Research  Open  Published:
Halpern iteration for strongly quasinonexpansive mappings on a geodesic space with curvature bounded above by one
Fixed Point Theory and Applicationsvolume 2013, Article number: 7 (2013)
Abstract
In this paper, we deal with the Halpern iterative scheme for a strongly quasinonexpansive mapping in the setting of a complete geodesic space with curvature bounded above by one. Our result can be applied to the image recovery problem. We also consider the approximation of a fixed point of a nonexpansive mapping and obtain convergence theorems, one of which is a supplement of the result by Pia̧tek with an additional sufficient condition.
MSC:47H09.
1 Introduction
Halpern’s iterative method [1] is one of the most effective methods to find a fixed point of a nonexpansive mapping, which guarantees strong convergence of the approximating sequence. A remarkable result for nonlinear mappings was obtained by Wittmann [2] in the setting of Hilbert spaces. Since then, it has been investigated by a large number of researchers, and they have obtained different types of strong convergence theorems for nonexpansive mappings and their variations.
On the other hand, the notion of a strongly nonexpansive mapping was first proposed by Bruck and Reich [3] as a generalization of firmly nonexpansive mappings. This mapping was later generalized to a strongly quasinonexpansive mapping by Bruck [4]. We propose a new definition of this mapping in the framework of $CAT(1)$ space by imposing a natural bound on the diameter of the space.
The first result of convergence of the Halpern iteration on a complete $CAT(0)$ space was obtained by Saejung [5] for nonexpansive mappings, and a similar result in the setting of a complete $CAT(1)$ space was proposed by Pia̧tek [6]. The combination of the Halpern iteration and strongly quasinonexpansive mappings was made recently. See [7, 8] and others.
In this paper, we deal with the Halpern iterative scheme for a strongly quasinonexpansive mapping in the setting of a complete $CAT(1)$ space. Then we show that the main result can be applied to the problem of image recovery. We also consider the approximation of a fixed point of a nonexpansive mapping. We point out that the proof of the result by Pia̧tek is not sufficient and we supplement it with an additional sufficient condition for the convergence of the iterative sequence.
2 Preliminaries
Let X be a metric space. An element z of X is said to be an asymptotic center of a sequence $\{{x}_{n}\}$ in X if
Moreover, $\{{x}_{n}\}$ is said to be Δconvergent and z is said to be its Δlimit if z is the unique asymptotic center of any subsequences of $\{{x}_{n}\}$.
A geodesic with endpoints $x,y\in X$ is defined as an isometric mapping from the closed segment $[0,l]$ of real numbers to X whose image connects x and y. If a geodesic exists for every $x,y\in X$, then X is called a geodesic space.
For a triangle $\mathrm{\u25b3}(x,y,z)$ in a geodesic space X satisfying $d(y,z)+d(z,x)+d(x,y)<2\pi $, we can find the comparison triangle $\mathrm{\u25b3}(\overline{x},\overline{y},\overline{z})$ in ${\mathbb{S}}^{2}$, that is, each corresponding edge has the same length as that of the original triangle. If every two points p, q on the edges of any $\mathrm{\u25b3}(x,y,z)$ and their corresponding points $\overline{p}$, $\overline{q}$ satisfy that
we call X a $CAT(1)$ space, where ${d}_{{\mathbb{S}}^{2}}$ is the spherical metric on ${\mathbb{S}}^{2}$.
In this paper, we deal with only $CAT(1)$ spaces; however, we remark that all the results can be easily generalized to $CAT(\kappa )$ spaces with positive κ by changing the scale of the space.
For two points x, y in a $CAT(1)$ space X with $d(x,y)<\pi $ and $t\in [0,1]$, we denote by $tx\oplus (1t)y$ the point z on a geodesic segment between x and y such that $d(y,z)=td(x,y)$ and $d(x,z)=(1t)d(x,y)$. A subset C of X is said to be πconvex if $tx\oplus (1t)y$ belongs to C for every $x,y\in C$ with $d(x,y)<\pi $ and $t\in [0,1]$.
We refer to [9] for more details on geodesic spaces including $CAT(1)$ spaces.
For three points x, y, z in a $CAT(1)$ space X with $d(y,z)+d(z,x)+d(x,y)<2\pi $ and $t\in [0,1]$, we know that the following inequality holds [10]:
where $v=ty\oplus (1t)z$. This simple inequality plays a very important role in this paper.
Let X be a complete $CAT(1)$ space, C a nonempty closed πconvex subset of X and suppose that $d(x,C)={inf}_{y\in C}d(x,y)<\pi /2$ for every $x\in X$. Then we can define the metric projection ${P}_{C}$ from X onto C; that is, for every $x\in X$, ${P}_{C}x\in C$ is the unique point satisfying
Let X be a $CAT(1)$ space. Let $T:X\to X$ and suppose that the set $F(T)=\{x\in X:x=Tx\}$ of fixed points is not empty. Then T is said to be quasinonexpansive if $d(Tx,p)\le d(x,p)$ for every $x\in X$ and $p\in F(T)$. T is said to be strongly quasinonexpansive if it is quasinonexpansive, and for every $p\in F(T)$ and every sequence $\{{x}_{n}\}$ in X satisfying that ${sup}_{n\in \mathbb{N}}d({x}_{n},p)<\pi /2$ and ${lim}_{n\to \mathrm{\infty}}(cosd({x}_{n},p)/cosd(T{x}_{n},p))=1$, it follows that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$. T is said to be Δdemiclosed if for any Δconvergent sequence $\{{x}_{n}\}$ in X, its Δlimit belongs to $F(T)$ whenever ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$.
The following lemmas are important for our main result.
Lemma 2.1 (Xu [11])
Let $\{{s}_{n}\}$, $\{{t}_{n}\}$ and $\{{u}_{n}\}$ be sequences of real numbers such that ${s}_{n}\ge 0$ and ${u}_{n}\ge 0$ for every $n\in \mathbb{N}$, ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{t}_{n}\le 0$, and ${\sum}_{n=0}^{\mathrm{\infty}}{u}_{n}<\mathrm{\infty}$. Let $\{{\gamma}_{n}\}$ be a sequence in $[0,1]$ such that ${\sum}_{n=0}^{\mathrm{\infty}}{\gamma}_{n}=\mathrm{\infty}$. If ${s}_{n+1}\le (1{\gamma}_{n}){s}_{n}+{\gamma}_{n}{t}_{n}+{u}_{n}$ for every $n\in \mathbb{N}$, then ${lim}_{n\to \mathrm{\infty}}{s}_{n}=0$.
Lemma 2.2 (SaejungYotkaew [12])
Let $\{{s}_{n}\}$ and $\{{t}_{n}\}$ be sequences of real numbers such that ${s}_{n}\ge 0$ for every $n\in \mathbb{N}$. Let $\{{\beta}_{n}\}$ be a sequence in $]0,1[$ such that ${\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty}$. Suppose that ${s}_{n+1}\le (1{\beta}_{n}){s}_{n}+{\beta}_{n}{t}_{n}$ for every $n\in \mathbb{N}$. If ${lim\hspace{0.17em}sup}_{k\to \mathrm{\infty}}{t}_{{n}_{k}}\le 0$ for every subsequence $\{{n}_{k}\}$ of ℕ satisfying ${lim\hspace{0.17em}inf}_{k\to \mathrm{\infty}}({s}_{{n}_{k}+1}{s}_{{n}_{k}})\ge 0$, then ${lim}_{n\to \mathrm{\infty}}{s}_{n}=0$.
Lemma 2.3 (HeFangLopezLi [13])
Let X be a complete $CAT(1)$ space and $p\in X$. If a sequence $\{{x}_{n}\}$ in X satisfies that ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}d({x}_{n},p)<\pi /2$ and that $\{{x}_{n}\}$ is Δconvergent to $x\in X$, then $d(x,p)\le {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}d({x}_{n},p)$.
3 Main result
As the main theorem of this paper, we prove strong convergence of the iterative sequence to a fixed point of a strongly quasinonexpansive mapping. We adopt the Halpern iterative scheme to generate the sequence. We begin with the following basic lemma, which is one of the main tools for our results.
Lemma 3.1 Let X be a $CAT(1)$ space such that $d(v,{v}^{\prime})<\pi $ for every $v,{v}^{\prime}\in X$. Let $\alpha \in [0,1]$ and $u,y,z\in X$. Then
where
Proof It is obvious if $u=y$. Otherwise, from the inequality
we have that
We also have that
and hence we obtain the desired result. □
Remark On the same assumption, we have
Indeed, it holds from the first inequality of the proof above together with
Now, we show the main theorem.
Theorem 3.2 Let X be a complete $CAT(1)$ space such that $d(v,{v}^{\prime})<\pi /2$ for every $v,{v}^{\prime}\in X$. Let $T:X\to X$ be a strongly quasinonexpansive and Δdemiclosed mapping, and suppose that $F(T)\ne \mathrm{\varnothing}$. Let $\{{\alpha}_{n}\}$ be a real sequence in $]0,1[$ such that ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$ and ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$. For given points $u,{x}_{0}\in X$, let $\{{x}_{n}\}$ be the sequence in X generated by
for $n\in \mathbb{N}$. Suppose that one of the following conditions holds:

(a)
${sup}_{v,{v}^{\prime}\in X}d(v,{v}^{\prime})<\pi /2$;

(b)
$d(u,{P}_{F(T)}u)<\pi /4$ and $d(u,{P}_{F(T)}u)+d({x}_{0},{P}_{F(T)}u)<\pi /2$;

(c)
${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}^{2}=\mathrm{\infty}$.
Then $\{{x}_{n}\}$ converges to ${P}_{F(T)}u$.
Proof Let $p={P}_{F(T)}u$ and let
for $n\in \mathbb{N}$. Then, since T is quasinonexpansive, it follows from Lemma 3.1 that
for every $n\in \mathbb{N}$. We also have that
for all $n\in \mathbb{N}$. Thus we obtain that
for all $n\in \mathbb{N}$ and hence ${sup}_{n\in \mathbb{N}}d({x}_{n},p)\le max\{d(u,p),d({x}_{0},p)\}<\pi /2$.
Now, we see that each of the conditions (a), (b) and (c) implies that ${\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty}$. For the cases of (a) and (b), let $M={sup}_{n\in \mathbb{N}}d(u,T{x}_{n})$. Then we show that $M<\pi /2$. For (a), it is trivial. For (b), since ${sup}_{n\in \mathbb{N}}d({x}_{n},p)\le max\{d(u,p),d({x}_{0},p)\}$, we have that
So, in each case of (a) and (b), we have
Since ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$, it follows that ${\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty}$. For the case of (c), we have that
for every $n\in \mathbb{N}$. Therefore, from the condition (c) we have that ${\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}=\mathrm{\infty}$.
For any subsequence $\{{s}_{{n}_{j}}\}$ of $\{{s}_{n}\}$ satisfying that ${lim\hspace{0.17em}inf}_{j\to \mathrm{\infty}}({s}_{{n}_{j}+1}{s}_{{n}_{j}})\ge 0$, we have that
Thus we have that ${lim}_{j\to \mathrm{\infty}}(cosd({x}_{{n}_{j}},p)cosd(T{x}_{{n}_{j}},p))=0$. Using the inequality ${sup}_{n\in \mathbb{N}}d(T{x}_{n},p)<\pi /2$, we also have ${lim}_{j\to \mathrm{\infty}}(cosd({x}_{{n}_{j}},p)/cosd(T{x}_{{n}_{j}},p))=1$. Since T is strongly quasinonexpansive, it follows that ${lim}_{j\to \mathrm{\infty}}d({x}_{{n}_{j}},T{x}_{{n}_{j}})=0$. Let $\{{v}_{k}\}$ be a Δconvergent subsequence of $\{{x}_{{n}_{j}}\}$ such that ${lim}_{k\to \mathrm{\infty}}d(u,{v}_{k})={lim\hspace{0.17em}inf}_{j\to \mathrm{\infty}}d(u,{x}_{{n}_{j}})$. Then, since T is Δdemiclosed and ${lim}_{k\to \mathrm{\infty}}d({v}_{k},T{v}_{k})=0$, the Δlimit z of $\{{v}_{k}\}$ belongs to $F(T)$. Using Lemma 2.3 and the definitions of the Δlimit and the metric projection, we have that
Therefore, we obtain that
By Lemma 2.2, we have that ${lim}_{n\to \mathrm{\infty}}{s}_{n}=0$, that is, $\{{x}_{n}\}$ converges to $p={P}_{F(T)}u$, and we finish the proof. □
4 Application to the image recovery problem
In the setting of Hilbert spaces, the image recovery problem can be formulated as to find the nearest point in the intersection of a family of closed convex subsets from a given point by using the metric projection of each subset. In this section, we consider this problem in the setting of complete $CAT(1)$ spaces. As the simplest case, we deal with only two closed convex subsets ${C}_{1}$ and ${C}_{2}$ such that ${C}_{1}\cap {C}_{2}\ne \mathrm{\varnothing}$ and generate an iterative sequence converging to the nearest point in ${C}_{1}\cap {C}_{2}$ from a given point.
First, we observe some properties of the metric projection defined on a $CAT(1)$ space. Let X be a complete $CAT(1)$ space, C a nonempty closed πconvex subset of X and suppose that $d(x,C)={inf}_{y\in C}d(x,y)<\pi /2$ for every $x\in X$. Then we can prove that the metric projection $P:X\to C$ is a strongly quasinonexpansive and Δdemiclosed mapping such that $F({P}_{C})=C$. Indeed, it is known that ${P}_{C}$ is quasinonexpansive; see [14]. Let $\{{x}_{n}\}$ be a sequence in X and $p\in C$ such that ${sup}_{n\in \mathbb{N}}d({x}_{n},p)<\pi /2$ and ${lim}_{n\to \mathrm{\infty}}(cosd({x}_{n},p)/cosd({P}_{C}{x}_{n},p))=1$. Then, from the property of metric projection, we have that
for every $n\in \mathbb{N}$. Therefore, we have
and thus ${lim}_{n\to \mathrm{\infty}}cosd({x}_{n},{P}_{C}{x}_{n})=1$, that is, ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{P}_{C}{x}_{n})=0$. Hence, ${P}_{C}$ is strongly quasinonexpansive.
On the other hand, let $\{{x}_{n}\}$ be such that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{P}_{C}{x}_{n})=0$ and assume that $\{{x}_{n}\}$ is Δconvergent to ${x}_{\mathrm{\infty}}\in X$. Then $\{{P}_{C}{x}_{n}\}$ is also Δconvergent to ${x}_{\mathrm{\infty}}$. Since $\{{P}_{C}{x}_{n}\}$ is a sequence in a closed πconvex subset C, we have that its Δlimit ${x}_{\mathrm{\infty}}$ belongs to C, that is, ${x}_{\mathrm{\infty}}\in F({P}_{C})$ [14]. It shows that ${P}_{C}$ is Δdemiclosed.
For two strongly quasinonexpansive and Δdemiclosed mappings having common fixed points, we can create a new strongly quasinonexpansive and Δdemiclosed mapping whose fixed points are common fixed points of given two mappings. For example, as we have seen above, metric projections to closed and convex sets are strongly quasinonexpansive and Δdemiclosed. Thus, for given two metric projections to closed convex sets whose intersection is nonempty, the following method is applicable. It is useful to solve the image recovery problem.
Lemma 4.1 Let X be a $CAT(1)$ space and ${y}_{0}$, ${y}_{1}$ and y elements of X such that $d({y}_{0},y)+d({y}_{1},y)+d({y}_{0},{y}_{1})<2\pi $. Then
Proof It is obvious if ${y}_{0}={y}_{1}$. Otherwise, we have that
Dividing above by $2sin(d({y}_{0},{y}_{1})/2)$, we get the conclusion. □
Corollary 4.2 Let ${T}_{0}$ and ${T}_{1}$ be quasinonexpansive mappings from X to X, ${x}_{0}$ and ${x}_{1}$ elements of X, and p an element of $F({T}_{0})\cap F({T}_{1})$. Then
Lemma 4.3 Let X be a complete $CAT(1)$ space such that $d(v,{v}^{\prime})<\pi /2$ for arbitrary v and ${v}^{\prime}$ of X, and ${T}_{0}$ and ${T}_{1}$ quasinonexpansive mappings from X to X such that $F({T}_{0})\cap F({T}_{1})\ne \mathrm{\varnothing}$. Then
Proof It is obvious that $F(\frac{1}{2}{T}_{0}\oplus \frac{1}{2}{T}_{1})\supset F({T}_{0})\cap F({T}_{1})$. We will show the opposite inclusion. We denote $T=\frac{1}{2}{T}_{0}\oplus \frac{1}{2}{T}_{1}$. Let $z\in F(T)$. Then, for arbitrary $p\in F({T}_{0})\cap F({T}_{1})$, from Corollary 4.2, we have that
that is, ${T}_{0}z={T}_{1}z$. Hence, $z=Tz={T}_{0}z={T}_{1}z$, which means $z\in F({T}_{0})\cap F({T}_{1})$. □
Lemma 4.4 Let X be a $CAT(1)$ space such that $d(v,{v}^{\prime})<\pi /2$ for arbitrary v and ${v}^{\prime}$ of X. Let ${T}_{0}$ and ${T}_{1}$ be mappings from X to X such that $F({T}_{0})\cap F({T}_{1})\ne \mathrm{\varnothing}$. If both ${T}_{0}$ and ${T}_{1}$ are strongly quasinonexpansive, then so is $\frac{1}{2}{T}_{0}\oplus \frac{1}{2}{T}_{1}$.
Proof We denote $T=\frac{1}{2}{T}_{0}\oplus \frac{1}{2}{T}_{1}$. By Corollary 4.2, for $x\in X$ and $p\in F({T}_{0})\cap F({T}_{1})$, we have
that is, $d(Tx,p)\le d(x,p)$ and hence T is quasinonexpansive. Moreover, for a sequence $\{{x}_{n}\}$ in X and a point p in $F(T)$ such that ${sup}_{n\in \mathbb{N}}d({x}_{n},p)<\pi /2$ and ${lim}_{n\to \mathrm{\infty}}(cosd({x}_{n},p)/cosd(T{x}_{n},p))=1$, by Lemma 4.1, we have
So, there exist two disjoint subsets $\{{m}_{i}\}$ and $\{{n}_{i}\}$ of ℕ such that $\mathbb{N}=\{{m}_{i}\}\cup \{{n}_{i}\}$ and
We may assume that both $\{{m}_{i}\}$ and $\{{n}_{i}\}$ are infinite sets without loss of generality. From Lemma 4.3, p is in $F({T}_{0})$ and thus
which means ${lim}_{i\to \mathrm{\infty}}(cosd({x}_{{m}_{i}},p)/cosd({T}_{0}{x}_{{m}_{i}},p))=1$. Since ${T}_{0}$ is strongly quasinonexpansive, we have that ${lim}_{i\to \mathrm{\infty}}d({T}_{0}{x}_{{m}_{i}},{x}_{{m}_{i}})=0$. By Corollary 4.2, we have
that is, ${lim}_{i\to \mathrm{\infty}}d({T}_{0}{x}_{{m}_{i}},{T}_{1}{x}_{{m}_{i}})=0$. Then it follows that ${lim}_{i\to \mathrm{\infty}}d({T}_{1}{x}_{{m}_{i}},{x}_{{m}_{i}})=0$. Similarly, we have that ${lim}_{i\to \mathrm{\infty}}d({T}_{1}{x}_{{n}_{i}},{x}_{{n}_{i}})=0$ and ${lim}_{i\to \mathrm{\infty}}d({T}_{0}{x}_{{n}_{i}},{x}_{{n}_{i}})=0$. Consequently, we have that ${lim}_{n\to \mathrm{\infty}}d({T}_{0}{x}_{n},{x}_{n})=0$ and ${lim}_{n\to \mathrm{\infty}}d({T}_{1}{x}_{n},{x}_{n})=0$. Hence, we obtain that ${lim}_{n\to \mathrm{\infty}}d(T{x}_{n},{x}_{n})=0$, which is the desired result. □
Lemma 4.5 Let X be a $CAT(1)$ space such that $d(v,{v}^{\prime})<\pi /2$ for arbitrary v and ${v}^{\prime}$ of X. Let ${T}_{0}$ and ${T}_{1}$ be mappings from X to X such that $F({T}_{0})\cap F({T}_{1})\ne \mathrm{\varnothing}$. If both ${T}_{0}$ and ${T}_{1}$ are Δdemiclosed, then so is $\frac{1}{2}{T}_{0}\oplus \frac{1}{2}{T}_{1}$.
Proof We denote $T=\frac{1}{2}{T}_{0}\oplus \frac{1}{2}{T}_{1}$. Let $\{{x}_{n}\}$ be a sequence in X and ${x}_{\mathrm{\infty}}$ an element of X such that $d(T{x}_{n},{x}_{n})\to 0$ and suppose that $\{{x}_{n}\}$ is Δconvergent to ${x}_{\mathrm{\infty}}$. Then, by Corollary 4.2, we have
that is, ${lim}_{n\to \mathrm{\infty}}d({T}_{0}{x}_{n},{T}_{1}{x}_{n})=0$. Thus we have
Since ${T}_{0}$ is Δdemiclosed, we have that ${T}_{0}{x}_{\mathrm{\infty}}={x}_{\mathrm{\infty}}$. In a similar fashion, we have that ${T}_{1}{x}_{\mathrm{\infty}}={x}_{\mathrm{\infty}}$. Hence $T{x}_{\mathrm{\infty}}={x}_{\mathrm{\infty}}$, that is, T is Δdemiclosed. □
Let X be a complete $CAT(1)$ space such that $d(v,{v}^{\prime})<\pi /2$ for every $v,{v}^{\prime}\in X$, and let ${C}_{0}$ and ${C}_{1}$ be closed convex subsets of X having the nonempty intersection. Then, for the metric projections ${P}_{{C}_{0}}$ and ${P}_{{C}_{1}}$, the mapping $\frac{1}{2}{P}_{{C}_{0}}\oplus \frac{1}{2}{P}_{{C}_{1}}$ is strongly quasinonexpansive and Δdemiclosed. Moreover, the set of its fixed points is ${C}_{0}\cap {C}_{1}$. Applying these facts to Theorem 3.2, we obtain the following result for the image recovery problem for two convex subsets.
Theorem 4.6 Let X be a complete $CAT(1)$ space such that $d(v,{v}^{\prime})<\pi /2$ for every $v,{v}^{\prime}\in X$. Let ${C}_{0}$ and ${C}_{1}$ be closed convex subsets of X such that ${C}_{0}\cap {C}_{1}\ne \mathrm{\varnothing}$. Let $\{{\alpha}_{n}\}$ be a real sequence in $]0,1[$ such that ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$ and ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$. For given points $u,{x}_{0}\in X$, let $\{{x}_{n}\}$ be the sequence in X generated by
for $n\in \mathbb{N}$. Suppose that one of the following conditions holds:

(a)
${sup}_{v,{v}^{\prime}\in X}d(v,{v}^{\prime})<\pi /2$;

(b)
$d(u,{P}_{{C}_{0}\cap {C}_{1}}u)<\pi /4$ and $d(u,{P}_{{C}_{0}\cap {C}_{1}}u)+d({x}_{0},{P}_{{C}_{0}\cap {C}_{1}}u)<\pi /2$;

(c)
${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}^{2}=\mathrm{\infty}$.
Then $\{{x}_{n}\}$ converges to ${P}_{{C}_{0}\cap {C}_{1}}u$.
5 Approximation to a fixed point of nonexpansive mappings
At the end of this paper, we prove two convergence theorems of iterative schemes which approximate a fixed point of a nonexpansive mapping. Firstly, we apply the main result Theorem 3.2 to this problem. We begin with the following lemmas.
Lemma 5.1 A nonexpansive mapping defined on a $CAT(1)$ space is Δdemiclosed.
Proof Let $S:X\to X$ be a nonexpansive mapping. Let $\{{x}_{n}\}$ be a Δconvergent sequence in X with the Δlimit ${x}_{\mathrm{\infty}}\in X$ and suppose that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},S{x}_{n})=0$. We will prove that ${x}_{\mathrm{\infty}}=S{x}_{\mathrm{\infty}}$. If ${x}_{\mathrm{\infty}}\ne S{x}_{\mathrm{\infty}}$, then, by the uniqueness of the asymptotic center, we have that
a contradiction. Hence, we have that S is Δdemiclosed. □
Lemma 5.2 Let X be a $CAT(1)$ space such that $d({v}^{\prime},{v}^{\u2033})+d({v}^{\u2033},v)+d(v,{v}^{\prime})<2\pi $ for every $v,{v}^{\prime},{v}^{\u2033}\in X$. Let $S:X\to X$ be a nonexpansive mapping with a nonempty set of fixed points $F(S)$. Then the mapping $T:X\to X$ defined by
for $x\in X$ is a strongly quasinonexpansive and Δdemiclosed mapping such that $F(T)=F(S)$.
Proof It is obvious that $F(T)=F(S)$ by definition and, since both the identity mapping I and S are quasinonexpansive, for $x\in X$ and $p\in F(T)=F(S)$, we have that
Thus T is quasinonexpansive. Let $\{{x}_{n}\}$ be a sequence in X such that ${sup}_{n\in \mathbb{N}}d({x}_{n},p)<\pi /2$, and suppose that ${lim}_{n\to \mathrm{\infty}}(cosd({x}_{n},p)/cosd(T{x}_{n},p))=1$. Then we have
for every $n\in \mathbb{N}$. It follows that
and since ${sup}_{n\in \mathbb{N}}d(T{x}_{n},p)\le {sup}_{n\in \mathbb{N}}d({x}_{n},p)<\pi /2$, we have
It implies that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$ and hence T is strongly quasinonexpansive.
For the Δdemiclosedness of T, use Lemmas 4.5 and 5.1 with the fact that the identity mapping is also Δdemiclosed. □
Applying this lemma and the results in the previous section to Theorem 3.2, we obtain the following convergence theorem of an iterative scheme approximating a fixed point of a nonexpansive mapping.
Theorem 5.3 Let X be a complete $CAT(1)$ space such that $d(v,{v}^{\prime})<\pi /2$ for every $v,{v}^{\prime}\in X$. Let $S:X\to X$ be a nonexpansive mapping and suppose that $F(S)\ne \mathrm{\varnothing}$. Let $\{{\alpha}_{n}\}$ be a real sequence in $]0,1[$ such that ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$ and ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$. For given points $u,{x}_{0}\in X$, let $\{{x}_{n}\}$ be the sequence in X generated by
for $n\in \mathbb{N}$. Suppose that one of the following conditions holds:

(a)
${sup}_{v,{v}^{\prime}\in X}d(v,{v}^{\prime})<\pi /2$;

(b)
$d(u,{P}_{F(S)}u)<\pi /4$ and $d(u,{P}_{F(S)}u)+d({x}_{0},{P}_{F(S)}u)<\pi /2$;

(c)
${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}^{2}=\mathrm{\infty}$.
Then $\{{x}_{n}\}$ converges to ${P}_{F(S)}u$.
The next convergence theorem of an iterative scheme on $CAT(1)$ spaces was first proposed by Pia̧tek [6]. The theorem deals with the Halperntype iterative sequence. Although the result itself is correct, a part of the proof does not seem to be exact. Precisely, in the proof of the convergence theorem for the explicit iteration process, the author makes use of Xu’s lemma, Lemma 2.1 in this paper. However, the conditions required for this lemma are not verified. We attempt to prove the following theorem as a supplement of the aforementioned result, and moreover, we find another coefficient condition which guarantees convergence of the iterative scheme.
Before showing the result, we need the following lemma which is analogous to [[6], Lemma 3.3]. The assumption for the length of the edges of the triangle is improved.
Lemma 5.4 Let X be a $CAT(1)$ space. For $M\in \phantom{\rule{0.2em}{0ex}}]0,\pi [$, let $u,v,w\in X$ be such that $d(u,v)\le M$ and $d(u,w)\le M$. For a given $\alpha \in \phantom{\rule{0.2em}{0ex}}]0,1[$, let ${v}^{\prime}=\alpha u\oplus (1\alpha )v$ and ${w}^{\prime}=\alpha u\oplus (1\alpha )w$. If $d(v,w)+d(w,u)+d(u,v)<2\pi $ and $sin((1\alpha )M)\le sinM$, then
Proof Consider the comparison triangle $\mathrm{\u25b3}(\overline{u},\overline{v},\overline{w})$ of $\mathrm{\u25b3}(u,v,w)$ on ${\mathbb{S}}^{2}$ and let ${\overline{v}}^{\prime}$ and ${\overline{w}}^{\prime}$ be the comparison points of ${v}^{\prime}$ and ${w}^{\prime}$, respectively. Let
and ${U}^{\prime}={d}_{{\mathbb{S}}^{2}}({\overline{v}}^{\prime},{\overline{w}}^{\prime})$. Then, letting ${\alpha}^{\prime}=1\alpha $, we have that
Since the functions ${f}_{V}(t)=sintV/sintM$, ${f}_{W}(t)=sintW/sintM$, and $g(t)=(1costVcostW)/(sintVsintW)$ are all increasing on $[0,1]$, it follows that
and
Therefore, we have that
Using the assumption that $sin{\alpha}^{\prime}M\le sinM$, we obtain that
and, by the $CAT(1)$ inequality, it follows that
which is the desired result. □
Theorem 5.5 Let X be a complete $CAT(1)$ space such that $d(v,{v}^{\prime})<\pi /2$ for every $v,{v}^{\prime}\in X$. Let $T:X\to X$ be a nonexpansive mapping and suppose that $F(T)\ne \mathrm{\varnothing}$. Let $\{{\alpha}_{n}\}$ be a real sequence in $[0,1]$ such that ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$, ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$, and ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n+1}{\alpha}_{n}<\mathrm{\infty}$. For given points $u,{x}_{0}\in X$, let $\{{x}_{n}\}$ be the sequence in X generated by
for $n\in \mathbb{N}$. Suppose that one of the following conditions holds:

(a)
${sup}_{v,{v}^{\prime}\in X}d(v,{v}^{\prime})<\pi /2$;

(b)
$d(u,{P}_{F(T)}u)<\pi /4$ and $d(u,{P}_{F(T)}u)+d({x}_{0},{P}_{F(T)}u)<\pi /2$;

(c)
${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}^{2}=\mathrm{\infty}$.
Then $\{{x}_{n}\}$ converges to ${P}_{F(T)}u$.
We employ the method used in [6] for some parts of the proof.
Proof From the definition of $\{{x}_{n}\}$, using Lemma 5.4, we have that
where ${M}_{n}=max\{d(u,T{x}_{n}),d(u,T{x}_{n1})\}$ for each $n\in \mathbb{N}$. Let
for all $n\in \mathbb{N}$. Then, as in the proof of Theorem 3.2, we have that each of the conditions (a), (b) and (c) implies that ${\sum}_{n=0}^{\mathrm{\infty}}{\gamma}_{n}=\mathrm{\infty}$. In particular, in cases of (a) and (b), for $M={sup}_{v,{v}^{\prime}\in X}d(v,{v}^{\prime})$ and $M=max\{2d(u,p),d(u,p)+d({x}_{0},p)\}$ with $p={P}_{F(T)}u$, respectively, it holds that
and in case of (c), it holds that
Then, by using Lemma 2.1 with the condition ${\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n+1}{\alpha}_{n}<\mathrm{\infty}$, we have that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}_{n+1})=0$. It follows that
and thus ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$. Let p, $\{{s}_{n}\}$, $\{{t}_{n}\}$, $\{{\beta}_{n}\}$ be as in the proof of Theorem 3.2 again. Then by Lemma 3.1, we have that
for every $n\in \mathbb{N}$. Since every nonexpansive mapping is Δdemiclosed, we can use the same technique as the proof of Theorem 3.2 and then we obtain that
Consequently, we have that ${lim}_{n\to \mathrm{\infty}}{s}_{n}=0$ by Lemma 2.1. Hence, $\{{x}_{n}\}$ converges to $p={P}_{F(T)}u$, which is the desired result. □
References
 1.
Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S000299041967118640
 2.
Wittmann R: Approximation of fixed points of nonexpansive mappings. Arch. Math. 1992, 58: 486–491. 10.1007/BF01190119
 3.
Bruck RE, Reich S: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houst. J. Math. 1977, 3: 459–470.
 4.
Bruck RE: Random products of contractions in metric and Banach spaces. J. Math. Anal. Appl. 1982, 88: 319–332. 10.1016/0022247X(82)901950
 5.
Saejung S:Halpern’s iteration in $CAT(0)$ spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 471781
 6.
Pia̧tek B:Halpern iteration in $CAT(\kappa )$ spaces. Acta Math. Sin. Engl. Ser. 2011, 27: 635–646. 10.1007/s1011401193127
 7.
Saejung S: Halpern’s iteration in Banach spaces. Nonlinear Anal. 2010, 73: 3431–3439. 10.1016/j.na.2010.07.031
 8.
Aoyama K, Kimura Y, Kohsaka F: Strong convergence theorems for strongly relatively nonexpansive sequences and applications. J. Nonlinear Anal. Optim. 2012, 3: 67–77.
 9.
Bridson MR, Haefliger A Grundlehren der Mathematischen Wissenschaften 319. In Metric Spaces of NonPositive Curvature. Springer, Berlin; 1999. [Fundamental Principles of Mathematical Sciences]
 10.
Kimura Y, Satô K: Convergence of subsets of a complete geodesic space with curvature bounded above. Nonlinear Anal. 2012, 75: 5079–5085. 10.1016/j.na.2012.04.024
 11.
Xu HK: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 2002, 65: 109–113. 10.1017/S0004972700020116
 12.
Saejung S, Yotkaew P: Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal. 2012, 75: 742–750. 10.1016/j.na.2011.09.005
 13.
He JS, Fang DH, López G, Li C:Mann’s algorithm for nonexpansive mappings in $CAT(\kappa )$ spaces. Nonlinear Anal. 2012, 75: 445–452. 10.1016/j.na.2011.07.070
 14.
Espínola R, FernándezLeón A:$CAT(k)$spaces, weak convergence and fixed points. J. Math. Anal. Appl. 2009, 353: 410–427. 10.1016/j.jmaa.2008.12.015
Acknowledgements
The first author is supported by GrantinAid for Scientific Research No. 22540175 from the Japan Society for the Promotion of Science.
Author information
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors have contributed in this work on an equal basis. All authors read and approved the final manuscript.
Rights and permissions
About this article
Received
Accepted
Published
DOI
Keywords
 $CAT(1)$ space
 Halpern iteration
 strongly quasinonexpansive
 Δdemiclosed
 fixed point