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On the strong convergence of a modified S-iteration process for asymptotically quasi-nonexpansive mappings in a space
Fixed Point Theory and Applications volume 2013, Article number: 12 (2013)
In this paper, we give strong convergence theorems for the modified S-iteration process of asymptotically quasi-nonexpansive mappings on a space which extend and improve many results in the literature.
MSC: Primary 47H09; secondary 47H10.
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. The initials of the term ‘CAT’ are in honor of E. Cartan, A. D. Alexanderov and V. A. Toponogov. A space is a generalization of the Hadamard manifold, which is a simply connected, complete Riemannian manifold such that the sectional curvature is non-positive. In fact, it is very well known that any complete simply connected Riemannian manifold with non-positive sectional curvature is a space. The complex Hilbert ball with a hyperbolic metric is a space (see ). Other examples include Pre-Hilbert spaces, R-trees (see ) and Euclidean buildings (see ). A space plays a fundamental role in various areas of mathematics (see Bridson and Haefliger , Burago, Burago and Ivanov , Gromov ). Moreover, there are applications in biology and computer science as well [6, 7].
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 many papers have appeared (see, e.g., [8–14]). 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]). Throughout the paper, ℕ and ℝ denote the set of natural numbers and the set of real numbers, respectively.
The Mann iteration process is defined by the sequence ,
where is a sequence in .
Further, the Ishikawa iteration process is defined as the sequence ,
where and are the sequences in . This iteration process reduces to the Mann iteration process when for all .
Agarwal, O’Regan and Sahu  introduced the S-iteration process in a Banach space,
where and are the sequences in . Note that (1.3) is independent of (1.2) (and hence of (1.1)). They showed that their process is independent of those of Mann and Ishikawa and converges faster than both of these (see [, Proposition 3.1]).
Schu , in 1991, considered the modified Mann iteration process which is a generalization of the Mann iteration process,
where is a sequence in .
Tan and Xu , in 1994, studied the modified Ishikawa iteration process which is a generalization of the Ishikawa iteration process,
where the sequences and are in . This iteration process reduces to the modified Mann iteration process when for all .
Recently, Agarwal, O’Regan and Sahu  introduced the modified S-iteration process in a Banach space,
where the sequences and are in . Note that (1.6) is independent of (1.5) (and hence of (1.4)). Also, (1.6) reduces to (1.3) when .
We now modify (1.6) in a space as follows.
Let K be a nonempty closed convex subset of a complete space X and be an asymptotically quasi-nonexpansive mapping with . Suppose that is a sequence generated iteratively by
where and throughout the paper , are the sequences such that for all .
In this paper, we study the modified S-iteration process for asymptotically quasi-nonexpansive mappings on the space and generalize some results of Khan and Abbas  which studied the S-iteration process in a space for nonexpansive mappings. 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. We give the main results related to the strong convergence theorems of the modified S-iteration process in a space in Section 3. Under some suitable condition, we obtain the main theorems which state that converges strongly to a fixed point of T. Our results can be applied to an S-iteration process since the modified S-iteration process reduces to the S-iteration process when .
2 Preliminaries and lemmas
Let us recall some definitions and known results in the existing literature on this concept.
Let be a metric space and K its nonempty subset. Let be a mapping. A point is called a fixed point of T if . We will also denote by the set of fixed points of T, that is, .
The concept of quasi-nonexpansiveness was introduced by Diaz and Metcalf  in 1967, the concept of asymptotically nonexpansiveness was introduced by Goebel and Kirk  in 1972. The iterative approximation problems for asymptotically quasi-nonexpansive mapping were studied by Liu , Fukhar-ud-din et al. , Khan et al.  and Beg et al.  in a Banach space and a space.
Definition 1 Let be a metric space and K be its nonempty subset. Then is said to be
nonexpansive if for all ,
asymptotically nonexpansive if there exists a sequence with the property and such that for all ,
quasi-nonexpansive if for all , ,
asymptotically quasi-nonexpansive if there exists a sequence with the property and such that for all , ,
semi-compact if for a sequence in K with , there exists a subsequence of such that .
Remark 1 From Definition 1, it is clear that the class of quasi-nonexpansive mappings and asymptotically nonexpansive mappings includes nonexpansive mappings, whereas the class of asymptotically quasi-nonexpansive mappings is larger than that of quasi-nonexpansive mappings and asymptotically nonexpansive mappings. The reverse of these implications may not be true.
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 uniquely geodesic if there is exactly one geodesic joining x to y for each . A subset is said to be convex if Y includes every geodesic segment joining any two of its points.
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 a 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 an 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 ,
A complete space is often called Hadamard space (see ).
Finally, we observe that if x, , are points of a space and if is the midpoint of the segment , which we will denote by , then the inequality implies
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 inequality (2.1) (which is known as the CN inequality of Bruhat and Tits ).
Let , by [, Lemma 2.1(iv)] for each , then 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 lemma which will be used frequently in the proof of our main results.
Lemma 1 Let X be a space. Then
for all and .
The following lemma can be found in .
Lemma 2 Let and be two sequences of positive real numbers satisfying
for all . If , then exists.
3 Main results
In this section we prove the strong convergence theorems of the modified S-iteration process in a space.
Theorem 1 Let K be a nonempty closed convex subset of a complete space X, be asymptotically quasi-nonexpansive mapping with and be a nonnegative real sequence with . Suppose that is defined by the iteration process (1.7). If
where , then the sequence converges strongly to a fixed point of T.
Proof Let . Since T is an asymptotically quasi-nonexpansive mapping, there exists a sequence with the property and such that
for all and . By combining this inequality and Lemma 1, we get
When and , we have . Thus,
Let . Thus, there exits a constant such that
for all and . By (3.2),
Since , we have . Lemma 2 and or gives that
Now, we show that is a Cauchy sequence in K. Since , for each , there exists such that
for all . Thus, there exists such that
and we obtain that
for all . Therefore, is a Cauchy sequence in K. Since the set K is complete, the sequence must be convergence to a point in K. Let . Here after, we show that p is a fixed point. By , for all , there exists such that
for all . From (3.3), for each , there exists such that
for all . In particular, . Thus, there must exist such that
From (3.4) and (3.5),
Since is arbitrary, so , i.e., . Therefore, . This completes the proof. □
Remark 2 Let the hypothesis of Theorem 1 be satisfied and be an asymptotically nonexpansive or quasi-nonexpansive mapping. From Remark 1, the class of asymptotically quasi-nonexpansive mappings includes quasi-nonexpansive mappings and asymptotically nonexpansive mappings. Then the sequence converges strongly to a fixed point of T.
Now, we give the following corollaries which have been proved by Theorem 1.
Corollary 1 Under the hypothesis of Theorem 1, T satisfies the following conditions:
If the sequence in K satisfies , then
Then the sequence converges strongly to a fixed point of T.
Proof It follows from the hypothesis that . From (2),
Therefore, the sequence must converge to a fixed point of T by Theorem 1. □
Corollary 2 Under the hypothesis of Theorem 1, T satisfies the following conditions:
There exists a function which is right-continuous at 0, and for all such that
Then the sequence converges strongly to a fixed point of T.
Proof It follows from the hypothesis that
Since is right-continuous at 0 and , therefore we have
Thus, . By Theorem 1, the sequence converges strongly to q, a fixed point of T. This completes the proof. □
Finally, we give the following theorem which has a different hypothesis from Theorem 1.
Theorem 2 Let K be a nonempty closed convex subset of a complete space X, be an asymptotically quasi-nonexpansive mapping with and be a nonnegative real sequence with . Suppose that is defined by the iteration process (1.7). If T is semi-compact and , then the sequence converges strongly to a fixed point of T.
Proof From the hypothesis, we have . Also, since T is semi-compact, there exists a subsequence of such that . Hence,
Thus, . By (3.2),
Since , we have . By Lemma 2, exists and gives that . This completes the proof. □
The class of quasi-nonexpansive mappings and asymptotically nonexpansive mappings includes nonexpansive mappings, where as the class of asymptotically quasi-nonexpansive mappings is larger than that of quasi-nonexpansive mappings and asymptotically nonexpansive mappings. Then these results presented in this paper extend and generalize some works for a space in the literature.
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This paper has been presented in The ‘International Conference on Applied Analysis and Algebra (ICAAA2012)’ in Yıldız Technical University, 20-24 June 2012, Istanbul, Turkey.
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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Şahin, A., Başarır, M. On the strong convergence of a modified S-iteration process for asymptotically quasi-nonexpansive mappings in a space. Fixed Point Theory Appl 2013, 12 (2013). https://doi.org/10.1186/1687-1812-2013-12
- asymptotically quasi-nonexpansive mapping
- strong convergence
- iterative process
- fixed point