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# Fixed point theorems for N-generalized hybrid mappings in uniformly convex metric spaces

Fixed Point Theory and Applications20132013:188

https://doi.org/10.1186/1687-1812-2013-188

• Accepted: 4 July 2013
• Published:

## Abstract

In this paper, we prove some fixed point theorems for N-generalized hybrid mappings in both uniformly convex metric spaces and $CAT\left(0\right)$ spaces. We also introduce a new iteration method for approximating a fixed point of N-generalized hybrid mappings in $CAT\left(0\right)$ spaces and obtain Δ-convergence to a fixed point of N-generalized hybrid mappings in such spaces. Our results improve and extend the corresponding results existing in the literature.

MSC:47H09, 47H10.

## Keywords

• fixed point
• uniformly convex metric spaces
• $CAT\left(0\right)$ spaces
• generalized hybrid mappings

## 1 Introduction and preliminaries

Let C be a nonempty closed subset of a metric space $\left(X,d\right)$ and let T be a mapping of C into itself. The set of all fixed points of T is denoted by $F\left(T\right)=\left\{x\in C:x=Tx\right\}$. In 1970, Takahashi [1] introduced the concept of convex metric spaces by using the convex structure as follows.

Definition 1.1 Let $\left(X,d\right)$ be a metric space. A mapping $W:X×X×\left[0,1\right]\to X$ is said to be a convex structure on X if for each $x,y\in X$ and $\lambda \in \left[0,1\right]$,
$d\left(z,W\left(x,y,\lambda \right)\right)\le \lambda d\left(z,x\right)+\left(1-\lambda \right)d\left(z,y\right)$

for all $z\in X$. A metric space $\left(X,d\right)$ together with a convex structure W is called a convex metric space which will be denoted by $\left(X,d,W\right)$.

A nonempty subset C of X is said to be convex if $W\left(x,y,\lambda \right)\in C$ for all $x,y\in C$ and $\lambda \in \left[0,1\right]$. Clearly, a normed space and each of its convex subsets are convex metric spaces, but the converse does not hold. For each $x,y\in X$ and $\lambda \in \left[0,1\right]$, it is known that a convex metric space has the following properties [1, 2]:
1. (i)

$W\left(x,x,\lambda \right)=x$, $W\left(x,y,0\right)=y$ and $W\left(x,y,1\right)=x$;

2. (ii)

$d\left(x,W\left(x,y,\lambda \right)\right)=\left(1-\lambda \right)d\left(x,y\right)$ and $d\left(y,W\left(x,y,\lambda \right)\right)=\lambda d\left(x,y\right)$.

In 1996, Shimizu and Takahashi [3] introduced the concept of uniform convexity in convex metric spaces and studied some properties of these spaces. A convex metric space $\left(X,d,W\right)$ is said to be uniformly convex if for any $\epsilon >0$, there exists ${\delta }_{\epsilon }>0$ such that for all $r>0$ and $x,y,z\in X$ with $d\left(z,x\right)\le r$, $d\left(z,y\right)\le r$ and $d\left(x,y\right)\ge r\epsilon$ imply that $d\left(z,W\left(x,y,\frac{1}{2}\right)\right)\le \left(1-{\delta }_{\epsilon }\right)r$. Obviously, uniformly convex Banach spaces are uniformly convex metric spaces.

Let C be a nonempty closed and convex subset of a convex metric space $\left(X,d,W\right)$ and let $\left\{{x}_{n}\right\}$ be a bounded sequence in X. For $x\in X$, we define a mapping $r\left(\cdot ,\left\{{x}_{n}\right\}\right):X\to \left[0,\mathrm{\infty }\right)$ by
$r\left(x,\left\{{x}_{n}\right\}\right)=\underset{n\to \mathrm{\infty }}{lim sup}d\left(x,{x}_{n}\right).$
Clearly, $r\left(\cdot ,\left\{{x}_{n}\right\}\right)$ is a continuous and convex function. The asymptotic radius of $\left\{{x}_{n}\right\}$ relative to C is given by
$r\left(C,\left\{{x}_{n}\right\}\right)=inf\left\{r\left(x,\left\{{x}_{n}\right\}\right):x\in C\right\},$
and the asymptotic center of $\left\{{x}_{n}\right\}$ relative to C is the set
$A\left(C,\left\{{x}_{n}\right\}\right)=\left\{x\in C:r\left(x,\left\{{x}_{n}\right\}\right)=r\left(C,\left\{{x}_{n}\right\}\right)\right\}.$

It is clear that the asymptotic center $A\left(C,\left\{{x}_{n}\right\}\right)$ is always closed and convex. It may either be empty or consist of one or many points. The asymptotic center $A\left(C,\left\{{x}_{n}\right\}\right)$ is singleton for uniformly convex Banach spaces [4, 5] or $CAT\left(0\right)$ spaces [6]. The following lemma obtained by Phuengrattana and Suantai [7] is useful for our results.

Lemma 1.2 Let C be a nonempty closed and convex subset of a complete uniformly convex metric space $\left(X,d,W\right)$ and let $\left\{{x}_{n}\right\}$ be a bounded sequence in X. Then $A\left(C,\left\{{x}_{n}\right\}\right)$ is a singleton set.

One of the special spaces of uniformly convex metric spaces is a $CAT\left(0\right)$ space; see [8]. It was noted in [9] that any $CAT\left(\kappa \right)$ space ($\kappa >0$) is uniformly convex in a certain sense but it is not a $CAT\left(0\right)$ space. Fixed point theory in $CAT\left(0\right)$ spaces was first studied by Kirk [9, 10]. He showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete $CAT\left(0\right)$ space always has a fixed point. Since then, the fixed point theory for single-valued and multivalued mappings in $CAT\left(0\right)$ spaces has been rapidly developed, and many papers have appeared (e.g., see [1127]).

Let $\left(X,d\right)$ be a metric space. A geodesic path joining $x\in X$ to $y\in X$ (or, more briefly, a geodesic from x to y) is a map c from a closed interval $\left[0,l\right]\subset \mathbb{R}$ to X such that $c\left(0\right)=x$, $c\left(l\right)=y$ and $d\left(c\left({t}_{1}\right),c\left({t}_{2}\right)\right)=|{t}_{1}-{t}_{2}|$ for all ${t}_{1},{t}_{2}\in \left[0,l\right]$. In particular, c is an isometry and $d\left(x,y\right)=l$. 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 $\left[x,y\right]$. Write $c\left(\alpha 0+\left(1-\alpha \right)l\right)=\alpha x\oplus \left(1-\alpha \right)y$ for $\alpha \in \left(0,1\right)$. The space $\left(X,d\right)$ is said to be a geodesic metric 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 and y for each $x,y\in X$. A subset Y of X is said to be convex if Y includes every geodesic segment joining any two of its points.

A geodesic triangle $\mathrm{△}\left({x}_{1},{x}_{2},{x}_{3}\right)$ in a geodesic metric space $\left(X,d\right)$ consists of three points ${x}_{1}$, ${x}_{2}$, ${x}_{3}$ in X (the vertices of ) and a geodesic segment between each pair of vertices (the edges of ). A comparison triangle for the geodesic triangle $\mathrm{△}\left({x}_{1},{x}_{2},{x}_{3}\right)$ in $\left(X,d\right)$ is a triangle $\overline{\mathrm{△}}\left({x}_{1},{x}_{2},{x}_{3}\right):=\mathrm{△}\left({\overline{x}}_{1},{\overline{x}}_{2},{\overline{x}}_{3}\right)$ in the Euclidean plane ${\mathbb{E}}^{2}$ such that ${d}_{{\mathbb{E}}^{2}}\left({\overline{x}}_{i},{\overline{x}}_{j}\right)=d\left({x}_{i},{x}_{j}\right)$ for $i,j\in \left\{1,2,3\right\}$.

A geodesic metric space is said to be a $CAT\left(0\right)$ space if all geodesic triangles satisfy the following comparison axiom: Let be a geodesic triangle in X and let $\overline{\mathrm{△}}$ be a comparison triangle for . Then is said to satisfy the $CAT\left(0\right)$ inequality if for all $x,y\in \mathrm{△}$ and all comparison points $\overline{x},\overline{y}\in \overline{\mathrm{△}}$,
$d\left(x,y\right)\le {d}_{{\mathbb{E}}^{2}}\left(\overline{x},\overline{y}\right).$

It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a $CAT\left(0\right)$ space. Other examples include pre-Hilbert spaces [8], -trees [16], the complex Hilbert ball with a hyperbolic metric [5], and many others.

If z, x, y are points in a $CAT\left(0\right)$ space and if $m\left[x,y\right]$ is the midpoint of the segment $\left[x,y\right]$, then the $CAT\left(0\right)$ inequality implies
$d{\left(z,m\left[x,y\right]\right)}^{2}\le \frac{1}{2}d{\left(z,x\right)}^{2}+\frac{1}{2}d{\left(z,y\right)}^{2}-\frac{1}{4}d{\left(x,y\right)}^{2}.$
(CN)
This is the (CN) inequality of Bruhat and Tits [28], which is equivalent to
for any $\lambda \in \left[0,1\right]$. The (CN*) inequality has appeared in [29]. Moreover, if X is a $CAT\left(0\right)$ space and $x,y\in X$, then for any $\lambda \in \left[0,1\right]$, there exists a unique point $\lambda x\oplus \left(1-\lambda \right)y\in \left[x,y\right]$ such that
$d\left(z,\lambda x\oplus \left(1-\lambda \right)y\right)\le \lambda d\left(z,x\right)+\left(1-\lambda \right)d\left(z,y\right)$

for any $z\in X$. It follows that $CAT\left(0\right)$ spaces have a convex structure $W\left(x,y,\lambda \right)=\lambda x\oplus \left(1-\lambda \right)y$.

Remark 1.3
1. (i)

By using the (CN) inequality, it is easy to see that $CAT\left(0\right)$ spaces are uniformly convex.

2. (ii)

A geodesic metric space is a $CAT\left(0\right)$ space if and only if it satisfies the (CN) inequality; see [8].

In 2012, Dhompongsa et al. [12] introduced the following notation in $CAT\left(0\right)$ spaces: Let ${x}_{1},\dots ,{x}_{N}$ be points in a $CAT\left(0\right)$ space X and ${\lambda }_{1},\dots ,{\lambda }_{N}\in \left(0,1\right)$ with ${\sum }_{i=1}^{N}{\lambda }_{i}=1$, we write
$\underset{i=1}{\overset{N}{⨁}}{\lambda }_{i}{x}_{i}:=\left(1-{\lambda }_{N}\right)\left(\frac{{\lambda }_{1}}{1-{\lambda }_{N}}{x}_{1}\oplus \frac{{\lambda }_{2}}{1-{\lambda }_{N}}{x}_{2}\oplus \cdots \oplus \frac{{\lambda }_{N-1}}{1-{\lambda }_{N}}{x}_{N-1}\right)\oplus {\lambda }_{N}{x}_{N}.$
(1.1)
The definition of is an ordered one in the sense that it depends on the order of points ${x}_{1},\dots ,{x}_{N}$. Under (1.1) we obtain that

In 1976, Lim [30] introduced the concept of Δ-convergence in a general metric space. Later in 2008, Kirk and Panyanak [15] extended the concept of Lim to a $CAT\left(0\right)$ space.

Definition 1.4 [15]

A sequence $\left\{{x}_{n}\right\}$ in a $CAT\left(0\right)$ space X is said to Δ-converge to $x\in X$ if x is the unique asymptotic center of $\left\{{u}_{n}\right\}$ for every subsequence $\left\{{u}_{n}\right\}$ of $\left\{{x}_{n}\right\}$. In this case, we write $\mathrm{\Delta }\text{-}{lim}_{n\to \mathrm{\infty }}{x}_{n}=x$ and call x the Δ-limit of $\left\{{x}_{n}\right\}$.

Lemma 1.5 [15]

Every bounded sequence in a complete $CAT\left(0\right)$ space has a Δ-convergent subsequence.

For any nonempty subset C of a $CAT\left(0\right)$ space X, let $\pi :={\pi }_{C}$ be the nearest point projection mapping from X to a subset C of X. In [8], it is known that if C is closed and convex, the mapping π is well defined, nonexpansive, and the following inequality holds:
$d{\left(x,y\right)}^{2}\ge d{\left(x,\pi x\right)}^{2}+d{\left(\pi x,y\right)}^{2}$

for all $x\in X$ and $y\in C$. By using the same argument as in [[31], Lemma 3.2], we can prove the following result for nearest point projection mappings in $CAT\left(0\right)$ spaces.

Lemma 1.6 Let C be a nonempty closed and convex subset of a complete $CAT\left(0\right)$ space X, let $\pi :X\to C$ be the nearest point projection mapping, and let $\left\{{x}_{n}\right\}$ be a sequence in X. If $d\left({x}_{n+1},p\right)\le d\left({x}_{n},p\right)$ for all $p\in C$ and $n\in \mathbb{N}$, then $\left\{\pi {x}_{n}\right\}$ converges strongly to some element in C.

Proof Let $m>n$. By the (CN) inequality and the property of π, it follows that
$\begin{array}{rcl}d{\left(\pi {x}_{m},\pi {x}_{n}\right)}^{2}& \le & 2d{\left({x}_{m},\pi {x}_{m}\right)}^{2}+2d{\left({x}_{m},\pi {x}_{n}\right)}^{2}-4d{\left({x}_{m},\frac{\pi {x}_{m}\oplus \pi {x}_{n}}{2}\right)}^{2}\\ \le & 2d{\left({x}_{m},\pi {x}_{m}\right)}^{2}+2d{\left({x}_{m},\pi {x}_{n}\right)}^{2}-4d{\left({x}_{m},\pi {x}_{m}\right)}^{2}\\ =& 2d{\left({x}_{m},\pi {x}_{n}\right)}^{2}-2d{\left({x}_{m},\pi {x}_{m}\right)}^{2}\\ \le & 2d{\left({x}_{n},\pi {x}_{n}\right)}^{2}-2d{\left({x}_{m},\pi {x}_{m}\right)}^{2}.\end{array}$
(1.2)
This implies that

Then ${lim}_{n\to \mathrm{\infty }}d{\left({x}_{n},\pi {x}_{n}\right)}^{2}$ exists. Letting $m,n\to \mathrm{\infty }$ in (1.2), we have that $\left\{\pi {x}_{n}\right\}$ is a Cauchy sequence in a closed subset C of a complete $CAT\left(0\right)$ space X, hence it converges to some element in C. □

Let C be a nonempty closed and convex subset of a Hilbert space H. A mapping $T:C\to C$ is called generalized hybrid if there exist $\alpha ,\beta \in \mathbb{R}$ such that
$\alpha {\parallel Tx-Ty\parallel }^{2}+\left(1-\alpha \right){\parallel x-Ty\parallel }^{2}\le \beta {\parallel Tx-y\parallel }^{2}+\left(1-\beta \right){\parallel x-y\parallel }^{2}$

for all $x,y\in C$. We note that the generalized hybrid mappings generalize several well-known mappings. For example, a generalized hybrid mapping is nonexpansive for $\alpha =1$ and $\beta =0$, nonspreading for $\alpha =2$ and $\beta =1$, and hybrid for $\alpha =\frac{3}{2}$ and $\beta =\frac{1}{2}$. In 2010, Kocourek et al. [32] proved the fixed point theorems for generalized hybrid mappings in Hilbert spaces. Later in 2011, Takahashi and Yao [33] extended the results of Kocourek et al. to uniformly convex Banach spaces.

Recently, Maruyama et al. [34] introduced a new nonlinear mapping in a Hilbert space as follows. Let $N\in \mathbb{N}$. A mapping $T:C\to C$ is called N-generalized hybrid if there are ${\alpha }_{1},\dots ,{\alpha }_{N},{\beta }_{1},\dots ,{\beta }_{N}\in \mathbb{R}$ such that
$\begin{array}{c}\sum _{k=1}^{N}{\alpha }_{k}{\parallel {T}^{N+1-k}x-Ty\parallel }^{2}+\left(1-\sum _{k=1}^{N}{\alpha }_{k}\right){\parallel x-Ty\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \sum _{k=1}^{N}{\beta }_{k}{\parallel {T}^{N+1-k}x-y\parallel }^{2}+\left(1-\sum _{k=1}^{N}{\beta }_{k}\right){\parallel x-y\parallel }^{2}\hfill \end{array}$

for all $x,y\in C$. They obtained the existence and weak convergence theorems for N-generalized hybrid mappings in Hilbert spaces. Hojo et al. [35] also studied the fixed point theorems for N-generalized hybrid mappings in Hilbert spaces and provided an example of N-generalized hybrid mappings which are not generalized hybrid mappings as follows.

Example 1.7 Let H be a Hilbert space, $A=\left\{x\in H:\parallel x\parallel \le 1\right\}$ and define a mapping $T:H\to H$ as follows:

We observe that the N-generalized hybrid mappings generalize several well-known mappings, for instance, nonexpansive mappings, nonspreading mappings, hybrid mappings, λ-hybrid mappings, generalized hybrid mappings, and 2-generalized hybrid mappings. Many researchers have studied the fixed point theorems of those mappings in both Hilbert spaces and Banach spaces (e.g., see [32, 33, 3638]). However, no researcher has studied the fixed point theorems for N-generalized hybrid mappings in more general spaces. So, in this paper, we are interested in studying and extending those mappings to both uniformly convex metric spaces and $CAT\left(0\right)$ spaces.

## 2 Fixed point theorems in uniformly convex metric spaces

We first define N-generalized hybrid mappings in convex metric spaces. Let C be a nonempty subset of a convex metric space $\left(X,d,W\right)$. Let $N\in \mathbb{N}$. A mapping $T:C\to C$ is called N-generalized hybrid if there are ${\alpha }_{1},\dots ,{\alpha }_{N},{\beta }_{1},\dots ,{\beta }_{N}\in \mathbb{R}$ such that
$\begin{array}{c}\sum _{k=1}^{N}{\alpha }_{k}d{\left({T}^{N+1-k}x,Ty\right)}^{2}+\left(1-\sum _{k=1}^{N}{\alpha }_{k}\right)d{\left(x,Ty\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \sum _{k=1}^{N}{\beta }_{k}d{\left({T}^{N+1-k}x,y\right)}^{2}+\left(1-\sum _{k=1}^{N}{\beta }_{k}\right)d{\left(x,y\right)}^{2}\hfill \end{array}$

for all $x,y\in C$. Now, we prove a fixed point theorem for N-generalized hybrid mappings in complete uniformly convex metric spaces.

Theorem 2.1 Let C be a nonempty closed and convex subset of a complete uniformly convex metric space $\left(X,d,W\right)$ and let $T:C\to C$ be an N-generalized hybrid mapping with ${\sum }_{k=1}^{N}{\alpha }_{k}\in \left(-\mathrm{\infty },0\right]\cup \left[1,\mathrm{\infty }\right)$ and ${\sum }_{k=1}^{N}{\beta }_{k}\in \left[0,1\right]$. Then T has a fixed point if and only if there exists an $x\in C$ such that $\left\{{T}^{n}x\right\}$ is bounded.

Proof The necessity is obvious. Conversely, we assume that there exists an $x\in C$ such that $\left\{{T}^{n}x\right\}$ is bounded. We will show that $F\left(T\right)$ is nonempty. From Lemma 1.2, $A\left(C,\left\{{T}^{n}x\right\}\right)$ is a singleton set. Let $A\left(C,\left\{{T}^{n}x\right\}\right)=\left\{z\right\}$. Since T is N-generalized hybrid, there are ${\alpha }_{1},\dots ,{\alpha }_{N},{\beta }_{1},\dots ,{\beta }_{N}\in \mathbb{R}$ such that
$\begin{array}{c}\sum _{k=1}^{N}{\alpha }_{k}d{\left({T}^{n+N+1-k}x,Tz\right)}^{2}+\left(1-\sum _{k=1}^{N}{\alpha }_{k}\right)d{\left({T}^{n}x,Tz\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \sum _{k=1}^{N}{\beta }_{k}d{\left({T}^{n+N+1-k}x,z\right)}^{2}+\left(1-\sum _{k=1}^{N}{\beta }_{k}\right)d{\left({T}^{n}x,z\right)}^{2}.\hfill \end{array}$
(2.1)
If ${\sum }_{k=1}^{N}{\alpha }_{k}\in \left[1,\mathrm{\infty }\right)$ and ${\sum }_{k=1}^{N}{\beta }_{k}\in \left[0,1\right]$, then (2.1) becomes
$\begin{array}{rcl}\sum _{k=1}^{N}{\alpha }_{k}d{\left({T}^{n+N+1-k}x,Tz\right)}^{2}& \le & \sum _{k=1}^{N}{\beta }_{k}d{\left({T}^{n+N+1-k}x,z\right)}^{2}+\left(1-\sum _{k=1}^{N}{\beta }_{k}\right)d{\left({T}^{n}x,z\right)}^{2}\\ +\left(\sum _{k=1}^{N}{\alpha }_{k}-1\right)d{\left({T}^{n}x,Tz\right)}^{2}.\end{array}$
This implies that
$\underset{n\to \mathrm{\infty }}{lim sup}d{\left({T}^{n}x,Tz\right)}^{2}\le \underset{n\to \mathrm{\infty }}{lim sup}d{\left({T}^{n}x,z\right)}^{2}.$
If ${\sum }_{k=1}^{N}{\alpha }_{k}\in \left(-\mathrm{\infty },0\right]$ and ${\sum }_{k=1}^{N}{\beta }_{k}\in \left[0,1\right]$, then (2.1) becomes
$\begin{array}{rcl}\left(1-\sum _{k=1}^{N}{\alpha }_{k}\right)d{\left({T}^{n}x,Tz\right)}^{2}& \le & \sum _{k=1}^{N}{\beta }_{k}d{\left({T}^{n+N+1-k}x,z\right)}^{2}+\left(1-\sum _{k=1}^{N}{\beta }_{k}\right)d{\left({T}^{n}x,z\right)}^{2}\\ -\sum _{k=1}^{N}{\alpha }_{k}d{\left({T}^{n+N+1-k}x,Tz\right)}^{2}.\end{array}$
This implies again that
$\underset{n\to \mathrm{\infty }}{lim sup}d{\left({T}^{n}x,Tz\right)}^{2}\le \underset{n\to \mathrm{\infty }}{lim sup}d{\left({T}^{n}x,z\right)}^{2}.$
Therefore, we have
$r\left(Tz,\left\{{T}^{n}x\right\}\right)\le r\left(z,\left\{{T}^{n}x\right\}\right).$

Since $Tz\in C$ and $r\left(z,\left\{{T}^{n}x\right\}\right)=inf\left\{r\left(y,\left\{{T}^{n}x\right\}\right):y\in C\right\}$, it implies that $Tz=z$. Hence, $F\left(T\right)$ is nonempty. □

As a direct consequence of Theorem 2.1, we obtain a fixed point theorem for N-generalized hybrid mappings in uniformly convex metric spaces as follows.

Theorem 2.2 Let C be a nonempty bounded closed and convex subset of a complete uniformly convex metric space $\left(X,d,W\right)$ and let $T:C\to C$ be an N-generalized hybrid mapping with ${\sum }_{k=1}^{N}{\alpha }_{k}\in \left(-\mathrm{\infty },0\right]\cup \left[1,\mathrm{\infty }\right)$ and ${\sum }_{k=1}^{N}{\beta }_{k}\in \left[0,1\right]$. Then T has a fixed point.

We can show that if T is an N-generalized hybrid mapping and $x=Tx$, then for any $y\in C$, we get
$\sum _{k=1}^{N}{\alpha }_{k}d{\left(x,Ty\right)}^{2}+\left(1-\sum _{k=1}^{N}{\alpha }_{k}\right)d{\left(x,Ty\right)}^{2}\le \sum _{k=1}^{N}{\beta }_{k}d{\left(x,y\right)}^{2}+\left(1-\sum _{k=1}^{N}{\beta }_{k}\right)d{\left(x,y\right)}^{2}$

and hence $d\left(x,Ty\right)\le d\left(x,y\right)$. This means that an N-generalized hybrid mapping with a fixed point is quasi-nonexpansive. Then, using the methods of the proof of Theorem 1.3 in [13], we can prove the following.

Corollary 2.3 Let C be a nonempty convex subset of a complete uniformly convex metric space $\left(X,d,W\right)$. Suppose that $T:C\to C$ is an N-generalized hybrid mapping and has a fixed point. Then $F\left(T\right)$ is closed and convex.

Remark 2.4
1. (i)

Theorems 2.1 and 2.2 extend and generalize the corresponding results in [17, 3234, 3638] to N-generalized hybrid mappings on uniformly convex metric spaces.

2. (ii)

In $CAT\left(0\right)$ spaces, if we set $W\left(x,y,\lambda \right):=\lambda x\oplus \left(1-\lambda \right)y$, then Theorems 2.1 and 2.2 can be applied to these spaces under the assumption that ${\sum }_{k=1}^{N}{\alpha }_{k}\in \left(-\mathrm{\infty },0\right]\cup \left[1,\mathrm{\infty }\right)$ and ${\sum }_{k=1}^{N}{\beta }_{k}\in \left[0,1\right]$.

## 3 Fixed point theorems in $CAT\left(0\right)$ spaces

In this section, we study the existence and Δ-convergence theorems for N-generalized hybrid mappings in complete $CAT\left(0\right)$ spaces.

We first recall the definition of a Banach limit. Let μ be a continuous linear functional on ${l}^{\mathrm{\infty }}$, the Banach space of bounded real sequences, and $\left({a}_{1},{a}_{2},\dots \right)\in {l}^{\mathrm{\infty }}$. We write ${\mu }_{n}\left({a}_{n}\right)$ instead of $\mu \left(\left({a}_{1},{a}_{2},\dots \right)\right)$. We call μ a Banach limit if μ satisfies $\parallel \mu \parallel =\mu \left(1,1,\dots \right)=1$ and ${\mu }_{n}\left({a}_{n}\right)={\mu }_{n}\left({a}_{n+1}\right)$ for each $\left({a}_{1},{a}_{2},\dots \right)\in {l}^{\mathrm{\infty }}$. For a Banach limit μ, we know that ${lim inf}_{n\to \mathrm{\infty }}{a}_{n}\le {\mu }_{n}\left({a}_{n}\right)\le {lim sup}_{n\to \mathrm{\infty }}{a}_{n}$ for all $\left({a}_{1},{a}_{2},\dots \right)\in {l}^{\mathrm{\infty }}$. So if $\left({a}_{1},{a}_{2},\dots \right)\in {l}^{\mathrm{\infty }}$ with ${lim}_{n\to \mathrm{\infty }}{a}_{n}=c$, then ${\mu }_{n}\left({a}_{n}\right)=c$; see [39] for more details.

Now, we obtain the following lemma in $CAT\left(0\right)$ spaces.

Lemma 3.1 Let C be a nonempty closed and convex subset of a complete $CAT\left(0\right)$ space X, let $\left\{{x}_{n}\right\}$ be a bounded sequence in X, and let μ be a Banach limit. If a function $f:C\to \mathbb{R}$ is defined by
$f\left(z\right)={\mu }_{n}d{\left({x}_{n},z\right)}^{2}\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.25em}{0ex}}z\in C,$
then there exists a unique ${z}_{0}\in C$ such that
$f\left({z}_{0}\right)=min\left\{f\left(z\right):z\in C\right\}.$
Proof It is easy to show that f is continuous. By (CN) inequality, we obtain that

This implies by Proposition 1.7 in [40] that there exists a unique ${z}_{0}\in C$ such that $f\left({z}_{0}\right)=min\left\{f\left(z\right):z\in C\right\}$. □

By using Lemma 3.1, we can prove the following fixed point theorem for N-generalized hybrid mappings in $CAT\left(0\right)$ spaces without the assumptions that ${\sum }_{k=1}^{N}{\alpha }_{k}\in \left(-\mathrm{\infty },0\right]\cup \left[1,\mathrm{\infty }\right)$ and ${\sum }_{k=1}^{N}{\beta }_{k}\in \left[0,1\right]$.

Theorem 3.2 Let C be a nonempty closed and convex subset of a complete $CAT\left(0\right)$ space X and let $T:C\to C$ be an N-generalized hybrid mapping. Then T has a fixed point if and only if there exists an $x\in C$ such that $\left\{{T}^{n}x\right\}$ is bounded.

Proof The necessity is obvious. Conversely, we assume that there exists an $x\in C$ such that $\left\{{T}^{n}x\right\}$ is bounded. Let μ be a Banach limit. Since T is N-generalized hybrid, there are ${\alpha }_{1},\dots ,{\alpha }_{N},{\beta }_{1},\dots ,{\beta }_{N}\in \mathbb{R}$ such that
$\begin{array}{c}\sum _{k=1}^{N}{\alpha }_{k}d{\left({T}^{n+N+1-k}x,Tz\right)}^{2}+\left(1-\sum _{k=1}^{N}{\alpha }_{k}\right)d{\left({T}^{n}x,Tz\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \sum _{k=1}^{N}{\beta }_{k}d{\left({T}^{n+N+1-k}x,z\right)}^{2}+\left(1-\sum _{k=1}^{N}{\beta }_{k}\right)d{\left({T}^{n}x,z\right)}^{2}\hfill \end{array}$
for any $z\in C$ and $n\in \mathbb{N}\cup \left\{0\right\}$. Since $\left\{{T}^{n}x\right\}$ is bounded, we have
$\begin{array}{c}\sum _{k=1}^{N}{\alpha }_{k}{\mu }_{n}d{\left({T}^{n+N+1-k}x,Tz\right)}^{2}+\left(1-\sum _{k=1}^{N}{\alpha }_{k}\right){\mu }_{n}d{\left({T}^{n}x,Tz\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \sum _{k=1}^{N}{\beta }_{k}{\mu }_{n}d{\left({T}^{n+N+1-k}x,z\right)}^{2}+\left(1-\sum _{k=1}^{N}{\beta }_{k}\right){\mu }_{n}d{\left({T}^{n}x,z\right)}^{2}.\hfill \end{array}$
This implies that
${\mu }_{n}d{\left({T}^{n}x,Tz\right)}^{2}\le {\mu }_{n}d{\left({T}^{n}x,z\right)}^{2}$

for all $z\in C$. It follows by Lemma 3.1 that $Tz=z$. Hence, $F\left(T\right)$ is nonempty. □

As a direct consequence of Theorem 3.2, we obtain a fixed point theorem for N-generalized hybrid mappings in $CAT\left(0\right)$ spaces as follows.

Theorem 3.3 Let C be a nonempty bounded closed and convex subset of a complete $CAT\left(0\right)$ space X and let $T:C\to C$ be an N-generalized hybrid mapping. Then T has a fixed point.

Remark 3.4 Theorems 3.2 and 3.3 extend and generalize the corresponding results in [17, 3234, 3638] to N-generalized hybrid mappings on $CAT\left(0\right)$ spaces.

Next, we study the Δ-convergence theorem for N-generalized hybrid mappings in $CAT\left(0\right)$ spaces. Before proving the theorem, we need the following lemma.

Lemma 3.5 Let C be a nonempty closed and convex subset of a complete $CAT\left(0\right)$ space X and let $T:C\to C$ be an N-generalized hybrid mapping with ${\sum }_{k=1}^{N}{\alpha }_{k}\in \left(-\mathrm{\infty },0\right]\cup \left[1,\mathrm{\infty }\right)$ and ${\sum }_{k=1}^{N}{\beta }_{k}\in \left[0,\mathrm{\infty }\right)$. If $\left\{{x}_{n}\right\}$ is a bounded sequence in C with $\mathrm{\Delta }\text{-}{lim}_{n\to \mathrm{\infty }}{x}_{n}=x$ and ${lim}_{n\to \mathrm{\infty }}d\left({x}_{n},{T}^{i}{x}_{n}\right)=0$ for all $i=1,2,\dots ,N$, then $x\in F\left(T\right)$.

Proof Since T is an N-generalized hybrid mapping, there are ${\alpha }_{1},\dots ,{\alpha }_{N},{\beta }_{1},\dots ,{\beta }_{N}\in \mathbb{R}$ such that
$\begin{array}{c}\sum _{k=1}^{N}{\alpha }_{k}d{\left({T}^{N+1-k}{x}_{n},Tx\right)}^{2}+\left(1-\sum _{k=1}^{N}{\alpha }_{k}\right)d{\left({x}_{n},Tx\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \sum _{k=1}^{N}{\beta }_{k}d{\left({T}^{N+1-k}{x}_{n},x\right)}^{2}+\left(1-\sum _{k=1}^{N}{\beta }_{k}\right)d{\left({x}_{n},x\right)}^{2}.\hfill \end{array}$
(3.1)
Case 1: ${\sum }_{k=1}^{N}{\alpha }_{k}\in \left[1,\mathrm{\infty }\right)$ and ${\sum }_{k=1}^{N}{\beta }_{k}\in \left[0,\mathrm{\infty }\right)$. It follows by (3.1) that
$\begin{array}{c}\sum _{k=1}^{N}{\alpha }_{k}d{\left({T}^{N+1-k}{x}_{n},Tx\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \sum _{k=1}^{N}{\beta }_{k}d{\left({T}^{N+1-k}{x}_{n},x\right)}^{2}+\left(1-\sum _{k=1}^{N}{\beta }_{k}\right)d{\left({x}_{n},x\right)}^{2}+\left(\sum _{k=1}^{N}{\alpha }_{k}-1\right)d{\left({x}_{n},Tx\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \sum _{k=1}^{N}{\beta }_{k}\left(d{\left({T}^{N+1-k}{x}_{n},{x}_{n}\right)}^{2}+2d\left({T}^{N+1-k}{x}_{n},{x}_{n}\right)d\left({x}_{n},x\right)+d{\left({x}_{n},x\right)}^{2}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\left(1-\sum _{k=1}^{N}{\beta }_{k}\right)d{\left({x}_{n},x\right)}^{2}+\left(\sum _{k=1}^{N}{\alpha }_{k}-1\right)\left(d{\left({x}_{n},{T}^{N+1-k}{x}_{n}\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+2d\left({x}_{n},{T}^{N+1-k}{x}_{n}\right)d\left({T}^{N+1-k}{x}_{n},Tx\right)+d{\left({T}^{N+1-k}{x}_{n},Tx\right)}^{2}\right)\hfill \\ \phantom{\rule{1em}{0ex}}=d{\left({x}_{n},x\right)}^{2}+\left(\sum _{k=1}^{N}{\beta }_{k}+\sum _{k=1}^{N}{\alpha }_{k}-1\right)d{\left({T}^{N+1-k}{x}_{n},{x}_{n}\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+2\sum _{k=1}^{N}{\beta }_{k}d\left({T}^{N+1-k}{x}_{n},{x}_{n}\right)d\left({x}_{n},x\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+2\left(\sum _{k=1}^{N}{\alpha }_{k}-1\right)d\left({x}_{n},{T}^{N+1-k}{x}_{n}\right)d\left({T}^{N+1-k}{x}_{n},Tx\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\left(\sum _{k=1}^{N}{\alpha }_{k}-1\right)d{\left({T}^{N+1-k}{x}_{n},Tx\right)}^{2}.\hfill \end{array}$
This implies that
$\begin{array}{c}d{\left({T}^{N+1-k}{x}_{n},Tx\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le d{\left({x}_{n},x\right)}^{2}+\left(\sum _{k=1}^{N}{\beta }_{k}+\sum _{k=1}^{N}{\alpha }_{k}-1\right)d{\left({T}^{N+1-k}{x}_{n},{x}_{n}\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+2\sum _{k=1}^{N}{\beta }_{k}d\left({T}^{N+1-k}{x}_{n},{x}_{n}\right)d\left({x}_{n},x\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+2\left(\sum _{k=1}^{N}{\alpha }_{k}-1\right)d\left({x}_{n},{T}^{N+1-k}{x}_{n}\right)d\left({T}^{N+1-k}{x}_{n},Tx\right).\hfill \end{array}$
Since $\left\{{x}_{n}\right\}$ is bounded and ${lim}_{n\to \mathrm{\infty }}d\left({x}_{n},{T}^{i}{x}_{n}\right)=0$ for all $i=1,2,\dots ,N$, we have that $\left\{T{x}_{n}\right\},\left\{{T}^{2}{x}_{n}\right\},\dots ,\left\{{T}^{N}{x}_{n}\right\}$ are bounded. So, we have
$\begin{array}{c}d{\left({T}^{N+1-k}{x}_{n},Tx\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le d{\left({x}_{n},x\right)}^{2}+\left(\sum _{k=1}^{N}{\beta }_{k}+\sum _{k=1}^{N}{\alpha }_{k}-1\right)d{\left({T}^{N+1-k}{x}_{n},{x}_{n}\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+2\sum _{k=1}^{N}{\beta }_{k}d\left({T}^{N+1-k}{x}_{n},{x}_{n}\right)M+2\left(\sum _{k=1}^{N}{\alpha }_{k}-1\right)d\left({x}_{n},{T}^{N+1-k}{x}_{n}\right)M\hfill \\ \phantom{\rule{1em}{0ex}}=d{\left({x}_{n},x\right)}^{2}+\left(\sum _{k=1}^{N}{\beta }_{k}+\sum _{k=1}^{N}{\alpha }_{k}-1\right)d\left({T}^{N+1-k}{x}_{n},{x}_{n}\right)\left(d\left({T}^{N+1-k}{x}_{n},{x}_{n}\right)+2M\right),\hfill \end{array}$

where $M={max}_{1\le k\le N}sup\left\{d\left({x}_{n},x\right),d\left({T}^{N+1-k}{x}_{n},Tx\right):n\in \mathbb{N}\right\}$.

Case 2: ${\sum }_{k=1}^{N}{\alpha }_{k}\in \left(-\mathrm{\infty },0\right]$ and ${\sum }_{k=1}^{N}{\beta }_{k}\in \left[0,\mathrm{\infty }\right)$. In the same way as Case 1, we can show that
$\begin{array}{c}d{\left({T}^{N+1-k}{x}_{n},Tx\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le d{\left({x}_{n},x\right)}^{2}+\left(\sum _{k=1}^{N}{\beta }_{k}-\sum _{k=1}^{N}{\alpha }_{k}\right)d\left({T}^{N+1-k}{x}_{n},{x}_{n}\right)\left(d\left({T}^{N+1-k}{x}_{n},{x}_{n}\right)+2M\right).\hfill \end{array}$
By Case 1, Case 2, and the assumption ${lim}_{n\to \mathrm{\infty }}d\left({x}_{n},{T}^{i}{x}_{n}\right)=0$ for all $i=1,2,\dots ,N$, we obtain
$\underset{n\to \mathrm{\infty }}{lim sup}d\left({x}_{n},Tx\right)\le \underset{n\to \mathrm{\infty }}{lim sup}d\left({x}_{n},x\right).$

Since $\mathrm{\Delta }\text{-}{lim}_{n\to \mathrm{\infty }}{x}_{n}=x$, it follows by the uniqueness of asymptotic centers that $Tx=x$. Hence, $x\in F\left(T\right)$. □

Fixed point iteration methods are very useful for approximating a fixed point of various nonlinear mappings such as Mann iteration, Ishikawa iteration, Noor iteration and so on. We now introduce a new iteration method for approximating a fixed point of mappings in a $CAT\left(0\right)$ space X as follows: Let C be a nonempty closed and convex subset of X, let $T:C\to C$ be a mapping and $N\in \mathbb{N}$. For ${x}_{1}\in C$, the sequence $\left\{{x}_{n}\right\}$ generated by
(3.2)

where $\left\{{\lambda }_{n}^{\left(i\right)}\right\}$ is a sequence in $\left[0,1\right]$ for all $i=0,1,\dots ,N$ with ${\sum }_{i=0}^{N}{\lambda }_{n}^{\left(i\right)}=1$.

Remark 3.6 If we put
${W}_{n}^{\left(N\right)}=\underset{i=0}{\overset{N}{⨁}}\frac{{\lambda }_{n}^{\left(i\right)}}{{\sum }_{j=0}^{N}{\lambda }_{n}^{\left(j\right)}}{T}^{i}{x}_{n},$
then by (1.1) we get
${W}_{n}^{\left(N\right)}=\frac{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{N}{\lambda }_{n}^{\left(j\right)}}{W}_{n}^{\left(N-1\right)}\oplus \frac{{\lambda }_{n}^{\left(N\right)}}{{\sum }_{j=0}^{N}{\lambda }_{n}^{\left(j\right)}}{T}^{N}{x}_{n}.$
(3.3)
Indeed, we put ${\delta }_{n}^{\left(i,N\right)}=\frac{{\lambda }_{n}^{\left(i\right)}}{{\sum }_{j=0}^{N}{\lambda }_{n}^{\left(j\right)}}$ for $i=0,1,\dots ,N$. Thus
$\begin{array}{rcl}{W}_{n}^{\left(N\right)}& =& \underset{i=0}{\overset{N}{⨁}}\frac{{\lambda }_{n}^{\left(i\right)}}{{\sum }_{j=0}^{N}{\lambda }_{n}^{\left(j\right)}}{T}^{i}{x}_{n}=\underset{i=0}{\overset{N}{⨁}}{\delta }_{n}^{\left(i,N\right)}{T}^{i}{x}_{n}\\ =& \left(1-{\delta }_{n}^{\left(N,N\right)}\right)\left(\frac{{\delta }_{n}^{\left(0,N\right)}}{1-{\delta }_{n}^{\left(N,N\right)}}{x}_{n}\oplus \frac{{\delta }_{n}^{\left(1,N\right)}}{1-{\delta }_{n}^{\left(N,N\right)}}T{x}_{n}\oplus \cdots \oplus \frac{{\delta }_{n}^{\left(N-1,N\right)}}{1-{\delta }_{n}^{\left(N,N\right)}}{T}^{N-1}{x}_{n}\right)\\ \oplus {\delta }_{n}^{\left(N,N\right)}{T}^{N}{x}_{n}\\ =& \left(1-{\delta }_{n}^{\left(N,N\right)}\right)\left({\delta }_{n}^{\left(0,N-1\right)}{x}_{n}\oplus {\delta }_{n}^{\left(1,N-1\right)}T{x}_{n}\oplus \cdots \oplus {\delta }_{n}^{\left(N-1,N-1\right)}{T}^{N-1}{x}_{n}\right)\oplus {\delta }_{n}^{\left(N,N\right)}{T}^{N}{x}_{n}\\ =& \left(1-{\delta }_{n}^{\left(N,N\right)}\right)\left(\frac{{\lambda }_{n}^{\left(0\right)}}{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}{x}_{n}\oplus \frac{{\lambda }_{n}^{\left(1\right)}}{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}T{x}_{n}\oplus \cdots \oplus \frac{{\lambda }_{n}^{\left(N-1\right)}}{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}{T}^{N-1}{x}_{n}\right)\\ \oplus {\delta }_{n}^{\left(N,N\right)}{T}^{N}{x}_{n}\\ =& \left(1-{\delta }_{n}^{\left(N,N\right)}\right){W}_{n}^{\left(N-1\right)}\oplus {\delta }_{n}^{\left(N,N\right)}{T}^{N}{x}_{n}\\ =& \frac{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{N}{\lambda }_{n}^{\left(j\right)}}{W}_{n}^{\left(N-1\right)}\oplus \frac{{\lambda }_{n}^{\left(N\right)}}{{\sum }_{j=0}^{N}{\lambda }_{n}^{\left(j\right)}}{T}^{N}{x}_{n}.\end{array}$

Therefore, (3.3) is justified.

Using Lemma 3.5, we can prove the Δ-convergence theorem for N-generalized hybrid mappings in complete $CAT\left(0\right)$ spaces as follows.

Theorem 3.7 Let C be a nonempty closed and convex subset of a complete $CAT\left(0\right)$ space X and let $T:C\to C$ be an N-generalized hybrid mapping with $F\left(T\right)\ne \mathrm{\varnothing }$ and ${\sum }_{k=1}^{N}{\alpha }_{k}\in \left(-\mathrm{\infty },0\right]\cup \left[1,\mathrm{\infty }\right)$ and ${\sum }_{k=1}^{N}{\beta }_{k}\in \left[0,\mathrm{\infty }\right)$. Let $\pi :C\to F\left(T\right)$ be the nearest point projection mapping. Suppose that $\left\{{x}_{n}\right\}$ is a sequence in C defined by (3.2) with $0 for all $i=0,1,\dots ,N$. Then $\left\{{x}_{n}\right\}$ Δ-converges to a fixed point u of T, where $u={lim}_{n\to \mathrm{\infty }}\pi {x}_{n}$.

Proof Since T is N-generalized hybrid and $F\left(T\right)\ne \mathrm{\varnothing }$, we get T is quasi-nonexpansive. Then, for $p\in F\left(T\right)$, we have
$\begin{array}{rcl}d\left({x}_{n+1},p\right)& =& d\left(\underset{i=0}{\overset{N}{⨁}}{\lambda }_{n}^{\left(i\right)}{T}^{i}{x}_{n},p\right)\\ \le & \sum _{i=0}^{N}{\lambda }_{n}^{\left(i\right)}d\left({T}^{i}{x}_{n},p\right)\\ \le & \sum _{i=0}^{N}{\lambda }_{n}^{\left(i\right)}d\left({x}_{n},p\right)\\ =& d\left({x}_{n},p\right).\end{array}$

Therefore, ${lim}_{n\to \mathrm{\infty }}d\left({x}_{n},p\right)$ exists and hence $\left\{{x}_{n}\right\}$ is bounded.

For each $p\in F\left(T\right)$, we obtain, by (3.2), (3.3), and the (CN*) inequality, that
$\begin{array}{c}d{\left({x}_{n+1},p\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}=d{\left(\underset{i=0}{\overset{N}{⨁}}\frac{{\lambda }_{n}^{\left(i\right)}}{{\sum }_{j=0}^{N}{\lambda }_{n}^{\left(j\right)}}{T}^{i}{x}_{n},p\right)}^{2}=d{\left({W}_{n}^{\left(N\right)},p\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}=d{\left(\frac{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{N}{\lambda }_{n}^{\left(j\right)}}{W}_{n}^{\left(N-1\right)}\oplus \frac{{\lambda }_{n}^{\left(N\right)}}{{\sum }_{j=0}^{N}{\lambda }_{n}^{\left(j\right)}}{T}^{N}{x}_{n},p\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{N}{\lambda }_{n}^{\left(j\right)}}d{\left({W}_{n}^{\left(N-1\right)},p\right)}^{2}+\frac{{\lambda }_{n}^{\left(N\right)}}{{\sum }_{j=0}^{N}{\lambda }_{n}^{\left(j\right)}}d{\left({T}^{N}{x}_{n},p\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\frac{{\lambda }_{n}^{\left(N\right)}}{{\sum }_{j=0}^{N}{\lambda }_{n}^{\left(j\right)}}\frac{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{N}{\lambda }_{n}^{\left(j\right)}}d{\left({W}_{n}^{\left(N-1\right)},{T}^{N}{x}_{n}\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}d{\left({W}_{n}^{\left(N-1\right)},p\right)}^{2}+{\lambda }_{n}^{\left(N\right)}d{\left({T}^{N}{x}_{n},p\right)}^{2}-{\lambda }_{n}^{\left(N\right)}\sum _{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}d{\left({W}_{n}^{\left(N-1\right)},{T}^{N}{x}_{n}\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}d{\left(\frac{{\sum }_{j=0}^{N-2}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}{W}_{n}^{\left(N-2\right)}\oplus \frac{{\lambda }_{n}^{\left(N-1\right)}}{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}{T}^{N-1}{x}_{n},p\right)}^{2}+{\lambda }_{n}^{\left(N\right)}d{\left({T}^{N}{x}_{n},p\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-{\lambda }_{n}^{\left(N\right)}\sum _{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}d{\left({W}_{n}^{\left(N-1\right)},{T}^{N}{x}_{n}\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \sum _{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}\left(\frac{{\sum }_{j=0}^{N-2}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}d{\left({W}_{n}^{\left(N-2\right)},p\right)}^{2}+\frac{{\lambda }_{n}^{\left(N-1\right)}}{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}d{\left({T}^{N-1}{x}_{n},p\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\frac{{\sum }_{j=0}^{N-2}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}\frac{{\lambda }_{n}^{\left(N-1\right)}}{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}d{\left({W}_{n}^{\left(N-2\right)},{T}^{N-1}{x}_{n}\right)}^{2}\right)+{\lambda }_{n}^{\left(N\right)}d{\left({T}^{N}{x}_{n},p\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-{\lambda }_{n}^{\left(N\right)}\sum _{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}d{\left({W}_{n}^{\left(N-1\right)},{T}^{N}{x}_{n}\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{j=0}^{N-2}{\lambda }_{n}^{\left(j\right)}d{\left({W}_{n}^{\left(N-2\right)},p\right)}^{2}+{\lambda }_{n}^{\left(N-1\right)}d{\left({T}^{N-1}{x}_{n},p\right)}^{2}+{\lambda }_{n}^{\left(N\right)}d{\left({T}^{N}{x}_{n},p\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\frac{{\lambda }_{n}^{\left(N-1\right)}{\sum }_{j=0}^{N-2}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}d{\left({W}_{n}^{\left(N-2\right)},{T}^{N-1}{x}_{n}\right)}^{2}-{\lambda }_{n}^{\left(N\right)}\sum _{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}d{\left({W}_{n}^{\left(N-1\right)},{T}^{N}{x}_{n}\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \sum _{j=0}^{N-3}{\lambda }_{n}^{\left(j\right)}d{\left({W}_{n}^{\left(N-3\right)},p\right)}^{2}+{\lambda }_{n}^{\left(N-2\right)}d{\left({T}^{N-2}{x}_{n},p\right)}^{2}+{\lambda }_{n}^{\left(N-1\right)}d{\left({T}^{N-1}{x}_{n},p\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\lambda }_{n}^{\left(N\right)}d{\left({T}^{N}{x}_{n},p\right)}^{2}-\frac{{\lambda }_{n}^{\left(N-2\right)}{\sum }_{j=0}^{N-3}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{N-2}{\lambda }_{n}^{\left(j\right)}}d{\left({W}_{n}^{\left(N-3\right)},{T}^{N-2}{x}_{n}\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\frac{{\lambda }_{n}^{\left(N-1\right)}{\sum }_{j=0}^{N-2}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}}d{\left({W}_{n}^{\left(N-2\right)},{T}^{N-1}{x}_{n}\right)}^{2}-{\lambda }_{n}^{\left(N\right)}\sum _{j=0}^{N-1}{\lambda }_{n}^{\left(j\right)}d{\left({W}_{n}^{\left(N-1\right)},{T}^{N}{x}_{n}\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}⋮\hfill \\ \phantom{\rule{1em}{0ex}}\le {\lambda }_{n}^{\left(0\right)}d{\left({W}_{n}^{\left(0\right)},p\right)}^{2}+\sum _{k=1}^{N}{\lambda }_{n}^{\left(k\right)}d{\left({T}^{k}{x}_{n},p\right)}^{2}-\sum _{k=1}^{N}\frac{{\lambda }_{n}^{\left(k\right)}{\sum }_{j=0}^{k-1}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{k}{\lambda }_{n}^{\left(j\right)}}d{\left({W}_{n}^{\left(k-1\right)},{T}^{k}{x}_{n}\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le \sum _{k=0}^{N}{\lambda }_{n}^{\left(k\right)}d{\left({x}_{n},p\right)}^{2}-\sum _{k=1}^{N}\frac{{\lambda }_{n}^{\left(k\right)}{\sum }_{j=0}^{k-1}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{k}{\lambda }_{n}^{\left(j\right)}}d{\left({W}_{n}^{\left(k-1\right)},{T}^{k}{x}_{n}\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}=d{\left({x}_{n},p\right)}^{2}-\sum _{k=1}^{N}\frac{{\lambda }_{n}^{\left(k\right)}{\sum }_{j=0}^{k-1}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{k}{\lambda }_{n}^{\left(j\right)}}d{\left({W}_{n}^{\left(k-1\right)},{T}^{k}{x}_{n}\right)}^{2}.\hfill \end{array}$
This implies that
$\sum _{k=1}^{N}\frac{{\lambda }_{n}^{\left(k\right)}{\sum }_{j=0}^{k-1}{\lambda }_{n}^{\left(j\right)}}{{\sum }_{j=0}^{k}{\lambda }_{n}^{\left(j\right)}}d{\left({W}_{n}^{\left(k-1\right)},{T}^{k}{x}_{n}\right)}^{2}\le d{\left({x}_{n},p\right)}^{2}-d{\left({x}_{n+1},p\right)}^{2}.$
Since ${lim}_{n\to \mathrm{\infty }}d\left({x}_{n},p\right)$ exists and $0 for all $i=0,1,\dots ,N$, we get that
(3.4)
For $k=2,3,\dots ,N$, we have
$\begin{array}{rcl}d\left({x}_{n},{T}^{k}{x}_{n}\right)& \le & d\left({x}_{n},{W}_{n}^{\left(k-1\right)}\right)+d\left({W}_{n}^{\left(k-1\right)},{T}^{k}{x}_{n}\right)\\ =& d\left({x}_{n},\underset{i=0}{\overset{k-1}{⨁}}\frac{{\lambda }_{n}^{\left(i\right)}}{{\sum }_{j=0}^{k-1}{\lambda }_{n}^{\left(j\right)}}{T}^{i}{x}_{n}\right)+d\left({W}_{n}^{\left(k-1\right)},{T}^{k}{x}_{n}\right)\\ \le & \sum _{i=0}^{k-1}\frac{{\lambda }_{n}^{\left(i\right)}}{{\sum }_{j=0}^{k-1}{\lambda }_{n}^{\left(j\right)}}d\left({x}_{n},{T}^{i}{x}_{n}\right)+d\left({W}_{n}^{\left(k-1\right)},{T}^{k}{x}_{n}\right)\\ =& \sum _{i=1}^{k-1}\frac{{\lambda }_{n}^{\left(i\right)}}{{\sum }_{j=0}^{k-1}{\lambda }_{n}^{\left(j\right)}}d\left({x}_{n},{T}^{i}{x}_{n}\right)+d\left({W}_{n}^{\left(k-1\right)},{T}^{k}{x}_{n}\right).\end{array}$
This implies by (3.4) that
(3.5)
We now let ${\omega }_{\mathrm{\Delta }}\left({x}_{n}\right):=\bigcup A\left(C,\left\{{u}_{n}\right\}\right)$, where the union is taken over all subsequences $\left\{{u}_{n}\right\}$ of $\left\{{x}_{n}\right\}$. We claim that ${\omega }_{\mathrm{\Delta }}\left({x}_{n}\right)\subset F\left(T\right)$. Let $u\in {\omega }_{\mathrm{\Delta }}\left({x}_{n}\right)$. Then there exists a subsequence $\left\{{u}_{n}\right\}$ of $\left\{{x}_{n}\right\}$ such that $A\left(C,\left\{{u}_{n}\right\}\right)=\left\{u\right\}$. By Lemma 1.5, there exists a subsequence $\left\{{u}_{{n}_{k}}\right\}$ of $\left\{{u}_{n}\right\}$ such that $\mathrm{\Delta }\text{-}{lim}_{k\to \mathrm{\infty }}{u}_{{n}_{k}}=y\in C$. It implies by (3.5) and Lemma 3.5 that $y\in F\left(T\right)$. Then, ${lim}_{n\to \mathrm{\infty }}d\left({x}_{n},y\right)$ exists. Suppose that $u\ne y$. By the uniqueness of asymptotic centers, we get
$\begin{array}{rcl}\underset{k\to \mathrm{\infty }}{lim sup}d\left({u}_{{n}_{k}},y\right)& <& \underset{k\to \mathrm{\infty }}{lim sup}d\left({u}_{{n}_{k}},u\right)\\ \le & \underset{n\to \mathrm{\infty }}{lim sup}d\left({u}_{n},u\right)\\ <& \underset{n\to \mathrm{\infty }}{lim sup}d\left({u}_{n},y\right)\\ =& \underset{n\to \mathrm{\infty }}{lim sup}d\left({x}_{n},y\right)\\ =& \underset{k\to \mathrm{\infty }}{lim sup}d\left({u}_{{n}_{k}},y\right).\end{array}$

This is a contradiction, hence $u=y\in F\left(T\right)$. This shows that ${\omega }_{\mathrm{\Delta }}\left({x}_{n}\right)\subset F\left(T\right)$.

Next, we show that ${\omega }_{\mathrm{\Delta }}\left({x}_{n}\right)$ consists of exactly one point. Let $\left\{{u}_{n}\right\}$ be a subsequence of $\left\{{x}_{n}\right\}$ with $A\left(C,\left\{{u}_{n}\right\}\right)=\left\{u\right\}$ and let $A\left(C,\left\{{x}_{n}\right\}\right)=\left\{z\right\}$. Since $u\in {\omega }_{\mathrm{\Delta }}\left({x}_{n}\right)\subset F\left(T\right)$, it follows that ${lim}_{n\to \mathrm{\infty }}d\left({x}_{n},u\right)$ exists. We will show that $z=u$. To show this, suppose not. By the uniqueness of asymptotic centers, we get
$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim sup}d\left({u}_{n},u\right)& <& \underset{n\to \mathrm{\infty }}{lim sup}d\left({u}_{n},z\right)\\ \le & \underset{n\to \mathrm{\infty }}{lim sup}d\left({x}_{n},z\right)\\ <& \underset{n\to \mathrm{\infty }}{lim sup}d\left({x}_{n},u\right)\\ =& \underset{n\to \mathrm{\infty }}{lim sup}d\left({u}_{n},u\right),\end{array}$
which is a contradiction, and so $z=u$. Hence, $\left\{{x}_{n}\right\}$ Δ-converges to a fixed point u of T. Since $F\left(T\right)$ is a closed convex subset of X and $d\left({x}_{n+1},p\right)\le d\left({x}_{n},p\right)$ for all $p\in F\left(T\right)$ and $n\in \mathbb{N}$, we obtain by Lemma 1.6 that $\left\{\pi {x}_{n}\right\}$ converges strongly to some element in $F\left(T\right)$, say q. Thus, by the property of π, we obtain that
$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim sup}d\left({x}_{n},q\right)& \le & \underset{n\to \mathrm{\infty }}{lim sup}\left(d\left({x}_{n},\pi {x}_{n}\right)+d\left(\pi {x}_{n},q\right)\right)\\ =& \underset{n\to \mathrm{\infty }}{lim sup}d\left({x}_{n},\pi {x}_{n}\right)\\ \le & \underset{n\to \mathrm{\infty }}{lim sup}d\left({x}_{n},u\right).\end{array}$

This implies, by the uniqueness of asymptotic centers, that $q=u$. This means $u={lim}_{n\to \mathrm{\infty }}\pi {x}_{n}$. □

Taking $N=2$ in Theorem 3.7, we obtain the following Δ-convergence theorem of a 2-generalized hybrid mapping in $CAT\left(0\right)$ spaces.

Theorem 3.8 Let C be a nonempty closed and convex subset of a complete $CAT\left(0\right)$ space X. Let $T:C\to C$ be a 2-generalized hybrid mapping, i.e., there are ${\alpha }_{1},{\alpha }_{2},{\beta }_{1},{\beta }_{2}\in \mathbb{R}$ such that
$\begin{array}{c}{\alpha }_{1}d{\left({T}^{2}x,Ty\right)}^{2}+{\alpha }_{2}d{\left(Tx,Ty\right)}^{2}+\left(1-{\alpha }_{1}-{\alpha }_{2}\right)d{\left(x,Ty\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le {\beta }_{1}d{\left({T}^{2}x,y\right)}^{2}+{\beta }_{2}d{\left(Tx,y\right)}^{2}+\left(1-{\beta }_{1}-{\beta }_{2}\right)d{\left(x,y\right)}^{2}\hfill \end{array}$
for all $x,y\in C$. Assume that $F\left(T\right)\ne \mathrm{\varnothing }$ and ${\alpha }_{1}+{\alpha }_{2}\in \left(-\mathrm{\infty },0\right]\cup \left[1,\mathrm{\infty }\right)$ and ${\beta }_{1}+{\beta }_{2}\in \left[0,\mathrm{\infty }\right)$. Let $\pi :C\to F\left(T\right)$ be the nearest point projection mapping. For ${x}_{1}\in C$, let $\left\{{x}_{n}\right\}$ be a sequence defined by
${x}_{n+1}=\underset{i=0}{\overset{2}{⨁}}{\lambda }_{n}^{\left(i\right)}{T}^{i}{x}_{n}\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.25em}{0ex}}n\in \mathbb{N},$

where $\left\{{\lambda }_{n}^{\left(i\right)}\right\}$ is a sequence in $\left[0,1\right]$ with $0 for all $i=0,1,2$ and ${\sum }_{i=0}^{2}{\lambda }_{n}^{\left(i\right)}=1$. Then $\left\{{x}_{n}\right\}$ Δ-converges to a fixed point u of T, where $u={lim}_{n\to \mathrm{\infty }}\pi {x}_{n}$.

Taking $N=1$ in Theorem 3.7, we obtain the following Δ-convergence theorem of a generalized hybrid mapping in $CAT\left(0\right)$ spaces.

Theorem 3.9 Let C be a nonempty closed and convex subset of a complete $CAT\left(0\right)$ space X. Let $T:C\to C$ be a generalized hybrid mapping, i.e., there are $\alpha ,\beta \in \mathbb{R}$ such that
$\alpha d{\left(Tx,Ty\right)}^{2}+\left(1-\alpha \right)d{\left(x,Ty\right)}^{2}\le \beta d{\left(Tx,y\right)}^{2}+\left(1-\beta \right)d{\left(x,y\right)}^{2}$
for all $x,y\in C$. Assume that $F\left(T\right)\ne \mathrm{\varnothing }$ and $\alpha \in \left(-\mathrm{\infty },0\right]\cup \left[1,\mathrm{\infty }\right)$ and $\beta \in \left[0,\mathrm{\infty }\right)$. Let $\pi :C\to F\left(T\right)$ be the nearest point projection mapping. For ${x}_{1}\in C$, let $\left\{{x}_{n}\right\}$ be a sequence defined by
${x}_{n+1}={\lambda }_{n}^{\left(0\right)}{x}_{n}\oplus {\lambda }_{n}^{\left(1\right)}T{x}_{n}\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.25em}{0ex}}n\in \mathbb{N},$

where $\left\{{\lambda }_{n}^{\left(0\right)}\right\}$ and $\left\{{\lambda }_{n}^{\left(1\right)}\right\}$ are sequences in $\left[0,1\right]$ with $0 and ${\lambda }_{n}^{\left(0\right)}+{\lambda }_{n}^{\left(1\right)}=1$. Then $\left\{{x}_{n}\right\}$ Δ-converges to a fixed point u of T, where $u={lim}_{n\to \mathrm{\infty }}\pi {x}_{n}$.

## Declarations

### Acknowledgements

The author would like to thank the referees for valuable suggestions on the paper.

## Authors’ Affiliations

(1)
Mathematics Program, Faculty of Science and Technology, Nakhon Pathom Rajabhat University, Nakhon Pathom, 73000, Thailand

## References

1. Takahashi W: A convexity in metric space and nonexpansive mappings. Kodai Math. Semin. Rep. 1970, 22: 142–149. 10.2996/kmj/1138846111
2. Aoyama K, Eshita K, Takahashi W: Iteration processes for nonexpansive mappings in convex metric spaces. Proceedings of the International Conference on Nonlinear and Convex Analysis 2005, 31–39.Google Scholar
3. Shimizu T, Takahashi W: Fixed points of multivalued mappings in certain convex metric spaces. Topol. Methods Nonlinear Anal. 1996, 8: 197–203.
4. Goebel K, Kirk WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.
5. Goebel K, Reich S Monographs and Textbooks in Pure and Applied Mathematics 83. In Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York; 1984.Google Scholar
6. Dhompongsa S, Kirk WA, Sims B: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal. 2006, 65: 762–772. 10.1016/j.na.2005.09.044
7. Phuengrattana W, Suantai S: Existence theorems for generalized asymptotically nonexpansive mappings in uniformly convex metric spaces. J. Convex Anal. 2013, 20: 753–761.
8. Bridson M, Haefliger A: Metric Spaces of Non-Positive Curvature. Springer, Berlin; 1999.
9. Kirk WA: Geodesic geometry and fixed point theory II. In International Conference on Fixed Point Theory and Applications. Yokohama Publishers, Yokohama; 2004:113–142.Google Scholar
10. Kirk WA: Geodesic geometry and fixed point theory. Colecc. Abierta 64. In Seminar of Mathematical Analysis. Univ. Sevilla Secr. Publ., Seville; Malaga/Seville 2002/2003 2003:195–225.Google Scholar
11. Dhompongsa S, Kaewkhao A, Panyanak B:Lim’s theorems for multivalued mappings in$CAT\left(0\right)$ spaces. J. Math. Anal. Appl. 2005, 312: 478–487. 10.1016/j.jmaa.2005.03.055
12. Dhompongsa S, Kaewkhao A, Panyanak B:On Kirk’s strong convergence theorem for multivalued nonexpansive mappings on $CAT\left(0\right)$ spaces. Nonlinear Anal. 2012, 75: 459–468. 10.1016/j.na.2011.08.046
13. Chaoha P, Phon-on A:A note on fixed point sets in $CAT\left(0\right)$ spaces. J. Math. Anal. Appl. 2006, 320: 983–987. 10.1016/j.jmaa.2005.08.006
14. Leuştean L:A quadratic rate of asymptotic regularity for $CAT\left(0\right)$-spaces. J. Math. Anal. Appl. 2007, 325: 386–399. 10.1016/j.jmaa.2006.01.081
15. Kirk WA, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal. 2008, 68: 3689–3696. 10.1016/j.na.2007.04.011
16. Kirk WA:Fixed point theorems in $CAT\left(0\right)$ spaces and -trees. Fixed Point Theory Appl. 2004, 2004: 309–316.
17. Lin LJ, Chuang CS, Yu ZT:Fixed point theorems and Δ-convergence theorems for generalized hybrid mappings on $CAT\left(0\right)$ spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 49Google Scholar
18. Nanjaras B, Panyanak B, Phuengrattana W:Fixed point theorems and convergence theorems for Suzuki-generalized nonexpansive mappings in $CAT\left(0\right)$ spaces. Nonlinear Anal. Hybrid Syst. 2010, 4: 25–31. 10.1016/j.nahs.2009.07.003
19. Phuengrattana W: Approximating fixed points of Suzuki-generalized nonexpansive mappings. Nonlinear Anal. Hybrid Syst. 2011, 5: 583–590. 10.1016/j.nahs.2010.12.006
20. Phuengrattana W, Suantai S: Strong convergence theorems for a countable family of nonexpansive mappings in convex metric spaces. Abstr. Appl. Anal. 2011., 2011: Article ID 929037Google Scholar
21. Cho YJ, Ćirić L, Wang SH:Convergence theorems for nonexpansive semigroups in $CAT\left(0\right)$ spaces. Nonlinear Anal. 2011, 74: 6050–6059. 10.1016/j.na.2011.05.082
22. Kirk WA:Fixed point theorems in $CAT\left(0\right)$ spaces and -trees. Fixed Point Theory Appl. 2004, 2004: 309–316.
23. Saejung S:Halpern’s iteration in $CAT\left(0\right)$ spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 471781Google Scholar
24. Fukhar-ud-din H, Khan AR, Akhtar Z: Fixed point results for generalized nonexpansive maps in uniformly convex metric spaces. Nonlinear Anal. 2012, 75: 4747–4760. 10.1016/j.na.2012.03.025
25. Khan AR, Khamsi MA, Fukhar-ud-din H:Strong convergence of a general iteration scheme in $CAT\left(0\right)$ spaces. Nonlinear Anal. 2011, 74: 783–791. 10.1016/j.na.2010.09.029
26. Khan AR, Fukhar-ud-din H, Domlo AA: Approximating fixed points of some maps in uniformly convex metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 385986Google Scholar
27. Fukhar-ud-din H, Domlo AA, Khan AR:Strong convergence of an implicit algorithm in $CAT\left(0\right)$ spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 173621Google Scholar
28. Bruhat F, Tits J: Groupes réductifs sur un corps local. Publ. Math. Inst. Hautes Études Sci. 1972, 41: 5–251. 10.1007/BF02715544
29. Dhompongsa S, Panyanak B:On Δ-convergence theorems in $CAT\left(0\right)$ spaces. Comput. Math. Appl. 2008, 56: 2572–2579. 10.1016/j.camwa.2008.05.036
30. Lim TC: Remark on some fixed point theorems. Proc. Am. Math. Soc. 1976, 60: 179–182. 10.1090/S0002-9939-1976-0423139-X
31. Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2003, 118: 417–428. 10.1023/A:1025407607560
32. Kocourek P, Takahashi W, Yao JC: Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces. Taiwan. J. Math. 2010, 14: 2497–2511.
33. Takahashi W, Yao JC: Weak convergence theorems for generalized hybrid mappings in Banach spaces. J. Nonlinear Anal. Optim. 2011, 2(1):147–158.
34. Maruyama T, Takahashi W, Yao M: Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces. J. Nonlinear Convex Anal. 2011, 12: 185–197.
35. Hojo M, Takahashi W, Termwuttipong I: Strong convergence theorems for 2-generalized hybrid mappings in Hilbert spaces. Nonlinear Anal. 2012, 75: 2166–2176. 10.1016/j.na.2011.10.017
36. Aoyama K, Iemoto S, Kohsaka F, Takahashi W: Fixed point and ergodic theorems for λ -hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal. 2010, 11: 335–343.
37. Kohsaka F, Takahashi W: Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch. Math. 2008, 91: 166–177. 10.1007/s00013-008-2545-8
38. Takahashi W: Fixed point theorems for new nonlinear mappings in a Hilbert space. J. Nonlinear Convex Anal. 2010, 11: 79–88.
39. Takahashi W: Nonlinear Function Analysis. Yokahama Publishers, Yokahama; 2000.Google Scholar
40. Sturm KT: Probability measures on metric spaces of nonpositive curvature. Contemp. Math. 338. In Heat Kernels and Analysis on Manifolds, Graphs and Metric Spaces. Am. Math. Soc., Providence; 2003:357–390. (Paris, 2002)