Open Access

# Approximating fixed points for nonself mappings in CAT(0) spaces

Fixed Point Theory and Applications20112011:65

https://doi.org/10.1186/1687-1812-2011-65

Accepted: 13 October 2011

Published: 13 October 2011

## Abstract

Suppose K is a nonempty closed convex subset of a complete CAT(0) space X with the nearest point projection P from X onto K. Let T : KX be a nonself mapping, satisfying Condition (E) with F(T): = {x K : Tx = x} . Suppose {x n } is generated iteratively by x1 K, x n +1 = P ((1 - α n )x n α n TP [(1 - β n )x n β n Tx n ]),n ≥ 1, where {α n } and {β n } are real sequences in [ε, 1 - ε] for some ε (0, 1). Then, {x n } Δ-converges to some point x in F(T). This extends a result of Laowang and Panyanak [Fixed Point Theory Appl. 367274, 11 (2010)] for nonself mappings satisfying Condition (E).

## Keywords

CAT(0) spacesfixed pointcondition (E)nonself mappings

## 1 Introduction

In 2010, Laowang and Panyanak [1] studied an iterative scheme and proved the following result: let K be a nonempty closed convex subset of a complete CAT(0) space X, (the initials of term "CAT" are in honor of E. Cartan, A.D. Alexanderov and V.A. Toponogov) with the nearest point projection P from X onto K. Let T : KX be a nonexpansive nonself mapping with nonempty fixed point set. If {x n } is generated iteratively by
${x}_{1}\in K,\phantom{\rule{1em}{0ex}}{x}_{n+1}=P\left(\left(1-{\alpha }_{n}\right){x}_{n}\oplus {\alpha }_{n}TP\left[\left(1-{\beta }_{n}\right){x}_{n}\oplus {\beta }_{n}T{x}_{n}\right]\right),$
(1.1)

where {α n } and {β n } are real sequences in [ε, 1 - ε] for some ε (0, 1), then {x n } is Δ-convergent to a fixed point of T. In this article, this result is extended for nonself mappings satisfying Condition (E).

Let K be a nonempty subset of a CAT(0) space X and T : KX be a mapping. A point x K is called a fixed point of T, if x = Tx. We shall denote the fixed point set of T by F(T). Moreover, T is called nonexpansive if for each x, y K, d(Tx, Ty) ≤ d(x, y).

In 2011, Falset et al. [2] introduced Condition (E) as follows:

Definition 1.1. Let K be a bounded closed convex subset of a complete CAT(0) space X. A mapping T : KX is called to satisfy Condition (E μ ) on C, if there exists μ ≥ 1 such that
$\mathsf{\text{d}}\left(x,Ty\right)\le \mu \mathsf{\text{d}}\left(Tx,x\right)+\mathsf{\text{d}}\left(x,y\right)$

holds, for all x, y K. It is called, T satisfies Condition (E) on C whenever T satisfies (E μ ) for some μ ≥ 1.

Proposition 1.2 [2]. Every nonexpansive mapping satisfies Condition (E), but the inverse is not true.

Now, we need some fact about CAT(0) spaces as follows:

Let (X, d) be a metric space. A geodesic path joining x X to y X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0, l] R to X such that c(0) = x, c(l) = y and d(c(t), c(t')) = ||t - t'|| for all t, t' [0, l]. In particular, c is an isometry and d(x, y) = l. The image α 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 uniquely geodesic if there is exactly one geodesic joining x to y, for each x, y X. A subset Y X is said to be convex if Y includes every geodesic segment joining any two of its points. A geodesic triangle Δ(x1, x2, x3) in a geodesic metric space (X, d) 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 Δ(x1, x2, x3) in (X, d) is a triangle $\stackrel{̄}{\Delta }\left({x}_{1},{x}_{2},{x}_{3}\right):=\Delta \left({\stackrel{̄}{x}}_{1},{\stackrel{̄}{x}}_{2},{\stackrel{̄}{x}}_{3}\right)$ in the Euclidean plane E2 such that ${d}_{{E}^{2}}\left({\stackrel{̄}{x}}_{i},{\stackrel{̄}{x}}_{j}\right)=d\left({x}_{i},{x}_{j}\right)$ for i, j {1, 2, 3}. A geodesic metric space is said to be a CAT(0) space [3], if all geodesic triangles of appropriate size satisfy the following comparison axiom. Let Δ be a geodesic triangle in X and $\stackrel{̄}{\Delta }$ be a comparison triangle for Δ. Then Δ is said to satisfy the CAT(0) inequality if for all x, y Δ and all comparison points $\stackrel{̄}{x},ȳ\in \stackrel{̄}{\Delta }$,
$\mathsf{\text{d}}\left(x,y\right)\le {\mathsf{\text{d}}}_{{E}^{2}}\left(\stackrel{̄}{x},ȳ\right).$
(1.2)
If x, y1, y2 are points in a CAT(0) space and y0 is the midpoint of the segment [y1, y2], then the CAT(0) inequality implies
$\mathsf{\text{d}}{\left(x,{y}_{0}\right)}^{2}\le \frac{1}{2}\mathsf{\text{d}}{\left(x,{y}_{1}\right)}^{2}+\frac{1}{2}\mathsf{\text{d}}{\left(x,{y}_{2}\right)}^{2}-\frac{1}{4}\mathsf{\text{d}}{\left({y}_{1},{y}_{2}\right)}^{2}.$
(CN)

In fact, a geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality (Courbure negative)[[3], p. 163].

Lemma 1.3. Let (X, d) be a CAT(0) space.

1. [[3], Proposition 2.4] Let K be a convex subset of X which is complete in the induced metric. Then for every x X, there exists a unique point P(x) K such that d(x, P(x)) = inf{d(x, y): y K}. Moreover, the map xP(x) is a nonexpansive retract from X onto K.

2. [[4], Lemma 2.1] For x, y X and t [0, 1], there exists a unique point z [x, y] such that
$\mathsf{\text{d}}\left(x,z\right)=t\mathsf{\text{d}}\left(x,y\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathsf{\text{d}}\left(y,z\right)=\left(1-t\right)\mathsf{\text{d}}\left(x,y\right)$

one uses the notation (1 - t)x ty for the unique point z.

3. [[4], Lemma 2.4] For x, y, z X and t [0, 1], one has
$\mathsf{\text{d}}\left(\left(1-t\right)x\oplus ty,z\right)\le \left(1-t\right)\mathsf{\text{d}}\left(x,z\right)+t\mathsf{\text{d}}\left(y,z\right).$
[[4], Lemma 2.5] For x, y, z X and t [0, 1], one has
$d{\left(\left(1-t\right)x\oplus ty,z\right)}^{2}\le \left(1-t\right)\mathsf{\text{d}}{\left(x,z\right)}^{2}+t\mathsf{\text{d}}{\left(y,z\right)}^{2}-t\left(1-t\right)\mathsf{\text{d}}{\left(x,y\right)}^{2}.$
Let {x n } be a bounded sequence in a CAT(0) space X. For x X, we set
$r\left(\left\{{x}_{n}\right\}\right)=inf\left\{r\left(x,\left\{{x}_{n}\right\}\right):x\in X\right\},$
and the asymptotic center A({x n }) of {x n } is the set
$A\left(\left\{{x}_{n}\right\}\right)=\left\{x\in X:r\left(x,\left\{{x}_{n}\right\}\right)=r\left(\left\{{x}_{n}\right\}\right)\right\}.$

It is known [[5], Proposition 7], in a CAT(0) space X, A({x n }) consists of exactly one point.

Definition 1.4. [[6], Definition 3.1] A sequence {x n } in a CAT(0) space X is said Δ-converges to x X, if x is the unique asymptotic center of {u n } for every subsequence {u n } of {x n }. In this case, one can write Δ - lim n x n = x and call x the Δ - lim of {x n }.

Lemma 1.5. Let (X, d) be a CAT(0) space.

1. [[6], p. 3690] Every bounded sequence in X has a Δ-convergent subsequence.

2. [[7], Proposition 2.1] If K is a closed convex subset of X and if {x n } is a bounded sequence in K, then the asymptotic center of {x n } is in K.

3. [[4], Lemma 2.8] If {x n } is a bounded sequence in X with A({x n }) = {x} and {u n } is a subsequence of {x n } with A({u n }) = {u} and the sequence {d(x n , u)} converges, then x = u.

## 2 Main results

The following lemma was proved by Dhompongsa and Panyanak in the case of nonexpansive [[4], Lemma 2.10].

Lemma 2.1. Let K be a nonempty closed convex subset of a complete CAT(0) space X, and T : KX be a nonself mapping, satisfying Condition (E). Suppose {x n } is a bounded sequence in K such that lim n d(x n , Tx n ) = 0 and {d(x n , v)} converges for all v F (T). Then
${\omega }_{w}\left({x}_{n}\right)\subset F\left(T\right),$

where ${\omega }_{w}\left({x}_{n}\right):=\bigcup A\left(\left\{{u}_{n}\right\}\right)$ and the union is taken over all subsequences {u n } of {x n }. Moreover, ω w (x n ) consists of exactly one point.

Proof. Let u ω w (x n ), then there exists a subsequence {u n } of {x n } such that A({u n }) = {u}. By part (1) and (2) of Lemma 1.5, there exists a subsequence {v n } of {u n } such that Δ - lim n v n = v K. We show v F (T). In order to prove this, by Condition (E), one can write
$\mathsf{\text{d}}\left({x}_{n},Tv\right)\le \mu \mathsf{\text{d}}\left(T{x}_{n},{x}_{n}\right)+\mathsf{\text{d}}\left({x}_{n},v\right)$
for some μ ≥ 1. Therefore
$\begin{array}{ll}\hfill \mathsf{\text{lim}}\phantom{\rule{2.77695pt}{0ex}}{sup}_{n}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{d}}\left({x}_{n},Tv\right)& \le \mathsf{\text{lim}}\phantom{\rule{2.77695pt}{0ex}}{sup}_{n}\left(\mu \mathsf{\text{d}}\left(T{x}_{n},{x}_{n}\right)+\mathsf{\text{d}}\left({x}_{n},v\right)\right)\phantom{\rule{2em}{0ex}}\\ =\mathsf{\text{lim}}\phantom{\rule{2.77695pt}{0ex}}{sup}_{n}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{d}}\left({x}_{n},v\right).\phantom{\rule{2em}{0ex}}\end{array}$

The uniqueness of asymptotic center, implies v K and T(v) = v. By part (3) Lemma 1.5, u = v. Therefore ω w (x n ) F(T). Let {u n } be a subsequence of {x n } with A({u n }) = {u} and A({x n }) = {x}. Since u ω w (x n ) F(T), {d(x n , v)} converges. By part (3) Lemma 1.5, x = u. This shows that ω w (x n ) consists of exactly one point. □

Theorem 2.2. Let K be a nonempty closed convex subset of a complete CAT(0) space X, and T : KX be a nonself mapping, satisfying Condition (E) with x F(T) = {x K : Tx = x}. Let {α n } and {β n } be sequences in [ε, 1 - ε] for some ε (0, 1). Starting from arbitrary x1 K, define the sequence {x n } by x n +1 = P((1 - α n )x n α n TP[(1 - β n )x n β n Tx n ]), n ≥ 1. Then lim n →∞ d(x n , x) exists.

Proof. By part (1) of Lemma 1.3, the nearest point projection P from X onto K is nonexpansive. Then,
$\begin{array}{ll}\hfill \mathsf{\text{d}}\left({x}_{n+1},{x}^{\star }\right)& =\mathsf{\text{d}}\left(P\left(\left(1-{\alpha }_{n}\right){x}_{n}\oplus {\alpha }_{n}TP\left[\left(1-{\beta }_{n}\right){x}_{n}\oplus {\beta }_{n}T{x}_{n}\right]\right),P{x}^{\star }\right)\phantom{\rule{2em}{0ex}}\\ \le \mathsf{\text{d}}\left(\left(1-{\alpha }_{n}\right){x}_{n}\oplus {\alpha }_{n}TP\left[\left(1-{\beta }_{n}\right){x}_{n}\oplus {\beta }_{n}T{x}_{n}\right],{x}^{\star }\right)\phantom{\rule{2em}{0ex}}\\ =\left(1-{\alpha }_{n}\right)\mathsf{\text{d}}\left({x}_{n},{x}^{\star }\right)+{\alpha }_{n}\mathsf{\text{d}}\left(TP\left[\left(1-{\beta }_{n}\right){x}_{n}\oplus {\beta }_{n}T{x}_{n}\right],{x}^{\star }\right).\phantom{\rule{2em}{0ex}}\end{array}$
But by Condition (E), for some μ ≥ 1, we have
$\begin{array}{c}\left(1-{\alpha }_{n}\right)\mathsf{\text{d}}\left({x}_{n},{x}^{\star }\right)+{\alpha }_{n}\mathsf{\text{d}}\left(TP\left[\left(1-{\beta }_{n}\right){x}_{n}\oplus {\beta }_{n}T{x}_{n}\right],{x}^{\star }\right)\\ \le \left(1-{\alpha }_{n}\right)\mathsf{\text{d}}\left({x}_{n},{x}^{\star }\right)+{\alpha }_{n}\left(\mu \mathsf{\text{d}}\left(T{x}^{\star },{x}^{\star }\right)+\mathsf{\text{d}}\left(P\left[\left(1-{\beta }_{n}\right){x}_{n}\oplus {\beta }_{n}T{x}_{n}\right],{x}^{\star }\right)\right)\\ \le \left(1-{\alpha }_{n}\right)\mathsf{\text{d}}\left({x}_{n},{x}^{\star }\right)+{\alpha }_{n}\left[\left(1-{\beta }_{n}\right)\mathsf{\text{d}}\left({x}_{n},{x}^{\star }\right)+{\beta }_{n}\mathsf{\text{d}}\left({x}_{n},{x}^{\star }\right)\right]\\ =\mathsf{\text{d}}\left({x}_{n},{x}^{\star }\right).\end{array}$

Consequently, d(x n +1, x) ≤ d(x n , x). Then d(x n , x) ≤ d(x1, x) for all n ≥ 1. This implies ${\left\{\mathsf{\text{d}}\left({x}_{n},{x}^{\star }\right)\right\}}_{n=1}^{\infty }$ is bounded and decreasing. Hence, lim n →∞ d(x n , x) exists. □

Theorem 2.3. Let K be a nonempty closed convex subset of a complete CAT(0) space X, and T : KX be a nonself mapping, satisfying Condition (E) with F(T) ≠ . Let {α n } and {β n } be sequences in [ε, 1 - ε] for some ε (0, 1). Starting from arbitrary x1 K, define the sequence {x n } by x n +1 = P((1 - α n )x n α n TP [(1 - β n )x n β n Tx n ]), n ≥ 1. Then lim n →∞ d(x n , Tx n ) = 0.

Proof. Let x F(T). By Theorem 2.2, lim n →∞ d(x n , x) exists. Set
$\underset{n\to \infty }{lim}\mathsf{\text{d}}\left({x}_{n},{x}^{\star }\right)=r.$
If r = 0, by the Condition (E), for some μ ≥ 1,
$\begin{array}{ll}\hfill \mathsf{\text{d}}\left({x}_{n},T{x}_{n}\right)& \le \mathsf{\text{d}}\left({x}^{\star },{x}_{n}\right)+\mathsf{\text{d}}\left({x}^{\star },T{x}_{n}\right)\phantom{\rule{2em}{0ex}}\\ \le \mathsf{\text{d}}\left({x}^{\star },{x}_{n}\right)+\mu \mathsf{\text{d}}\left({x}^{\star },T{x}^{\star }\right)+\mathsf{\text{d}}\left({x}^{\star },{x}_{n}\right).\phantom{\rule{2em}{0ex}}\end{array}$

Therefore lim n →∞ d(x n , Tx n ) = 0.

If r > 0, set y n = P [(1 - β n )x n β n Tx n ]. By part (4) of Lemma 1.3,
$\begin{array}{ll}\hfill \mathsf{\text{d}}{\left({y}_{n},{x}^{\star }\right)}^{2}& =\mathsf{\text{d}}{\left(P\left[\left(1-{\beta }_{n}\right){x}_{n}\oplus {\beta }_{n}T{x}_{n}\right],P{x}^{\star }\right)}^{2}\phantom{\rule{2em}{0ex}}\\ \le \mathsf{\text{d}}{\left(\left[\left(1-{\beta }_{n}\right){x}_{n}\oplus {\beta }_{n}T{x}_{n}\right],{x}^{\star }\right)}^{2}\phantom{\rule{2em}{0ex}}\\ \le \left(1-{\beta }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},{x}^{\star }\right)}^{2}+{\beta }_{n}\mathsf{\text{d}}{\left(T{x}_{n},{x}^{\star }\right)}^{2}-{\beta }_{n}\left(1-{\beta }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},T{x}_{n}\right)}^{2}\phantom{\rule{2em}{0ex}}\\ \le \left(1-{\beta }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},{x}^{\star }\right)}^{2}+{\beta }_{n}\mathsf{\text{d}}{\left(T{x}_{n},{x}^{\star }\right)}^{2}.\phantom{\rule{2em}{0ex}}\end{array}$
(2.3)
Using Condition (E), for some μ ≥ 1,
$\begin{array}{ll}\hfill \left(1-{\beta }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},{x}^{\star }\right)}^{2}& +{\beta }_{n}\mathsf{\text{d}}{\left(T{x}_{n},{x}^{\star }\right)}^{2}\phantom{\rule{2em}{0ex}}\\ \le \left(1-{\beta }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},{x}^{\star }\right)}^{2}+{\beta }_{n}{\left(\mu \mathsf{\text{d}}\left(T{x}^{\star },{x}^{\star }\right)+\mathsf{\text{d}}\left({x}_{n},{x}^{\star }\right)\right)}^{2}\phantom{\rule{2em}{0ex}}\\ =\mathsf{\text{d}}{\left({x}_{n},{x}^{\star }\right)}^{2}.\phantom{\rule{2em}{0ex}}\end{array}$
(2.4)
Therefore by inequities (2.3) and (2.4), one can get
$\mathsf{\text{d}}\left({y}_{n},{x}^{\star }\right)\le \mathsf{\text{d}}\left({x}_{n},{x}^{\star }\right).$
(2.5)
Part (4) of Lemma 1.3, shows
$\begin{array}{ll}\hfill \mathsf{\text{d}}{\left({x}_{n+1},{x}^{\star }\right)}^{2}& =\mathsf{\text{d}}{\left(P\left[\left(1-{\alpha }_{n}\right){x}_{n}\oplus {\alpha }_{n}T{y}_{n}\right],P{x}^{\star }\right)}^{2}\phantom{\rule{2em}{0ex}}\\ \le \mathsf{\text{d}}{\left(\left(1-{\alpha }_{n}\right){x}_{n}\oplus {\alpha }_{n}T{y}_{n},{x}^{\star }\right)}^{2}\phantom{\rule{2em}{0ex}}\\ \le \left(1-{\alpha }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},{x}^{\star }\right)}^{2}+{\alpha }_{n}\mathsf{\text{d}}{\left(T{y}_{n},{x}^{\star }\right)}^{2}-{\alpha }_{n}\left(1-{\alpha }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},T{y}_{n}\right)}^{2}\phantom{\rule{2em}{0ex}}\\ \le \left(1-{\alpha }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},{x}^{\star }\right)}^{2}+{\alpha }_{n}{\left(\mu \mathsf{\text{d}}\left(T{x}^{\star },{x}^{\star }\right)+\mathsf{\text{d}}\left({y}_{n},{x}^{\star }\right)\right)}^{2}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-{\alpha }_{n}\left(1-{\alpha }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},T{y}_{n}\right)}^{2}\phantom{\rule{2em}{0ex}}\\ =\left(1-{\alpha }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},{x}^{\star }\right)}^{2}+{\alpha }_{n}\mathsf{\text{d}}{\left({y}_{n},{x}^{\star }\right)}^{2}-{\alpha }_{n}\left(1-{\alpha }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},T{y}_{n}\right)}^{2}\phantom{\rule{2em}{0ex}}\\ \le \left(1-{\alpha }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},{x}^{\star }\right)}^{2}+{\alpha }_{n}\mathsf{\text{d}}{\left({x}_{n},{x}^{\star }\right)}^{2}-{\alpha }_{n}\left(1-{\alpha }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},T{y}_{n}\right)}^{2}\phantom{\rule{2em}{0ex}}\\ =\mathsf{\text{d}}{\left({x}_{n},{x}^{\star }\right)}^{2}-{\alpha }_{n}\left(1-{\alpha }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},T{y}_{n}\right)}^{2}.\phantom{\rule{2em}{0ex}}\end{array}$
Therefore
$\mathsf{\text{d}}{\left({x}_{n+1},{x}^{\star }\right)}^{2}\le \mathsf{\text{d}}{\left({x}_{n},{x}^{\star }\right)}^{2}-W\left({\alpha }_{n}\right)\mathsf{\text{d}}{\left({x}_{n},T{y}_{n}\right)}^{2},$

where W(α) = α(1 - α). Since α [ε, 1 - ε], W(α n ) ≥ ε2.

Therefore
${\epsilon }^{2}\sum _{n=1}^{\infty }\mathsf{\text{d}}{\left({x}_{n},T{y}_{n}\right)}^{2}\le \mathsf{\text{d}}{\left({x}_{1},{x}^{\star }\right)}^{2}<\infty .$

This implies lim n →∞ d(x n , Ty n ) = 0.

By Condition (E), for some μ ≥ 1, we have
$\begin{array}{ll}\hfill \mathsf{\text{d}}\left({x}_{n},{x}^{\star }\right)& \le \mathsf{\text{d}}\left({x}_{n},T{y}_{n}\right)+\mathsf{\text{d}}\left(T{y}_{n},{x}^{\star }\right)\phantom{\rule{2em}{0ex}}\\ \le \mathsf{\text{d}}\left({x}_{n},T{y}_{n}\right)+\mu \mathsf{\text{d}}\left(T{x}^{\star },{x}^{\star }\right)+\mathsf{\text{d}}\left({y}_{n},{x}^{\star }\right)\phantom{\rule{2em}{0ex}}\\ =\mathsf{\text{d}}\left({x}_{n},T{y}_{n}\right)+\mathsf{\text{d}}\left({y}_{n},{x}^{\star }\right).\phantom{\rule{2em}{0ex}}\end{array}$
Hence
On the other hand, from (2.5),
This implies
$\underset{n\to \infty }{lim}\mathsf{\text{d}}\left({y}_{n},{x}^{\star }\right)=r.$
Thus (2.5) shows
$\underset{n\to \infty }{\mathrm{lim}}\text{d}\left(\left(1-{\beta }_{n}\right){x}_{n}\oplus {\beta }_{n}T{x}_{n}\right],{x}^{\star }\right)=r.$
Since T satisfies Condition (E), we have
$\begin{array}{ll}\hfill \mathsf{\text{d}}\left(T{x}_{n},{x}^{\star }\right)& \le \mu \mathsf{\text{d}}\left(T{x}^{\star },{x}^{\star }\right)+\mathsf{\text{d}}\left({x}_{n},{x}^{\star }\right)\phantom{\rule{2em}{0ex}}\\ =\mathsf{\text{d}}\left({x}_{n},{x}^{\star }\right)\phantom{\rule{2em}{0ex}}\end{array}$
Thus

Now, by [[1], Lemma 2.9], lim n →∞ d(x n , Tx n ) = 0. □

Theorem 2.4. Let K be a nonempty closed convex subset of a complete CAT(0) space X, and T : KX be a nonself mapping, satisfying Condition (E) with F(T) ≠ . Assume {α n } and {β n } are sequences in [ε, 1 - ε] for some ε (0, 1). Starting from arbitrary x1 K, define the sequence {x n } by x n +1 = P((1 - α n )x n α n TP[(1 - β n )x n β n Tx n ]), n ≥ 1. Then {x n } is Δ-convergent to some point x in F(T).

Proof. By Theorem 2.3, lim n →∞ d(x n , Tx n ) = 0. The proof of Theorem 2.2 shows {d(x n , v)} is bounded and decreasing for each v F (T), and so it is convergent. By Lemma 2.1, ω w (x n ) consists exactly one point which is a fixed point of T. Consequently, the sequence {x n } is Δ-convergent to some point x in F(T). □

The following definition is recalled from [8].

Definition 2.5. A mapping T : KX is said to satisfy Condition I, if there exists a nondecreasing function f : [0, ∞) → [0, ∞) with f(0) = 0 and f(r) > 0 for all r > 0 such that
$\mathsf{\text{d}}\left(x,Tx\right)\ge f\left(\mathsf{\text{d}}\left(x,F\left(T\right)\right)\right),$

where x K.

With respect to the above definition, we have the following theorem [[1], Theorem 3.4].

Theorem 2.6. Let K be a nonempty closed convex subset of a complete CAT(0) space X, and T : KX be a nonself mapping, satisfying condition (E) with F(T) ≠ . Assume {α n } and {β n } are sequences in [ε, 1 - ε] for some ε (0, 1). Starting from arbitrary x1 K, define the sequence {x n } by x n +1 = P((1 - α n )x n α n TP [(1 - β n )x n β n Tx n ]), n ≥ 1. If T satisfies condition I, then {x n } converges strongly to a fixed point of T.

We state another strong convergence theorem [[1], Theorem 3.5] as follows:

Theorem 2.7. Let K be a nonempty compact convex subset of a complete CAT(0) space X, and T : KX be a nonself mapping, satisfying condition (E) with F(T) ≠ . Assume {α n } and {β n } are sequences in [ε, 1 - ε] for some ε (0, 1). Starting from arbitrary x1 K, define the sequence {x n } by x n +1 = P((1 - α n )x n α n TP[(1 - β n )x n β n Tx n ]),n ≥ 1. Then, {x n } converges strongly to a fixed point of T.

Another result in [1] is to obtain the Δ-convergence of a defined sequence, to a common fixed point of two nonexpansive self-mappings. According to the present setting, we can state the following result.

Theorem 2.8. Let K be a nonempty closed convex subset of a complete CAT(0) space X, and S, T : KX be two nonself mappings, satisfying Condition (E) with F(S) ∩ F(T) ≠ . Assume {α n } and {β n } are sequences in [ε, 1 - ε] for some ε (0, 1). Starting from arbitrary x1 K, define the sequence {x n } by x n +1 = (1 - α n )x n α n S[(1 - β n )x n β n Tx n ], n ≥ 1. Then {x n } is Δ-convergent to a common fixed point of S and T.

## Authors’ Affiliations

(1)
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

## References

1. Laowang W, Panyanak B: Approximating fixed points of nonexpansive nonself mappings in CAT(0) spaces. Fixed Point Theory Appl 2010, 367274: 11.
2. Garcia-Falset J, Liorens-Fuster E, Suzuki T: Fixed point theory for a class of generalized nonexpansive mapping. J Math Anal Appl 2011, 375: 185–195. 10.1016/j.jmaa.2010.08.069
3. Bridson M, Haefliger A: Metric Spaces of Non-Positive Curvature, Fundamental Principles of Mathematical Sciences. Volume 319. Springer, Berlin; 1999.
4. Dhompongsa S, Panyanak B: On Δ-convergence theorems in CAT(0) spaces. Comput Math Appl 2008, 56: 2572–2579. 10.1016/j.camwa.2008.05.036
5. Dhompongsa S, Kirk WA, Sims B: Fixed point of uniformly lipschitzian mappings. Nonlinear Anal 2006, 65: 762–772. 10.1016/j.na.2005.09.044
6. Kirk W, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal 2008, 68: 3689–3696. 10.1016/j.na.2007.04.011
7. Dhompongsa S, Kirk WA, Panyanak B: Nonexpansive set-valued mappings in metric and Banach spaces. J Nonlinear Convex Anal 2007, 8: 35–45.
8. 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