Convergence analysis of a general iteration schema of nonlinear mappings in hyperbolic spaces
© Fukhar-ud-din and Khan; licensee Springer. 2013
Received: 13 February 2013
Accepted: 21 August 2013
Published: 4 October 2013
Iterative schemas are ubiquitous in the area of abstract nonlinear analysis and still remain as a main tool for approximation of fixed points of generalizations of nonexpansive maps. The analysis of general iterative schemas, in a more general setup, is a problem of interest in theoretical numerical analysis. Therefore, we propose and analyze a general iterative schema for two finite families of asymptotically quasi-nonexpansive maps in hyperbolic spaces. Results concerning △-convergence as well as strong convergence of the proposed iteration are proved. It is instructive to compare the proposed general iteration schema and the consequent convergence results with that of several recent results in spaces and uniformly convex Banach spaces.
MSC:47H09, 47H10, 49M05.
1 Introduction and preliminaries
Iterative schemas play a key role in approximating fixed points for nonlinear mappings. Structural properties of the space under consideration are very important in establishing the fixed point property of the space, for example, strict convexity, uniform convexity and uniform smoothness etc. Hyperbolic spaces are general in nature and have rich geometrical structures for different results with applications in topology, graph theory, multivalued analysis and metric fixed point theory. The study of hyperbolic spaces has been largely motivated and dominated by questions about hyperbolic groups, one of the main objects of study in geometric group theory. Throughout the paper, we work in the setting of hyperbolic spaces, introduced by Kohlenbach , which are prominent among non-positively curved spaces and play a significant role in many branches of mathematics.
Nonexpansive mappings are Lipschitzian mappings with the Lipschitz constant equal to 1. Moreover, the class of nonexpansive mappings is closely related to the class of strict pseudo-contractions as nonexpansive mappings are 0-strictly pseudo-contractive. The class of nonexpansive mappings enjoys the fixed point property and the approximate fixed point property in various settings of spaces. The importance of this class lies in its powerful applications in initial value problems of differential equations, game-theoretic model, image recovery and minimax problems.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk  as an important generalization of the class of nonexpansive mappings. Therefore, it is natural to extend such powerful results to the class of asymptotically nonexpansive mappings as a means of testing the limit of the theory of nonexpansive mappings. Most of the results in fixed point theory guarantee that a fixed point exists, but they do not help in finding the fixed point. As a consequence, iterative construction of fixed points emerged as the most powerful tool for solving such nonlinear problems. It is worth mentioning that iteration schemas are the only main tool for approximation of fixed points of various generalizations of nonexpansive mappings. Several authors have studied approximation of fixed points of several generalizations of nonexpansive mappings using Mann and Ishikawa iterations (see, e.g., [3–15]).
Moreover, finding common fixed points of a finite family of mappings acting on a Hilbert space is a problem that often arises in applied mathematics, for instance, in convex minimization problems and systems of simultaneous equations. One of the most elegant ways to prove that a partial differential equation or integral equation has a solution is to pose it as a fixed point problem. Hence, the analysis of a general iteration schema, in a more general setup, is a problem of interest in theoretical numerical analysis. Therefore, considerable research efforts have been devoted to developing iterations for the approximation of common fixed points of several classes of nonlinear mappings with a nonempty set of common fixed points.
In 1991, Schu  established weak and strong convergence results for asymptotically nonexpansive mappings using a modified Mann iteration. A unified treatment regarding weak convergence theorems for asymptotically nonexpansive mappings was analyzed by Chang et al.  and consequently improved and generalized the results of Schu  and many more. See, for example, Bose , Tan and Xu  and many others.
In 2000, Osilike and Aniagbosor  obtained weak and strong convergence results for asymptotically nonexpansive mappings using a modified Ishikawa iteration. Since the case for two mappings has a direct link to minimization problems , so this fact motivated Khan and Takahashi  to approximate common fixed points of two asymptotically nonexpansive mappings. For this purpose, they used a modified Ishikawa iteration. See also  and .
In 2008, Khan et al.  introduced a general iteration schema for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces. Khan et al.  also proposed and analyzed a general iteration schema for strong convergence results in spaces. Inspired by the work of Khan et al. , Kettapun et al.  introduced a new iterative schema for finding a common fixed point of a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces. Quite recently, Sahin and Basarir  approximated common fixed points of a finite family of asymptotically quasi-nonexpansive mappings by a modified general iteration schema in spaces. Recently, Yildirim and Özdemir  approximated a common fixed point of a finite family of asymptotically quasi-nonexpansive mappings using a new general iteration in a Banach space setting as follows.
nonexpansive if for ;
quasi-nonexpansive if for and for ;
asymptotically nonexpansive if there exists a sequence and and for , ;
asymptotically quasi-nonexpansive if there exists a sequence and and for , , ;
uniformly L-Lipschitzian if there exists a constant such that for and .
It follows from the above definitions that a nonexpansive mapping is quasi-nonexpansive and that an asymptotically nonexpansive mapping is asymptotically quasi-nonexpansive. Moreover, an asymptotically nonexpansive mapping is uniformly L-Lipschitzian. However, the converse of these statements is not true, in general.
for all and .
Note that the above example can be extended from ℂ to general complex Hilbert spaces as follows.
defines a metric on (also known as the Kobayashi distance). The open unit ball together with this metric is coined as a Hilbert ball. Since is a unique geodesic space, so one can define W in a similar way for the corresponding hyperbolic space .
A metric space satisfying only (1) is a convex metric space introduced by Takahashi . A subset K of a hyperbolic space X is convex if for all and . For more on hyperbolic spaces and a comparison between different notions of hyperbolic space present in the literature, we refer to [, p.384].
A hyperbolic space is uniformly convex  if for all , and , there exists such that whenever , , .
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). spaces are uniformly convex hyperbolic spaces with modulus of uniform convexity . Therefore, the class of uniformly convex hyperbolic spaces includes both uniformly convex normed spaces and spaces as special cases.
Inspired and motivated by Khan and Takahashi , Sahin and Basarir , Shahzad and Udomene , Yildirim and Özdemir  and Khan et al. , we introduce a general iteration schema in hyperbolic spaces and approximate common fixed points of two finite families of asymptotically quasi-nonexpansive mappings as follows.
where for each .
In 1976, Lim  introduced the notion of asymptotic center and, consequently, coined the concept of △-convergence in a general setting of a metric space. In 2008, Kirk and Panyanak  proposed an analogous version of convergence in geodesic spaces, namely △-convergence, which was originally introduced by Lim . They showed that △-convergence coincides with the usual weak convergence in Banach spaces and both concepts share many useful properties.
This is the set of minimizers of the functional . If the asymptotic center is taken with respect to X, then it is simply denoted by . It is known that uniformly convex Banach spaces and even 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 1.1 
Let be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity. Then every bounded sequence in X has a unique asymptotic center with respect to any nonempty closed convex subset K of X.
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 a △-limit of . A mapping is semi-compact if every bounded sequence , satisfying , has a convergent subsequence.
holds for at least one or one , where .
In the sequel, we shall need the following results.
Lemma 1.2 
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 , and for some , then .
Lemma 1.3 
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 1.4 
for all and , then exists. Moreover, if there exists a subsequence of such that as , then as .
2 Some preparatory lemmas
From now onward, we denote for two finite families and of asymptotically quasi-nonexpansive self-mappings on K with sequences and respectively. If we put , then is a sequence in and .
We start with the following lemma.
Lemma 2.1 Let K be a nonempty, closed and convex subset of a hyperbolic space X, and let and be two finite families of asymptotically quasi-nonexpansive self-mappings on K with a sequence satisfying , . Then, for the sequence in (1.2), exists for all .
Applying Lemma 1.4 to the above inequality, we have exists for each . Consequently, exists. □
Proof It follows from Lemma 2.1 that exists for each . Assume that . Otherwise the proof is trivial.
This completes the proof. □
Remark 2.3 (i) It is worth mentioning that the asymptotic regularity (2.33)-(2.34) of the iteration schema (1.2) can easily be extended to a more general class of weakly asymptotically quasi-nonexpansive (short: w.aq.n.) mappings. That is, is a w.aq.n. mapping if for all , there exists such that , where is a sequence in with . Obviously, all self-mappings having a zero vector and satisfying are w.aq.n. mappings. On the other hand, if we define by , then . However, quasi-nonexpansivity fails for 1, and hence quasi-nonexpansive mappings are properly included in the class of w.aq.n. mappings. The aforementioned class of w.aq.n. mappings was introduced by Kohlenbach and Lambov  as it has nice logical behavior w.r.t. metatheorems .
(ii) The above derived results (2.33)-(2.34) can also be achieved if the hypothesis regarding the existence of a common fixed point is weakened by the existence of common approximate fixed points in some neighborhood of the starting point .
(iii) The seminal work of Kohlenbach and Leustean  gives a comprehensive logical treatment of asymptotically nonexpansive mappings in the more general setup of uniformly convex hyperbolic spaces and generalizes the corresponding results announced in . They extract explicit rates Φ of metastability (in the sense of Tao) for the asymptotic regularity for the Krasnoselskii-Mann iteration schema. For more on rates of asymptotic regularity in the context of spaces, we refer to [31, 32]. Following the procedure in  and [, Theorem 3.5], one should be able to get such rates Φ also for (2.33)-(2.34) which will - as the rates in [28, 30] - only depend on a (monotone) modulus of uniform convexity for X, an upper bound , the Lipschitz constant L, an upper bound and such that . Thus Φ will be largely independent of X, T or . We intend to carry out the extraction of such Φ in another paper.
3 Convergence of approximants to fixed points
In this section, we approximate common fixed points of two finite families of asymptotically nonexpansive mappings in a hyperbolic space. More briefly, we establish △-convergence and strong convergence of the iteration schema (1.2).
Theorem 3.1 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η, and let and be two finite families of uniformly L-Lipschitzian asymptotically quasi-nonexpansive self-mappings on K. Then the sequence defined in (1.2) △-converges to a common fixed point of .
We claim that u is the common fixed point of and .
For each , we define a sequence in K by .
This implies that as . It follows from Lemma 1.3 that . Utilizing the uniform continuity of , we have that . From the arbitrariness of m, we conclude that v is the common fixed point of . Similarly, we can show that v is the common fixed point of . Therefore .
Next, we claim that the common fixed point ‘v’ is the unique asymptotic center for each subsequence of .
Assume contrarily, that is, .
a contradiction. Hence . Since is an arbitrary subsequence of , therefore for all subsequences of . This proves that △-converges to a common fixed point of and . □
Remark 3.2 It follows from the uniqueness of the asymptotic center and a common fixed point of the two families of mappings that Theorem 3.1 can also be generalized to the class of mappings as mentioned in Remark 2.3(i).
Theorem 3.3 Let K, X, , and be as in Theorem 3.1. Then converges strongly to some if and only if .
Proof If converges to , then . Since , we have .
Conversely, suppose that . It follows from Lemma 2.1 that exists. Now reveals that .
Then we have . Since ϵ is arbitrary, we have . Similarly, we can show that . Hence . □
We now establish strong convergence of the iteration schema (1.2) based on Lemma 2.2.
Theorem 3.4 Let K, X, , and be as in Theorem 3.1. Suppose that a pair of mappings T and S in and , respectively, satisfies condition (A). Then the sequence defined in (1.2) converges strongly to some .
Proof It follows from Lemma 2.1 that exists. Moreover, Lemma 2.2 implies that for each . So, condition (A) guarantees that . Since f is nondecreasing with , it follows that . Then Theorem 3.3 implies that converges strongly to a point p in F. □
Theorem 3.5 Let K, X, , and be as in Theorem 3.1. Suppose that either or is semi-compact. Then the sequence defined in (1.2) converges strongly to .
This implies that q is the common fixed point of . Similarly, we can show that q is the common fixed point of . Therefore . The rest of the proof is similar to Theorem 3.1 and is, therefore, omitted. □
Remark 3.6 Compactness of the underlying sequence space is useful for establishing strong convergence of an approximant of a fixed point. Sequential compactness (every sequence has a convergent subsequence), among other notions of compactness, is a widely used tool in this regard. Moreover, if K (or just ) is compact, then the approximate sequence strongly converges to a fixed point. Using a logical analysis of the classical compactness argument, it is shown in [, Theorem 4.7] how to convert an approximate fixed point bound Φ for Krasnoselskii-Mann iteration schema of asymptotically nonexpansive mappings (and hence a fortiori any rate of metastability for the asymptotic regularity of ) into a rate Ψ of metastability for the strong convergence of in the case of compact K. This rate Ψ depends - in addition to the data on which Φ depends (see Remark 2.3(iii)) - only on a modulus of total boundedness for K. We intend to carry out the extraction of a suitable Ψ for our more general iteration schema with two finite families of mappings in another paper. Combined with the Φ discussed in Remark 2.3(iii), this then yields a highly uniform rate of metastability of Theorem 3.5.
The authors are very grateful to the editor and anonymous referees for their helpful comments. We are indebted to Prof. Dr. Ulrich Kohlenbach for various constructive comments to improve the content of the manuscript. The author H. Fukhar-ud-din is grateful to King Fahd University of Petroleum & Minerals for supporting the research project IN 121023. The author M.A.A. Khan gratefully acknowledges the support of Higher Education Commission of Pakistan.
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