- Research
- Open access
- Published:
Strong convergence of an Ishikawa-type algorithm in spaces
Fixed Point Theory and Applications volume 2013, Article number: 207 (2013)
Abstract
We study strong convergence of an Ishikawa-type algorithm of two asymptotically nonexpansive type maps to their common fixed point on a space. Our work provides an affirmative answer to the question of Tan and Xu (Proc. Am. Math. Soc. 122:733-739, 1994); in particular, strong convergence of an Ishikawa-type algorithm of two asymptotically nonexpansive maps without the rate of convergence condition is obtained on a nonlinear domain.
MSC:47H09, 47H10, 49M05.
1 Introduction
A space is simply a geodesic metric space whose each geodesic triangle is at least as thin as its comparison triangle in the Euclidean plane. In 2004, Kirk [1] proved a fixed point theorem for a nonexpansive map defined on a subset of a space. Since then, approximation of fixed points of nonlinear maps on a space has rapidly developed (see, e.g., [2–5]).
We describe briefly the needed details for a space. A metric space is said to be a length space if any two points of X are joined by a rectifiable path (that is, a path of finite length) and the distance between any two points of X is taken to be the infimum of the lengths of all rectifiable paths joining them. In this case, d is said to be a length metric (otherwise known as an inner metric or intrinsic metric). In case no rectifiable path joins two points of the space, the distance between them is taken to be ∞.
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 Δ) and a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for geodesic triangle in is a triangle in such that for . Such a triangle always exists (see [6]).
A geodesic metric space is said to be a space if all geodesic triangles of appropriate size satisfy the following comparison axiom.
Let Δ be a geodesic triangle in X and let be a comparison triangle for Δ. Then Δ is said to satisfy the inequality if for all and all comparison points ,
If x, , are points of a space and is the midpoint of the segment , which we will denote by , then the inequality implies
The above inequality is the (CN) inequality of Bruhat and Titz [7] and it was extended in [8] as follows:
for any and .
Let us recall that a geodesic metric space is a space if and only if it satisfies the (CN) inequality (see [6], 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 C of a space X is convex if for any , we have .
Complete spaces are known as Hadamard spaces (see [9]). The reader interested in a more general nonlinear domain, namely 2-uniformly convex hyperbolic space containing a space as a special case, is referred to Dehaish [10] and Dehaish et al. [11].
Let C be a nonempty subset of a metric space . Then a selfmap T on C is:
-
(i)
uniformly L-Lipschitzian if for some , for , ;
-
(ii)
uniformly Hölder continuous if for some positive constants L and α, for , ;
-
(iii)
uniformly equicontinuous if for any , there exists such that whenever for , or, equivalently, T is uniformly equicontinuous if and only if whenever as ;
-
(iv)
asymptotically nonexpansive if there is a sequence with such that for , ;
-
(v)
asymptotically nonexpansive in the intermediate sense provided T is uniformly continuous and for , and
-
(vi)
of asymptotically nonexpansive type in the sense of Xu [12] if for each , ;
-
(vii)
of asymptotically nonexpansive type in the sense of Chang et al. [13] if for each , .
The map T is semi-compact if for any bounded sequence in C with as , there is a subsequence of such that as .
It is not difficult to see that nonexpansive map, asymptotically nonexpansive map, asymptotically nonexpansive map in the intermediate sense and asymptotically nonexpansive type map in the sense of Xu [12] all are special cases of asymptotically nonexpansive type map in the sense of Chang et al. [13]. Moreover, a uniformly L-Lipschitzian map is uniformly Hölder continuous, and a uniformly Hölder continuous map is uniformly equicontinuous. However, the converse statements are not true as indicated below.
Example 1.1 Take and . Define by for all . Then T is uniformly equicontinuous, but it is neither uniformly L-Lipschitzian nor uniformly Hölder continuous.
In uniformly convex Banach spaces, the convergence of an Ishikawa-type algorithm and a Mann-type algorithm of nonexpansive maps, asymptotically nonexpansive maps and asymptotically nonexpansive maps in the intermediate sense to their fixed points have been studied by a number of researchers [12, 14–24]. For the iterative construction of fixed points of some other classes of nonlinear maps, see [25–27].
The sequence in definition (iv) satisfies the rate of convergence condition if . This condition has been extensively used in iterative construction of fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces and spaces (see, e.g., [4, 5, 21, 28, 29]).
Chang et al. [13] established strong convergence of an Ishikawa-type algorithm as well as a Mann-type algorithm to a fixed point of an asymptotically nonexpansive type map.
We shall follow the idea of a geodesic path, namely, there exists a unique point for any and , to construct an Ishikawa-type algorithm of two asymptotically nonexpansive type maps on a nonempty subset C of a space.
where .
When (the identity map) in (1.1), it reduces to the following Mann-type algorithm:
where .
The purpose of this paper is to approximate a common fixed point of asymptotically nonexpansive type maps in a special kind of a metric space, namely a space. Our work is a significant generalization of the corresponding results in [5], and it provides analogues of the related results of Chang et al. [13] in uniformly convex Banach spaces. One of our results (Theorem 2.4) gives an affirmative answer to a famous question of Tan and Xu [30] on a nonlinear domain for common fixed points.
2 Fixed point approximation
We begin with the following asymptotic regularity result.
Lemma 2.1 Let C be a nonempty bounded closed convex subset of a space X. Let be uniformly equicontinuous. Then for the sequence in (1.1) satisfying
we have that
Proof Since S is uniformly equicontinuous and
therefore,
Now
gives that
Clearly,
applying limsup to both sides of (2.2), using the uniformly equicontinuous property of S and (2.1), we get that
and hence
Similarly,
That is,
 □
Our main result is as follows.
Theorem 2.2 Let C be a nonempty, bounded, closed and convex subset of a space X. Let be uniformly equicontinuous and asymptotically nonexpansive type maps such that . Suppose that for some , where and are the control parameters of the iteration scheme in (1.1). If S or T is semi-compact, then converges strongly to a common fixed point of S and T.
Proof For any , by the (CN)-inequality, we have
That is,
Next we consider the third term on the right side of (2.3):
That is,
Substituting (2.4) into (2.3) and using , we have
Next we prove that
Assume that and .
Then there exist subsequences (we use the same notation for a subsequence as well) of , and , such that and .
Now from (2.5) it follows that
For an asymptotically nonexpansive type map T, we have that
That is,
Hence, for given (), there exists a positive integer such that
Since and are sequences in C, therefore, for , it follows that
and
In the light of the two inequalities above, (2.6) reduces to
Let be any positive integer. Obtain inequalities from (2.7) and then, summing up these inequalities, we get
If , then
a contradiction.
This proves that .
That is,
As
and S is uniformly equicontinuous. So, by taking lim sup on both sides, we get
Now, Lemma 2.1 implies that
Since T is semi-compact, therefore there exists a subsequence of and such that
Now, by the uniform equicontinuity of S and T and hence continuity, it follows from (2.8) that
This gives that q is a common fixed point of S and T.
We now proceed to establish strong convergence of to q.
Since
therefore
Clearly,
Therefore, from (2.9) and (2.10), it follows that
Next we prove that as .
Since is of asymptotically nonexpansive type and is a sequence in C, therefore we have
As as , it follows from (2.11) that
That is,
Replace p by q in (2.5) to get
which gives that as .
Continuing in this way, by induction, we can prove that for any ,
By induction, one can prove that converges to q as ; in fact gives that as . □
We need the following lemma to approximate a common fixed point of two asymptotically nonexpansive maps.
Lemma 2.3 Every asymptotically nonexpansive selfmap T on a nonempty bounded subset C of a metric space X is uniformly equicontinuous and of asymptotically nonexpansive type.
Proof Let be an asymptotically nonexpansive map with a sequence such that . Let . Then, for each , there exists a positive integer such that for all . Put . Then for , . Choose . Then whenever for , , proving that T is uniformly equicontinuous.
The second part of the lemma follows from
 □
By Theorem 2.2 and Lemma 2.3, we have the following result which is new in the literature and sets an analogue of Theorem 2 in [21] without the rate of convergence condition.
Theorem 2.4 Let C be a nonempty, bounded, closed and convex subset of a space X. Let be asymptotically nonexpansive maps with sequences , respectively and . Suppose that for some , where and are the control parameters of the sequence in (1.1). If S or T is semi-compact, then converges strongly to a common fixed point of S and T.
As every uniformly equicontinuous map is uniformly L-Lipschitzian, so the following result is immediate and it unifies Theorem 2.1 and Theorem 2.2 of Chang et al. [13] in Hadamard spaces.
Theorem 2.5 Let C be a nonempty, bounded, closed and convex subset of a space X. Let be uniformly L-Lipschitzian and asymptotically nonexpansive type maps such that . Suppose that for some , where and are the control parameters of the sequence in (1.1). If S or T is semi-compact, then converges strongly to a common fixed point of S and T.
For , Theorem 2.5 sets an analogue of Theorem 2.1 in [13].
Theorem 2.6 Let C be a nonempty, bounded, closed and convex subset of a space X. Let be a uniformly L-Lipschitzian and asymptotically nonexpansive type map such that . Suppose that for some , where and are the control parameters of the sequence in (1.1) with . If T is semi-compact, then converges strongly to a fixed point of T.
On taking (the identity map) in Theorem 2.5, we obtain an analogue of Theorem 2.2 in [13].
Theorem 2.7 Let C be a nonempty, bounded, closed and convex subset of a space X. Let be a uniformly L-Lipschitzian and asymptotically nonexpansive type map such that . Suppose that for some , where and are the control parameters of the sequence in (1.2). If T is semi-compact, then converges strongly to a fixed point of T.
Remark 2.8 (1) Tan and Xu [30] obtained only weak convergence theorems for asymptotically nonexpansive maps satisfying the rate of convergence condition and remarked, ‘We do not know whether our weak convergence Theorem 3.1 remains valid if is allowed to approach 1 slowly enough so that diverges’. Our Theorem 2.4 gives an affirmative answer to their question in spaces.
(2) Our results are generalizations in spaces of the corresponding basic results in [16, 21, 28, 29].
(3) Theorem 2.2 improves and generalizes Theorems 4.2-4.3 in [5].
References
Kirk WA: Geodesic geometry and fixed point theory. In Seminar of Mathematical Analysis. Univ. Sevilla Secr. Publ., Seville; Malaga/Seville 2002/2003 2003:195–225.
Dhompongsa S, Fupinwong W, Kaewkhao A: Common fixed points of a nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces. Nonlinear Anal. 2009, 70: 4268–4273. 10.1016/j.na.2008.09.012
Leustean L:A quadratic rate of asymptotic regularity for -spaces. J. Math. Anal. Appl. 2007, 325: 386–399. 10.1016/j.jmaa.2006.01.081
Nanjaras B, Panyanak B:Demiclosedness principle for asymptotically nonexpansive mappings in spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 268780
Niwongsa Y, Panyanak B:Noor iterations for asymptotically nonexpansive mappings in spaces. Int. J. Math. Anal. 2010, 4: 645–656.
Bridson M, Haefliger A: Metric Spaces of Non-Positive Curvature. Springer, Berlin; 1999.
Bruhat F, Tits J: Groupes réductifs sur un corps local I: Données radicielles valuées. Publ. Math. Inst. Hautes Études Sci. 1972, 41: 5–251. 10.1007/BF02715544
Dhompongsa S, Panyanak B:On Δ-convergence theorems in spaces. Comput. Math. Appl. 2008, 56: 2572–2579. 10.1016/j.camwa.2008.05.036
Khamsi MA, Kirk WA: An Introduction to Metric Spaces and Fixed Point Theory. Wiley-Interscience, New York; 2001.
Ibn Dehaish BA: Ishikawa iteration process for asymptotic pointwise nonexpansive mappings in metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 98 10.1186/1687-1812-2013-98
Ibn Dehaish BA, Khamsi MA, Khan AR: Mann iteration process for asymptotic pointwise nonexpansive mappings in metric spaces. J. Math. Anal. Appl. 2013, 397: 861–868. 10.1016/j.jmaa.2012.08.013
Xu HK: Existence and convergence for fixed points of mappings of asymptotically nonexpansive type. Nonlinear Anal. 1991, 16: 1139–1146. 10.1016/0362-546X(91)90201-B
Chang SS, Chu YJ, Kim JK, Kim KH: Iterative approximation of fixed points for asymptotically nonexpansive type mappings in Banach spaces. Panam. Math. J. 2001, 11: 53–63.
Cho YJ, Zhou H, Hyun HG, Kim JK: Convergence theorems of iterative sequences for asymptotically nonexpansive type mappings in Banach spaces. Commun. Appl. Nonlinear Anal. 2003, 10: 85–92.
Fukhar-ud-din H, Khan SH: Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications. J. Math. Anal. Appl. 2007, 328: 821–829. 10.1016/j.jmaa.2006.05.068
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.2007.01.008
Huang JC: On common fixed points of asymptotically nonexpansive mappings in the intermediate sense. Czechoslov. Math. J. 2004, 54: 1055–1063.
Khan AR: On modified Noor iterations for asymptotically nonexpansive mappings. Bull. Belg. Math. Soc. Simon Stevin 2010, 17: 127–140.
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 10.1186/1687-1812-2012-54
Khan SH, Fukhar-ud-din H: Weak and strong convergence of a scheme with errors for two nonexpansive mappings. Nonlinear Anal. 2005, 61: 1295–1301. 10.1016/j.na.2005.01.081
Khan SH, Takahashi W: Approximating common fixed points of two asymptotically nonexpansive mappings. Sci. Math. Jpn. 2001, 53: 143–148.
Kim GE, Kiuchi H, Takahashi W: Weak and strong convergence of Ishikawa iterations for asymptotically nonexpansive mappings in the intermediate sense. Sci. Math. Jpn. 2004, 60: 95–106.
Li G: Asymptotic behavior for commutative semigroups of asymptotically nonexpansive type mappings. Nonlinear Anal. 2000, 42: 175–183. 10.1016/S0362-546X(98)00338-1
Li G, Sims B: Fixed point theorems for mappings of asymptotically nonexpansive type. Nonlinear Anal. 2002, 50: 1085–1091. 10.1016/S0362-546X(01)00744-1
Berinde V: On the convergence of the Ishikawa iteration in the class of quasi contractive operators. Acta Math. Univ. Comen. 2004, 73(1):119–126.
Berinde V: A convergence theorem for Mann iteration in the class of Zamfirescu operators. An. Univ. Vest. Timiş., Ser. Mat.-Inf. 2007, 45(1):33–41.
Berinde V: Iterative Approximation of Fixed Points. Springer, Berlin; 2007.
Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884
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-U
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
Acknowledgements
The author is grateful to King Fahd University of Petroleum & Minerals for supporting research project IN 121023.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author did not provide this information.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Fukhar-ud-din, H. Strong convergence of an Ishikawa-type algorithm in spaces. Fixed Point Theory Appl 2013, 207 (2013). https://doi.org/10.1186/1687-1812-2013-207
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-207