Approximating fixed points for generalized nonexpansive mapping in $CAT(0)$ spaces

Abstract

Takahashi and Kim (Math. Jpn. 48:1-9, 1998) used the Ishikawa iteration process to prove some convergence theorems for nonexpansive mappings in Banach spaces. The aim of this paper is to prove similar results in $CAT(0)$ spaces for generalized nonexpansive mappings, which, in turn, generalize the corresponding results of Takahashi and Kim (Math. Jpn. 48:1-9, 1998), Laokul and Panyanak (Int. J. Math. Anal. 3(25-28):1305-1315, 2009), Razani and Salahifard (Bull. Iran. Math. Soc. 37(1):235-246, 2011) and some others.

MSC:47H10, 54H25.

1 Introduction

A self-mapping T defined on a bounded closed and convex subset K of a Banach space X is said to be nonexpansive if

$$\parallel Tx-Ty\parallel \le \parallel x-y\parallel \text{for all }x,y\in K.$$

In an attempt to construct a convergent sequence of iterates with respect to a nonexpansive mapping, Mann [1] defined an iteration method as follows: (for any $x_1\in K$)

$$x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n,n\in \mathbb{N},$$

where ${\alpha }_n\in (0,1)$.

In 1974, with a view to approximate the fixed point of pseudo-contractive compact mappings in Hilbert spaces, Ishikawa [2] introduced a new iteration procedure as follows (for $x_1\in K$):

$$\{\begin{array}{c}{y}_n=(1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n,\hfill \\ x_{n+1}=(1-{\beta }_n)x_n+{\beta }_{n}T{y}_n,n\in \mathbb{N},\hfill \end{array}$$

where ${\alpha }_n,{\beta }_n\in (0,1)$.

For a comparison of the preceding two iterative schemes in one-dimensional case, we refer the reader to Rhoades [3] wherein it is shown that under suitable conditions (see part (a) of Theorem 3) the rate of convergence of the Ishikawa iteration is faster than that of the Mann iteration procedure. Iterative techniques for approximating fixed points of nonexpansive single-valued mappings have been investigated by various authors using the Mann as well as Ishikawa iteration schemes. By now, there exists an extensive literature on the iterative fixed points for various classes of mappings. For an up-to-date account of the literature on this theme, we refer the readers to Berinde [4].

In 1998, Takahashi and Kim [5] consider the Ishikawa iteration procedure described as

$$\{\begin{array}{c}{y}_n={\alpha }_{n}T{x}_n+(1-{\alpha }_n)x_n,\hfill \\ x_{n+1}={\beta }_{n}T{y}_n+(1-{\beta }_n)x_n\text{for }n\ge 1,\hfill \end{array}$$

(1)

where the sequences $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are sequences in $[0,1]$ such that one of the following holds:

• (i) ${\alpha }_n\in [a,b]$ and ${\beta }_n\in [0,b]$ for some a, b with $0<a\le b<1$,

• (ii) ${\alpha }_n\in [a,1]$ and ${\beta }_n\in [a,b]$ for some a, b with $0<a\le b<1$.

Utilizing the forgoing iterative scheme, Takahashi and Kim [5] proved weak as well as strong convergence theorems for a nonexpansive mapping in Banach spaces.

Recently, García-Falset et al. [6] introduced two generalizations of nonexpansive mappings which in turn include Suzuki generalized nonexpansive mappings contained in [7] and also utilized the same to prove some fixed point theorems.

The following definitions are relevant to our subsequent discussions.

Definition 1.1 ([6])

Let C be the nonempty subset of a Banach space X and $T:C\to X$ be a single-valued mapping. Then T is said to satisfy the condition $(E_{\mu })$ (for some $\mu \ge 1$) if for all $x,y\in C$

$$\parallel x-Ty\parallel \le \mu \parallel x-Tx\parallel +\parallel x-y\parallel .$$

We say that T satisfies the condition $(E)$ whenever T satisfies the condition $(E_{\mu })$ for some $\mu \ge 1$.

Definition 1.2 ([6])

Let C be the nonempty subset of a Banach space X and $T:C\to X$ be a single-valued mapping. Then T is said to satisfy the condition $(C_{\lambda })$ (for some $\lambda \in (0,1)$) if for all $x,y\in C$

$$\lambda \parallel x-Tx\parallel \le \parallel x-y\parallel \Rightarrow \parallel Tx-Ty\parallel \le \parallel x-y\parallel .$$

The following theorem is essentially due to García-Falset et al. [6].

Theorem 1.3 ([6])

Let C be a convex subset of a Banach space X and $T:C\to C$ be a mapping satisfying conditions $(E)$ and $(C_{\lambda })$ for some $\lambda \in (0,1)$. Further, assume that either of the following holds.

• (a) C is weakly compact and X satisfies the Opial condition.

• (b) C is compact.

• (c) C is weakly compact and X is (UCED).

Then there exists $z\in C$ such that $Tz=z$.

A metric space $(X,d)$ is a $CAT(0)$ space if it is geodesically connected and 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. Other examples are pre-Hilbert spaces, R-trees, the Hilbert ball (see [8]) and many others. For more details on $CAT(0)$ spaces, one can consult [9–11]. Fixed point theory in $CAT(0)$ spaces was initiated by Kirk [12] who proved some theorems for nonexpansive mappings. Since then the fixed point theory for single-valued as well as multi-valued mappings has intensively been developed in $CAT(0)$ spaces (e.g. [13–16]). Further relevant background material on $CAT(0)$ spaces is included in the next section.

The purpose of this paper is to prove some weak and strong convergence theorems of the iterative scheme (1) in $CAT(0)$ spaces for generalized nonexpansive mappings enabling us to enlarge the class of underlying mappings as well as the class of spaces in the corresponding results of Takahashi and Kim [5].

2 Preliminaries

In this section, to make our presentation self-contained, we collect relevant definitions and results. In a metric space $(X,d)$, a geodesic path joining $x\in X$ and $y\in X$ is a map c from a closed interval $[0,r]\subset R$ to X such that $c(0)=x$, $c(r)=y$ and $d(c(t),c(s))=|s-t|$ for all $s,t\in [0,r]$. In particular, the mapping c is an isometry and $d(x,y)=r$. The image of c is called a geodesic segment joining x and y, which is denoted by $[x,y]$, whenever such a segment exists uniquely. For any $x,y\in X$, we denote the point $z\in [x,y]$ by $z=(1-\alpha )x\oplus \alpha y$, where $0\le \alpha \le 1$ if $d(x,z)=\alpha d(x,y)$ and $d(z,y)=(1-\alpha )d(x,y)$. The space $(X,d)$ is called a geodesic space if any 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 and y for each $x,y\in X$. A subset C of X is called convex if C contains every geodesic segment joining any two points in C.

A geodesic triangle $△(x_1,x_2,x_3)$ in a geodesic metric space $(X,d)$ consists of three points of X (as the vertices of △) and a geodesic segment between each pair of points (as the edges of △). A comparison triangle for $△(x_1,x_2,x_3)$ in $(X,d)$ is a triangle $\overline{△}(x_1,x_2,x_3):=△({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3)$ 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 point $\overline{x}\in [{\overline{x}}_1,{\overline{x}}_2]$ is said to be comparison point for $x\in [x_1,x_2]$ if $d(x_1,x)=d({\overline{x}}_1,\overline{x})$. Comparison points on $[{\overline{x}}_2,{\overline{x}}_3]$ and $[{\overline{x}}_3,{\overline{x}}_1]$ are defined in the same way.

A geodesic metric space X is called a $CAT(0)$ space if all geodesic triangles satisfy the following comparison axiom ($CAT(0)$ inequality):

Let △ be a geodesic triangle in X and let $\overline{△}$ be its comparison triangle in ${\mathbb{R}}^2$. Then △ is said to satisfy the $CAT(0)$ inequality if for all $x,y\in △$ and all comparison points $\overline{x},\overline{y}\in \overline{△}$,

$$d(x,y)\le d_{{\mathbb{R}}^2}(\overline{x},\overline{y}).$$

If x, $y_1$ and $y_2$ are points of $CAT(0)$ space and $y_0$ is the midpoint of the segment $[y_1,y_2]$, then the $CAT(0)$ inequality implies

$$d{(x,y_0)}^2\le \frac{1}{2}d{(x,y_1)}^2+\frac{1}{2}d{(x,y_2)}^2-\frac{1}{4}d{(y_1,y_2)}^2.$$

The above inequality is known as the (CN) inequality and was given by Bruhat and Tits [17]. A geodesic space is a $CAT(0)$ space if and only if it satisfies the (CN) inequality. The following are some examples of $CAT(0)$ spaces:

• (i) Any convex subset of a Euclidean space ${\mathbb{R}}^n$, when endowed with the induced metric is a $CAT(0)$ space.

• (ii) Every pre-Hilbert space is a $CAT(0)$ space.

• (iii) If a normed real vector space X is $CAT(0)$ space, then it is a pre-Hilbert space.

• (iv) The Hilbert ball with the hyperbolic metric is a $CAT(0)$ space.

• (v) If $X_1$ and $X_2$ are $CAT(0)$ spaces, then $X_1\times X_2$ is also a $CAT(0)$ space.

For detailed information regarding these spaces, one can refer to [9–11, 17].

Now, we collect some basic geometric properties which are instrumental throughout our subsequent discussions. Let X be a complete $CAT(0)$ space and $\{x_n\}$ be a bounded sequence in X. For $x\in X$ set

$$r(x,\{x_n\})=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x,x_n).$$

The asymptotic radius $r(\{x_n\})$ is given by

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

and the asymptotic center $A(\{x_n\})$ of $\{x_n\}$ is defined as

$$A(\{x_n\})=\{x\in X:r(x,x_n)=r(\{x_n\})\}.$$

It is well known that in a $CAT(0)$ space, $A(\{x_n\})$ consists of exactly one point (see Proposition 5 of [18]).

In 2008, Kirk and Panyanak [19] gave a concept of convergence in $CAT(0)$ spaces which is the analog of the weak convergence in Banach spaces and a restriction of Lim’s concepts of convergence [20] to $CAT(0)$ spaces.

Definition 2.1 A sequence $\{x_n\}$ in X is said to Δ-converge 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 $\mathrm{\Delta }\text{-}{lim}_n{x}_n=x$ and x is the Δ-limit of $\{x_n\}$.

Notice that given $\{x_n\}\subset X$ such that $\{x_n\}$ Δ-converges to x and, given $y\in X$ with $y\ne x$, by uniqueness of the asymptotic center we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_n,x)<\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_n,y).$$

Thus every $CAT(0)$ space satisfies the Opial property. Now we collect some basic facts about $CAT(0)$ spaces which will be used throughout the text frequently.

Lemma 2.2 ([19])

Every bounded sequence in a complete $CAT(0)$ space admits a Δ-convergent subsequence.

Lemma 2.3 ([21])

If C is closed convex subset of a complete $CAT(0)$ space and if $\{x_n\}$ is a bounded sequence in C, then the asymptotic center of $\{x_n\}$ is in C.

Lemma 2.4 ([22])

Let $(X,d)$ be a $CAT(0)$ space. For $x,y\in X$ and $t\in [0,1]$, there exists a unique $z\in [x,y]$ such that

$$d(x,z)=td(x,y)\mathit{\text{and}}d(y,z)=(1-t)d(x,y).$$

We use the notation $(1-t)x\oplus ty$ for the unique point z of the above lemma.

Lemma 2.5 For $x,y,z\in X$ and $t\in [0,1]$ we have

$$d((1-t)x\oplus ty,z)\le (1-t)d(x,z)+td(y,z).$$

Recently García-Falset et al. [6] introduced two generalizations of the condition $(C)$ in Banach spaces. Now, we state their condition in the framework of $CAT(0)$ spaces.

Definition 2.6 Let T be a mapping defined on a subset C of $CAT(0)$ space X and $\mu \ge 1$, then T is said to satisfy the condition $(E_{\mu })$, if (for all $x,y\in C$)

$$d(x,Ty)\le \mu d(x,Tx)+d(x,y).$$

We say that T satisfies the condition $(E)$ whenever T satisfies the condition $(E_{\mu })$ for some $\mu \ge 1$.

Definition 2.7 Let T be a mapping defined on a subset C of a $CAT(0)$ space X and $\lambda \in (0,1)$, then T is said to satisfy the condition $(C_{\lambda })$ if (for all $x,y\in C$)

$$\lambda d(x,Tx)\le d(x,y)\Rightarrow d(Tx,Ty)\le d(x,y).$$

In the case $0<{\lambda }_1<{\lambda }_2<1$, then the condition $(C_{{\lambda }_1})$ implies the condition $(C_{{\lambda }_2})$. The following example shows that the class of mappings satisfying the conditions $(E)$ and $(C_{\lambda })$ (for some $\lambda \in (0,1)$) is larger than the class of mappings satisfying the condition $(C)$.

Example 2.8 ([6])

For a given $\lambda \in (0,1)$, define a mapping T on $[0,1]$ by

$$T(x)=\{\begin{array}{cc}\frac{x}{2}\hfill & \text{if }x\ne 1,\hfill \\ \frac{1+\lambda }{2+\lambda }\hfill & \text{if }x=1.\hfill \end{array}$$

Then the mapping T satisfies the condition $(C_{\lambda })$ but it fails the condition $(C_{{\lambda }_1})$ whenever $0<{\lambda }_1<\lambda$. Moreover, T satisfies the condition $(E_{\mu })$ for $\mu =\frac{2+\lambda }{2}$.

The following theorem is an analog of Theorem 4 in [6] to $CAT(0)$ spaces.

Theorem 2.9 Let C be a bounded convex subset of a complete $CAT(0)$ space X. If $T:C\to C$ satisfies the condition $(C_{\lambda })$ on C for some $\lambda \in (0,1)$, then there exists an approximating fixed point sequence for T.

Now, we present the Ishikawa iterative scheme in the framework of $CAT(0)$ spaces. For $x_1\in C$, the Ishikawa iteration is defined as

$$\{\begin{array}{c}{y}_n={\alpha }_{n}T{x}_n\oplus (1-{\alpha }_n)x_n,\hfill \\ x_{n+1}={\beta }_{n}T{y}_n\oplus (1-{\beta }_n)x_n\text{for }n\ge 1,\hfill \end{array}$$

(2)

where the sequences ${\alpha }_n$ and ${\beta }_n$ are sequences in $[0,1]$.

The following lemma is a consequence of Lemma 2.9 of [23] which will be used to prove our main results.

Lemma 2.10 Let X be a complete $CAT(0)$ space and let $x\in X$. Suppose $\{t_n\}$ is a sequence in $[b,c]$ for some $b,c\in (0,1)$ and $\{x_n\}$, $\{y_n\}$ are sequences in X such that ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}d(x_n,x)\le r$, ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}d(y_n,x)\le r$, and ${lim}_{n\to \mathrm{\infty }}d((1-t_n)x_n\oplus t_n{y}_n,x)=r$ for some $r\ge 0$. Then

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,y_n)=0.$$

In this paper, we prove that the sequence $\{x_n\}$ described by (2) Δ-converges to a fixed point of T if one of the following conditions holds:

$$\{\begin{array}{cc}\text{(i)}\hfill & {\alpha }_n\in [a,b]\text{and}{\beta }_n\in [0,b]\text{for some }a,b\text{ with }0<a\le b<1\text{or}\hfill \\ \text{(ii)}\hfill & {\alpha }_n\in [a,1]\text{and}{\beta }_n\in [a,b]\text{for some }a,b\text{ with }0<a\le b<1.\hfill \end{array}$$

(3)

This result is an analog of a result of the weak convergence theorem of Takahashi and Kim [5] for a generalized nonexpansive mapping in a Banach space. A strong convergence theorem is also proved. In the process, the corresponding results of Takahashi and Kim [5], Laokul and Panyanak [24], Razani and Salahifard [25], and others are generalized and improved.

3 Main results

Before proving our main results, firstly we re-write Theorem 1.3 in the setting of $CAT(0)$ spaces which is essentially Theorem 3.2 of [13].

Theorem 3.1 Let C be a bounded, closed, and convex subset of a complete $CAT(0)$ space X. If $T:C\to C$ satisfies the conditions $(E)$ and $(C_{\lambda })$ for some $\lambda \in (0,1)$, then T has a fixed point.

Now, to accomplish our main results, we prove the following lemma.

Lemma 3.2 Let C be a nonempty closed convex subset of a complete $CAT(0)$ space X and $T:C\to C$ be a mapping which satisfies the condition $(C_{\lambda })$ for some $\lambda \in (0,1)$. If $\{x_n\}$ is a sequence defined by (2) and the sequences $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ satisfy the condition described in (3), then ${lim}_{n\to \mathrm{\infty }}d(x_n,x)$ exists for all $x\in F(T)$.

Proof Since T satisfies the condition $(C_{\lambda })$ (for some $\lambda \in (0,1)$) and $x\in F(T)$, we have

$$\lambda d(x,Tx)=0\le d(x,y_n)$$

and

$$\lambda d(x,Tx)=0\le d(x,x_n)\text{for all }n\ge 1,$$

so that

$$d(Tx,T{y}_n)\le d(x,y_n)\text{and}d(Tx,T{x}_n)\le d(x,x_n).$$

Now consider

$$\begin{array}{rcl}d(x_{n+1},x)& =& d({\beta }_{n}T{y}_n\oplus (1-{\beta }_n)x_n,x)\\ \le & {\beta }_{n}d(T{y}_n,x)+(1-{\beta }_n)d(x_n,x)\\ \le & {\beta }_{n}d(y_n,x)+(1-{\beta }_n)d(x_n,x)\\ =& {\beta }_{n}d({\alpha }_{n}T{x}_n\oplus (1-{\alpha }_n)x_n,x)+(1-{\beta }_n)d(x_n,x)\\ \le & {\beta }_n{\alpha }_{n}d(T{x}_n,x)+{\beta }_n(1-{\alpha }_n)d(x_n,x)+(1-{\beta }_n)d(x_n,x)\\ \le & d(x_n,x),\end{array}$$

which shows that the sequence $d(x_n,x)$ is decreasing and bounded below so that ${lim}_{n\to \mathrm{\infty }}d(x_n,x)$ exists. □

Lemma 3.3 Let C be a nonempty closed convex subset of a complete $CAT(0)$ space X and let $T:C\to C$ satisfy the conditions $(C_{\lambda })$ and $(E)$ on C. If $\{x_n\}$ is a sequence defined by (2) and the sequences $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ satisfy the condition described in (3), then $F(T)$ is nonempty if and only if $\{x_n\}$ is bounded and ${lim}_{n\to \mathrm{\infty }}d(T{x}_n,x_n)=0$.

Proof Suppose that the fixed point set $F(T)$ is nonempty and $x\in F(T)$. Then by Lemma 3.2, ${lim}_{n\to \mathrm{\infty }}d(x_n,x)$ exists and let us take it to be c besides that $\{x_n\}$ is bounded. We have

$$\lambda d(x,Tx)=0\le d(x,y_n)$$

and

$$\lambda d(x,Tx)=0\le d(x,x_n)\text{for all }n\ge 1.$$

Owing to the condition $(C_{\lambda })$, we have $d(Tx,T{y}_n)\le d(x,y_n)$ and $d(Tx,T{x}_n)\le d(x,x_n)$.

Therefore, we have

$$\begin{array}{rcl}d(T{y}_n,x)& \le & d(y_n,x)\\ =& d({\alpha }_{n}T{x}_n\oplus (1-{\alpha }_n)x_n,x)\\ \le & {\alpha }_{n}d(T{x}_n,x)+(1-{\alpha }_n)d(x_n,x)\\ \le & {\alpha }_{n}d(x_n,x)+(1-{\alpha }_n)d(x_n,x)\\ =& d(x_n,x),\end{array}$$

so that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(T{y}_n,x)\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(y_n,x)\le c.$$

Also, we observe that

$$\underset{n\to \mathrm{\infty }}{lim}d({\beta }_{n}T{y}_n\oplus (1-{\beta }_n)x_n,x)=\underset{n\to \mathrm{\infty }}{lim}d(x_{n+1},x)=c.$$

Case I: If $0<a\le {\beta }_n\le b<1$ and $0\le {\alpha }_n\le 1$, then by the foregoing discussion and by Lemma 2.10, we have ${lim}_{n\to \mathrm{\infty }}d(T{y}_n,x_n)=0$.

For each ${\alpha }_n\in [0,b]$,

$$\begin{array}{rcl}d(T{x}_n,x_n)& \le & d(T{x}_n,y_n)+d(y_n,x_n)\\ =& d(T{x}_n,{\alpha }_{n}T{x}_n\oplus (1-{\alpha }_n)x_n)+d(y_n,x_n)\\ \le & (1-{\alpha }_n)d(T{x}_n,x_n)+d(y_n,x_n),\end{array}$$

so that

$${\alpha }_{n}d(T{x}_n,x_n)\le d(y_n,x_n).$$

As ${\alpha }_n\in [0,b]$, in view of the condition $(C_{\lambda })$ we have $d(T{x}_n,T{y}_n)\le d(x_n,y_n)$.

Now, consider

$$\begin{array}{rcl}d(T{x}_n,x_n)& \le & d(T{x}_n,T{y}_n)+d(T{y}_n,x_n)\\ \le & d(x_n,y_n)+d(T{y}_n,x_n)\\ \le & {\alpha }_{n}d(x_n,T{x}_n)+d(T{y}_n,x_n).\end{array}$$

Making use of the observation $(1-b)d(T{x}_n,x_n)\le (1-{\alpha }_n)d(x_n,T{x}_n)\le d(T{y}_n,x_n)$, we have ${lim}_{n\to \mathrm{\infty }}d(T{x}_n,x_n)\le \frac{1}{1-b}{lim}_{n\to \mathrm{\infty }}d(T{y}_n,x_n)=0$.

Case 2: $0<a\le {\beta }_n\le 1$ and $0<a\le {\alpha }_n\le b<1$.

Since we have $d(T{x}_n,x)\le d(x_n,x)$, for all $n\ge 1$, we get

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(T{x}_n,x)\le c.$$

Consider

$$\begin{array}{rcl}d(x_{n+1},x)& \le & {\beta }_{n}d(T{y}_n,x)+(1-{\beta }_n)d(x_n,x)\\ \le & {\beta }_{n}d(y_n,x)+(1-{\beta }_n)d(x_n,x),\end{array}$$

which amounts to saying that

$$\frac{d(x_{n+1},x)-d(x_n,x)}{{\alpha }_n}\le d(y_n,x)-d(x_n,x).$$

Taking ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}$ of both sides of the above inequality, we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\frac{d(x_{n+1},x)-d(x_n,x)}{{\alpha }_n}\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}(d(y_n,x)-d(x_n,x)).$$

As ${lim}_{n\to \mathrm{\infty }}d(x_{n+1},x)={lim}_{n\to \mathrm{\infty }}d(x_n,x)=c$, we have

$$0\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}(d(y_n,x)-d(x_n,x)),$$

while, owing to $d(y_n,x)-d(x_n,x)\le 0$, we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}(d(y_n,x)-d(x_n,x))\le 0.$$

Therefore, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}(d(y_n,x)-d(x_n,x))=0$. Thus, we get

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}d(x_n,x)\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}d(y_n,x).$$

That is, $c\le {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}d(y_n,x)$. By combining the foregoing observations, we have

$$c\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}d(y_n,x)\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(y_n,x)\le c,$$

so that

$$c=\underset{n\to \mathrm{\infty }}{lim}d(y_n,x)=\underset{n\to \mathrm{\infty }}{lim}d({\alpha }_{n}T{x}_n\oplus (1-{\alpha }_n)x_n,x).$$

Again in view of Lemma 2.10, we have ${lim}_{n\to \mathrm{\infty }}d(T{x}_n,x_n)=0$.

Conversely, suppose that $\{x_n\}$ is bounded and ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$. Let $A(\{x_n\})=\{x\}$. Then, by Lemma 2.3, $x\in C$. As T satisfies the condition $(E_{\mu })$ on C, there exists $\mu >1$ such that

$$d(x_n,Tx)\le \mu d(x_n,T{x}_n)+d(x_n,x),$$

which implies that

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_n,Tx)& \le & \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\{\mu d(x_n,T{x}_n)+d(x_n,x)\}\\ =& \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_n,x).\end{array}$$

Owing to the uniqueness of asymptotic center, $Tx=x$, so that x is fixed point of T. □

Theorem 3.4 Let C be a nonempty closed convex subset of a complete $CAT(0)$ space X and $T:C\to C$ be a mapping which satisfies conditions $(C_{\lambda })$ for some $\lambda \in (0,1)$ and $(E)$ on C with $F(T)\ne \mathrm{\varnothing }$. If the sequences $\{x_n\}$, $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are described as in (2) and (3), then the sequence $\{x_n\}$ Δ-converges to a fixed point of T.

Proof By Lemma 3.3, we observe that $\{x_n\}$ is a bounded sequence and

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,T{x}_n)=0.$$

Let $W_{\omega }(\{x_n\})=:\bigcup A(\{u_n\})$, where the union is taken over all subsequence $\{u_n\}$ over $\{x_n\}$. To show the Δ-convergence of $\{x_n\}$ to a fixed point of T, we show that $W_{\omega }(\{x_n\})\subset F(T)$ and $W_{\omega }(\{x_n\})$ is a singleton set. To show that $W_{\omega }(\{x_n\})\subset F(T)$ let $y\in W_{\omega }(\{x_n\})$. Then there exists a subsequence $\{y_n\}$ of $\{x_n\}$ such that $A(\{y_n\})=y$. By Lemmas 2.2 and 2.3, there exists a subsequence $\{z_n\}$ of $\{y_n\}$ such that $\mathrm{\Delta }\text{-}{lim}_n{z}_n=z$ and $z\in C$. Since ${lim}_{n\to \mathrm{\infty }}d(z_n,T{z}_n)=0$ and T satisfies the condition $(E)$, there exists a $\mu \ge 1$ such that

$$d(z_n,Tz)\le \mu d(z_n,T{z}_n)+d(z_n,z).$$

By taking the lim sup of both sides, we have

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(z_n,Tz)& \le & \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\{\mu d(z_n,T{z}_n)+d(z_n,z)\}\\ \le & \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(z_n,z).\end{array}$$

As $\mathrm{\Delta }\text{-}{lim}_n{z}_n=z$, by the Opial property

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(z_n,z)\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(z_n,Tz).$$

Hence $Tz=z$, i.e. $z\in F(T)$. Now, we claim that $z=y$. If not, by Lemma 3.2, ${lim}_{n}d(x_n,z)$ exists and, owing to the uniqueness of the asymptotic centers,

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(z_n,z)& <& \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(z_n,y)\\ \le & \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(y_n,y)\\ <& \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(y_n,z)\\ =& \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_n,z)\\ =& \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(z_n,z),\end{array}$$

which is a contradiction. Hence $y=z$. To assert that $W_{\omega }(\{x_n\})$ is a singleton let $\{y_n\}$ be a subsequence of $\{x_n\}$. In view of Lemmas 2.2 and 2.3, there exists a subsequence $\{z_n\}$ of $\{y_n\}$ such that $\mathrm{\Delta }\text{-}{lim}_n{z}_n=z$. Let $A(\{y_n\})=y$ and $A(\{x_n\})=x$. Earlier, we have shown that $y=z$. Therefore it is enough to show $z=x$. If $z\ne x$, then in view of Lemma 3.2 $\{d(x_n,z)\}$ is convergent. By uniqueness of the asymptotic centers

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(z_n,z)& <& \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(z_n,x)\\ \le & \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_n,x)\\ <& \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_n,z)\\ =& \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(z_n,z),\end{array}$$

which is a contradiction so that the conclusion follows. □

In view of Theorem 3.1, we have the following corollary of the preceding theorem.

Corollary 3.5 Let C be a nonempty bounded, closed and convex subset of a complete $CAT(0)$ space X and $T:C\to C$ be a mapping which satisfies conditions $(C_{\lambda })$ for some $\lambda \in (0,1)$ and $(E)$ on C. If sequences $\{x_n\}$, $\{{\alpha }_n\}$, and $\{{\beta }_n\}$ are described by (2) and (3), then the sequence $\{x_n\}$ Δ-converges to a fixed point of T.

Theorem 3.6 Let C be a nonempty closed convex subset of a complete $CAT(0)$ space X and $T:C\to C$ be a mapping which satisfies conditions $(C_{\lambda })$ for some $\lambda \in (0,1)$ and $(E)$ on C. Moreover, T satisfies the condition $(I)$ with $F(T)\ne \mathrm{\varnothing }$. If sequences $\{x_n\}$, $\{{\alpha }_n\}$, and $\{{\beta }_n\}$ are defined as in (2) and (3), respectively, then $\{x_n\}$ converges strongly to some fixed point of T.

Proof To show that the fixed point set $F(T)$ is closed, let $\{x_n\}$ be a sequence in $F(T)$ which converges to some point $z\in C$. As

$$\lambda d(x_n,T{x}_n)=0\le d(x_n,z),$$

in view of the condition $(C_{\lambda })$, we have

$$d(x_n,Tz)=d(T{x}_n,Tz)\le d(x_n,z).$$

By taking the limit of both sides we obtain

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,Tz)\le \underset{n\to \mathrm{\infty }}{lim}d(x_n,z)=0.$$

In view of the uniqueness of the limit, we have $z=Tz$, so that $F(T)$ is closed. Observe that by Lemma 3.2, we have ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$. It follows from the condition $(I)$ that

$$\underset{n\to \mathrm{\infty }}{lim}f\left(d(x_n,F(T))\right)\le \underset{n\to \mathrm{\infty }}{lim}d(x_n,T{x}_n)=0,$$

so that ${lim}_{n\to \mathrm{\infty }}f(d(x_n,F(T)))=0$. Since $f:[0,\mathrm{\infty }]\to [0,\mathrm{\infty })$ is a nondecreasing mapping satisfying $f(0)=0$ for all $r\in (0,\mathrm{\infty })$, we have ${lim}_{n\to \mathrm{\infty }}d(x_n,F(T))=0$. This implies that there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that

$$d(x_{n_k},p_k)\le \frac{1}{2^k}\text{for all }k\ge 1$$

wherein $\{p_k\}$ is in $F(T)$. By Lemma 3.2, we have

$$d(x_{n_{k+1}},p_k)\le d(x_{n_k},p_k)\le \frac{1}{2^k},$$

so that

$$\begin{array}{rcl}d(p_{k+1},p_k)& \le & d(p_{k+1},x_{n_{k+1}})+d(x_{n_{k+1}},p_k)\\ \le & \frac{1}{2^{k+1}}+\frac{1}{2^k}<\frac{1}{2^{k-1}},\end{array}$$

which implies that $\{p_k\}$ is a Cauchy sequence. Since $F(T)$ is closed, $\{p_k\}$ is a convergent sequence. Write ${lim}_{k\to \mathrm{\infty }}{p}_k=p$. Now, in order to show that $\{x_n\}$ converges to p let us proceed as follows:

$$d(x_{n_k},p)\le d(x_{n_k},p_k)+d(p_k,p)\to 0\text{as }k\to \mathrm{\infty },$$

so that ${lim}_{k\to \mathrm{\infty }}d(x_{n_k},p)=0$. Since ${lim}_{n\to \mathrm{\infty }}d(x_n,p)$ exists, we have $x_n\to p$. □

Corollary 3.7 Let C be a nonempty bounded, closed, and convex subset of a complete $CAT(0)$ space X and let $T:C\to C$ be a mapping which satisfies the conditions $(C_{\lambda })$ for some $\lambda \in (0,1)$ and $(E)$ on C. Moreover, T satisfies the condition $(I)$. If the sequences $\{x_n\}$, $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are described by (2) and (3), then $\{x_n\}$ converges strongly to some fixed point of T.