On the strong and Δ-convergence of new multi-step and S-iteration processes in a space
© Başarır and Şahin; licensee Springer. 2013
Received: 1 August 2013
Accepted: 19 September 2013
Published: 7 November 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.
By taking in (1.1), we define a new class of mappings as follows.
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.
Therefore T is a generalized nonexpansive mapping, but T is not nonexpansive mapping.
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.
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.
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 .
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.
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 ).
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.
for all and .
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.
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 .
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 .
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 .
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.
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.
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.
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.
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.
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).
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