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On the strong and Δ-convergence of new multi-step and S-iteration processes in a space
Journal of Inequalities and Applications volume 2013, Article number: 482 (2013)
In this paper, we introduce a new class of mappings and prove the demiclosedness principle for mappings of this type in a space. Also, we obtain the strong and Δ-convergence theorems of new multi-step and S-iteration processes in a space. Our results extend and improve the corresponding recent results announced by many authors in the literature.
MSC:47H09, 47H10, 54E40, 58C30.
Contractive mappings and iteration processes are some of the main tools in the study of fixed point theory. There are many contractive mappings and iteration processes that have been introduced and developed by several authors to serve various purposes in the literature (see [1–6]).
Imoru and Olantiwo  gave the following contractive definition.
Definition 1 Let T be a self-mapping on a metric space X. The mapping T is called a contractive-like mapping if there exist a constant and a strictly increasing and continuous function with such that, for all ,
By taking in (1.1), we define a new class of mappings as follows.
Definition 2 The mapping T is called a generalized nonexpansive mapping if there exists a non-decreasing and continuous function with such that, for all ,
Remark 1 For in (1.2), we have
If X is an interval of ℝ, then is convex. The same is also true in each space with unique geodesic for each pair of points (e.g., metric trees or spaces).
In the case for all , it is easy to show that every nonexpansive mapping satisfies (1.2), but the inverse is not necessarily true.
Example 1 Let , , and define T by
By taking and , we have
Therefore T is a generalized nonexpansive mapping, but T is not nonexpansive mapping.
Both a contractive-like mapping and a generalized nonexpansive mapping need not have a fixed point, even if X is complete. For example, let , and define T by
It is proved in Gürsoy et al.  that T is a contractive-like mapping. Similarly, one can prove that T is a generalized nonexpansive mapping. But the mapping T has no fixed point.
By using (1.1), it is obvious that if a contractive-like mapping has a fixed point, then it is unique. However, if a generalized nonexpansive mapping has a fixed point, then it need not be unique. For example, let ℝ be the real line with the usual norm , and let . Define a mapping by
Now, we show that T is a nonexpansive mapping. In fact, if or , then we have
If and or and , then we have
This implies that T is a nonexpansive mapping and so T is a generalized nonexpansive mapping with for all . But .
Gürsoy et al.  introduced a new multi-step iteration process in a Banach space. We modify this iteration process in a space as follows.
For an arbitrary fixed order ,
or, in short,
In this paper, motivated by the above results, we prove demiclosedness principle for a new class of mappings and the Δ-convergence theorems of the new multi-step iteration and the S-iteration processes for mappings of this type in a space. Also, we present the strong convergence theorems of these iteration processes for contractive-like mappings in a space.
2 Preliminaries on a space
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. Fixed point theory in a space was first studied by Kirk (see [14, 15]). 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 many papers have appeared (see [14–20]). It is worth mentioning that the results in a space can be applied to any space with since any space is a space for every (see [, p.165]).
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 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 . Such a triangle always exists (see ).
A geodesic metric space is said to be a space  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 ,
We observe that if x, , are points of a space and if is the midpoint of the segment , then the inequality implies that
The equality holds for the Euclidean metric. In fact (see [, p.163]), a geodesic metric space is a space if and only if it satisfies the inequality (2.1) (which is known as the CN inequality of Bruhat and Tits ).
Let , by [, Lemma 2.1(iv)] for each , there exists a unique point such that
From now on, we will use the notation for the unique point z satisfying (2.2). By using this notation, Dhompongsa and Panyanak  obtained the following lemmas which will be used frequently in the proof of our main results.
Lemma 1 Let X be a space. Then
for all and .
Lemma 2 Let X be a space. Then
for all and .
3 Demiclosedness principle for a new class of mappings
In 1976 Lim  introduced the concept of convergence in a general metric space setting which is called △-convergence. Later, Kirk and Panyanak  used the concept of △-convergence introduced by Lim  to prove on a space analogs of some Banach space results which involve weak convergence. Also, Dhompongsa and Panyanak  obtained the △-convergence theorems for the Picard, Mann and Ishikawa iterations in a space for nonexpansive mappings under some appropriate conditions.
Now, we recall some definitions.
Let be a bounded sequence in a space X. For , we set
The asymptotic radius of is given by
and 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
Proposition 1 ([, Proposition 3.2])
Let be a bounded sequence in a complete space X, and let K be a closed convex subset of X, then and are singletons.
Definition 3 ([, Definition 3.1])
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 .
Every bounded sequence in a complete space always has a △-convergent subsequence (see [, p.3690]).
Let K be a nonempty closed convex subset of a complete space, and let be a bounded sequence in K. Then the asymptotic center of is in K (see [, Proposition 2.1]).
Lemma 4 ([, Lemma 2.8])
If is a bounded sequence in a complete space with , is a subsequence of with and the sequence converges, then .
Let be a bounded sequence in a space X, and let K be a closed convex subset of X which contains . We denote the notation
We note that if and only if (see ).
Nanjaras and Panyanak  gave a connection between the ‘⇀’ convergence and Δ-convergence.
Proposition 2 ([, Proposition 3.12])
Let be a bounded sequence in a space X, and let K be a closed convex subset of X which contains . Then implies that .
By using the convergence defined in (3.1), we obtain the demiclosedness principle for the new class of mappings in a space.
Theorem 1 Let K be a nonempty closed convex subset of a complete space X, and let be a generalized nonexpansive mapping with . Let be a bounded sequence in K such that and . Then .
Proof By the hypothesis, . Then we have . By Lemma 3(ii), we obtain . Since , then we have
for all . By taking in (3.2), we have
The rest of the proof closely follows the pattern of Proposition 3.14 in Nanjaras and Panyanak . Hence as desired. □
Now, we prove the Δ-convergence of the new multi-step iteration process for the new class of mappings in a space.
Theorem 2 Let K be a nonempty closed convex subset of a complete space X, let be a generalized nonexpansive mapping with , and let be a sequence defined by (1.4) such that , and for some . Then the sequence Δ-converges to the fixed point of T.
Proof Let . From (1.2), (1.4) and Lemma 1, we have
Also, we obtain
Continuing the above process, we have
This inequality guarantees that the sequence is non-increasing and bounded below, and so exists for all . Let . By using (3.3), we get
By Lemma 2, we also have
which implies that
Thus . To show that the sequence △-converges to a fixed point of T, we prove that
and consists of exactly one point. Let . Then there exists a subsequence of such that . By Lemma 3, there exists a subsequence of such that . By Proposition 2 and Theorem 1, . By Lemma 4, we have . This shows that . Now, we prove that consists of exactly one point. Let be a subsequence of with , and let . We have already seen that and . Finally, since converges, by Lemma 4, . This shows that . This completes the proof. □
We give the following theorem related to the Δ-convergence of the S-iteration process for the new class of mappings in a space.
Theorem 3 Let K be a nonempty closed convex subset of a complete space X, let be a generalized nonexpansive mapping with , and let be a sequence defined by (1.3) such that for some . Then the sequence Δ-converges to the fixed point of T.
Proof Let . Using (1.2), (1.3) and Lemma 1, we have
Also, we obtain
From (3.4) and (3.5), we have
This inequality guarantees that the sequence is non-increasing and bounded below, and so exists for all . Let
Now, we prove that . By (3.4), we have
This gives that
By (3.5) and (3.6), we obtain
Then we get
By Lemma 2, we also have
which implies that
Thus . The rest of the proof follows the pattern of the above theorem and is therefore omitted. □
4 Strong convergence theorems for a contractive-like mapping
Now, we prove the strong convergence of the new multi-step iteration process for a contractive-like mapping in a space.
Theorem 4 Let K be a nonempty closed convex subset of a complete space X, let be a contractive-like mapping with , and let be a sequence defined by (1.4) such that , and , . Then the sequence converges strongly to the unique fixed point of T.
Proof Let p be the unique fixed point of T. From (1.1), (1.4) and Lemma 1, we have
Also, we obtain
In a similar fashion, we can get
Continuing the above process, we have
In addition, we obtain
From (4.1) and (4.2), we have
Using the fact that , and , we get that
This together with (4.3) implies that
Consequently, and this completes the proof. □
Remark 2 In Theorem 4 the condition may be replaced with for a fixed .
Finally, we give the strong convergence theorem of the S-iteration process for a contractive-like mapping in a space as follows.
Theorem 5 Let K be a nonempty closed convex subset of a complete space X, let be a contractive-like mapping with , and let be a sequence defined by (1.3) such that . Then the sequence converges strongly to the unique fixed point of T.
Proof Let p be the unique fixed point of T. From (1.1), (1.3) and Lemma 1, we have
Similarly, we obtain
Then, from (4.4) and (4.5), we get that
If , we obtain
Thus we have . If , the result is clear. This completes the proof. □
The new multi-step iteration reduces to the two-step iteration and the SP-iteration processes. Also, the class of generalized nonexpansive mappings includes nonexpansive mappings. Then these results presented in this paper extend and generalize some works for a space in the literature.
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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 Conference ‘Anatolian Communications in Nonlinear Analysis (ANCNA 2013)’ in Abant Izzet Baysal University, Bolu, Turkey, July 03-06, 2013. This paper was supported by Sakarya University Scientific Research Foundation (Project number: 2013-02-00-003).
The authors declare that they have no competing interests.
Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.
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Başarır, M., Şahin, A. On the strong and Δ-convergence of new multi-step and S-iteration processes in a space. J Inequal Appl 2013, 482 (2013). https://doi.org/10.1186/1029-242X-2013-482
- contractive-like mapping
- strong convergence
- new multi-step iteration
- fixed point