- Open Access
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.
- asymptotically nonexpansive mapping
- Kuhfittig iteration
- common fixed point
- strong convergence
- modulus of uniform convexity
- hyperbolic space
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.
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.
- Kohlenbach U: Some logical metatheorems with applications in functional analysis. Trans. Am. Math. Soc. 2005, 357: 89–128. 10.1090/S0002-9947-04-03515-9MathSciNetView ArticleGoogle Scholar
- Goebel K, Kirk WA: Iteration processes for nonexpansive mappings. Contemp. Math. 21. In Topological Methods in Nonlinear Functional Analysis. Edited by: Singh SP, Thomeier S, Watson B. Am. Math. Soc., Providence; 1983:115–123.View ArticleGoogle Scholar
- Reich S, Shafrir I: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 1990, 15: 537–558. 10.1016/0362-546X(90)90058-OMathSciNetView ArticleGoogle Scholar
- Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York; 1984.Google Scholar
- Bridson M, Haefliger A: Metric Spaces of Non-Positive Curvature. Springer, Berlin; 1999.View ArticleGoogle Scholar
- Kohlenbach U Springer Monogr. Math. In Applied Proof Theory: Proof Interpretations and Their Use in Mathematics. Springer, Berlin; 2008.Google Scholar
- Leustean L:A quadratic rate of asymptotic regularity for -spaces. J. Math. Anal. Appl. 2007, 325: 386–399. 10.1016/j.jmaa.2006.01.081MathSciNetView ArticleGoogle Scholar
- Goebel K, Kirk WA: A fixed point theorem for asymptotically non-expansive mappings. Proc. Am. Math. Soc. 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView ArticleGoogle Scholar
- Kuhfittig PKF: Common fixed points of nonexpansive mappings by iteration. Pac. J. Math. 1981, 97(1):137–139. 10.2140/pjm.1981.97.137MathSciNetView ArticleGoogle Scholar
- Rhoades BE: Finding common fixed point of nonexpansive mappings by iteration. Bull. Aust. Math. Soc. 2000, 62: 307–310. 10.1017/S0004972700018785MathSciNetView ArticleGoogle Scholar
- Chang SS, Cho YJ, Zhou H: Demiclosed principal and weak convergence problems for asymptotically nonexpansive mappings. J. Korean Math. Soc. 2001, 38(6):1245–1260.MathSciNetGoogle Scholar
- Chidume CE, Ali B: Convergence theorems for finite families of asymptotically quasi-nonexpansive mappings. J. Inequal. Appl. 2007., 2007: Article ID 68616 10.1155/2007/68616Google Scholar
- Fukhar-ud-din H, Khan AR, Khan MAA: A new implicit algorithm of asymptotically quasi-nonexpansive mappings in uniformly convex Banach spaces. IAENG Int. J. Appl. Math. 2012., 42: Article ID 3Google Scholar
- Fukhar-ud-din H, Khan AR: Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces. Comput. Math. Appl. 2007, 53: 1349–1360. 10.1016/j.camwa.2007.01.008MathSciNetView ArticleGoogle Scholar
- Gu F, Fu Q: Strong convergence theorems for common fixed points of multistep iterations with errors in Banach spaces. J. Inequal. Appl. 2009., 2009: Article ID 819036 10.1155/2009/819036Google Scholar
- Khan AR, Domlo AA, Fukhar-ud-din H: Common fixed point Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 2008, 341: 1–11. 10.1016/j.jmaa.2007.06.051MathSciNetView ArticleGoogle Scholar
- Khan AR, Khamsi MA, Fukhar-ud-din H:Strong convergence of a general iteration scheme in -spaces. Nonlinear Anal. 2011, 74: 783–791. 10.1016/j.na.2010.09.029MathSciNetView ArticleGoogle Scholar
- Fukhar-ud-din, H, Khan, MAA: Convergence analysis of a general iteration schema of nonlinear mappings in hyperbolic spaces. Fixed Point Theory Appl. (in press)Google Scholar
- Osilike MO, Aniagbosor SC: Weak and strong convergence theorems for fixed points of asymptotically non-expansive mappings. Math. Comput. Model. 2000, 32: 1181–1191. 10.1016/S0895-7177(00)00199-0MathSciNetView ArticleGoogle Scholar
- Sahin A, Basarır M: On the strong convergence of a modified S -iteration process for asymptotically quasi-nonexpansive mappings in a space. Fixed Point Theory Appl. 2013., 2013: Article ID 12 10.1186/1687-1812-2013-12Google Scholar
- Sahin A, Basarır M: On the strong convergence and △-convergence of SP -iteration on space. J. Inequal. Appl. 2013., 2013: Article ID 311 10.1186/1029-242X-2013-311Google Scholar
- Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884MathSciNetView ArticleGoogle Scholar
- Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 1991, 158: 407–413. 10.1016/0022-247X(91)90245-UMathSciNetView ArticleGoogle Scholar
- Tan KK, Xu HK: Fixed point iteration process for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1994, 122(3):733–739. 10.1090/S0002-9939-1994-1203993-5MathSciNetView ArticleGoogle Scholar
- Yao Y, Liou Y-C: New iterative schemes for asymptotically quasi-nonexpansive mappings. J. Inequal. Appl. 2010., 2010: Article ID 934692 10.1155/2010/934692Google Scholar
- Lim TC: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 1976, 60: 179–182. 10.1090/S0002-9939-1976-0423139-XView ArticleGoogle Scholar
- Kirk W, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal. 2008, 68: 3689–3696. 10.1016/j.na.2007.04.011MathSciNetView ArticleGoogle Scholar
- Leustean L: Nonexpansive iterations in uniformly convex W -hyperbolic spaces. Contemporary Mathematics 513. In Nonlinear Analysis and Optimization I: Nonlinear Analysis. Edited by: Leizarowitz A, Mordukhovich BS, Shafrir I, Zaslavski A. Am. Math. Soc., Providence; 2010:193–209.View ArticleGoogle Scholar
- Khan AR, Fukhar-ud-din H, Khan MAA: An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 54 10.1186/1687-1812-2012-54Google Scholar
- Bruck R, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach space with uniform Opial property. Colloq. Math. 1993, 65(2):169–179.MathSciNetGoogle Scholar
- Kohlenbach U, Lambov B, et al.: Bounds on iterations of asymptotically quasi-nonexpansive mappings. In International Conference on Fixed Point Theory and Applications Valencia, 2003. Edited by: Garcia Falset J. Yokohama Publishers, Yokohama; 2004:143–172.Google Scholar
- Khan MAA, Kohlenbach U: Bounds on Kuhfittig’s iteration schema in uniformly convex hyperbolic spaces. J. Math. Anal. Appl. 2013, 403: 633–642. 10.1016/j.jmaa.2013.02.058MathSciNetView ArticleGoogle Scholar
- Tao T: Soft analysis, hard analysis, and the finite convergence principle. 298. In Structure and Randomness: Pages from Year One of a Mathematical Blog. Am. Math. Soc., Providence; 2008.Google Scholar
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