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Some convergence results for modified SP-iteration scheme in hyperbolic spaces
Fixed Point Theory and Applications volume 2014, Article number: 133 (2014)
In this paper, we prove some strong and Δ-convergence theorems for a modified SP-iteration scheme for total asymptotically nonexpansive mappings in hyperbolic spaces by employing recent technical results of Khan et al. (Fixed Point Theory Appl. 2012:54, 2012). The results presented here extend and improve some well-known results in the current literature.
1 Introduction and preliminaries
Iterative schemes play a key role in approximating fixed points of nonlinear mappings. Structural properties of the underlying space, such as strict convexity and uniform convexity, are very much needed for the development of iterative fixed point theory in it. Hyperbolic spaces are general in nature and inherit rich geometrical structure suitable to obtain new results in topology, graph theory, multi-valued analysis and metric fixed point theory.
Kohlenbach  introduced the hyperbolic spaces, defined below, which play a significant role in many branches of mathematics.
A hyperbolic space is a metric space together with a mapping satisfying
for all and .
A subset K of a hyperbolic space X is convex if for all and . If a space satisfies only (W1), it coincides with the convex metric space introduced by Takahashi . The concept of hyperbolic spaces in  is more restrictive than the hyperbolic type introduced by Goebel et al.  since (W1)-(W3) together are equivalent to being a space of hyperbolic type in . Also it is slightly more general than the hyperbolic space defined by Reich et al. . The class of hyperbolic spaces in  contains all normed linear spaces and convex subsets thereof, ℝ-trees, the Hilbert ball with the hyperbolic metric (see ), Cartesian products of Hilbert balls, Hadamard manifolds and CAT(0) spaces (see ) as special cases. Recently, the concept of p-uniformly convexity has been defined by Naor et al.  and its nonlinear version for in hyperbolic spaces was studied by Khan . Any CAT(0) space is 2-uniformly convex (see ).
The following example accentuates the importance of hyperbolic spaces.
Let be an open unit ball in a complex Hilbert space w.r.t. the metric (also known as the Kobayashi distance)
Then is a hyperbolic space where defines a unique point z in a unique geodesic segment for all . For more on hyperbolic spaces and a detailed treatment of examples, we refer the readers to .
A hyperbolic space is said to be
strictly convex  if for any and , there exists a unique element such that and ;
uniformly convex  if for all , and , there exists such that whenever , and .
A mapping providing such for given and is called modulus of uniform convexity. We call η monotone if it decreases with r (for a fixed ε). A uniformly convex hyperbolic space is strictly convex (see ).
Let K be a nonempty subset of a metric space and T be a self-mapping on K. Denote by the set of fixed points of T and . A self-mapping T is said to be
nonexpansive if for all ;
asymptotically nonexpansive if there exists a sequence with such that for all and ;
uniformly L-Lipschitzian if there exists a constant such that for all and ;
total asymptotically nonexpansive if there exist non-negative real sequences , with , and a strictly increasing continuous function with such that
for all and (see [, Definition 2.1]).
It follows from the above definitions that each nonexpansive mapping is an asymptotically nonexpansive mapping with , and that each asymptotically nonexpansive mapping is a total asymptotically nonexpansive mapping with , , , , . Moreover, each asymptotically nonexpansive mapping is a uniformly L-Lipschitzian mapping with . However, the converse of these statements is not true, in general.
It has been shown that every total asymptotically nonexpansive mapping defined on a nonempty bounded closed convex subset of a complete uniformly convex hyperbolic space always has a fixed point (see [, Theorem 3.1]).
The following iteration process is a translation of the SP-iteration scheme introduced in  from Banach spaces to hyperbolic spaces. The SP-iteration is equivalent to Mann, Ishikawa, Noor iterations and converges faster than the others for the class of continuous and non-decreasing functions (see ).
where K is a nonempty, closed and convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity and is a uniformly L-Lipschitzian and total asymptotically nonexpansive mapping.
Inspired and motivated by Khan et al. , Khan , Fukhar-ud-din and Khan , Wan , Zhao et al.  and Zhao et al. , we prove some strong and △-convergence theorems of the modified SP-iteration process for approximating a fixed point of total asymptotically nonexpansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results given in [14, 21].
The concept of △-convergence in a metric space was introduced by Lim  and its analogue in CAT(0) spaces was investigated by Dhompongsa and Panyanak . In order to define the concept of △-convergence in the general setup of hyperbolic spaces, we first collect some basic concepts.
Let be a bounded sequence in a hyperbolic space X. For , we define a continuous functional by
The asymptotic radius of is given by
The asymptotic center of with respect to a subset K of X is defined as follows:
This is the set of minimizer of the functional . If the asymptotic center is taken with respect to X, then it is simply denoted by .
Recall that a sequence in X is said to △-converge to if x is the unique asymptotic center of for every subsequence of . In this case, we write and call x as △-limit of .
It is known that uniformly convex Banach spaces and even CAT(0) spaces enjoy the property that ‘bounded sequences have unique asymptotic centers with respect to closed convex subsets.’ The following lemma is due to Leustean  and ensures that this property also holds in a complete uniformly convex hyperbolic space.
Lemma 1 [, Proposition 3.3]
Let be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Then every bounded sequence in X has a unique asymptotic center with respect to any nonempty closed convex subset K of X.
In the sequel, we shall need the following results.
Lemma 2 [, Lemma 2.5]
Let be a uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Let and be a sequence in for some . If and are sequences in X such that
for some , then
Lemma 3 [, Lemma 2.6]
Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space and let be a bounded sequence in K such that and . If is another sequence in K such that , then .
Lemma 4 [, Lemma 2]
Let , and be sequences of non-negative real numbers such that
If and , then exists.
2 Main results
We begin with Δ-convergence of the modified SP-iterative sequence defined by (1.1) for total asymptotically nonexpansive mappings in hyperbolic spaces.
Theorem 1 Let K be a nonempty, closed and convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let be a uniformly L-Lipschitzian and total asymptotically nonexpansive mapping with . If the following conditions are satisfied:
there exist constants such that ;
there exists a constant such that , ,
then the sequence defined by (1.1), Δ-converges to a fixed point of T.
Proof We divide our proof into three steps.
Step 1. First we prove that the following limits exist:
Since T is a total asymptotically nonexpansive mapping, by condition (iii), we get
Substituting (2.2) into (2.3) and simplifying it, we have
Similarly, we obtain
Combining (2.4) and (2.5), we have
where and . By virtue of condition (i),
By Lemma 4, and exist for each .
Step 2. Next we prove that .
In fact, it follows from (2.1) that exists for each given . We may assume that . The case is trivial. Next, we deal with the case . Taking lim sup on both sides in inequality (2.4), we have
With the help of (2.7)-(2.9) and Lemma 2, we have
On the other hand, since
we have . Combined with (2.7), it yields that . This implies that
Taking lim sup on both sides in inequality (2.2), we have
With the help of (2.11)-(2.13) and Lemma 2, we have
By the same method, we can also prove that
By (2.10), we get
In a similar way, we have
It follows that
Since T is uniformly L-Lipschitzian, therefore we obtain
Hence, (2.15) and (2.16) imply that
Step 3. Now we prove that the sequence △-converges to a fixed point of T.
In fact, for each , exists. This implies that the sequence is bounded. Hence by virtue of Lemma 1, has a unique asymptotic center . Let be any subsequence of such that . Then, by (2.17), we have
We claim that . In fact, we define a sequence in K by . So, we calculate
Since T is uniformly L-Lipschitzian, from (2.19), we have
Taking lim sup on both sides of the above estimate and using (2.18), we have
This implies that as . It follows from Lemma 3 that . Utilizing the uniform continuity of T, we have that
Hence . Moreover, exists by (2.1). Suppose that . By the uniqueness of asymptotic centers, we have
a contradiction. Hence . Since is an arbitrary subsequence of , therefore for all subsequences of , that is, △-converges to . The proof is completed. □
We now discuss the strong convergence of the modified SP-iteration for total asymptotically nonexpansive mappings in hyperbolic spaces.
Theorem 2 Let K, X, T and be the same as in Theorem 1. Suppose that conditions (i)-(iii) in Theorem 1 are satisfied. Then converges strongly to some if and only if .
Proof If converges to , then . Since , we have .
Conversely, suppose that . It follows from (2.1) that exists. Thus by hypothesis .
Next, we show that is a Cauchy sequence. In fact, it follows from (2.6) that for any ,
where and . Hence for any positive integers n, m, we have
Since for each , , we have
where . Therefore we have
This shows that is a Cauchy sequence in K. Since K is a closed subset in a complete hyperbolic space X, it is complete. We can assume that converges strongly to some point . It is easy to prove that is a closed subset in K, so is . Since , we obtain . This completes the proof. □
Remark 1 In Theorem 2, the condition may be replaced with .
Example 1 Let ℝ be the real line with the usual norm and let . Define two mappings by
It is proved in [, Example 3.1] that both and are asymptotically nonexpansive mappings with , . Therefore they are total asymptotically nonexpansive mappings with , , , . Additionally, they are uniformly L-Lipschitzian mappings with . Clearly, and . Set
Thus, the conditions of Theorem 1 are fulfilled. Therefore the results of Theorem 1 and Theorem 2 can be easily seen.
Example 2 Let ℝ be the real line with the usual norm and let . Define two mappings by and . It is proved in [, Example 1] that both and are total asymptotically nonexpansive mappings with , , . Additionally, they are uniformly L-Lipschitzian mappings with . Clearly, and . Let , and be the same as in (2.20). Similarly, the conditions of Theorem 1 are satisfied. So, the results of Theorem 1 and Theorem 2 also can be received.
Recall that a mapping T from a subset K of a metric space into itself is semi-compact if every bounded sequence satisfying as has a strongly convergent subsequence.
Senter and Dotson [, p.375] introduced the concept of condition (I) as follows.
A mapping with is said to satisfy condition (I) if there exists a non-decreasing function with and for all such that
By using the above definitions, we obtain the following strong convergence theorems.
Theorem 3 Under the assumptions of Theorem 1, if T is semi-compact, then converges strongly to a fixed point of T.
Proof It follows from (2.1) that is a bounded sequence. Also, by (2.17), we have . Then, by the semi-compactness of T, there exists a subsequence such that converges strongly to some point . Moreover, by the uniform continuity of T, we have
This implies that . Again, by (2.1), exists. Hence p is the strong limit of the sequence . As a result, converges strongly to a fixed point p of T. □
Theorem 4 Under the assumptions of Theorem 1, if T satisfies condition (I), then converges strongly to a fixed point of T.
Proof By virtue of (2.1), exists. Further, by condition (I) and (2.17), we have
That is, . Since f is a non-decreasing function satisfying and for all , it follows that . Now Theorem 2 implies that converges strongly to a point p in . □
Remark 2 Theorems 1-4 contain the corresponding theorems proved for asymptotically nonexpansive mappings when , , , , .
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The authors would like to thank the editor and referees for their careful reading and valuable comments and suggestions which led to the present form of the paper. 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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Cite this article
Şahin, A., Başarır, M. Some convergence results for modified SP-iteration scheme in hyperbolic spaces. Fixed Point Theory Appl 2014, 133 (2014). https://doi.org/10.1186/1687-1812-2014-133
- hyperbolic space
- fixed point
- total asymptotically nonexpansive mapping
- modified SP-iteration scheme
- strong convergence