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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

• Received: 22 March 2013
• 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  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  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 . The following lemma obtained by Phuengrattana and Suantai  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 . It was noted in  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 ).

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 , -trees , the complex Hilbert ball with a hyperbolic metric , 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)
for any $\lambda \in \left[0,1\right]$. The (CN*) inequality has appeared in . 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 .

In 2012, Dhompongsa et al.  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  introduced the concept of Δ-convergence in a general metric space. Later in 2008, Kirk and Panyanak  extended the concept of Lim to a $CAT\left(0\right)$ space.

Definition 1.4 

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 

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 , 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 [, 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.  proved the fixed point theorems for generalized hybrid mappings in Hilbert spaces. Later in 2011, Takahashi and Yao  extended the results of Kocourek et al. to uniformly convex Banach spaces.

Recently, Maruyama et al.  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.  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 , 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  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  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 