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

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 : K → X 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).

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 : K → X be a nonexpansive nonself mapping with nonempty fixed point set. If {x _n } is generated iteratively by

$$x_1\in K,x_{n+1}=P\left(\left(1-{\alpha }_n\right)x_n\oplus {\alpha }_{n}TP\left[\left(1-{\beta }_n\right)x_n\oplus {\beta }_{n}T{x}_n\right]\right),$$

(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 : K → X 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 : K → X 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 $\bar{\Delta }\left(x_1,x_2,x_3\right):=\Delta \left({\bar{x}}_1,{\bar{x}}_2,{\bar{x}}_3\right)$ in the Euclidean plane E² such that $d_{E^2}\left({\bar{x}}_i,{\bar{x}}_j\right)=d\left(x_i,x_j\right)$ 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 $\bar{\Delta }$ 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 $\bar{x},ȳ\in \bar{\Delta }$,

$$\mathsf{\text{d}}\left(x,y\right)\le {\mathsf{\text{d}}}_{E^2}\left(\bar{x},ȳ\right).$$

(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 x → P(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

$$\mathsf{\text{d}}\left(x,z\right)=t\mathsf{\text{d}}\left(x,y\right),\mathsf{\text{d}}\left(y,z\right)=\left(1-t\right)\mathsf{\text{d}}\left(x,y\right)$$

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

$$r\left(x,\left\{x_n\right\}\right)=\underset{n\to \infty }{lim\; sup}\mathsf{\text{ d}}\left(x,x_n\right).$$

The asymptotic radius

$$r\left(\left\{x_n\right\}\right)=inf\left\{r\left(x,\left\{x_n\right\}\right):x\in X\right\},$$

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 : K → X 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

$$\begin{array}{ll}\hfill \mathsf{\text{lim}}{sup}_n\mathsf{\text{d}}\left(x_n,Tv\right)& \le \mathsf{\text{lim}}{sup}_n\left(\mu \mathsf{\text{d}}\left(T{x}_n,x_n\right)+\mathsf{\text{d}}\left(x_n,v\right)\right)\\ =\mathsf{\text{lim}}{sup}_n\mathsf{\text{d}}\left(x_n,v\right).\end{array}$$

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 : K → X 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,

$$\begin{array}{ll}\hfill \mathsf{\text{d}}\left(x_{n+1},x^{\star }\right)& =\mathsf{\text{d}}\left(P\left(\left(1-{\alpha }_n\right)x_n\oplus {\alpha }_{n}TP\left[\left(1-{\beta }_n\right)x_n\oplus {\beta }_{n}T{x}_n\right]\right),P{x}^{\star }\right)\\ \le \mathsf{\text{d}}\left(\left(1-{\alpha }_n\right)x_n\oplus {\alpha }_{n}TP\left[\left(1-{\beta }_n\right)x_n\oplus {\beta }_{n}T{x}_n\right],x^{\star }\right)\\ =\left(1-{\alpha }_n\right)\mathsf{\text{d}}\left(x_n,x^{\star }\right)+{\alpha }_n\mathsf{\text{d}}\left(TP\left[\left(1-{\beta }_n\right)x_n\oplus {\beta }_{n}T{x}_n\right],x^{\star }\right).\end{array}$$

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 : K → X 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,

$$\begin{array}{ll}\hfill \mathsf{\text{d}}\left(x_n,T{x}_n\right)& \le \mathsf{\text{d}}\left(x^{\star },x_n\right)+\mathsf{\text{d}}\left(x^{\star },T{x}_n\right)\\ \le \mathsf{\text{d}}\left(x^{\star },x_n\right)+\mu \mathsf{\text{d}}\left(x^{\star },T{x}^{\star }\right)+\mathsf{\text{d}}\left(x^{\star },x_n\right).\end{array}$$

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,

$$\begin{array}{ll}\hfill \mathsf{\text{d}}{\left(y_n,x^{\star }\right)}^2& =\mathsf{\text{d}}{\left(P\left[\left(1-{\beta }_n\right)x_n\oplus {\beta }_{n}T{x}_n\right],P{x}^{\star }\right)}^2\\ \le \mathsf{\text{d}}{\left(\left[\left(1-{\beta }_n\right)x_n\oplus {\beta }_{n}T{x}_n\right],x^{\star }\right)}^2\\ \le \left(1-{\beta }_n\right)\mathsf{\text{d}}{\left(x_n,x^{\star }\right)}^2+{\beta }_n\mathsf{\text{d}}{\left(T{x}_n,x^{\star }\right)}^2-{\beta }_n\left(1-{\beta }_n\right)\mathsf{\text{d}}{\left(x_n,T{x}_n\right)}^2\\ \le \left(1-{\beta }_n\right)\mathsf{\text{d}}{\left(x_n,x^{\star }\right)}^2+{\beta }_n\mathsf{\text{d}}{\left(T{x}_n,x^{\star }\right)}^2.\end{array}$$

(2.3)

Using Condition (E), for some μ ≥ 1,

$$\begin{array}{ll}\hfill \left(1-{\beta }_n\right)\mathsf{\text{d}}{\left(x_n,x^{\star }\right)}^2& +{\beta }_n\mathsf{\text{d}}{\left(T{x}_n,x^{\star }\right)}^2\\ \le \left(1-{\beta }_n\right)\mathsf{\text{d}}{\left(x_n,x^{\star }\right)}^2+{\beta }_n{\left(\mu \mathsf{\text{d}}\left(T{x}^{\star },x^{\star }\right)+\mathsf{\text{d}}\left(x_n,x^{\star }\right)\right)}^2\\ =\mathsf{\text{d}}{\left(x_n,x^{\star }\right)}^2.\end{array}$$

(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

$$\begin{array}{ll}\hfill \mathsf{\text{d}}{\left(x_{n+1},x^{\star }\right)}^2& =\mathsf{\text{d}}{\left(P\left[\left(1-{\alpha }_n\right)x_n\oplus {\alpha }_{n}T{y}_n\right],P{x}^{\star }\right)}^2\\ \le \mathsf{\text{d}}{\left(\left(1-{\alpha }_n\right)x_n\oplus {\alpha }_{n}T{y}_n,x^{\star }\right)}^2\\ \le \left(1-{\alpha }_n\right)\mathsf{\text{d}}{\left(x_n,x^{\star }\right)}^2+{\alpha }_n\mathsf{\text{d}}{\left(T{y}_n,x^{\star }\right)}^2-{\alpha }_n\left(1-{\alpha }_n\right)\mathsf{\text{d}}{\left(x_n,T{y}_n\right)}^2\\ \le \left(1-{\alpha }_n\right)\mathsf{\text{d}}{\left(x_n,x^{\star }\right)}^2+{\alpha }_n{\left(\mu \mathsf{\text{d}}\left(T{x}^{\star },x^{\star }\right)+\mathsf{\text{d}}\left(y_n,x^{\star }\right)\right)}^2\\ -{\alpha }_n\left(1-{\alpha }_n\right)\mathsf{\text{d}}{\left(x_n,T{y}_n\right)}^2\\ =\left(1-{\alpha }_n\right)\mathsf{\text{d}}{\left(x_n,x^{\star }\right)}^2+{\alpha }_n\mathsf{\text{d}}{\left(y_n,x^{\star }\right)}^2-{\alpha }_n\left(1-{\alpha }_n\right)\mathsf{\text{d}}{\left(x_n,T{y}_n\right)}^2\\ \le \left(1-{\alpha }_n\right)\mathsf{\text{d}}{\left(x_n,x^{\star }\right)}^2+{\alpha }_n\mathsf{\text{d}}{\left(x_n,x^{\star }\right)}^2-{\alpha }_n\left(1-{\alpha }_n\right)\mathsf{\text{d}}{\left(x_n,T{y}_n\right)}^2\\ =\mathsf{\text{d}}{\left(x_n,x^{\star }\right)}^2-{\alpha }_n\left(1-{\alpha }_n\right)\mathsf{\text{d}}{\left(x_n,T{y}_n\right)}^2.\end{array}$$

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

$$\begin{array}{ll}\hfill \mathsf{\text{d}}\left(x_n,x^{\star }\right)& \le \mathsf{\text{d}}\left(x_n,T{y}_n\right)+\mathsf{\text{d}}\left(T{y}_n,x^{\star }\right)\\ \le \mathsf{\text{d}}\left(x_n,T{y}_n\right)+\mu \mathsf{\text{d}}\left(T{x}^{\star },x^{\star }\right)+\mathsf{\text{d}}\left(y_n,x^{\star }\right)\\ =\mathsf{\text{d}}\left(x_n,T{y}_n\right)+\mathsf{\text{d}}\left(y_n,x^{\star }\right).\end{array}$$

Hence

$$r\le \underset{n\to \infty }{lim\; inf}\mathsf{\text{ d}}\left(y_n,x^{\star }\right).$$

On the other hand, from (2.5),

$$\underset{n\to \infty }{lim\; sup}\mathsf{\text{ d}}\left(y_n,x^{\star }\right)\le r.$$

This implies

$$\underset{n\to \infty }{lim}\mathsf{\text{d}}\left(y_n,x^{\star }\right)=r.$$

Thus (2.5) shows

$$\underset{n\to \infty }{\lim }\text{d}((1-{\beta }_n)x_n\oplus {\beta }_{n}T{x}_n],x^{\star })=r.$$

Since T satisfies Condition (E), we have

$$\begin{array}{ll}\hfill \mathsf{\text{d}}\left(T{x}_n,x^{\star }\right)& \le \mu \mathsf{\text{d}}\left(T{x}^{\star },x^{\star }\right)+\mathsf{\text{d}}\left(x_n,x^{\star }\right)\\ =\mathsf{\text{d}}\left(x_n,x^{\star }\right)\end{array}$$

Thus

$$\underset{n\to \infty }{lim\; sup}\mathsf{\text{ d}}\left(T{x}_n,x^{\star }\right)\le r.$$

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 : K → X 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 : K → X 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 : K → X 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 : K → X 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 : K → X 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.