Convergence analysis of a multi-step iteration for a finite family of asymptotically quasi-nonexpansive mappings
© Khan; licensee Springer. 2013
Received: 15 April 2013
Accepted: 20 August 2013
Published: 8 September 2013
This paper is a continuation of the analysis of classical Kuhfittig iteration involving a finite family of asymptotically quasi-nonexpansive mappings in the general setup of uniformly convex hyperbolic spaces. We establish strong and △-convergence results of Kuhfittig iteration, which subsequently help to apply proof mining techniques for the extraction of rates of metastability in the sense of Tao. Additionally, our proposed convergence results extend and improve various results in the current literature.
MSC:47H09, 47H10, 49M05.
Most of the problems in various disciplines of science are nonlinear in nature, whereas fixed point theory proposed in the setting of normed linear spaces or Banach spaces majorly depends on the corresponding linear structures of those spaces. A nonlinear framework for fixed point theory is a metric space embedded with a ‘convex structure.’ It is remarked that the non-positively curved spaces play a significant role in many branches of mathematics. The class of hyperbolic spaces - nonlinear in nature - is prominent among non-positively curved spaces and provides 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 this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach  which is more restrictive than the hyperbolic type introduced in  and more general than the concept of hyperbolic space in .
for all and . A nonempty subset K of a hyperbolic space X is convex if for all and . The class of hyperbolic spaces contains normed spaces and convex subsets thereof, the Hilbert ball equipped with the hyperbolic metric , ℝ-trees, Hadamard manifolds as well as spaces in the sense of Gromov (see  for a detailed treatment).
The following example accentuates the importance of hyperbolic spaces.
Then is a hyperbolic space where defines a unique point z in a unique geodesic segment for all . The above example is of importance for metric fixed point theory of holomorphic mappings which are -nonexpansive. For a detailed discussion of the topic, we refer to .
provided , and .
A map , which provides such a for given and , is known as a modulus of uniform convexity of X. We call η monotone if it decreases with r (for a fixed ϵ), i.e., , ().
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.
The fixed point property (fpp) of various nonlinear mappings has relevant applications in many branches of nonlinear analysis and topology. On the other hand, there are certain situations where it is hard to derive conditions for the (fpp) of certain nonlinear mappings. In such situations, the approximate fixed point property (afpp) is more desirable. Moreover, in a nonlinear domain, the (afpp) of various generalization nonexpansive mappings is still being developed.
The problem of finding a common fixed point of a finite family of nonlinear mappings acting on a nonempty convex domain often arises in applied mathematics, for instance, in convex minimization problems and systems of simultaneous equations. A fundamental result in the construction of common fixed points of a finite family of nonexpansive mappings is essentially due to Kuhfittig . The following iteration is a translation of classical Kuhfittig iteration for a finite family of nonexpansive mappings in hyperbolic spaces.
where for some .
The classical Kuhfittig iteration converges strongly under the compactness condition of K, whereas the weak convergence is established through Opial’s condition. Kuhfittig iteration is comparatively less developed for various nonlinear mappings in a more general setup of spaces with non-positive sectional curvature such as hyperbolic spaces. To the best of our knowledge, Kuhfittig iteration has never been used as a tool for the approximation of common fixed points of a finite family of asymptotically quasi-nonexpansive mappings. Moreover, Rhoades  mentioned that one can replace λ in the Kuhfittig iteration with a sequence . Here a natural question arises:
Question Is Kuhfittig iteration valid for the class of asymptotically quasi-nonexpansive mappings with a general sequence of control parameters for some in the general setup of hyperbolic spaces?
The purpose of this paper is to provide an affirmative answer to the above question. Our convergence results not only can be viewed as an analogue of various existing results but also improve and generalize various results in the current literature; see, for example, [11–25] and the references cited therein.
We start this section with the concept of △-convergence which is essentially due to Lim  in the general setting of metric spaces. In 2008, Kirk and Panyanak  investigated △-convergence in spaces and showed that △-convergence coincides with the usual weak convergence in Banach spaces. Moreover, both concepts share many useful properties in uniformly convex spaces.
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 2.1 
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 the △-limit of . A sequence is said to be quasi-Fejér monotone w.r.t. a set K if for all and for all . This concept generalizes the classical concept of Fejér monotone sequence in a sense that it satisfies the standard Fejér monotonicity property within an additional error term . A mapping is semi-compact if every bounded sequence satisfying has a convergent subsequence.
In the sequel, we need the following useful results.
Lemma 2.2 
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 2.3 
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 2.4 
Let and be two sequences of non-negative real numbers such that . If , , then exists.
3 Main results
Throughout this section, we assume that the mappings are nonexpansive and satisfy . We are now in a position to prove our main convergence results.
△-converges to a common fixed point of .
where and is finite.
Hence, estimate (3.3) implies that is quasi-Fejér monotone w.r.t. F. Therefore, is bounded, and hence Lemma 2.1 implies that has a unique asymptotic center .
For the △-convergence of , we first show that the sequence is asymptotic regular w.r.t. the k th-mapping , that is, . For this, we reason as follows.
Next, we show that . For this, we define a sequence in K by .
This implies that as . It follows from Lemma 2.3 that . As is uniformly continuous, so we get that . That is, and hence u is the common fixed point of and . Reasoning as above - by utilizing the uniqueness of asymptotic centers - we get that . This infers that u is the unique asymptotic center of for every subsequence of .
For this, we reason as follows.
u is the common fixed point of and .
Continuing in a similar fashion, we can show that u is the common fixed point of , , …, . Hence . This completes the proof. □
The strong convergence of iteration (3.1) can easily be established under compactness condition of K or . Next, we give a necessary and sufficient condition for the strong convergence of iteration (3.1).
Theorem 3.2 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 be a finite family of uniformly Lipschitzian asymptotically quasi-nonexpansive self-mappings of K with a sequence such that and . Assume that , then the sequence defined in (3.1) converges strongly to a common fixed point of if and only if .
Proof The necessity of the conditions is obvious. Thus, we only prove the sufficiency. It follows from estimate (3.2) that converges. Moreover, implies that . This completes the proof. □
Since the class of asymptotically nonexpansive mappings is properly contained in the class of asymptotically quasi-nonexpansive mappings, therefore, we now list the following useful corollaries of Theorems (3.1)-(3.2).
Corollary 3.3 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 be a finite family of uniformly Lipschitzian asymptotically nonexpansive self-mappings of K with a sequence such that and . Assume that , then the sequence defined in (3.1) △-converges to a common fixed point of .
Corollary 3.4 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 be a finite family of uniformly Lipschitzian asymptotically nonexpansive self-mappings of K with a sequence such that and . Assume that , then the sequence defined in (3.1) converges strongly to a common fixed point of if and only if .
generalized asymptotically-quasi-nonexpansive mappings (i.e., , where and );
asymptotically nonexpansive mappings in the intermediate sense .
It is worth mentioning that Kuhfittig iteration for a finite family of nonexpansive mappings is analyzed in the general setup of uniformly convex hyperbolic spaces resulting in explicit and uniform rates of asymptotical regularity ; whereas for iteration (3.1), there does not seem to exist a computable rate of asymptotic regularity, let alone a rate of metastability (in the sense of Tao ) in cases where strong convergence holds.
Future work We intend to extract explicit and effective rates of metastability of Kuhfittig iteration involving a finite family of asymptotically quasi-nonexpansive mappings in the general setup of uniformly convex hyperbolic spaces.
The author would like to thank the editor and referees for their helpful comments and suggestions.
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