) and Euclidean buildings (see [7]).
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Fixed point theory in a space has been first studied by Kirk (see [8, 9]). He showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete space always has a fixed point. Since then the fixed point theory in a space has been rapidly developed and a lot of papers have appeared (see, e.g., [8‐16]).
for all , where , and are sequences in . If we take for all n, (1.1) reduces to the Mann iteration (see [18]), and we take for all n, (1.1) reduces to the Ishikawa iteration (see [19]).
The new two-step iteration (see [20]) is defined by and
(1.2)
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for all , where and are sequences in .
Recently, Phuengrattana and Suantai (see [21]) defined the SP-iteration as follows:
(1.3)
for all , where , , and are sequences in . They showed that the Mann, Ishikawa, Noor and SP-iterations are equivalent and the SP-iteration converges better than the others for the class of continuous and nondecreasing functions. Clearly, the new two-step and Mann iterations are special cases of the SP-iteration.
Now, we apply SP-iteration (1.3) in a space for nonexpansive mappings as follows:
(1.4)
for all , where K is a nonempty convex subset of a space, , , and are sequences in .
In this paper, we study the SP-iteration for a nonexpansive mapping in a space. This paper contains three sections. In Section 2, we first collect some known preliminaries and lemmas that will be used in the proofs of our main theorems. In Section 3, we give the main results which are related to the strong and △-convergence theorems of the SP-iteration in a space. It is worth mentioning that our results in a space can be applied to any space with since any space is a space for every (see [1], p.165).
2 Preliminaries and lemmas
Let us recall some definitions and known results in the existing literature on this concept.
Let K be a nonempty subset of a space X and let be a mapping. A point is called a fixed point of T if . We will denote the set of fixed points of T by . The mapping T is said to be nonexpansive if
Let be a metric space. A geodesic path joining to (or more briefly, a geodesic from x to y) is a map c from a closed interval to X such that , and for all . In particular, c is an isometry and . The image of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denoted by . The space is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be a uniquely geodesic space if there is exactly one geodesic joining x to y for each .
A geodesic triangle in a geodesic metric space consists of three points in X (the vertices of △) and a geodesic segment between each pair of vertices (the edges of △). A comparison triangle for the geodesic triangle in is a triangle in the Euclidean plane such that for .
A geodesic metric space is said to be a space [1] if all geodesic triangles of appropriate size satisfy the following comparison axiom.
: Let △ be a geodesic triangle in X and let be a comparison triangle for △. Then △ is said to satisfy the inequality if for all and all comparison points ,
Finally, we observe that if x, , are points of a space and if is the midpoint of the segment , then the inequality implies
(2.1)
The equality holds for the Euclidean metric. In fact (see [1], p.163), a geodesic metric space is a space if and only if it satisfies inequality (2.1) (which is known as the CN inequality of Bruhat and Tits [22]).
Let X be a complete space and let be a bounded sequence in X. For , set
The asymptotic radius of is given by
The asymptotic center of is the set
It is known that in a complete space, consists of exactly one point ([10], Proposition 7). Also, every space has the property, i.e., if is a sequence in K and , then for each ,
A sequence in a space X is said to be △-convergent to if x is the unique asymptotic center of for every subsequence of . In this case, we write and x is called the △-limit of .
The notion of △-convergence in a general metric space was introduced by Lim [23]. Recently, Kirk and Panyanak [16] used the concept of △-convergence introduced by Lim [23] to prove on the space analogous of some Banach space results which involve weak convergence. Further, Dhompongsa and Panyanak [12] obtained △-convergence theorems for the Picard, Mann and Ishikawa iterations in a space.
Every bounded sequence in a completespace always has a △-convergent subsequence.
(ii)
LetKbe a nonempty closed convex subset of a completespace and letbe a bounded sequence inK. Then the asymptotic center ofis inK.
(iii)
LetKbe a nonempty closed convex subset of a completespaceXand letbe a nonexpansive mapping. Then the conditions, △-converges toxand , implyand .
3 Main results
We start with proving the lemma for later use in this section.
Lemma 4LetKbe a nonempty closed convex subset of a completespaceXand letbe a nonexpansive mapping with . Letandbe sequences in , be a sequence infor someandbe defined by the iteration process (1.4). Then
(i)
exists for all .
(ii)
.
Proof (i) Let . By (1.4) and Lemma 1, we have
(3.1)
Also, we get
(3.2)
Then we obtain
(3.3)
Using (1.4) and Lemma 1, we have
(3.4)
Combining (3.3) and (3.4), we get
This implies that the sequence is nonincreasing and bounded below, and so exists for all . This completes the proof of part (i).
(ii)
Let
(3.5)
Firstly, we will prove that . By (3.4) and (3.5),
Also, from (3.3) and (3.5),
Then we obtain
(3.6)
Secondly, we will prove that . From (3.1) and (3.2), we have
This gives
(3.7)
Next, by Lemma 2,
Thus,
so that
Now using (3.5) and (3.7), and hence,
This completes the proof of part (ii). □
Now, we give the △-convergence theorem of the SP-iteration on a space.
Theorem 1LetX, K, T, , , , satisfy the hypotheses of Lemma 4. Then the sequence △-converges to a fixed point ofT.
Proof By Lemma 4, we have . Also, exists for all . Thus is bounded. Let , where the union is taken over all subsequences of . We claim that . Let . Then there exists a subsequence of such that . By Lemma 3(i) and (ii), there exists a subsequence of such that . By Lemma 3(iii), . By Lemma 4(i), exists. Now, we claim that . On the contrary, assume that . Then, by the uniqueness of asymptotic centers, we have
(3.8)
This is a contradiction. Thus and . To show that the sequence △-converges to a fixed point of T, we will show that consists of exactly one point. Let be a subsequence of with and let . We have already seen that and . Finally, we claim that . If not, then the existence of and the uniqueness of asymptotic centers imply that there exists a contradiction as (3.8) and hence . Therefore, . As a result, the sequence △-converges to a fixed point of T. □
We give the strong convergence theorem on a space as follows.
Theorem 2LetX, K, T, , , , satisfy the hypotheses of Lemma 4. Then the sequenceconverges strongly to a fixed point ofTif and only if
where .
Proof Necessity is obvious. Conversely, suppose that . As proved in Lemma 4(i),
for all . This implies that
Since the sequence is nonincreasing and bounded below, exists. Thus, by the hypothesis, .
Next, we will show that is a Cauchy sequence in K. Let be arbitrarily chosen. Since , there exists a constant such that for all , we have
In particular, . Thus there exists such that
Now, for all , we have
Hence is a Cauchy sequence in a closed subset K of a complete space X, it must be convergent to a point in K. Let . Now, gives that and the closedness of forces to be in . Therefore, the sequence converges strongly to a fixed point of T. □
Senter and Dotson [24] introduced Condition as follows.
A mapping is said to satisfy Condition if there exists a nondecreasing function with and for all such that
With respect to the above definition, we have the following theorem.
Theorem 3LetX, K, , , , satisfy the hypotheses of Lemma 4 and letbe a nonexpansive mapping satisfying Condition . Then the sequenceconverges strongly to a fixed point ofT.
Proof By Lemma 4(i), exists for all . Let this limit be c, where . If , there is nothing to prove. Suppose that . Now,
gives
which means that and so exists. Also, by Lemma 4(ii), we have . It follows from Condition that
That is,
Since is a nondecreasing function satisfying , for all , therefore we obtain
The conclusion now follows from Theorem 2. □
It is worth noting that, in the case of a nonexpansive mapping, Condition is weaker than the compactness of K.
Since the SP-iteration reduces to the new two-step iteration when for all and to the Mann iteration when for all , we have the following corollaries.
Corollary 1LetX, K, T, satisfy the hypotheses of Lemma 4 and letbe defined by the iteration process (1.2). Then the sequence △-converges to a fixed point ofT. Further, ifis defined by the iteration process (1.1), the sequence △-converges to a fixed point ofT.
Corollary 2LetX, K, satisfy the hypotheses of Lemma 4, letbe a nonexpansive mapping satisfying Conditionand letbe defined by the iteration process (1.2). Then the sequenceconverges strongly to a fixed point ofT. Also, ifis defined by the iteration process (1.1), the sequenceconverges strongly to a fixed point ofT.
Conclusions
The SP-iteration reduces to the new two-step and Mann iterations. Then these results presented in this paper extend and generalize some works for space in the literature.
Acknowledgements
This paper was supported by Sakarya University BAPK Project No. 2013-02-00-003. The authors are grateful to the referee for his/her careful reading and valuable comments and suggestions which led to the present form of the paper. This paper has been presented in International Congress in Honour of Professor H. M. Srivastava in Uludağ University, Bursa, Turkey, 23-26 August 2012.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.