Open Access

On an open problem of Kyung Soo Kim

Fixed Point Theory and Applications20152015:186

https://doi.org/10.1186/s13663-015-0438-7

Received: 5 August 2015

Accepted: 8 October 2015

Published: 15 October 2015

Abstract

We prove a convergence theorem of the Mann iteration scheme for a uniformly L-Lipschitzian asymptotically demicontractive mapping in a \(\operatorname{CAT}(\kappa)\) space with \(\kappa>0\). We also obtain a convergence theorem of the Ishikawa iteration scheme for a uniformly L-Lipschitzian asymptotically hemicontractive mapping. Our results provide a complete solution to an open problem raised by Kim (Abstr. Appl. Anal. 2013:381715, 2013).

Keywords

Mann iterationIshikawa iterationstrong convergencefixed point \(\operatorname{CAT}(\kappa)\) space

MSC

47H0949J53

1 Introduction

Roughly speaking, \(\operatorname{CAT}(\kappa)\) spaces are geodesic spaces of bounded curvature and generalizations of Riemannian manifolds of sectional curvature bounded above. The precise definition is given below. The letters C, A, and T stand for Cartan, Alexandrov, and Toponogov, who have made important contributions to the understanding of curvature via inequalities for the distance function, and κ is a real number that we impose it as the curvature bound of the space.

Fixed point theory in \(\operatorname{CAT}(\kappa)\) spaces was first studied by Kirk [1, 2]. His work was followed by a series of new works by many authors, mainly focusing on \(\operatorname{CAT}(0)\) spaces (see e.g., [325]). Since any \(\operatorname{CAT}(\kappa )\) space is a \(\operatorname{CAT}(\kappa')\) space for \(\kappa' \geq\kappa\), all results for \(\operatorname{CAT}(0)\) spaces immediately apply to any \(\operatorname {CAT}(\kappa)\) space with \(\kappa\leq0\). However, there are only a few articles that contain fixed point results in the setting of \(\operatorname{CAT}(\kappa)\) spaces with \(\kappa>0\), because in this case the proof seems to be more complicated.

The notion of uniformly L-Lipschitzian mappings, which is more general than the notion of asymptotically nonexpansive mappings, was introduced by Goebel and Kirk [26]. In 1991, Schu [27] proved the strong convergence of Mann iteration for asymptotically nonexpansive mappings in Hilbert spaces. Qihou [28] extended Schu’s result to the general setting of asymptotically demicontractive mappings and also obtained the strong convergence of Ishikawa iteration for asymptotically hemicontractive mappings. Recently, Kim [29] proved the analogous results of Qihou in the framework of the so-called \(\operatorname{CAT}(0)\) spaces. Precisely, Kim obtained the following theorems.

Theorem A

Let \((X, \rho)\) be a complete \(\operatorname {CAT}(0)\) space, C be a nonempty bounded closed convex subset of X, and \(T:C\to C\) be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive mapping with constant \(k\in[0,1)\) and sequence \(\{a_{n}\}\) in \([1,\infty)\) such that \(\sum^{\infty}_{n=1}(a_{n}^{2}-1)<\infty\). Let \(\{\alpha_{n}\}\) be a sequence in \([\varepsilon, 1-k-\varepsilon]\) for some \(\varepsilon>0\). Given \(x_{1}\in C\), define the iteration scheme \(\{x_{n}\}\) by
$$x_{n+1}=(1-\alpha_{n})x_{n}\oplus \alpha_{n} T^{n}x_{n}, \quad n\geq1. $$
Then \(\{x_{n}\}\) converges strongly to a fixed point of T.

Theorem B

Let \((X, \rho)\) be a complete \(\operatorname {CAT}(0)\) space, let C be a nonempty bounded closed convex subset of X, and let \(T:C\to C\) be a completely continuous and uniformly L-Lipschitzian asymptotically hemicontractive mapping with sequence \(\{a_{n}\}\) in \([1,\infty)\) such that \(\sum_{n=1}^{\infty}(a_{n}-1)<\infty\). Let \(\{\alpha_{n}\}, \{\beta_{n}\}\subset[0,1]\) be such that \(\varepsilon \leq\alpha_{n}\leq\beta_{n}\leq b\) for some \(\varepsilon>0\) and \(b\in (0,\frac{\sqrt{1+L^{2}}-1}{L^{2}} )\). Given \(x_{1}\in C\), define the iteration scheme \(\{x_{n}\}\) by
$$\begin{aligned}& x_{n+1}=(1-\alpha_{n})x_{n}\oplus \alpha_{n}T^{n}y_{n}, \\& y_{n}=(1-\beta_{n})x_{n}\oplus\beta_{n} T^{n}x_{n},\quad n\geq1. \end{aligned}$$
Then \(\{x_{n}\}\) converges strongly to a fixed point of T.

In [29], the author raised the following problem.

Problem

Can we extend Theorems A and B to the general setting of \(\operatorname{CAT}(\kappa)\) spaces with \(\kappa>0\)?

The purpose of the paper is to solve this problem. Our main discoveries are Theorems 3.2 and 3.6.

2 Preliminaries

Let \((X,\rho)\) 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 \([0,l]\subset \mathbb{R}\) to X such that \(c(0)=x\), \(c(l)=y\), and \(\rho(c(t),c(t^{\prime}))=|t-t^{\prime}|\) for all \(t,t^{\prime}\in[0,l]\). In particular, c is an isometry and \(\rho(x,y)=l\). The image \(c([0,l])\) of c is called a geodesic segment joining x and y. When it is unique this geodesic segment is denoted by \([x,y]\). This means that \(z\in[x, y]\) if and only if there exists \(\alpha\in[0, 1]\) such that
$$\rho(x,z)=(1-\alpha)\rho(x,y) \quad \mbox{and} \quad \rho(y,z)=\alpha\rho(x,y). $$
In this case, we write \(z=\alpha x\oplus(1-\alpha)y\). The space \((X,\rho)\) is said to be a geodesic space (D-geodesic space) if every two points of X (every two points of distance smaller than D) are joined by a geodesic, and X is said to be uniquely geodesic (D-uniquely geodesic) if there is exactly one geodesic joining x and y for each \(x, y\in X\) (for \(x, y \in X\) with \(\rho(x, y) < D\)). A subset C of X is said to be convex if C includes every geodesic segment joining any two of its points. The set C is said to be bounded if
$$\operatorname{diam}(C) := \sup\bigl\{ \rho(x,y) : x, y\in C\bigr\} < \infty. $$
Now we introduce the model spaces \(M^{n}_{\kappa}\), for more details on these spaces the reader is referred to [30, 31]. Let \(n\in\mathbb{N}\). We denote by \(\mathbb{E}^{n}\) the metric space \(\mathbb{R}^{n}\) endowed with the usual Euclidean distance. We denote by \((\cdot|\cdot)\) the Euclidean scalar product in \(\mathbb{R}^{n}\), that is,
$$( x|y )=x_{1}y_{1}+\cdots+x_{n}y_{n},\quad \mbox{where } x=(x_{1},\ldots,x_{n}), y=(y_{1}, \ldots,y_{n}). $$
Let \(\mathbb{S}^{n}\) denote the n-dimensional sphere defined by
$$\mathbb{S}^{n}= \bigl\{ x=(x_{1},\ldots,x_{n+1})\in \mathbb{R}^{n+1} : ( x|x )=1 \bigr\} , $$
with metric \(d_{\mathbb{S}^{n}}(x,y)=\arccos( x|y )\), \(x,y\in \mathbb{S}^{n}\).
Let \(\mathbb{E}^{n,1}\) denote the vector space \(\mathbb{R}^{n+1}\) endowed with the symmetric bilinear form which associates to vectors \(u = (u_{1},\ldots, u_{n+1})\) and \(v = (v_{1},\ldots, v_{n+1})\) the real number \(\langle u|v\rangle\) defined by
$$\langle u|v\rangle= - u_{n+1} v_{n+1}+\sum _{i=1}^{n}u_{i} v_{i}. $$
Let \(\mathbb{H}^{n}\) denote the hyperbolic n-space defined by
$$\mathbb{H}^{n}= \bigl\{ u=(u_{1},\ldots,u_{n+1})\in \mathbb{E}^{n,1} : \langle u|u \rangle= -1, u_{n+1}> 0 \bigr\} , $$
with metric \(d_{\mathbb{H}^{n}}\) such that
$$\cosh d_{\mathbb{H}^{n}}(x,y)= -\langle x|y \rangle,\quad x,y\in \mathbb{H}^{n}. $$

Definition 2.1

Given \(\kappa\in\mathbb{R}\), we denote by \(M^{n}_{\kappa}\) the following metric spaces:
  1. (i)

    if \(\kappa= 0\) then \(M^{n}_{0}\) is the Euclidean space \(\mathbb{E}^{n}\);

     
  2. (ii)

    if \(\kappa> 0\) then \(M^{n}_{\kappa}\) is obtained from the spherical space \(\mathbb{S}^{n}\) by multiplying the distance function by the constant \(1/\sqrt{\kappa}\);

     
  3. (iii)

    if \(\kappa< 0\) then \(M^{n}_{\kappa}\) is obtained from the hyperbolic space \(\mathbb{H}^{n}\) by multiplying the distance function by the constant \(1/\sqrt{-\kappa}\).

     
A geodesic triangle \(\triangle(x, y, z)\) in a geodesic space \((X,\rho)\) consists of three points x, y, z in X (the vertices of ) and three geodesic segments between each pair of vertices (the edges of ). A comparison triangle for a geodesic triangle \(\triangle(x, y, z)\) in \((X,\rho)\) is a triangle \(\overline{\triangle}(\bar{x}, \bar{y}, \bar{z})\) in \(M^{2}_{\kappa}\) such that
$$\rho(x,y)=d_{M_{\kappa}^{2}}(\bar{x},\bar{y}), \qquad \rho(y,z)=d_{M_{\kappa }^{2}}( \bar{y},\bar{z})\quad \mbox{and} \quad \rho(z,x)=d_{M_{\kappa}^{2}}(\bar {z}, \bar{x}). $$
If \(\kappa\leq0\) then such a comparison triangle always exists in \(M^{2}_{\kappa}\). If \(\kappa> 0\) then such a triangle exists whenever \(\rho(x, y) + \rho(y, z) + \rho(z, x) < 2D_{\kappa}\), where \(D_{\kappa}=\pi/\sqrt{\kappa}\). A point \(\bar{p}\in[\bar{x}, \bar{y}]\) is called a comparison point for \(p\in[x, y]\) if \(\rho(x, p) = d_{M_{\kappa}^{2}}(\bar{x}, \bar{p})\).
A geodesic triangle \(\triangle(x, y, z)\) in X is said to satisfy the \(\operatorname{CAT}(\kappa)\) inequality if for any \(p,q\in \triangle(x, y, z)\) and for their comparison points \(\bar{p}, \bar{q}\in \overline{\triangle}(\bar{x}, \bar{y}, \bar{z})\), one has
$$\rho(p,q)\leq d_{M_{\kappa}^{2}}(\bar{p}, \bar{q}). $$

Definition 2.2

If \(\kappa\leq0\), then X is called a \(\operatorname{CAT}(\kappa)\) space if X is a geodesic space such that all of its geodesic triangles satisfy the \(\operatorname{CAT}(\kappa)\) inequality.

If \(\kappa> 0\), then X is called a \(\operatorname {CAT}(\kappa)\) space if X is \(D_{\kappa}\)-geodesic and any geodesic triangle \(\triangle(x, y, z)\) in X with \(\rho(x, y) + \rho(y, z) + \rho(z, x) < 2D_{\kappa}\) satisfies the \(\operatorname{CAT}(\kappa)\) inequality.

Notice that in a \(\operatorname{CAT}(0)\) space \((X,\rho)\), if \(x,y,z\in X\) then the \(\operatorname{CAT}(0)\) inequality implies
$$(\mathrm{CN})\quad \rho^{2} \biggl(x,\frac{1}{2}y\oplus \frac{1}{2}z \biggr)\leq\frac{1}{2}\rho ^{2}(x,y)+ \frac{1}{2} \rho^{2}(x,z)-\frac{1}{4}\rho^{2}(y,z). $$
This is the (CN) inequality of Bruhat and Tits [32]. This inequality is extended by Dhompongsa and Panyanak [9] as
$$(\mathrm{CN}^{*})\quad \rho^{2}\bigl(x,(1-\alpha)y\oplus\alpha z\bigr)\leq(1-\alpha)\rho^{2}(x,y)+\alpha \rho^{2}(x,z)- \alpha(1-\alpha)\rho^{2}(y,z) $$
for all \(\alpha\in[0,1]\) and \(x, y, z\in X\). In fact, if X is a geodesic space then the following statements are equivalent:
  1. (i)

    X is a \(\operatorname{CAT}(0)\) space;

     
  2. (ii)

    X satisfies (CN);

     
  3. (iii)

    X satisfies (CN).

     
Let \(R\in(0,2]\). Recall that a geodesic space \((X, \rho)\) is said to be R-convex for R [33] if for any three points \(x, y, z \in X\), we have
$$ \rho^{2}\bigl(x,(1-\alpha)y\oplus\alpha z\bigr)\leq (1- \alpha)\rho^{2}(x,y)+\alpha\rho^{2}(x,z)-\frac{R}{2} \alpha(1-\alpha)\rho^{2}(y,z). $$
(1)

It follows from (CN) that a geodesic space \((X, \rho)\) is a \(\operatorname{CAT}(0)\) space if and only if \((X, \rho)\) is R-convex for \(R=2\). The following lemma generalizes Proposition 3.1 of Ohta [33].

Lemma 2.3

Let κ be an arbitrary positive real number and \((X, \rho)\) be a \(\operatorname{CAT}(\kappa)\) space with \(\operatorname{diam}(X)\leq\frac{\pi/2-\eta}{\sqrt{\kappa}}\) for some \(\eta\in(0,\pi/2)\). Then \((X, \rho)\) is R-convex for \(R=(\pi-2\eta)\tan(\eta)\).

Proof

Let \(x,y,z\in X\). Since \(\operatorname{diam}(X) < \frac{\pi}{2\sqrt {\kappa}}\), \(\rho(x, y) + \rho(x, z) + \rho(y, z) < 2D_{\kappa}\) where \(D_{\kappa}=\frac{\pi}{\sqrt{\kappa}}\). Let \(\triangle(x, y, z)\) be the geodesic triangle constructed from x, y, z and \(\overline{\triangle}(\bar{x}, \bar{y}, \bar{z})\) its comparison triangle. Then
$$ \rho(x,y)=d_{M_{\kappa}^{2}}(\bar{x},\bar {y}), \qquad \rho(x,z)=d_{M_{\kappa}^{2}}(\bar{x},\bar{z})\quad \mbox{and}\quad \rho(y,z)=d_{M_{\kappa}^{2}}(\bar{y},\bar{z}). $$
(2)
It is sufficient to prove (1) only the case of \(\alpha=1/2\). Let \(a=d_{\mathbb{S}^{2}}(\bar{x},\bar{y})\), \(b=d_{\mathbb{S}^{2}}(\bar{x},\bar{z})\), \(c=d_{\mathbb{S}^{2}}(\bar{y},\bar{z})/2\), and \(d=d_{\mathbb{S}^{2}} (\bar{x},\frac{1}{2}\bar{y}\oplus\frac{1}{2}\bar {z} )\) and define
$$f(a,b,c):=\frac{2}{c^{2}} \biggl(\frac{1}{2}a^{2}+ \frac{1}{2}b^{2}-d^{2} \biggr). $$
By using the same argument in the proof of Proposition 3.1 in [33], we obtain
$$d^{2}_{\mathbb{S}^{2}} \biggl(\bar{x},\frac{1}{2}\bar{y}\oplus \frac{1}{2}\bar {z} \biggr)\leq \frac{1}{2}d^{2}_{\mathbb{S}^{2}}( \bar{x},\bar{y})+\frac{1}{2} d^{2}_{\mathbb{S}^{2}}(\bar{x}, \bar{z})- \biggl(\frac{R}{2} \biggr) \biggl(\frac {1}{4} \biggr)d^{2}_{\mathbb{S}^{2}}(\bar{y},\bar{z}), $$
where \(R=(\pi-2\eta)\tan(\eta)\). This implies that
$$ d^{2}_{M_{\kappa}^{2}} \biggl(\bar{x},\frac {1}{2} \bar{y}\oplus\frac{1}{2}\bar{z} \biggr)\leq \frac{1}{2}d^{2}_{M_{\kappa}^{2}}( \bar{x},\bar{y})+\frac{1}{2} d^{2}_{M_{\kappa}^{2}}(\bar{x}, \bar{z})- \biggl(\frac{R}{2} \biggr) \biggl(\frac {1}{4} \biggr)d^{2}_{M_{\kappa}^{2}}(\bar{y},\bar{z}). $$
(3)
By (2) and (3), we get
$$\rho^{2} \biggl(x,\frac{1}{2}y\oplus \frac{1}{2}z \biggr) \leq\frac{1}{2}\rho^{2}(x,y)+\frac{1}{2} \rho^{2}(x,z)- \biggl(\frac{R}{2} \biggr) \biggl(\frac{1}{4} \biggr)\rho^{2}(y,z). $$
This completes the proof. □

The following lemma is also needed.

Lemma 2.4

Let \(\{s_{n}\}\) and \(\{t_{n}\}\) be sequences of nonnegative real numbers satisfying
$$s_{n+1}\leq s_{n} + t_{n} \quad \textit{for all } n \in\mathbb{N}. $$
If \(\sum_{n=1}^{\infty} t_{n}<\infty\) and \(\{s_{n}\}\) has a subsequence converging to 0, then \(\lim_{n\to\infty} s_{n}=0\).

Definition 2.5

Let C be a nonempty subset of a \(\operatorname{CAT}(\kappa)\) space \((X,\rho)\) and \(T:C\to C\) be a mapping. We denote by \(F(T)\) the set of all fixed points of T, i.e., \(F(T)=\{x\in C: x=Tx\}\). Then T is said to
  1. (i)

    be completely continuous if T is continuous and for any bounded sequence \(\{x_{n}\}\) in C, \(\{Tx_{n}\}\) has a convergent subsequence in C;

     
  2. (ii)
    be uniformly L-Lipschitzian if there exists a constant \(L>0\) such that
    $$\rho\bigl(T^{n}x,T^{n}y\bigr)\leq L \rho(x,y) \quad \text{for all } x, y\in C \text{ and all } n\in\mathbb{N}; $$
     
  3. (iii)
    be asymptotically demicontractive if \(F(T)\neq \emptyset\) and there exist \(k\in[0,1)\) and a sequence \(\{a_{n}\}\) with \(\lim_{n\to\infty}a_{n}=1\) such that
    $$\rho^{2}\bigl(T^{n}x, p\bigr)\leq a^{2}_{n} \rho^{2}(x,p)+k \rho^{2}\bigl(x,T^{n}x\bigr) \quad \text{for all } x\in C, p\in F(T) \text{ and } n\in\mathbb{N}; $$
     
  4. (iv)
    be asymptotically hemicontractive if \(F(T)\neq \emptyset\) and there exists a sequence \(\{a_{n}\}\) with \(\lim_{n\to\infty}a_{n}=1\) such that
    $$\rho^{2}\bigl(T^{n}x, p\bigr)\leq a_{n} \rho^{2}(x,p)+ \rho^{2}\bigl(x,T^{n}x\bigr) \quad \text{for all } x\in C, p\in F(T) \text{ and } n\in\mathbb{N}. $$
     

It follows from the definition that every asymptotically demicontractive mapping is asymptotically hemicontractive. For more details as regards these classes of mappings the reader is referred to [27, 28].

Let C be a nonempty convex subset of a \(\operatorname{CAT}(\kappa)\) space \((X,\rho)\) and \(T:C\to C\) be a mapping. Given \(x_{1}\in C\).

Algorithm 1

The sequence \(\{x_{n}\}\) defined by
$$\begin{aligned}& x_{n+1}=(1-\alpha_{n})x_{n}\oplus \alpha_{n}T^{n}y_{n}, \\& y_{n}=(1-\beta_{n})x_{n}\oplus\beta_{n} T^{n}x_{n},\quad n\geq1, \end{aligned}$$
is called an Ishikawa iterative sequence (see [34]).

If \(\beta_{n} = 0\) for all \(n\in\mathbb{N}\), then Algorithm 1 reduces to the following.

Algorithm 2

The sequence \(\{x_{n}\}\) defined by
$$x_{n+1}=(1-\alpha_{n})x_{n}\oplus \alpha_{n} T^{n}x_{n}, \quad n\geq1, $$
is called a Mann iterative sequence (see [35]).

3 Main results

We first discuss the strong convergence of Mann iteration for uniformly L-Lipschitzian asymptotically demicontractive mappings. The following lemma follows immediately from Lemma 6 of [29] and [30], p.176.

Lemma 3.1

Let \(\kappa>0\) and \((X, \rho)\) be a \(\operatorname{CAT}(\kappa)\) space with \(\operatorname{diam}(X)\leq \frac{\pi/2-\eta}{\sqrt{\kappa}}\) for some \(\eta\in(0,\pi/2)\). Let C be a nonempty convex subset of X, \(T:C\to C\) be a uniformly L-Lipschitzian mapping, and \(\{\alpha _{n}\}\), \(\{\beta_{n}\}\) be sequences in \([0, 1]\). Given \(x_{1}\in C\), define the iteration scheme \(\{x_{n}\}\) by
$$\begin{aligned}& x_{n+1}=(1-\alpha_{n})x_{n}\oplus \alpha_{n}T^{n}y_{n}, \\& y_{n}=(1-\beta_{n})x_{n}\oplus\beta_{n} T^{n}x_{n}, \quad n\geq1. \end{aligned}$$
Then
$$\rho(x_{n},Tx_{n})\leq\rho\bigl(x_{n},T^{n}x_{n} \bigr)+ L\bigl(1+2L+L^{2}\bigr)\rho\bigl(x_{n-1},T^{n-1}x_{n-1} \bigr) $$
for all \(n\geq1\).

The following theorem is one of our main results.

Theorem 3.2

Let \(\kappa>0\) and \((X, \rho)\) be a \(\operatorname{CAT}(\kappa)\) space with \(\operatorname{diam}(X)\leq \frac{\pi/2-\eta}{\sqrt{\kappa}}\) for some \(\eta\in(0,\pi/2)\). Let C be a nonempty closed convex subset of X, and \(T:C\to C\) be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive mapping with constant \(k\in[0,1)\) and sequence \(\{a_{n}\}\) in \([1,\infty)\) such that \(\sum^{\infty}_{n=1}(a_{n}^{2}-1)<\infty\). Let \(\{\alpha_{n}\}\) be a sequence in \([\varepsilon, R/2-k-\varepsilon]\) for some \(\varepsilon>0\) where \(R=(\pi-2\eta)\tan(\eta)\). Given \(x_{1}\in C\), define the iteration scheme \(\{x_{n}\}\) by
$$x_{n+1}=(1-\alpha_{n})x_{n}\oplus \alpha_{n} T^{n}x_{n},\quad n\geq1. $$
Then \(\{x_{n}\}\) converges strongly to a fixed point of T.

Proof

Let \(p\in F(T)\). By (1), we have
$$\rho^{2}(x_{n+1},p)\leq(1-\alpha_{n}) \rho^{2}(x_{n},p)+\alpha_{n}\rho ^{2} \bigl(T^{n}x_{n},p\bigr)-\frac{R}{2} \alpha_{n}(1-\alpha_{n})\rho^{2} \bigl(x_{n},T^{n}x_{n}\bigr). $$
It follows from the asymptotically demicontractiveness of T that
$$\begin{aligned} \rho^{2}(x_{n+1},p) \leq&(1- \alpha_{n})\rho^{2}(x_{n},p)+\alpha_{n} \bigl[a_{n}^{2}\rho^{2}(x_{n},p)+k \rho^{2}\bigl(x_{n},T^{n}x_{n}\bigr) \bigr] \\ &{}-\frac{R}{2}\alpha _{n}(1-\alpha_{n}) \rho^{2}\bigl(x_{n},T^{n}x_{n}\bigr) \\ =& \rho^{2}(x_{n},p)+\alpha_{n} \bigl(a^{2}_{n}-1\bigr) \rho^{2}(x_{n},p)- \alpha_{n} \biggl(\frac{R}{2}-\frac{R}{2} \alpha_{n}-k \biggr)\rho^{2}\bigl(x_{n},T^{n}x_{n} \bigr) \\ \leq&\rho^{2}(x_{n},p)+\alpha_{n} \bigl(a^{2}_{n}-1\bigr) \rho^{2}(x_{n},p)- \alpha_{n} \biggl(\frac{R}{2}-\alpha_{n}-k \biggr)\rho ^{2}\bigl(x_{n},T^{n}x_{n}\bigr). \end{aligned}$$
(4)
Since \(\varepsilon\leq\alpha_{n}\leq R/2-k-\varepsilon\), we have \(\varepsilon\leq R/2-\alpha_{n}-k\). Thus,
$$ \varepsilon^{2}\leq\alpha_{n} (R/2- \alpha_{n}-k). $$
(5)
By (4) and (5), we have
$$\begin{aligned} \rho^{2}(x_{n+1},p) \leq&\rho^{2}(x_{n},p)+ \alpha_{n} \bigl(a^{2}_{n}-1\bigr) \rho ^{2}(x_{n},p)- \varepsilon^{2}\rho^{2} \bigl(x_{n},T^{n}x_{n}\bigr) \\ \leq&\rho^{2}(x_{n},p)+ \frac{\pi^{2}(a^{2}_{n}-1)}{4\kappa} - \varepsilon^{2}\rho^{2}\bigl(x_{n},T^{n}x_{n} \bigr). \end{aligned}$$
(6)
Therefore,
$$\varepsilon^{2}\rho^{2}\bigl(x_{n},T^{n}x_{n} \bigr)\leq\rho^{2}(x_{n},p)-\rho^{2}(x_{n+1},p) + \frac{\pi^{2}(a^{2}_{n}-1)}{4\kappa}. $$
Since \(\sum^{\infty}_{n=1}(a_{n}^{2}-1)<\infty\), \(\sum^{\infty}_{n=1}\rho^{2}(x_{n},T^{n}x_{n})<\infty\), which implies that \(\lim_{n\to\infty}\rho(x_{n}, T^{n}x_{n})=0\). By Lemma 3.1, we have
$$ \lim_{n\to\infty}\rho(x_{n},Tx_{n})=0. $$
(7)
Since T is completely continuous, \(\{Tx_{n}\}\) has a convergent subsequence in C. By (7), \(\{x_{n}\}\) has a convergent subsequence, say \(x_{n_{k}}\to q\in C\). Moreover,
$$\rho(q,Tq)\leq\rho(q,x_{n_{k}})+\rho(x_{n_{k}},Tx_{n_{k}})+ \rho (Tx_{n_{k}},Tq)\to0 \quad \text{as } k\to\infty. $$
That is \(q\in F(T)\). It follows from (6) that
$$\rho^{2}(x_{n+1},p)\leq\rho^{2}(x_{n},p)+ \frac{\pi^{2}(a^{2}_{n}-1)}{4\kappa}. $$
Since \(\sum^{\infty}_{n=1}(a_{n}^{2}-1)<\infty\), by Lemma 2.4 we have \(x_{n}\to q\). This completes the proof. □

Corollary 3.3

(Theorem 7 of [29])

Let \((X, \rho)\) be a \(\operatorname{CAT}(0)\) space, C be a nonempty bounded closed convex subset of X, and \(T:C\to C\) be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive mapping with constant \(k\in[0,1)\) and sequence \(\{a_{n}\}\) in \([1,\infty)\) such that \(\sum^{\infty}_{n=1}(a_{n}^{2}-1)<\infty\). Let \(\{\alpha_{n}\}\) be a sequence in \([\varepsilon, 1-k-\varepsilon]\) for some \(\varepsilon>0\). Given \(x_{1}\in C\), define the iteration scheme \(\{x_{n}\}\) by
$$x_{n+1}=(1-\alpha_{n})x_{n}\oplus \alpha_{n} T^{n}x_{n}, \quad n\geq1. $$
Then \(\{x_{n}\}\) converges strongly to a fixed point of T.

Proof

It is well known that every convex subset of a \(\operatorname {CAT}(0)\) space, equipped with the induced metric, is a \(\operatorname{CAT}(0)\) space. Then \((C,\rho)\) is a \(\operatorname{CAT}(0)\) space and hence it is a \(\operatorname{CAT}(\kappa)\) space for all \(\kappa>0\). Notice also that C is R-convex for \(R=2\). Since C is bounded, we can choose \(\eta\in(0,\pi /2)\) and \(\kappa>0\) so that \(\operatorname{diam}(C)\leq \frac{\pi/2-\eta}{\sqrt{\kappa}}\). The conclusion follows from Theorem 3.2. □

Next, we prove the strong convergence of Ishikawa iteration for uniformly L-Lipschitzian asymptotically hemicontractive mappings. The following lemmas are also needed.

Lemma 3.4

Let \(\kappa>0\) and \((X, \rho )\) be a \(\operatorname{CAT}(\kappa)\) space with \(\operatorname{diam}(X)\leq \frac{\pi/2-\eta}{\sqrt{\kappa}}\) for some \(\eta\in(0,\pi/2)\). Let \(R=(\pi-2\eta)\tan(\eta)\), C be a nonempty convex subset of X, and \(T:C\to C\) be a uniformly L-Lipschitzian and asymptotically hemicontractive mapping with sequence \(\{a_{n}\}\) in \([1,\infty)\). Given \(x_{1}\in C\), define the iteration scheme \(\{x_{n}\}\) by
$$\begin{aligned}& x_{n+1}=(1-\alpha_{n})x_{n}\oplus \alpha_{n}T^{n}y_{n}, \\& y_{n}=(1-\beta_{n})x_{n}\oplus\beta_{n} T^{n}x_{n},\quad n\geq1, \end{aligned}$$
where \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are sequences in \([0,1]\). Then the following inequality holds:
$$\begin{aligned} \rho^{2}(x_{n+1},p) \leq& \bigl[1+\alpha_{n}(a_{n}-1) (1+a_{n}\beta_{n}) \bigr]\rho ^{2}(x_{n},p) \\ &{} -\alpha_{n}\beta_{n} \biggl[\frac{R}{2}(1-\beta _{n}) (1+a_{n})-\bigl(a_{n}+L^{2} \beta_{n}^{2}\bigr) \biggr]\rho^{2} \bigl(x_{n},T^{n}x_{n}\bigr) \\ &{} -\alpha_{n} \biggl[\frac{R}{2}(1-\alpha_{n})-(1- \beta_{n}) \biggr]\rho^{2}\bigl(x_{n},T^{n}y_{n} \bigr) \end{aligned}$$
for all \(p\in F(T)\).

Proof

Let \(p\in F(T)\). By (1), we have
$$ \rho^{2}(x_{n+1},p)\leq(1-\alpha_{n}) \rho ^{2}(x_{n},p)+\alpha_{n}\rho^{2} \bigl(T^{n}y_{n},p\bigr)-\frac{R}{2} \alpha_{n}(1-\alpha _{n})\rho^{2} \bigl(x_{n},T^{n}y_{n}\bigr) $$
(8)
and
$$ \rho^{2}(y_{n},p)\leq(1-\beta_{n}) \rho ^{2}(x_{n},p)+\beta_{n}\rho^{2} \bigl(T^{n}x_{n},p\bigr)-\frac{R}{2} \beta_{n}(1-\beta_{n})\rho ^{2}\bigl(x_{n},T^{n}x_{n} \bigr). $$
(9)
Since T is asymptotically hemicontractive,
$$ \rho^{2}\bigl(T^{n}y_{n},p\bigr) \leq a_{n}\rho ^{2}(y_{n},p)+\rho^{2} \bigl(y_{n},T^{n}y_{n}\bigr) $$
(10)
and
$$ \rho^{2}\bigl(T^{n}x_{n},p\bigr) \leq a_{n}\rho ^{2}(x_{n},p)+\rho^{2} \bigl(x_{n},T^{n}x_{n}\bigr). $$
(11)
It follows from (9) and (11) that
$$\begin{aligned} \begin{aligned}[b] \rho^{2}(y_{n},p) \leq{}&(1- \beta_{n})\rho^{2}(x_{n},p)+\beta_{n} \bigl[a_{n}\rho ^{2}(x_{n},p)+\rho^{2} \bigl(x_{n},T^{n}x_{n}\bigr) \bigr] \\ &{}- \frac{R}{2}\beta_{n}(1-\beta _{n})\rho^{2} \bigl(x_{n},T^{n}x_{n}\bigr) \\ ={}& \bigl(1+\beta_{n}(a_{n}-1) \bigr)\rho ^{2}(x_{n},p)+\beta_{n} \biggl(1- \frac{R}{2}(1-\beta_{n}) \biggr)\rho ^{2} \bigl(x_{n},T^{n}x_{n}\bigr). \end{aligned} \end{aligned}$$
(12)
Substituting (12) into (10) and using (1), we get
$$\begin{aligned} \rho^{2}\bigl(T^{n}y_{n},p\bigr) \leq& a_{n} \bigl(1+\beta_{n}(a_{n}-1) \bigr)\rho ^{2}(x_{n},p) \\ &{}+a_{n}\beta_{n} \biggl(1- \frac{R}{2}(1-\beta_{n}) \biggr)\rho ^{2} \bigl(x_{n},T^{n}x_{n}\bigr)+\rho^{2} \bigl(y_{n}, T^{n}y_{n}\bigr) \\ \leq& a_{n} \bigl(1+\beta_{n}(a_{n}-1) \bigr) \rho^{2}(x_{n},p)+a_{n}\beta_{n} \biggl(1- \frac{R}{2}(1-\beta_{n}) \biggr)\rho^{2} \bigl(x_{n},T^{n}x_{n}\bigr) \\ &{} +(1-\beta_{n})\rho^{2}\bigl(x_{n},T^{n}y_{n} \bigr)+\beta_{n}\rho ^{2}\bigl(T^{n}x_{n},T^{n}y_{n} \bigr) \\ &{}-\frac{R}{2}\beta_{n}(1-\beta_{n}) \rho^{2}\bigl(x_{n},T^{n}x_{n}\bigr) \\ \leq& a_{n} \bigl(1+\beta_{n}(a_{n}-1) \bigr) \rho^{2}(x_{n},p) \\ &{}+ \biggl[a_{n}\beta _{n}-a_{n}\beta_{n}\frac{R}{2}(1- \beta_{n})-\beta_{n}\frac{R}{2}(1-\beta_{n}) \biggr]\rho^{2}\bigl(x_{n},T^{n}x_{n} \bigr) \\ &{} +(1-\beta_{n})\rho^{2}\bigl(x_{n},T^{n}y_{n} \bigr)+\beta_{n}L^{2}\rho ^{2}(x_{n},y_{n}) \\ \leq& a_{n} \bigl(1+\beta_{n}(a_{n}-1) \bigr) \rho^{2}(x_{n},p) \\ &{} + \biggl[a_{n}\beta_{n}-a_{n} \beta_{n}\frac{R}{2}(1-\beta_{n})-\beta _{n} \frac{R}{2}(1-\beta_{n})+\beta^{3}L^{2} \biggr]\rho^{2}\bigl(x_{n},T^{n}x_{n} \bigr) \\ &{} +(1-\beta_{n})\rho^{2}\bigl(x_{n},T^{n}y_{n} \bigr). \end{aligned}$$
(13)
Substituting (13) into (8), we obtain
$$\begin{aligned} \rho^{2}(x_{n+1},p) \leq&(1-\alpha_{n}) \rho^{2}(x_{n},p)+\alpha_{n}a_{n} \bigl(1+\beta_{n}(a_{n}-1) \bigr)\rho^{2}(x_{n}, p) \\ &{}+\alpha_{n} \biggl[a_{n}\beta_{n}-a_{n} \beta_{n}\frac{R}{2}(1-\beta _{n})-\beta_{n} \frac{R}{2}(1-\beta_{n})+\beta^{3}L^{2} \biggr]\rho^{2}\bigl(x_{n},T^{n}x_{n} \bigr) \\ &{}+\alpha_{n}(1-\beta_{n})\rho^{2} \bigl(x_{n},T^{n}y_{n}\bigr)-\frac{R}{2} \alpha _{n}(1-\alpha_{n})\rho^{2} \bigl(x_{n},T^{n}y_{n}\bigr) \\ =& \bigl[1+\alpha_{n}(a_{n}-1) (1+a_{n} \beta_{n}) \bigr]\rho^{2}(x_{n},p) \\ &{} -\alpha_{n}\beta_{n} \biggl[\frac{R}{2}(1-\beta _{n}) (1+a_{n})-\bigl(a_{n}+L^{2} \beta_{n}^{2}\bigr) \biggr]\rho^{2} \bigl(x_{n},T^{n}x_{n}\bigr) \\ &{} -\alpha_{n} \biggl[\frac{R}{2}(1-\alpha_{n})-(1- \beta_{n}) \biggr]\rho^{2}\bigl(x_{n},T^{n}y_{n} \bigr). \end{aligned}$$
This completes the proof. □

Lemma 3.5

Let \(\kappa>0\) and \((X, \rho)\) be a \(\operatorname{CAT}(\kappa)\) space with \(\operatorname{diam}(X)\leq\frac{\pi/2-\eta}{\sqrt{\kappa}}\) for some \(\eta\in(0,\pi/2)\). Let C be a nonempty convex subset of X, and \(T:C\to C\) be a uniformly L-Lipschitzian and asymptotically hemicontractive mapping with sequence \(\{a_{n}\}\) in \([1,\infty)\) such that \(\sum_{n=1}^{\infty}(a_{n}-1)<\infty\). Let \(\{\alpha_{n}\}, \{\beta_{n}\}\subset[0,1]\) be such that \(\frac{1-\beta_{n}}{1-\alpha_{n}}\leq\frac{R}{2}\) where \(R=(\pi-2\eta)\tan(\eta)\) and \(\alpha_{n}, \beta_{n}\in[\varepsilon, b]\) for some \(\varepsilon>0\) and \(b\in (0,\frac{\sqrt{R^{2}+4RL^{2}-4L^{2}}-R}{2L^{2}} )\). Given \(x_{1}\in C\), define the iteration scheme \(\{x_{n}\}\) by
$$\begin{aligned}& x_{n+1}=(1-\alpha_{n})x_{n}\oplus \alpha_{n}T^{n}y_{n}, \\& y_{n}=(1-\beta_{n})x_{n}\oplus\beta_{n} T^{n}x_{n},\quad n\geq1. \end{aligned}$$
Then
$$ \lim_{n\to\infty}\rho(x_{n},Tx_{n})=0. $$
(14)

Proof

First, we prove that \(\lim_{n\to\infty}\rho(x_{n},T^{n}x_{n})=0\). Since \(\frac{1-\beta_{n}}{1-\alpha_{n}}\leq\frac{R}{2}\), by Lemma 3.4 we have
$$\begin{aligned} \rho^{2}(x_{n+1},p)-\rho^{2}(x_{n},p) \leq&\alpha_{n} (a_{n}-1) (1+a_{n}\beta _{n})\rho^{2}(x_{n},p) \\ &{} -\alpha_{n}\beta_{n} \biggl[\frac{R}{2}(1-\beta _{n}) (1+a_{n})-\bigl(a_{n}+L^{2} \beta_{n}^{2}\bigr) \biggr]\rho^{2} \bigl(x_{n},T^{n}x_{n}\bigr). \end{aligned}$$
Since \(\{\alpha_{n}(1+a_{n}\beta_{n})\rho^{2}(x_{n},p) \}^{\infty }_{n=1}\) is a bounded sequence, there exists \(M>0\) such that
$$\begin{aligned} \rho^{2}(x_{n+1},p)-\rho^{2}(x_{n},p) \leq& (a_{n}-1) M \\ &{}-\alpha_{n}\beta_{n} \biggl[\frac {R}{2}(1- \beta_{n}) (1+a_{n})-\bigl(a_{n}+L^{2} \beta_{n}^{2}\bigr) \biggr]\rho^{2} \bigl(x_{n},T^{n}x_{n}\bigr). \end{aligned}$$
(15)
Let \(D=R(1-b)-(1+L^{2}b^{2})>0\). Since \(\lim_{n\to\infty}a_{n}=1\), there exists a natural number N such that
$$ \frac{R}{2}(1-\beta_{n}) (1+a_{n})- \bigl(a_{n}+L^{2}\beta _{n}^{2}\bigr)\geq \frac{R}{2}(1-b) (1+a_{n})-\bigl(a_{n}+L^{2}b^{2} \bigr)\geq\frac{D}{2}>0 $$
(16)
for all \(n\geq N\). Suppose that \(\lim_{n\to\infty}\rho(x_{n},T^{n}x_{n})\neq0\). Then there exist \(\varepsilon_{0}>0\) and a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that
$$ \rho^{2}\bigl(x_{n_{i}},T^{n_{i}}x_{n_{i}} \bigr)\geq \varepsilon_{0}. $$
(17)
Without loss of generality, we let \(n_{1}\geq N\). From (15), we have
$$\alpha_{n}\beta_{n} \biggl[\frac{R}{2}(1- \beta_{n}) (1+a_{n})-\bigl(a_{n}+L^{2} \beta _{n}^{2}\bigr) \biggr]\rho^{2} \bigl(x_{n},T^{n}x_{n}\bigr)\leq(a_{n}-1)M+ \rho^{2}(x_{n},p)-\rho^{2}(x_{n+1},p). $$
Then
$$\begin{aligned}& \sum_{l=1}^{i} \alpha_{n_{l}} \beta_{n_{l}} \biggl[\frac{R}{2}(1-\beta _{n_{l}}) (1+a_{n_{l}})-\bigl(a_{n_{l}}+L^{2}\beta_{n_{l}}^{2} \bigr) \biggr]\rho ^{2}\bigl(x_{n_{l}},T^{n_{l}}x_{n_{l}} \bigr) \\& \quad =\sum_{m=n_{1}}^{n_{i}} \alpha_{m}\beta_{m} \biggl[\frac{R}{2}(1-\beta _{m}) (1+a_{m})-\bigl(a_{m}+L^{2} \beta_{m}^{2}\bigr) \biggr]\rho^{2} \bigl(x_{m},T^{m}x_{m}\bigr) \\& \quad \leq\sum_{m=n_{1}}^{n_{i}} (a_{m}-1)M+\rho^{2}(x_{n_{1}},p)- \rho^{2}(x_{n_{i}+1},p). \end{aligned}$$
From this, together with (16), (17) and the fact that \(\varepsilon\leq\alpha_{n}\leq\beta_{n}\), we obtain
$$ i\cdot\varepsilon^{2}\cdot\frac{D}{2}\cdot \varepsilon_{0} \leq\sum_{m=n_{1}}^{n_{i}} (a_{m}-1)M+\rho^{2}(x_{n_{1}},p)-\rho ^{2}(x_{n_{i}+1},p). $$
(18)
If we take \(i\to\infty\), the right side of (18) is bounded while the left side is unbounded. This is a contradiction. Therefore \(\lim_{n\to\infty}\rho(x_{n},T^{n}x_{n})=0\), and hence \(\lim_{n\to\infty}\rho (x_{n},Tx_{n})=0\) by Lemma 3.1. □

Theorem 3.6

Let \(\kappa>0\) and \((X, \rho)\) be a \(\operatorname{CAT}(\kappa)\) space with \(\operatorname{diam}(X)\leq\frac{\pi/2-\eta}{\sqrt{\kappa}}\) for some \(\eta\in(0,\pi/2)\). Let C be a nonempty closed convex subset of X, and \(T:C\to C\) be a completely continuous and uniformly L-Lipschitzian asymptotically hemicontractive mapping with sequence \(\{a_{n}\}\) in \([1,\infty)\) such that \(\sum_{n=1}^{\infty}(a_{n}-1)<\infty\). Let \(\{\alpha_{n}\}, \{\beta_{n}\}\subset[0,1]\) be such that \(\frac{1-\beta_{n}}{1-\alpha_{n}}\leq\frac{R}{2}\) where \(R=(\pi-2\eta)\tan(\eta)\) and \(\alpha_{n}, \beta_{n}\in[\varepsilon, b]\) for some \(\varepsilon>0\) and \(b\in (0,\frac{\sqrt{R^{2}+4RL^{2}-4L^{2}}-R}{2L^{2}} )\). Given \(x_{1}\in C\), define the iteration scheme \(\{x_{n}\}\) by
$$\begin{aligned}& x_{n+1}=(1-\alpha_{n})x_{n}\oplus \alpha_{n}T^{n}y_{n}, \\& y_{n}=(1-\beta_{n})x_{n}\oplus\beta_{n} T^{n}x_{n}, \quad n\geq1. \end{aligned}$$
Then \(\{x_{n}\}\) converges strongly to a fixed point of T.

Proof

Since T is completely continuous, \(\{Tx_{n}\}\) has a convergent subsequence in C. By using Lemma 3.5, we can show that \(\{x_{n}\}\) has a convergent subsequence, say \(x_{n_{k}}\to q\in C\). Hence \(q\in F(T)\) by (14) and the continuity of T. It follows from (15) and (16) that
$$\rho^{2}(x_{n+1},p)\leq\rho^{2}(x_{n},p)+ (a_{n}-1)M. $$
Since \(\sum^{\infty}_{n=1}(a_{n}-1)<\infty\), by Lemma 2.4 we have \(x_{n}\to q\). This completes the proof. □

As consequences of Theorem 3.6, we obtain the following.

Corollary 3.7

Let \(\kappa>0\) and \((X, \rho)\) be a \(\operatorname {CAT}(\kappa)\) space with \(\operatorname{diam}(X)\leq\frac{\pi/2-\eta}{\sqrt{\kappa}}\) for some \(\eta\in(0,\pi/2)\). Let C be a nonempty closed convex subset of X, and \(T:C\to C\) be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive mapping with sequence \(\{a_{n}\}\) in \([1,\infty)\) such that \(\sum_{n=1}^{\infty}(a_{n}^{2}-1)<\infty\). Let \(\{\alpha_{n}\}, \{\beta_{n}\}\subset[0,1]\) be such that \(\frac{1-\beta_{n}}{1-\alpha_{n}}\leq\frac{R}{2}\) where \(R=(\pi-2\eta)\tan(\eta)\) and \(\alpha_{n}, \beta_{n}\in[\varepsilon, b]\) for some \(\varepsilon>0\) and \(b\in (0,\frac{\sqrt{R^{2}+4RL^{2}-4L^{2}}-R}{2L^{2}} )\). Given \(x_{1}\in C\), define the iteration scheme \(\{x_{n}\}\) by
$$\begin{aligned}& x_{n+1}=(1-\alpha_{n})x_{n}\oplus \alpha_{n}T^{n}y_{n}, \\& y_{n}=(1-\beta_{n})x_{n}\oplus\beta_{n} T^{n}x_{n}, \quad n\geq1. \end{aligned}$$
Then \(\{x_{n}\}\) converges strongly to a fixed point of T.

Corollary 3.8

(Theorem 11 of [29])

Let \((X, \rho)\) be a \(\operatorname{CAT}(0)\) space, let C be a nonempty bounded closed convex subset of X, and let \(T:C\to C\) be a completely continuous and uniformly L-Lipschitzian asymptotically hemicontractive mapping with sequence \(\{a_{n}\}\) in \([1,\infty)\) such that \(\sum_{n=1}^{\infty}(a_{n}-1)<\infty\). Let \(\{\alpha_{n}\}, \{\beta_{n}\}\subset[0,1]\) be such that \(\varepsilon \leq\alpha_{n}\leq\beta_{n}\leq b\) for some \(\varepsilon>0\) and \(b\in (0,\frac{\sqrt{1+L^{2}}-1}{L^{2}} )\). Given \(x_{1}\in C\), define the iteration scheme \(\{x_{n}\}\) by
$$\begin{aligned}& x_{n+1}=(1-\alpha_{n})x_{n}\oplus \alpha_{n}T^{n}y_{n}, \\& y_{n}=(1-\beta_{n})x_{n}\oplus\beta_{n} T^{n}x_{n},\quad n\geq1. \end{aligned}$$
Then \(\{x_{n}\}\) converges strongly to a fixed point of T.

Declarations

Acknowledgements

The author thanks a referee for his/her careful reading and valuable comments and suggestions which led to the present form of the paper. This research was supported by Chiang Mai University.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Chiang Mai University

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