On the strong and Δ-convergence of new multi-step and S-iteration processes in a $CAT(0)$ space

Abstract

In this paper, we introduce a new class of mappings and prove the demiclosedness principle for mappings of this type in a $CAT(0)$ space. Also, we obtain the strong and Δ-convergence theorems of new multi-step and S-iteration processes in a $CAT(0)$ space. Our results extend and improve the corresponding recent results announced by many authors in the literature.

MSC:47H09, 47H10, 54E40, 58C30.

1 Introduction

Contractive mappings and iteration processes are some of the main tools in the study of fixed point theory. There are many contractive mappings and iteration processes that have been introduced and developed by several authors to serve various purposes in the literature (see [1–6]).

Imoru and Olantiwo [7] gave the following contractive definition.

Definition 1 Let T be a self-mapping on a metric space X. The mapping T is called a contractive-like mapping if there exist a constant $\delta \in [0,1)$ and a strictly increasing and continuous function $\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\phi (0)=0$ such that, for all $x,y\in X$,

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

(1.1)

This mapping is more general than those considered by Berinde [8, 9], Harder and Hicks [10], Zamfirescu [11], Osilike and Udomene [12].

By taking $\delta =1$ in (1.1), we define a new class of mappings as follows.

Definition 2 The mapping T is called a generalized nonexpansive mapping if there exists a non-decreasing and continuous function $\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\phi (0)=0$ such that, for all $x,y\in X$,

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

(1.2)

Remark 1 For $x\in F(T)$ in (1.2), we have

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

If X is an interval of ℝ, then $F(T)$ is convex. The same is also true in each space with unique geodesic for each pair of points (e.g., metric trees or $CAT(0)$ spaces).

In the case $\phi (t)=0$ for all $t\in [0,\mathrm{\infty })$, it is easy to show that every nonexpansive mapping satisfies (1.2), but the inverse is not necessarily true.

Example 1 Let $X=[0,2]$, $d(x,y)=|x-y|$, $\phi (t)=t$ and define T by

$$T(x)=\{\begin{array}{cc}0\hfill & \text{if }x\ne 2,\hfill \\ 1\hfill & \text{if }x=2.\hfill \end{array}$$

By taking $x=2$ and $y=1.5$, we have

$$d(T(2),T(1.5))=1<1.5=d(2,1.5)+\phi \left(d(2,T(2))\right)$$

but

$$d(T(2),T(1.5))=1\nleqq 0.5=d(2,1.5).$$

Therefore T is a generalized nonexpansive mapping, but T is not nonexpansive mapping.

Both a contractive-like mapping and a generalized nonexpansive mapping need not have a fixed point, even if X is complete. For example, let $X=[0,\mathrm{\infty })$, $d(x,y)=|x-y|$ and define T by

$$Tx=\{\begin{array}{cc}1\hfill & \text{if }0\le x\le 0.8,\hfill \\ 0.6\hfill & \text{if }0.8<x<+\mathrm{\infty }.\hfill \end{array}$$

It is proved in Gürsoy et al. [13] that T is a contractive-like mapping. Similarly, one can prove that T is a generalized nonexpansive mapping. But the mapping T has no fixed point.

By using (1.1), it is obvious that if a contractive-like mapping has a fixed point, then it is unique. However, if a generalized nonexpansive mapping has a fixed point, then it need not be unique. For example, let ℝ be the real line with the usual norm $|\cdot |$, and let $K=[-1,1]$. Define a mapping $T:K\to K$ by

$$Tx=\{\begin{array}{cc}x\hfill & \text{if }x\in [0,1],\hfill \\ -x\hfill & \text{if }x\in [-1,0).\hfill \end{array}$$

Now, we show that T is a nonexpansive mapping. In fact, if $x,y\in [0,1]$ or $x,y\in [-1,0)$, then we have

$$|Tx-Ty|=|x-y|.$$

If $x\in [0,1]$ and $y\in [-1,0)$ or $x\in [-1,0)$ and $y\in [0,1]$, then we have

$$|Tx-Ty|=|x+y|\le |x-y|.$$

This implies that T is a nonexpansive mapping and so T is a generalized nonexpansive mapping with $\phi (t)=0$ for all $t\in [0,\mathrm{\infty })$. But $F(T)=\{x\in K;0\le x\le 1\}$.

Agarwal et al. [1] introduced the S-iteration process which is independent of those of Mann [3] and Ishikawa [2] and converges faster than both of these. We apply this iteration process in a $CAT(0)$ space as

$$\{\begin{array}{c}{x}_0\in K,\hfill \\ x_{n+1}=(1-{\alpha }_n)T{x}_n\oplus {\alpha }_{n}T{y}_n,\hfill \\ y_n=(1-{\beta }_n)x_n\oplus {\beta }_{n}T{x}_n,n\ge 0.\hfill \end{array}$$

(1.3)

Gürsoy et al. [13] introduced a new multi-step iteration process in a Banach space. We modify this iteration process in a $CAT(0)$ space as follows.

For an arbitrary fixed order $k\ge 2$,

$$\{\begin{array}{c}{x}_0\in K,\hfill \\ x_{n+1}=(1-{\alpha }_n)y_n^1\oplus {\alpha }_{n}T{y}_n^1,\hfill \\ y_n^1=(1-{\beta }_n^1)y_n^2\oplus {\beta }_n^{1}T{y}_n^2,\hfill \\ y_n^2=(1-{\beta }_n^2)y_n^3\oplus {\beta }_n^{2}T{y}_n^3,\hfill \\ \cdots \hfill \\ y_n^{k-2}=(1-{\beta }_n^{k-2})y_n^{k-1}\oplus {\beta }_n^{k-2}T{y}_n^{k-1},\hfill \\ y_n^{k-1}=(1-{\beta }_n^{k-1})x_n\oplus {\beta }_n^{k-1}T{x}_n,n\ge 0,\hfill \end{array}$$

or, in short,

$$\{\begin{array}{c}{x}_0\in K,\hfill \\ x_{n+1}=(1-{\alpha }_n)y_n^1\oplus {\alpha }_{n}T{y}_n^1,\hfill \\ y_n^i=(1-{\beta }_n^i)y_n^{i+1}\oplus {\beta }_n^{i}T{y}_n^{i+1},i=1,2,\dots ,k-2,\hfill \\ y_n^{k-1}=(1-{\beta }_n^{k-1})x_n\oplus {\beta }_n^{k-1}T{x}_n,n\ge 0.\hfill \end{array}$$

(1.4)

By taking $k=3$ and $k=2$ in (1.4), we obtain the SP-iteration process of Phuengrattana and Suantai [4] and the two-step iteration process of Thianwan [6], respectively.

In this paper, motivated by the above results, we prove demiclosedness principle for a new class of mappings and the Δ-convergence theorems of the new multi-step iteration and the S-iteration processes for mappings of this type in a $CAT(0)$ space. Also, we present the strong convergence theorems of these iteration processes for contractive-like mappings in a $CAT(0)$ space.

2 Preliminaries on a $CAT(0)$ space

A metric space X is a $CAT(0)$ space if it is geodesically connected and if every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane. Fixed point theory in a $CAT(0)$ space was first studied by Kirk (see [14, 15]). He showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete $CAT(0)$ space always has a fixed point. Since then the fixed point theory in a $CAT(0)$ space has been rapidly developed and many papers have appeared (see [14–20]). It is worth mentioning that the results in a $CAT(0)$ space can be applied to any $CAT(k)$ space with $k\le 0$ since any $CAT(k)$ space is a $CAT(k^{\prime })$ space for every $k^{\prime }\ge k$ (see [[21], p.165]).

Let $(X,d)$ 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 $[0,l]\subset R$ to X such that $c(0)=x$, $c(l)=y$ and $d(c(t),c(t^{\prime }))=|t-t^{\prime }|$ for all $t,t^{\prime }\in [0,l]$. In particular, c is an isometry and $d(x,y)=l$. The image of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denoted by $[x,y]$. The space $(X,d)$ is said to be a geodesic space if every two points of X are joined by a geodesic and X is said to be a uniquely geodesic if there is exactly one geodesic joining x to y for each $x,y\in X$.

A geodesic triangle $△(x_1,x_2,x_3)$ in a geodesic metric space $(X,d)$ consists of three points in X (the vertices of △) and a geodesic segment between each pair of vertices (the edges of △). A comparison triangle for the geodesic triangle $△(x_1,x_2,x_3)$ in $(X,d)$ is a triangle $\overline{△}(x_1,x_2,x_3)=△({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3)$ in the Euclidean plane ${\mathbb{R}}^2$ such that

$$d_{{\mathbb{R}}^2}({\overline{x}}_i,{\overline{x}}_j)=d(x_i,x_j)$$

for $i,j\in \{1,2,3\}$. Such a triangle always exists (see [21]).

A geodesic metric space is said to be a $CAT(0)$ space [21] if all geodesic triangles of appropriate size satisfy the following comparison axiom.

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

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

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

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

(2.1)

The equality holds for the Euclidean metric. In fact (see [[21], p.163]), a geodesic metric space is a $CAT(0)$ space if and only if it satisfies the inequality (2.1) (which is known as the CN inequality of Bruhat and Tits [22]).

Let $x,y\in X$, by [[18], Lemma 2.1(iv)] for each $t\in [0,1]$, there exists a unique point $z\in [x,y]$ such that

$$d(x,z)=td(x,y),d(y,z)=(1-t)d(x,y).$$

(2.2)

From now on, we will use the notation $(1-t)x\oplus ty$ for the unique point z satisfying (2.2). By using this notation, Dhompongsa and Panyanak [18] obtained the following lemmas which will be used frequently in the proof of our main results.

Lemma 1 Let X be a $CAT(0)$ space. Then

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

for all $t\in [0,1]$ and $x,y,z\in X$.

Lemma 2 Let X be a $CAT(0)$ space. Then

$$d{((1-t)x\oplus ty,z)}^2\le (1-t)d{(x,z)}^2+td{(y,z)}^2-t(1-t)d{(x,y)}^2$$

for all $t\in [0,1]$ and $x,y,z\in X$.

3 Demiclosedness principle for a new class of mappings

In 1976 Lim [23] introduced the concept of convergence in a general metric space setting which is called △-convergence. Later, Kirk and Panyanak [24] used the concept of △-convergence introduced by Lim [23] to prove on a $CAT(0)$ space analogs of some Banach space results which involve weak convergence. Also, Dhompongsa and Panyanak [18] obtained the △-convergence theorems for the Picard, Mann and Ishikawa iterations in a $CAT(0)$ space for nonexpansive mappings under some appropriate conditions.

Now, we recall some definitions.

Let $\{x_n\}$ be a bounded sequence in a $CAT(0)$ space X. For $x\in X$, we set

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

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

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

and the asymptotic radius $r_K(\{x_n\})$ of $\{x_n\}$ with respect to $K\subset X$ is given by

$$r_K(\{x_n\})=inf\{r(x,\{x_n\}):x\in K\}.$$

The asymptotic center $A(\{x_n\})$ of $\{x_n\}$ is the set

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

and the asymptotic center $A_K(\{x_n\})$ of $\{x_n\}$ with respect to $K\subset X$ is the set

$$A_K(\{x_n\})=\{x\in K:r(x,\{x_n\})=r_K(\{x_n\})\}.$$

Proposition 1 ([[25], Proposition 3.2])

Let $\{x_n\}$ be a bounded sequence in a complete $CAT(0)$ space X, and let K be a closed convex subset of X, then $A(\{x_n\})$ and $A_K(\{x_n\})$ are singletons.

Definition 3 ([[24], Definition 3.1])

A sequence $\{x_n\}$ in a $CAT(0)$ space X is said to be △-convergent to $x\in X$ if x is the unique asymptotic center of $\{u_n\}$ for every subsequence $\{u_n\}$ of $\{x_n\}$. In this case, we write $△\text{-}{lim}_n{x}_n=x$ and x is called the △-limit of $\{x_n\}$.

Lemma 3

• (i) Every bounded sequence in a complete $CAT(0)$ space always has a △-convergent subsequence (see [[24], p.3690]).

• (ii) Let K be a nonempty closed convex subset of a complete $CAT(0)$ space, and let $\{x_n\}$ be a bounded sequence in K. Then the asymptotic center of $\{x_n\}$ is in K (see [[17], Proposition 2.1]).

Lemma 4 ([[18], Lemma 2.8])

If $\{x_n\}$ is a bounded sequence in a complete $CAT(0)$ space with $A(\{x_n\})=\{x\}$, $\{u_n\}$ is a subsequence of $\{x_n\}$ with $A(\{u_n\})=\{u\}$ and the sequence $\{d(x_n,u)\}$ converges, then $x=u$.

Let $\{x_n\}$ be a bounded sequence in a $CAT(0)$ space X, and let K be a closed convex subset of X which contains $\{x_n\}$. We denote the notation

$$\{x_n\}\rightharpoonup w\iff \mathrm{\Phi }(w)=\underset{x\in K}{inf}\mathrm{\Phi }(x),$$

(3.1)

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

We note that $\{x_n\}\rightharpoonup w$ if and only if $A_K(\{x_n\})=\{w\}$ (see [25]).

Nanjaras and Panyanak [25] gave a connection between the ‘⇀’ convergence and Δ-convergence.

Proposition 2 ([[25], Proposition 3.12])

Let $\{x_n\}$ be a bounded sequence in a $CAT(0)$ space X, and let K be a closed convex subset of X which contains $\{x_n\}$. Then $\mathrm{\Delta }\text{-}{lim}_{n\to \mathrm{\infty }}{x}_n=p$ implies that $\{x_n\}\rightharpoonup p$.

By using the convergence defined in (3.1), we obtain the demiclosedness principle for the new class of mappings in a $CAT(0)$ space.

Theorem 1 Let K be a nonempty closed convex subset of a complete $CAT(0)$ space X, and let $T:K\to K$ be a generalized nonexpansive mapping with $F(T)\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be a bounded sequence in K such that $\{x_n\}\rightharpoonup w$ and ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$. Then $Tw=w$.

Proof By the hypothesis, $\{x_n\}\rightharpoonup w$. Then we have $A_K(\{x_n\})=\{w\}$. By Lemma 3(ii), we obtain $A(\{x_n\})=\{w\}$. Since ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$, then we have

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

(3.2)

for all $x\in K$. By taking $x=Tw$ in (3.2), we have

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

The rest of the proof closely follows the pattern of Proposition 3.14 in Nanjaras and Panyanak [25]. Hence $Tw=w$ as desired. □

Now, we prove the Δ-convergence of the new multi-step iteration process for the new class of mappings in a $CAT(0)$ space.

Theorem 2 Let K be a nonempty closed convex subset of a complete $CAT(0)$ space X, let $T:K\to K$ be a generalized nonexpansive mapping with $F(T)\ne \mathrm{\varnothing }$, and let $\{x_n\}$ be a sequence defined by (1.4) such that $\{{\alpha }_n\},\{{\beta }_n^i\}\subset [0,1]$, $i=1,2,\dots ,k-2$ and $\{{\beta }_n^{k-1}\}\subset [a,b]$ for some $a,b\in (0,1)$. Then the sequence $\{x_n\}$ Δ-converges to the fixed point of T.

Proof Let $p\in F(T)$. From (1.2), (1.4) and Lemma 1, we have

$$\begin{array}{rcl}d(x_{n+1},p)& =& d((1-{\alpha }_n)y_n^1\oplus {\alpha }_{n}T{y}_n^1,p)\\ \le & (1-{\alpha }_n)d(y_n^1,p)+{\alpha }_{n}d(T{y}_n^1,p)\\ \le & (1-{\alpha }_n)d(y_n^1,p)+{\alpha }_n\{d(y_n^1,p)+\phi (d(p,Tp))\}\\ =& d(y_n^1,p).\end{array}$$

Also, we obtain

$$\begin{array}{rcl}d(y_n^1,p)& =& d((1-{\beta }_n^1)y_n^2\oplus {\beta }_n^{1}T{y}_n^2,p)\\ \le & (1-{\beta }_n^1)d(y_n^2,p)+{\beta }_n^{1}d(T{y}_n^2,p)\\ \le & (1-{\beta }_n^1)d(y_n^2,p)+{\beta }_n^1\{d(y_n^2,p)+\phi (d(p,Tp))\}\\ =& d(y_n^2,p).\end{array}$$

Continuing the above process, we have

$$d(x_{n+1},p)\le d(y_n^1,p)\le d(y_n^2,p)\le \cdots \le d(y_n^{k-1},p)\le d(x_n,p).$$

(3.3)

This inequality guarantees that the sequence $\{d(x_n,p)\}$ is non-increasing and bounded below, and so ${lim}_{n\to \mathrm{\infty }}d(x_n,p)$ exists for all $p\in F(T)$. Let ${lim}_{n\to \mathrm{\infty }}d(x_n,p)=r$. By using (3.3), we get

$$\underset{n\to \mathrm{\infty }}{lim}d(y_n^{k-1},p)=r.$$

By Lemma 2, we also have

$$\begin{array}{rcl}d{(y_n^{k-1},p)}^2& =& d{((1-{\beta }_n^{k-1})x_n\oplus {\beta }_n^{k-1}T{x}_n,p)}^2\\ \le & (1-{\beta }_n^{k-1})d{(x_n,p)}^2+{\beta }_n^{k-1}d{(T{x}_n,p)}^2-{\beta }_n^{k-1}(1-{\beta }_n^{k-1})d{(x_n,T{x}_n)}^2\\ \le & (1-{\beta }_n^{k-1})d{(x_n,p)}^2+{\beta }_n^{k-1}{\{d(x_n,p)+\phi (d(p,Tp))\}}^2\\ -{\beta }_n^{k-1}(1-{\beta }_n^{k-1})d{(x_n,T{x}_n)}^2\\ =& d{(x_n,p)}^2-{\beta }_n^{k-1}(1-{\beta }_n^{k-1})d{(x_n,T{x}_n)}^2,\end{array}$$

which implies that

$$d{(x_n,T{x}_n)}^2\le \frac{1}{a(1-b)}[d{(x_n,p)}^2-d{(y_n^{k-1},p)}^2].$$

Thus ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$. To show that the sequence $\{x_n\}$ △-converges to a fixed point of T, we prove that

$$W_{△}(x_n)=\bigcup _{\{u_n\}\subset \{x_n\}}A(\{u_n\})\subseteq F(T)$$

and $W_{△}(x_n)$ consists of exactly one point. Let $u\in W_{△}(x_n)$. Then there exists a subsequence $\{u_n\}$ of $\{x_n\}$ such that $A(\{u_n\})=\{u\}$. By Lemma 3, there exists a subsequence $\{v_n\}$ of $\{u_n\}$ such that $△\text{-}{lim}_{n\to \mathrm{\infty }}{v}_n=v\in K$. By Proposition 2 and Theorem 1, $v\in F(T)$. By Lemma 4, we have $u=v\in F(T)$. This shows that $W_{△}(x_n)\subseteq F(T)$. Now, we prove that $W_{△}(x_n)$ consists of exactly one point. Let $\{u_n\}$ be a subsequence of $\{x_n\}$ with $A(\{u_n\})=\{u\}$, and let $A(\{x_n\})=\{x\}$. We have already seen that $u=v$ and $v\in F(T)$. Finally, since $\{d(x_n,v)\}$ converges, by Lemma 4, $x=v\in F(T)$. This shows that $W_{△}(x_n)=\{x\}$. This completes the proof. □

We give the following theorem related to the Δ-convergence of the S-iteration process for the new class of mappings in a $CAT(0)$ space.

Theorem 3 Let K be a nonempty closed convex subset of a complete $CAT(0)$ space X, let $T:K\to K$ be a generalized nonexpansive mapping with $F(T)\ne \mathrm{\varnothing }$, and let $\{x_n\}$ be a sequence defined by (1.3) such that $\{{\alpha }_n\},\{{\beta }_n\}\subset [a,b]$ for some $a,b\in (0,1)$. Then the sequence $\{x_n\}$ Δ-converges to the fixed point of T.

Proof Let $p\in F(T)$. Using (1.2), (1.3) and Lemma 1, we have

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

(3.4)

Also, we obtain

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

(3.5)

From (3.4) and (3.5), we have

$$d(x_{n+1},p)\le d(x_n,p).$$

This inequality guarantees that the sequence $\{d(x_n,p)\}$ is non-increasing and bounded below, and so ${lim}_{n\to \mathrm{\infty }}d(x_n,p)$ exists for all $p\in F(T)$. Let

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

(3.6)

Now, we prove that ${lim}_{n\to \mathrm{\infty }}d(y_n,p)=r$. By (3.4), we have

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

This gives that

$${\alpha }_{n}d(x_n,p)\le d(x_n,p)+{\alpha }_{n}d(y_n,p)-d(x_{n+1},p)$$

or

$$\begin{array}{rcl}d(x_n,p)& \le & d(y_n,p)+\frac{1}{{\alpha }_n}[d(x_n,p)-d(x_{n+1},p)]\\ \le & d(y_n,p)+\frac{1}{a}[d(x_n,p)-d(x_{n+1},p)].\end{array}$$

This gives

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

By (3.5) and (3.6), we obtain

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(y_n,p)\le r.$$

Then we get

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

By Lemma 2, we also have

$$\begin{array}{rcl}d{(y_n,p)}^2& =& d{((1-{\beta }_n)x_n\oplus {\beta }_{n}T{x}_n,p)}^2\\ \le & (1-{\beta }_n)d{(x_n,p)}^2+{\beta }_{n}d{(T{x}_n,p)}^2-{\beta }_n(1-{\beta }_n)d{(x_n,T{x}_n)}^2\\ \le & (1-{\beta }_n)d{(x_n,p)}^2+{\beta }_n{\{d(x_n,p)+\phi (d(p,Tp))\}}^2-{\beta }_n(1-{\beta }_n)d{(x_n,T{x}_n)}^2\\ =& d{(x_n,p)}^2-{\beta }_n(1-{\beta }_n)d{(x_n,T{x}_n)}^2,\end{array}$$

which implies that

$$d{(x_n,T{x}_n)}^2\le \frac{1}{a(1-b)}[d{(x_n,p)}^2-d{(y_n,p)}^2].$$

Thus ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$. The rest of the proof follows the pattern of the above theorem and is therefore omitted. □

4 Strong convergence theorems for a contractive-like mapping

Now, we prove the strong convergence of the new multi-step iteration process for a contractive-like mapping in a $CAT(0)$ space.

Theorem 4 Let K be a nonempty closed convex subset of a complete $CAT(0)$ space X, let $T:K\to K$ be a contractive-like mapping with $F(T)\ne \mathrm{\varnothing }$, and let $\{x_n\}$ be a sequence defined by (1.4) such that $\{{\alpha }_n\}\subset [0,1)$, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$ and $\{{\beta }_n^i\}\subset [0,1)$, $i=1,2,\dots ,k-1$. Then the sequence $\{x_n\}$ converges strongly to the unique fixed point of T.

Proof Let p be the unique fixed point of T. From (1.1), (1.4) and Lemma 1, we have

$$\begin{array}{rcl}d(x_{n+1},p)& =& d((1-{\alpha }_n)y_n^1\oplus {\alpha }_{n}T{y}_n^1,p)\\ \le & (1-{\alpha }_n)d(y_n^1,p)+{\alpha }_{n}d(T{y}_n^1,p)\\ \le & (1-{\alpha }_n)d(y_n^1,p)+{\alpha }_n\{\delta d(y_n^1,p)+\phi (d(p,Tp))\}\\ =& (1-{\alpha }_n)d(y_n^1,p)+{\alpha }_n\delta d(y_n^1,p)\\ =& [1-{\alpha }_n(1-\delta )]d(y_n^1,p).\end{array}$$

Also, we obtain

$$\begin{array}{rcl}d(y_n^1,p)& =& d((1-{\beta }_n^1)y_n^2\oplus {\beta }_n^{1}T{y}_n^2,p)\\ \le & (1-{\beta }_n^1)d(y_n^2,p)+{\beta }_n^{1}d(T{y}_n^2,p)\\ \le & (1-{\beta }_n^1)d(y_n^2,p)+{\beta }_n^1\{\delta d(y_n^2,p)+\phi (d(p,Tp))\}\\ =& (1-{\beta }_n^1)d(y_n^2,p)+{\beta }_n^1\delta d(y_n^2,p)\\ =& [1-{\beta }_n^1(1-\delta )]d(y_n^2,p).\end{array}$$

In a similar fashion, we can get

$$d(y_n^2,p)\le [1-{\beta }_n^2(1-\delta )]d(y_n^3,p).$$

Continuing the above process, we have

$$\begin{array}{rcl}d(x_{n+1},p)& \le & [1-{\alpha }_n(1-\delta )][1-{\beta }_n^1(1-\delta )][1-{\beta }_n^2(1-\delta )]\cdots \\ [1-{\beta }_n^{k-2}(1-\delta )]d(y_n^{k-1},p).\end{array}$$

(4.1)

In addition, we obtain

$$\begin{array}{rcl}d(y_n^{k-1},p)& =& d((1-{\beta }_n^{k-1})x_n\oplus {\beta }_n^{k-1}T{x}_n,p)\\ \le & (1-{\beta }_n^{k-1})d(x_n,p)+{\beta }_n^{k-1}d(T{x}_n,p)\\ \le & (1-{\beta }_n^{k-1})d(x_n,p)+{\beta }_n^{k-1}\{\delta d(x_n,p)+\phi (d(p,Tp))\}\\ =& (1-{\beta }_n^{k-1})d(x_n,p)+{\beta }_n^{k-1}\delta d(x_n,p)\\ =& [1-{\beta }_n^{k-1}(1-\delta )]d(x_n,p).\end{array}$$

(4.2)

From (4.1) and (4.2), we have

$$\begin{array}{rcl}d(x_{n+1},p)& \le & [1-{\alpha }_n(1-\delta )][1-{\beta }_n^1(1-\delta )][1-{\beta }_n^2(1-\delta )]\cdots \\ [1-{\beta }_n^{k-2}(1-\delta )][1-{\beta }_n^{k-1}(1-\delta )]d(x_n,p)\\ \le & [1-{\alpha }_n(1-\delta )]d(x_n,p)\\ \le & \prod _{j=0}^n[1-{\alpha }_j(1-\delta )]d(x_0,p)\\ \le & e^{-(1-\delta ){\sum }_{j=0}^n{\alpha }_j}d(x_0,p).\end{array}$$

(4.3)

Using the fact that $0\le \delta <1$, ${\alpha }_j\in [0,1]$ and ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, we get that

$$\underset{n\to \mathrm{\infty }}{lim}{e}^{-(1-\delta ){\sum }_{j=0}^n{\alpha }_j}=0.$$

This together with (4.3) implies that

$$\underset{n\to \mathrm{\infty }}{lim}d(x_{n+1},p)=0.$$

Consequently, $x_n\to p\in F(T)$ and this completes the proof. □

Remark 2 In Theorem 4 the condition ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$ may be replaced with ${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n^i=\mathrm{\infty }$ for a fixed $i=1,2,\dots ,k-1$.

Finally, we give the strong convergence theorem of the S-iteration process for a contractive-like mapping in a $CAT(0)$ space as follows.

Theorem 5 Let K be a nonempty closed convex subset of a complete $CAT(0)$ space X, let $T:K\to K$ be a contractive-like mapping with $F(T)\ne \mathrm{\varnothing }$, and let $\{x_n\}$ be a sequence defined by (1.3) such that $\{{\alpha }_n\},\{{\beta }_n\}\subset [0,1]$. Then the sequence $\{x_n\}$ converges strongly to the unique fixed point of T.

Proof Let p be the unique fixed point of T. From (1.1), (1.3) and Lemma 1, we have

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

(4.4)

Similarly, we obtain

$$\begin{array}{rcl}d(y_n,p)& =& d((1-{\beta }_n)x_n\oplus {\beta }_{n}T{x}_n,p)\\ \le & (1-{\beta }_n)d(x_n,p)+{\beta }_{n}d(T{x}_n,p)\\ \le & (1-{\beta }_n)d(x_n,p)+{\beta }_n\{\delta d(x_n,p)+\phi (d(p,Tp))\}\\ =& (1-{\beta }_n)d(x_n,p)+{\beta }_n\delta d(x_n,p)\\ =& (1-{\beta }_n(1-\delta ))d(x_n,p)\\ \le & d(x_n,p).\end{array}$$

(4.5)

Then, from (4.4) and (4.5), we get that

$$\begin{array}{rcl}d(x_{n+1},p)& \le & (1-{\alpha }_n)\delta d(x_n,p)+{\alpha }_n\delta d(y_n,p)\\ \le & (1-{\alpha }_n)\delta d(x_n,p)+{\alpha }_n\delta d(x_n,p)\\ \le & \delta d(x_n,p)\\ ⋮\\ \le & {\delta }^{n+1}d(x_0,p).\end{array}$$

If $\delta \in (0,1)$, we obtain

$$\underset{n\to \mathrm{\infty }}{lim}d(x_{n+1},p)=0.$$

Thus we have $x_n\to p\in F(T)$. If $\delta =0$, the result is clear. This completes the proof. □

5 Conclusions

The new multi-step iteration reduces to the two-step iteration and the SP-iteration processes. Also, the class of generalized nonexpansive mappings includes nonexpansive mappings. Then these results presented in this paper extend and generalize some works for a $CAT(0)$ space in the literature.