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On an open problem of Kyung Soo Kim

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).

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.

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]).

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.

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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.

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Panyanak, B. On an open problem of Kyung Soo Kim. Fixed Point Theory Appl 2015, 186 (2015). https://doi.org/10.1186/s13663-015-0438-7

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MSC

  • 47H09
  • 49J53

Keywords

  • Mann iteration
  • Ishikawa iteration
  • strong convergence
  • fixed point
  • \(\operatorname{CAT}(\kappa)\) space