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On the strong and △convergence of SPiteration on CAT(0) space
Journal of Inequalities and Applications volume 2013, Article number: 311 (2013)
Abstract
In this paper, we study the strong and △convergence theorems of SPiteration for nonexpansive mappings on a CAT(0) space. Our results extend and improve many results in the literature.
MSC:47H09, 47H10.
Dedication
Dedicated to Professor Hari M Srivastava
1 Introduction
A CAT(0) space plays a fundamental role in various areas of mathematics (see Bridson and Haefliger [1], Burago et al. [2], Gromov [3]). Moreover, there are applications in biology and computer science as well [4, 5]. A metric space X is a CAT(0) 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 CAT(0) space. The complex Hilbert ball with a hyperbolic metric is a CAT(0) space (see [6]). Other examples include preHilbert spaces, Rtrees (see [1]) and Euclidean buildings (see [7]).
Fixed point theory in a CAT(0) space has been first studied by Kirk (see [8, 9]). He showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory in a CAT(0) space has been rapidly developed and a lot of papers have appeared (see, e.g., [8–16]).
The Noor iteration (see [17]) is defined by {x}_{1}\in K and
for all n\ge 1, where \{{\alpha}_{n}\}, \{{\beta}_{n}\} and \{{\gamma}_{n}\} are sequences in [0,1]. If we take {\beta}_{n}={\gamma}_{n}=0 for all n, (1.1) reduces to the Mann iteration (see [18]), and we take {\gamma}_{n}=0 for all n, (1.1) reduces to the Ishikawa iteration (see [19]).
The new twostep iteration (see [20]) is defined by {x}_{1}\in K and
for all n\ge 1, where \{{\alpha}_{n}\} and \{{\beta}_{n}\} are sequences in [0,1].
Recently, Phuengrattana and Suantai (see [21]) defined the SPiteration as follows:
for all n\ge 1, where {x}_{1}\in K, \{{\alpha}_{n}\}, \{{\beta}_{n}\} and \{{\gamma}_{n}\} are sequences in [0,1]. They showed that the Mann, Ishikawa, Noor and SPiterations are equivalent and the SPiteration converges better than the others for the class of continuous and nondecreasing functions. Clearly, the new twostep and Mann iterations are special cases of the SPiteration.
Now, we apply SPiteration (1.3) in a CAT(0) space for nonexpansive mappings as follows:
for all n\ge 1, where K is a nonempty convex subset of a CAT(0) space, {x}_{1}\in K, \{{\alpha}_{n}\}, \{{\beta}_{n}\} and \{{\gamma}_{n}\} are sequences in [0,1].
In this paper, we study the SPiteration for a nonexpansive mapping in a CAT(0) space. This paper contains three sections. In Section 2, we first collect some known preliminaries and lemmas that will be used in the proofs of our main theorems. In Section 3, we give the main results which are related to the strong and △convergence theorems of the SPiteration in a CAT(0) space. It is worth mentioning that our results in a CAT(0) space can be applied to any CAT(k) space with k\le 0 since any CAT(k) space is a CAT({k}^{\mathrm{\prime}}) space for every {k}^{\mathrm{\prime}}\ge k (see [1], p.165).
2 Preliminaries and lemmas
Let us recall some definitions and known results in the existing literature on this concept.
Let K be a nonempty subset of a CAT(0) space X and let T:K\to K be a mapping. A point x\in K is called a fixed point of T if Tx=x. We will denote the set of fixed points of T by F(T). The mapping T is said to be nonexpansive if
Let (X,d) be a metric space. A geodesic path joining x\in X to y\in X (or more briefly, a geodesic from x to y) is a map c from a closed interval [0,l]\subset R to X such that c(0)=x, c(l)=y and d(c(t),c({t}^{\mathrm{\prime}}))=t{t}^{\mathrm{\prime}} for all t,{t}^{\mathrm{\prime}}\in [0,l]. In particular, c is an isometry and d(x,y)=l. The image of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denoted by [x,y]. The space (X,d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be a uniquely geodesic space if there is exactly one geodesic joining x to y for each x,y\in X.
A geodesic triangle \u25b3({x}_{1},{x}_{2},{x}_{3}) in a geodesic metric space (X,d) 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 \u25b3({x}_{1},{x}_{2},{x}_{3}) in (X,d) is a triangle \overline{\u25b3}({x}_{1},{x}_{2},{x}_{3})=\u25b3({\overline{x}}_{1},{\overline{x}}_{2},{\overline{x}}_{3}) in the Euclidean plane {\mathbb{R}}^{2} such that {d}_{{\mathbb{R}}^{2}}({\overline{x}}_{i},{\overline{x}}_{j})=d({x}_{i},{x}_{j}) for i,j\in \{1,2,3\}.
A geodesic metric space is said to be a CAT(0) space [1] if all geodesic triangles of appropriate size satisfy the following comparison axiom.
CAT(0): Let △ be a geodesic triangle in X and let \overline{\u25b3} be a comparison triangle for △. Then △ is said to satisfy the CAT(0) inequality if for all x,y\in \u25b3 and all comparison points \overline{x},\overline{y}\in \overline{\u25b3},
Finally, we observe that if x, {y}_{1}, {y}_{2} are points of a CAT(0) space and if {y}_{0} is the midpoint of the segment [{y}_{1},{y}_{2}], then the CAT(0) inequality implies
The equality holds for the Euclidean metric. In fact (see [1], p.163), a geodesic metric space is a CAT(0) space if and only if it satisfies inequality (2.1) (which is known as the CN inequality of Bruhat and Tits [22]).
The following lemmas can be found in [12].
Lemma 1 ([12], Lemma 2.4)
Let X be a CAT(0) space. Then
for all t\in [0,1] and x,y,z\in X.
Lemma 2 ([12], Lemma 2.5)
Let X be a CAT(0) space. Then
for all t\in [0,1] and x,y,z\in X.
Now, we recall some definitions.
Let X be a complete CAT(0) space and let \{{x}_{n}\} be a bounded sequence in X. For x\in X, set
The asymptotic radius r(\{{x}_{n}\}) of \{{x}_{n}\} is given by
The asymptotic center A(\{{x}_{n}\}) of \{{x}_{n}\} is the set
It is known that in a complete CAT(0) space, A(\{{x}_{n}\}) consists of exactly one point ([10], Proposition 7). Also, every CAT(0) space has the Opial property, i.e., if \{{x}_{n}\} is a sequence in K and \u25b3\text{}{lim}_{n\to \mathrm{\infty}}{x}_{n}=x, then for each y(\ne x)\in K,
Definition 1 ([16], Definition 3.1)
A sequence \{{x}_{n}\} in a CAT(0) space X is said to be △convergent to x\in X if x is the unique asymptotic center of \{{u}_{n}\} for every subsequence \{{u}_{n}\} of \{{x}_{n}\}. In this case, we write \u25b3\text{}{lim}_{n\to \mathrm{\infty}}{x}_{n}=x and x is called the △limit of \{{x}_{n}\}.
The notion of △convergence in a general metric space was introduced by Lim [23]. Recently, Kirk and Panyanak [16] used the concept of △convergence introduced by Lim [23] to prove on the CAT(0) space analogous of some Banach space results which involve weak convergence. Further, Dhompongsa and Panyanak [12] obtained △convergence theorems for the Picard, Mann and Ishikawa iterations in a CAT(0) space.
Lemma 3 ([12], Lemma 2.7)

(i)
Every bounded sequence in a complete CAT(0) space always has a △convergent subsequence.

(ii)
Let K be a nonempty closed convex subset of a complete CAT(0) space and let \{{x}_{n}\} be a bounded sequence in K. Then the asymptotic center of \{{x}_{n}\} is in K.

(iii)
Let K be a nonempty closed convex subset of a complete CAT(0) space X and let f:K\to X be a nonexpansive mapping. Then the conditions, \{{x}_{n}\} △converges to x and d({x}_{n},f({x}_{n}))\to 0, imply x\in K and f(x)=x.
3 Main results
We start with proving the lemma for later use in this section.
Lemma 4 Let K be a nonempty closed convex subset of a complete CAT(0) space X and let T:K\to K be a nonexpansive mapping with F(T)\ne \mathrm{\varnothing}. Let \{{\alpha}_{n}\} and \{{\beta}_{n}\} be sequences in [0,1], \{{\gamma}_{n}\} be a sequence in [\u03f5,1\u03f5] for some \u03f5\in (0,1) and \{{x}_{n}\} be defined by the iteration process (1.4). Then

(i)
{lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}^{\star}) exists for all {x}^{\star}\in F(T).

(ii)
{lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0.
Proof (i) Let {x}^{\star}\in F(T). By (1.4) and Lemma 1, we have
Also, we get
Then we obtain
Using (1.4) and Lemma 1, we have
Combining (3.3) and (3.4), we get
This implies that the sequence \{d({x}_{n},{x}^{\star})\} is nonincreasing and bounded below, and so {lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}^{\star}) exists for all {x}^{\star}\in F(T). This completes the proof of part (i).

(ii)
Let
\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},{x}^{\star})=c.(3.5)
Firstly, we will prove that {lim}_{n\to \mathrm{\infty}}d({y}_{n},{x}^{\star})=c. By (3.4) and (3.5),
Also, from (3.3) and (3.5),
Then we obtain
Secondly, we will prove that {lim}_{n\to \mathrm{\infty}}d({z}_{n},{x}^{\star})=c. From (3.1) and (3.2), we have
This gives
Next, by Lemma 2,
Thus,
so that
Now using (3.5) and (3.7), \underset{n\to \mathrm{\infty}}{limsup}d({x}_{n},T{x}_{n})\le 0 and hence,
This completes the proof of part (ii). □
Now, we give the △convergence theorem of the SPiteration on a CAT(0) space.
Theorem 1 Let X, K, T, \{{\alpha}_{n}\}, \{{\beta}_{n}\}, \{{\gamma}_{n}\}, \{{x}_{n}\} satisfy the hypotheses of Lemma 4. Then the sequence \{{x}_{n}\} △converges to a fixed point of T.
Proof By Lemma 4, we have {lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0. Also, {lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}^{\star}) exists for all {x}^{\star}\in F(T). Thus \{{x}_{n}\} is bounded. Let {W}_{\u25b3}({x}_{n})=\cup A(\{{u}_{n}\}), where the union is taken over all subsequences \{{u}_{n}\} of \{{x}_{n}\}. We claim that {W}_{\u25b3}({x}_{n})\subseteq F(T). Let u\in {W}_{\u25b3}({x}_{n}). Then there exists a subsequence \{{u}_{n}\} of \{{x}_{n}\} such that A(\{{u}_{n}\})=\{u\}. By Lemma 3(i) and (ii), there exists a subsequence \{{v}_{n}\} of \{{u}_{n}\} such that \u25b3\text{}{lim}_{n\to \mathrm{\infty}}{v}_{n}=v\in K. By Lemma 3(iii), v\in F(T). By Lemma 4(i), {lim}_{n\to \mathrm{\infty}}d({x}_{n},v) exists. Now, we claim that u=v. On the contrary, assume that u\ne v. Then, by the uniqueness of asymptotic centers, we have
This is a contradiction. Thus u=v\in F(T) and {W}_{\u25b3}({x}_{n})\subseteq F(T). To show that the sequence \{{x}_{n}\} △converges to a fixed point of T, we will show that {W}_{\u25b3}({x}_{n}) consists of exactly one point. Let \{{u}_{n}\} be a subsequence of \{{x}_{n}\} with A(\{{u}_{n}\})=\{u\} and let A(\{{x}_{n}\})=\{x\}. We have already seen that u=v and v\in F(T). Finally, we claim that x=v. If not, then the existence of {lim}_{n\to \mathrm{\infty}}d({x}_{n},v) and the uniqueness of asymptotic centers imply that there exists a contradiction as (3.8) and hence x=v\in F(T). Therefore, {W}_{\u25b3}({x}_{n})=\{x\}. As a result, the sequence \{{x}_{n}\} △converges to a fixed point of T. □
We give the strong convergence theorem on a CAT(0) space as follows.
Theorem 2 Let X, K, T, \{{\alpha}_{n}\}, \{{\beta}_{n}\}, \{{\gamma}_{n}\}, \{{x}_{n}\} satisfy the hypotheses of Lemma 4. Then the sequence \{{x}_{n}\} converges strongly to a fixed point of T if and only if
where d(x,F(T))=inf\{d(x,p):p\in F(T)\}.
Proof Necessity is obvious. Conversely, suppose that lim{inf}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0. As proved in Lemma 4(i),
for all {x}^{\star}\in F(T). This implies that
Since the sequence \{d({x}_{n},F(T))\} is nonincreasing and bounded below, {lim}_{n\to \mathrm{\infty}}d({x}_{n},F(T)) exists. Thus, by the hypothesis, {lim}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0.
Next, we will show that \{{x}_{n}\} is a Cauchy sequence in K. Let \epsilon >0 be arbitrarily chosen. Since {lim}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0, there exists a constant {n}_{0} such that for all n\ge {n}_{0}, we have
In particular, inf\{d({x}_{{n}_{0}},p):p\in F(T)\}<\frac{\epsilon}{4}. Thus there exists {p}^{\star}\in F(T) such that
Now, for all m,n\ge {n}_{0}, we have
Hence \{{x}_{n}\} is a Cauchy sequence in a closed subset K of a complete CAT(0) space X, it must be convergent to a point in K. Let {lim}_{n\to \mathrm{\infty}}{x}_{n}={x}^{\star}\in K. Now, {lim}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0 gives that d({x}^{\star},F(T))=0 and the closedness of F(T) forces {x}^{\star} to be in F(T). Therefore, the sequence \{{x}_{n}\} converges strongly to a fixed point {x}^{\star} of T. □
Senter and Dotson [24] introduced Condition (I) as follows.
Definition 2 ([24], p.375)
A mapping T:K\to K is said to satisfy Condition (I) if there exists a nondecreasing function f:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with f(0)=0 and f(r)>0 for all r>0 such that
With respect to the above definition, we have the following theorem.
Theorem 3 Let X, K, \{{\alpha}_{n}\}, \{{\beta}_{n}\}, \{{\gamma}_{n}\}, \{{x}_{n}\} satisfy the hypotheses of Lemma 4 and let T:K\to K be a nonexpansive mapping satisfying Condition (I). Then the sequence \{{x}_{n}\} converges strongly to a fixed point of T.
Proof By Lemma 4(i), {lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}^{\star}) exists for all {x}^{\star}\in F(T). Let this limit be c, where c\ge 0. If c=0, there is nothing to prove. Suppose that c>0. Now,
gives
which means that d({x}_{n+1},F(T))\le d({x}_{n},F(T)) and so {lim}_{n\to \mathrm{\infty}}d({x}_{n},F(T)) exists. Also, by Lemma 4(ii), we have {lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0. It follows from Condition (I) that
That is,
Since f:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is a nondecreasing function satisfying f(0)=0, f(t)>0 for all t\in (0,\mathrm{\infty}), therefore we obtain
The conclusion now follows from Theorem 2. □
It is worth noting that, in the case of a nonexpansive mapping, Condition (I) is weaker than the compactness of K.
Since the SPiteration reduces to the new twostep iteration when {\alpha}_{n}=0 for all n\in \mathbb{N} and to the Mann iteration when {\alpha}_{n}={\beta}_{n}=0 for all n\in \mathbb{N}, we have the following corollaries.
Corollary 1 Let X, K, T, \{{\gamma}_{n}\} satisfy the hypotheses of Lemma 4 and let \{{x}_{n}\} be defined by the iteration process (1.2). Then the sequence \{{x}_{n}\} △converges to a fixed point of T. Further, if \{{x}_{n}\} is defined by the iteration process (1.1), the sequence \{{x}_{n}\} △converges to a fixed point of T.
Corollary 2 Let X, K, \{{\gamma}_{n}\} satisfy the hypotheses of Lemma 4, let T:K\to K be a nonexpansive mapping satisfying Condition (I) and let \{{x}_{n}\} be defined by the iteration process (1.2). Then the sequence \{{x}_{n}\} converges strongly to a fixed point of T. Also, if \{{x}_{n}\} is defined by the iteration process (1.1), the sequence \{{x}_{n}\} converges strongly to a fixed point of T.
Conclusions
The SPiteration reduces to the new twostep and Mann iterations. Then these results presented in this paper extend and generalize some works for CAT(0) space in the literature.
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Acknowledgements
This paper was supported by Sakarya University BAPK Project No. 20130200003. The authors are grateful to the referee for his/her careful reading and valuable comments and suggestions which led to the present form of the paper. This paper has been presented in International Congress in Honour of Professor H. M. Srivastava in Uludağ University, Bursa, Turkey, 2326 August 2012.
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Şahin, A., Başarır, M. On the strong and △convergence of SPiteration on CAT(0) space. J Inequal Appl 2013, 311 (2013). https://doi.org/10.1186/1029242X2013311
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DOI: https://doi.org/10.1186/1029242X2013311