Abstract
The purpose of this paper is to study modified S-iteration process and investigate the existence and convergence theorems in the setting of spaces for a class of mappings which is wider than that of asymptotically nonexpansive mappings. Our results generalize, unify and extend several comparable results in the existing literature.
MSC:54H25, 54E40.
Similar content being viewed by others
1 Introduction
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. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a space. Other examples include pre-Hilbert spaces (see [1]), ℝ-trees (see [2]), Euclidean buildings (see [3]), the complex Hilbert ball with a hyperbolic metric (see [4]), and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [1].
Fixed point theory in spaces has been first studied by Kirk (see [5, 6]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete space always has a fixed point. It is worth mentioning that the results in spaces can be applied to any space with since any space is a space for every (see, e.g., [1]).
The Mann iteration process is defined by the sequence ,
where is a sequence in .
Further, the Ishikawa iteration process is defined by the sequence ,
where and are the sequences in . This iteration process reduces to the Mann iteration process when for all .
In 2007, Agarwal, O’Regan and Sahu [7] introduced the S-iteration process in a Banach space,
where and are the sequences in . Note that (3) is independent of (2) (and hence (1)). They showed that their process is independent of those of Mann and Ishikawa and converges faster than both of these (see [[7], Proposition 3.1]).
In 1991, Schu [8] considered the modified Mann iteration process which is a generalization of the Mann iteration process,
where is a sequence in .
In 1994, Tan and Xu [9] studied the modified Ishikawa iteration process which is a generalization of the Ishikawa iteration process,
where and are the sequences in . This iteration process reduces to the modified Mann iteration process when for all .
Recently, Agarwal, O’Regan and Sahu [7] introduced the modified S-iteration process in a Banach space,
where and are the sequences in . Note that (6) is independent of (5) (and hence of (4)). Also (6) reduces to (3) when for all .
Very recently, Şahin and Başarir [10] modified iteration process (6) in a space as follows.
Let K be a nonempty closed convex subset of a complete space X, and let 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 . They studied modified S-iteration process for asymptotically quasi-nonexpansive mappings in a space and established some strong convergence results under some suitable conditions which generalize some results of Khan and Abbas [11].
Inspired and motivated by [10] and some others, we modify iteration scheme (7) for two mappings in a space as follows.
Let K be a nonempty closed convex subset of a complete space X, and let be two asymptotically nonexpansive mappings in the intermediate sense 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 newly defined modified S-iteration process (8) involving two asymptotically nonexpansive mappings in the intermediate sense and investigate the existence and convergence theorems for the above mentioned mappings and iteration scheme in the setting of spaces. Our results generalize, unify and extend several comparable results in the existing literature.
2 Preliminaries and lemmas
In order to prove the main results of this paper, we need the following definitions, concepts and lemmas.
Let be a metric space and K be its subset. Let be a mapping. A point is called a fixed point of T if . We will also denote by the set of common fixed points of S and T, that is, .
The concept of asymptotically nonexpansive mapping was introduced by Goebel and Kirk [12] in 1972. The iterative approximation problem for asymptotically nonexpansive and asymptotically quasi-nonexpansive mappings was studied by many authors in a Banach space and a space (see, e.g., [9, 13–19]).
Definition 2.1 Let be a metric space and K be its nonempty subset. Then is said to be
-
(1)
nonexpansive if for all ;
-
(2)
asymptotically nonexpansive if there exists a sequence with such that for all and ;
-
(3)
uniformly L-Lipschitzian if there exists a constant such that for all and ;
-
(4)
semi-compact if for a sequence in K with , there exists a subsequence of such that .
In 1993, Bruck, Kuczumow and Reich [20] introduced a notion of asymptotically nonexpansive mapping in the intermediate sense. A mapping is said to be asymptotically nonexpansive in the intermediate sense provided that T is uniformly continuous and
From the above definitions, it follows that an asymptotically nonexpansive mapping must be asymptotically nonexpansive mapping in the intermediate sense. But the converse does not hold as the following example.
Example 2.1 (See [21])
Let , and . For each , define
Then T is an asymptotically nonexpansive mapping in the intermediate sense but it is not an asymptotically nonexpansive mapping.
Remark 2.1 It is clear that the class of asymptotically nonexpansive mappings includes nonexpansive mappings, whereas the class of asymptotically nonexpansive mappings in the intermediate sense is larger than that of asymptotically nonexpansive mappings.
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 α of c is called a geodesic (or metric) segment joining x and y. We say that X is (i) a geodesic space if any two points of X are joined by a geodesic and (ii) uniquely geodesic if there is exactly one geodesic joining x and y for each , which we will denote by , called the segment joining x to y.
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 the geodesic triangle in is a triangle in such that for . Such a triangle always exists (see [1]).
space: A geodesic metric space is said to be a space if all geodesic triangles of 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 ,
Complete spaces are often called Hadamard spaces (see [21]). If x, , are points of a space and is the midpoint of the segment which we will denote by , then the inequality implies
Inequality (9) is the inequality of Bruhat and Tits [22]. The above inequality was extended in [23] as
for any and .
Let us recall that a geodesic metric space is a space if and only if it satisfies the inequality (see [[1], p.163]). Moreover, if X is a metric space and , then for any , there exists a unique point such that
for any and .
A subset K of a space X is convex if, for any , we have .
For the development of our main results, we recall some definitions, and some key results are listed in the form of lemmas.
Lemma 2.1 (See [17])
Let X be a space.
-
(i)
For and , there exists a unique point such that
(A)
We use the notation for the unique point z satisfying (A).
-
(ii)
For and , we have
Let be a bounded sequence in a closed convex subset K of a space X. For , set
The asymptotic radius of is given by
and the asymptotic center of is the set
It is known that, in a space, consists of exactly one point [[24], Proposition 7].
We now recall the definition of Δ-convergence and weak convergence (⇀) in a space.
Definition 2.2 (See [25])
A sequence in a space X is said to Δ-converge to if x is the unique asymptotic center of for every subsequence of .
In this case we write and call x the Δ-limit of .
Recall that a bounded sequence in X is said to be regular if for every subsequence of . In the Banach space it is known that every bounded sequence has a regular subsequence [[26], Lemma 15.2].
Since in a space every regular sequence Δ-converges, we see that every bounded sequence in X has a Δ-convergent subsequence, also it is noticed that [[25], p.3690].
Lemma 2.2 (See [27])
Given such that Δ-converges to x and given with , then
In a Banach space, the above condition is known as the Opial property.
Now, recall the definition of weak convergence in a space.
Definition 2.3 (See [28])
Let K be a closed convex subset of a space X. A bounded sequence in K is said to converge weakly to if and only if , where .
Note that if and only if .
Nanjaras and Panyanak [29] established the following relation between Δ-convergence and weak convergence in a space.
Lemma 2.3 (See [29], Proposition 3.12)
Let be a bounded sequence in a space X, and let K be a closed convex subset of X which contains . Then
-
(i)
implies .
-
(ii)
The converse of (i) is true if is regular.
Lemma 2.4 (See [[23], Lemma 2.8])
If is a bounded sequence in a space X with and is a subsequence of with and the sequence converges, then .
Lemma 2.5 (See [[30], Proposition 2.1])
If K is a closed convex subset of a space X and if is a bounded sequence in K, then the asymptotic center of is in K.
Lemma 2.6 (See [31])
Suppose that and are two sequences of nonnegative numbers such that for all . If converges, then exists.
Lemma 2.7 (See [[27], Theorem 3.1])
Let X be a complete space, K be a nonempty closed convex subset of X. If is an asymptotically nonexpansive mapping in the intermediate sense, then T has a fixed point.
Lemma 2.8 (See [[27], Theorem 3.2])
Let X be a complete space, K be a nonempty closed convex subset of X. If is an asymptotically nonexpansive mapping in the intermediate sense, then is closed and convex.
Lemma 2.9 (Demiclosed principle) (See [[27], Proposition 3.3])
Let K be a closed convex subset of a complete space X and be an asymptotically nonexpansive mapping in the intermediate sense. If is a bounded sequence in K such that and , then .
Lemma 2.10 (See [[27], Corollary 3.4])
Let K be a closed convex subset of a complete space X and be an asymptotically nonexpansive mapping in the intermediate sense. If is a bounded sequence in K Δ-converging to x and , then and .
3 Main results
Now, we prove the following lemmas using modified S-iteration scheme (8) needed in the sequel.
Lemma 3.1 Let K be a nonempty closed convex subset of a complete space X, and let be two asymptotically nonexpansive mappings in the intermediate sense with . Suppose that is defined by the iteration process (8). Put
such that . Suppose that and are real sequences in for some . Then
-
(i)
exists for all .
-
(ii)
exists.
Proof Let . From (8), (13) and Lemma 2.1(ii), we have
Again using (8), (13), (14) and Lemma 2.1(ii), we have
Taking infimum over all , we have
Since by the hypothesis of the theorem , it follows from Lemma 2.6, (15) and (16) that and exist. □
Lemma 3.2 Let K be a nonempty closed convex subset of a complete space X, and let be two asymptotically nonexpansive mappings in the intermediate sense with . Suppose that is defined by the iteration process (8) and is taken as in Lemma 3.1. Suppose that and are real sequences in for some . Then and .
Proof Using (8) and (11), we have
where , since by the hypothesis , it follows that . Again using (8), (11) and (17), we have
where and , since by the hypothesis , it follows that and . This implies that
and
Since , as and is convergent, therefore on taking limit as in (19) and (20), we get
and
Now using (8) and (21), we get
implies
Again using (21) and (22), we get
implies
Further using (23) and (24), we get
implies
Now using (8), (21) and (24), we get
implies
Let , by (21), we have as . Now, we have
by (21), (26), and the uniform continuity of T. Similarly, we can prove that
This completes the proof. □
Now we prove the Δ-convergence and strong convergence results.
Theorem 3.1 Let K be a nonempty closed convex subset of a complete space X, and let be two asymptotically nonexpansive mappings in the intermediate sense with . Suppose that is defined by the iteration process (8) and be taken as in Lemma 3.1. Suppose that and are real sequences in for some . Then the sequence is Δ-convergent to a point of .
Proof We first show that . Let , then there exists a subsequence of such that . By Lemma 2.5, there exists a subsequence of such that . By Lemma 2.10, and and so . By Lemma 3.1 exists, so by Lemma 2.4, we have , i.e., .
To show that Δ-converges to a point in , it is sufficient to show that consists of exactly one point.
Let be a subsequence of with , and let for some and converge. By Lemma 2.4, we have . Thus . This shows that is Δ-convergent to a point of . This completes the proof. □
Theorem 3.2 Let K be a nonempty closed convex subset of a complete space X, and let be two asymptotically nonexpansive mappings in the intermediate sense with . Suppose that is defined by the iteration process (8) and is taken as in Lemma 3.1. Suppose that and are real sequences in for some . If or , where , then the sequence converges strongly to a point in .
Proof From (16) of Lemma 3.1, we have
where . Since by the hypothesis of the theorem , by Lemma 2.6 and or gives that
Now, we show that is a Cauchy sequence in K.
From (15), we have
for the natural numbers m, n and . Since , therefore for any , there exists a natural number such that and for all . So, we can find such that . Hence, for all and , we have
This proves that is a Cauchy sequence in K. Thus, the completeness of X implies that must be convergent. Assume that . Since K is closed, therefore . Next, we show that . Since , we get , closedness of gives that . This completes the proof. □
Theorem 3.3 Let K be a nonempty closed convex subset of a complete space X, and let be two asymptotically nonexpansive mappings in the intermediate sense with . Suppose that is defined by the iteration process (8) and is taken as in Lemma 3.1. Suppose that and are real sequences in for some . If S and T satisfy the following conditions:
-
(i)
and ;
-
(ii)
If the sequence in K satisfies and , then or .
Then the sequence converges strongly to a point of .
Proof It follows from the hypothesis that and . From (ii), or . Therefore, the sequence must converge strongly to a point in by Theorem 3.2. This completes the proof. □
Theorem 3.4 Let K be a nonempty closed convex subset of a complete space X, and let be two asymptotically nonexpansive mappings in the intermediate sense with . Suppose that is defined by the iteration process (8) and is taken as in Lemma 3.1. Suppose that and are real sequences in for some . If either S or T is semi-compact, then the sequence converges strongly to a point of .
Proof Suppose that T is semi-compact. By Lemma 3.2, we have . So there exists a subsequence of such that . Now Lemma 3.2 guarantees that and so . Similarly, we can show that . Thus . By (16), we have
Since by the hypothesis , by Lemma 2.6, exists and gives that . This shows that converges strongly to a point of . This completes the proof. □
We recall the following definition.
A mapping , where K is a subset of a normed linear space E, is said to satisfy Condition (A) [32] if there exists a nondecreasing function with and for all such that for all , where .
We modify this definition for two mappings.
Two mappings , where K is a subset of a normed linear space E, are said to satisfy Condition (B) if there exists a nondecreasing function with and for all such that for all , where and and are two nonnegative real numbers such that . It is to be noted that Condition (B) is weaker than the compactness of the domain K.
Remark 3.1 Condition (B) reduces to Condition (A) when .
As an application of Theorem 3.2, we establish another strong convergence result employing Condition (B) as follows.
Theorem 3.5 Let K be a nonempty closed convex subset of a complete space X, and let be two asymptotically nonexpansive mappings in the intermediate sense with . Suppose that is defined by the iteration process (8) and is taken as in Lemma 3.1. Suppose that and are real sequences in for some . If S and T satisfy Condition (B), then the sequence converges strongly to a point of .
Proof By Lemma 3.2, we know that
From Condition (B) and (32), we get
i.e., . Since is a nondecreasing function satisfying , for all , therefore we have
Now all the conditions of Theorem 3.2 are satisfied, therefore by its conclusion converges strongly to a point of . This completes the proof. □
Remark 3.2 Our results generalize, unify and extend several comparable results in the existing literature.
References
Bridson MR, Haefliger A Grundlehren der Mathematischen Wissenschaften 319. In Metric Spaces of Non-positive Curvature. Springer, Berlin; 1999.
Kirk WA:Fixed point theory in spaces and ℝ-trees. Fixed Point Theory Appl. 2004,2004(4):309-316.
Brown KS: Buildings. Springer, New York; 1989.
Goebel K, Reich S Monograph and Textbooks in Pure and Applied Mathematics 83. In Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York; 1984.
Kirk WA: Geodesic geometry and fixed point theory. Colección Abierta 64. In Seminar of Mathematical Analysis. University of Seville Secretary of Publications, Seville; 2003: (Malaga/Seville, 2002/2003)195-225. (Malaga/Seville, 2002/2003)
Kirk WA: Geodesic geometry and fixed point theory. II. In International Conference on Fixed Point Theory and Applications. Yokohama Publishers, Yokohama; 2004:113-142.
Agarwal RP, O’Regan D, Sahu DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 2007,8(1):61-79.
Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991,43(1):153-159. 10.1017/S0004972700028884
Tan KK, Xu HK: Fixed point iteration processes for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1994, 122: 733-739. 10.1090/S0002-9939-1994-1203993-5
Şahin A, Başarir M: On the strong convergence of a modified S -iteration process for asymptotically quasi-nonexpansive mapping in space. Fixed Point Theory Appl. 2013. Article ID 12, 2013: Article ID 12
Khan SH, Abbas M:Strong and △-convergence of some iterative schemes in spaces. Comput. Math. Appl. 2011,61(1):109-116. 10.1016/j.camwa.2010.10.037
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1972, 35: 171-174. 10.1090/S0002-9939-1972-0298500-3
Fukhar-ud-din H, Khan SH: Convergence of iterates with errors of asymptotically quasi-nonexpansive and applications. J. Math. Anal. Appl. 2007, 328: 821-829. 10.1016/j.jmaa.2006.05.068
Khan AR, Khamsi MA, Fukhar-ud-din H:Strong convergence of a general iteration scheme in spaces. Nonlinear Anal., Theory Methods Appl. 2011,74(3):783-791. 10.1016/j.na.2010.09.029
Liu QH: Iterative sequences for asymptotically quasi-nonexpansive mappings. J. Math. Anal. Appl. 2001, 259: 1-7. 10.1006/jmaa.2000.6980
Liu QH: Iterative sequences for asymptotically quasi-nonexpansive mappings with error member. J. Math. Anal. Appl. 2001, 259: 18-24. 10.1006/jmaa.2000.7353
Niwongsa Y, Panyanak B:Noor iterations for asymptotically nonexpansive mappings in spaces. Int. J. Math. Anal. 2010,4(13):645-656.
Saluja GS: Strong convergence theorem for two asymptotically quasi-nonexpansive mappings with errors in Banach space. Tamkang J. Math. 2007,38(1):85-92.
Shahzad N, Udomene A: Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2006. Article ID 18909, 2006: Article ID 18909
Bruck R, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq. Math. 1993,65(2):169-179.
Khamsi MA, Kirk WA Pure Appl. Math. In An Introduction to Metric Spaces and Fixed Point Theory. Wiley-Interscience, New York; 2001.
Bruhat F, Tits J: Groupes réductifs sur un corps local. Publ. Math. IHES 1972, 41: 5-251. 10.1007/BF02715544
Dhompongsa S, Panyanak B:On △-convergence theorem in spaces. Comput. Math. Appl. 2008,56(10):2572-2579. 10.1016/j.camwa.2008.05.036
Dhompongsa S, Kirk WA, Sims B: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal. 2006,65(4):762-772. 10.1016/j.na.2005.09.044
Kirk WA, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal. 2008,68(12):3689-3696. 10.1016/j.na.2007.04.011
Goebel K, Kirk WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.
Abbas M, Thakur BS, Thakur D:Fixed points of asymptotically nonexpansive mappings in the intermediate sense in spaces. Commun. Korean Math. Soc. 2013,28(4):107-121.
Hussain N, Khamsi MA: On asymptotic pointwise contractions in metric spaces. Nonlinear Anal. 2009,71(10):4423-4429. 10.1016/j.na.2009.02.126
Nanjaras B, Panyanak B:Demiclosed principle for asymptotically nonexpansive mappings in spaces. Fixed Point Theory Appl. 2010. Article ID 268780, 2010: Article ID 268780
Dhompongsa S, Kirk WA, Panyanak B: Nonexpansive set-valued mappings in metric and Banach spaces. J. Nonlinear Convex Anal. 2007,8(1):35-45.
Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 1993, 178: 301-308. 10.1006/jmaa.1993.1309
Senter HF, Dotson WG: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 1974, 44: 375-380. 10.1090/S0002-9939-1974-0346608-8
Acknowledgements
The authors would like to thank the editors and anonymous referees for their valuable suggestions that helped to improve the manuscript. This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NUR No. 57000621).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally and significantly in this research work. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kumam, P., Saluja, G.S. & Nashine, H.K. Convergence of modified S-iteration process for two asymptotically nonexpansive mappings in the intermediate sense in spaces. J Inequal Appl 2014, 368 (2014). https://doi.org/10.1186/1029-242X-2014-368
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-368