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Viscosity iteration method in CAT(0) spaces without the nice projection property

Abstract

A complete CAT(0) space X is said to have the nice projection property (property \(\mathcal{N}\) for short) if its metric projection onto a geodesic segment preserves points on each geodesic segment, that is, for any geodesic segment L in X and \(x,y\in X\), \(m\in[x,y]\) implies \(P_{L}(m)\in[P_{L}(x), P_{L}(y)]\), where \(P_{L}\) denotes the metric projection from X onto L. In this paper, we prove a strong convergence theorem of a two-step viscosity iteration method for nonexpansive mappings in CAT(0) spaces without the condition on the property \(\mathcal{N}\). Our result gives an affirmative answer to a problem raised by Piatek (Numer. Funct. Anal. Optim. 34:1245-1264, 2013).

Introduction

A mapping T on a metric space \((X,\rho)\) is said to be a contraction if there exists a constant \(k\in[0,1)\) such that

$$ \rho\bigl(T(x),T(y)\bigr)\leq k \rho(x,y) \quad \mbox{for all } x, y\in X. $$
(1)

If (1) is valid when \(k=1\), then T is called nonexpansive. A point \(x\in X\) is called a fixed point of T if \(x=T(x)\). We shall denote by \(\operatorname{Fix}(T)\) the set of all fixed points of T.

One of the powerful iteration methods for finding fixed points of nonexpansive mappings was given by Moudafi [1]. More precisely, let C be a nonempty, closed, and convex subset of a Hilbert space H and \(T : C \to C\) be a nonexpansive mapping with \(\operatorname{Fix}(T)\neq\emptyset\), the following scheme is known as the viscosity iteration method:

$$\begin{aligned} &x_{1}=u\in C \mbox{ arbitrarily chosen}, \\ &x_{n+1}=\frac{\alpha_{n}}{1+\alpha _{n}}f(x_{n})+\frac{1}{1+\alpha_{n}}T(x_{n}), \end{aligned}$$
(2)

where \(f : C \to C\) is a contraction and \(\{\alpha_{n}\}\) is a sequence in \((0, 1)\) satisfying (i) \(\lim_{n\to\infty} \alpha_{n} =0\), (ii) \(\sum^{\infty}_{n=1} \alpha_{n} = \infty\), and (iii) \(\lim_{n\to\infty} (1/\alpha_{n} - 1/\alpha_{n+1})=0\). In [1], the author proved that the sequence \(\{x_{n}\}\) defined by (2) converges strongly to a fixed point z of T. The point z also satisfies the following variational inequality:

$$\bigl\langle f(z)-z, z-x \bigr\rangle \geq0,\quad x\in \operatorname{Fix}(T). $$

The first extension of Moudafi’s result to the so-called CAT(0) space was proved by Shi and Chen [2]. They assumed that the space \((X, \rho)\) must satisfy the property \(\mathcal{P}\), i.e., for \(x, u, y_{1}, y_{2}\in X\), one has

$$\rho(x, m_{1})\rho(x, y_{1})\leq\rho(x, m_{2}) \rho(x, y_{2}) + \rho(x, u)\rho (y_{1}, y_{2}), $$

where \(m_{1}\) and \(m_{2}\) are the unique nearest points of u on the segments \([x, y_{1}]\) and \([x, y_{2}]\), respectively. By using the concept of quasi-linearization introduced by Berg and Nikolaev [3], Wangkeeree and Preechasilp [4] could omit the property \(\mathcal{P}\) from Shi and Chen’s result as the following theorem.

Theorem A

([4], Theorem 3.4)

Let C be a nonempty, closed, and convex subset of a complete CAT(0) space X, \(T : C \to C\) be a nonexpansive mapping with \(\operatorname{Fix}(T)\neq\emptyset\), and \(f:C\to C\) be a contraction with \(k\in[0,1)\). For \(x_{1}\in C\), let \(\{x_{n}\}\) be generated by

$$x_{n+1}=\alpha_{n} f(x_{n})\oplus (1- \alpha_{n}) T(x_{n}), \quad \forall n\geq1, $$

where \(\{\alpha_{n}\}\subset(0,1)\) satisfies the conditions: (i) \(\lim_{n\to\infty} \alpha_{n} =0\), (ii) \(\sum^{\infty}_{n=1} \alpha_{n} = \infty\), (iii) either \(\sum^{\infty}_{n=1} |\alpha_{n+1}-\alpha_{n}|<\infty\) or \(\lim_{n\to\infty} (\alpha_{n+1}/\alpha_{n}) =1\). Then \(\{x_{n}\}\) converges strongly to such that \(\tilde{x}=P_{\operatorname{Fix}(T)}(f(\tilde{x}))\) which is equivalent to the variational inequality:

$$\bigl\langle \overrightarrow{\tilde{x}f(\tilde{x})}, \overrightarrow{x \tilde {x}}\bigr\rangle \geq0, \quad x\in \operatorname{Fix}(T). $$

Among other things, by using the geometric properties of CAT(0) spaces, Piatek [5] proved the strong convergence of a two-step viscosity iteration method as the following result.

Theorem B

([5], Theorem 4.3)

Let X be a complete CAT(0) space with the property \(\mathcal{N}\). Let \(T: X \to X\) be a nonexpansive mapping with \(\operatorname{Fix}(T)\neq\emptyset\) and \(f : X\to X\) be a contraction with \(k\in [0,\frac{1}{2} )\). Then there is a unique point \(q\in \operatorname{Fix}(T)\) such that \(q=P_{\operatorname{Fix}(T)}(f(q))\). Moreover, for each \(u\in X\) and for each couple of sequences \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) in \((0,1)\) satisfying (i) \(\lim_{n\to\infty} \alpha_{n} =0\), (ii) \(\sum^{\infty}_{n=1} \alpha_{n} = \infty\), and (iii) \(0<\liminf_{n} \beta_{n} \leq\limsup_{n} \beta_{n} <1\), the viscosity iterative sequence defined by \(x_{1}=u\),

$$\begin{aligned}& y_{n}=\alpha_{n} f(x_{n})\oplus(1- \alpha_{n}) T(x_{n}), \\& x_{n+1}=\beta_{n} x_{n}\oplus(1- \beta_{n}) y_{n}, \quad \forall n\geq1, \end{aligned}$$

converges to q.

In [5], the author provided an example of a CAT(0) space lacking property \(\mathcal{N}\) and also raised the following open problem.

Problem

Can we omit the property \(\mathcal{N}\) in Theorem B?

In this paper, by combining the ideas of [4] and [5] intensively, we can omit the property \(\mathcal{N}\) from Theorem B. This gives a complete solution to the problem mentioned above.

Preliminaries

Let \([0,l]\) be a closed interval in \(\mathbb{R}\) and x, y be two points in a metric space \((X,\rho)\). A geodesic joining x to y is a map \(\xi:[0,l]\to X\) such that \(\xi(0)=x\), \(\xi(l)=y\), and \(\rho(\xi(s),\xi(t))=|s-t|\) for all \(s, t\in[0,l]\). The image of ξ is called a geodesic segment joining x and y which when unique is denoted by \([x,y]\). The space \((X,\rho)\) is said to be a geodesic space if every two points in 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 C of X is said to be convex if every pair of points \(x,y\in C\) can be joined by a geodesic in X and the image of every such geodesic is contained in C.

A geodesic triangle \(\triangle(p, q, r)\) in a geodesic space \((X,\rho)\) consists of three points p, q, r in X and a choice of three geodesic segments \([p, q]\), \([q, r]\), \([r, p]\) joining them. A comparison triangle for the geodesic triangle \(\triangle(p, q, r)\) in X is a triangle \(\overline{\triangle}(\bar{p}, \bar{q}, \bar{r})\) in the Euclidean plane \(\mathbb{R}^{2}\) such that \(d_{\mathbb{R}^{2}} ( \bar{p},\bar{q} ) =\rho(p, q)\), \(d_{\mathbb{R}^{2}} ( \bar{q},\bar{r} ) =\rho(q, r)\), and \(d_{\mathbb{R}^{2}} ( \bar{r},\bar{p} ) =\rho(r, p)\). A point \(\bar{u}\in[\bar{p}, \bar{q}]\) is called a comparison point for \(u\in[p, q]\) if \(\rho(p, u)=d_{\mathbb{R}^{2}}(\bar{p},\bar{u})\). Comparison points on \([\bar{q}, \bar{r}]\) and \([\bar{r}, \bar{p}]\) are defined in the same way.

Definition 2.1

A geodesic triangle \(\triangle(p, q, r)\) in \((X,\rho)\) is said to satisfy the CAT(0) inequality if for any \(u,v\in\triangle(p, q, r)\) and for their comparison points \(\bar{u}, \bar{v}\in \overline{\triangle}(\bar{p}, \bar{q}, \bar{r})\), one has

$$\rho(u,v)\leq d_{\mathbb{R}^{2}}(\bar{u}, \bar{v}). $$

A geodesic space X is said to be a CAT(0) space if all of its geodesic triangles satisfy the CAT(0) inequality. For other equivalent definitions and basic properties of CAT(0) spaces, we refer the reader to standard texts, such as [6, 7]. It is well known that every CAT(0) space is uniquely geodesic. Notice also that pre-Hilbert spaces, \(\mathbb{R}\)-trees, Euclidean buildings are examples of CAT(0) spaces (see [6, 8]). Let C be a nonempty, closed, and convex subset of a complete CAT(0) space \((X,\rho)\). It follows from Proposition 2.4 of [6] that for each \(x\in X\), there exists a unique point \(x_{0}\in C\) such that

$$\rho(x, x_{0})=\inf\bigl\{ \rho(x,y) : y\in C\bigr\} . $$

In this case, \(x_{0}\) is called the unique nearest point of x in C. The metric projection of X onto C is the mapping \(P_{C}:X\to C\) defined by

$$P_{C}(x):= \mbox{the unique nearest point of } x \mbox{ in } C. $$

Definition 2.2

A complete CAT(0) space X is said to have the nice projection property [9] if for any geodesic segment L in X, it is the case that \(P_{L}(m)\in[P_{L}(x), P_{L}(y)]\) for any \(x,y\in X\) and \(m\in[x,y]\).

Let \((X,\rho)\) be a CAT(0) space. For each \(x,y\in X\) and \(t\in[0,1]\), there exists a unique point \(z\in[x,y]\) such that

$$ \rho(x,z)=(1-t)\rho(x,y) \quad\mbox{and}\quad \rho(y,z)=t \rho(x,y). $$
(3)

We shall denote by \(tx\oplus(1-t)y\) the unique point z satisfying (3). Now, we collect some elementary facts about CAT(0) spaces which will be used in the proof of our main theorem.

Lemma 2.3

([10], Lemma 2.4)

Let \((X,\rho)\) be a CAT(0) space. Then

$$\rho\bigl(tx\oplus(1-t)y,z\bigr)\leq t\rho(x,z)+(1-t)\rho(y,z) $$

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

Lemma 2.4

([10], Lemma 2.5)

Let \((X,\rho)\) be a CAT(0) space. Then

$$\rho^{2}\bigl(t x\oplus(1-t)y,z\bigr)\leq t \rho^{2}(x,z)+(1-t) \rho^{2}(y,z)-t(1-t)\rho^{2}(x,y) $$

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

Lemma 2.5

([11], Lemma 3)

Let \((X,\rho)\) be a CAT(0) space. Then

$$\rho\bigl(tx\oplus(1-t)z,ty\oplus(1-t)z\bigr)\leq t \rho(x,y) $$

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

Lemma 2.6

(cf. [12, 13])

Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be bounded sequences in a CAT(0) space \((X,\rho)\) and let \(\{\beta_{n}\}\) be a sequence in \([0,1]\) with \(0<\liminf_{n}\beta_{n}\leq\limsup_{n}\beta_{n}<1\). Suppose that \(x_{n+1}=\beta_{n}x_{n}\oplus(1-\beta_{n})y_{n}\) for all \(n\in \mathbb{N}\) and

$$\limsup_{n\to\infty} \bigl(\rho(y_{n+1},y_{n})- \rho(x_{n+1},x_{n}) \bigr)\leq0. $$

Then \(\lim_{n\to\infty} \rho(x_{n},y_{n})=0\).

Lemma 2.7

([14], Lemma 2.1)

Let \(\{s_{n}\}\) be a sequence of non-negative real numbers satisfying

$$s_{n+1}\leq(1-\alpha_{n})s_{n}+\alpha_{n} \beta_{n}, \quad \forall n\geq1, $$

where \(\{\alpha_{n}\}\subset(0,1)\) and \(\{\beta_{n}\}\subset \mathbb{R}\) such that

  1. (i)

    \(\sum_{n=1}^{\infty} \alpha_{n}=\infty\);

  2. (ii)

    \(\limsup_{n\to\infty}\beta_{n} \leq0\) or \(\sum_{n=1}^{\infty} |\alpha_{n}\beta_{n}| < \infty\).

Then \(\{s_{n}\}\) converges to zero as \(n\to\infty\).

We finish this section by recalling an important concept of quasi-linearization introduced by Berg and Nikolaev [3]. Let us denote a pair \((a, b) \in X\times X\) by \(\overrightarrow{ab}\) and call it a vector. The quasi-linearization is a map \(\langle\cdot, \cdot\rangle: (X \times X)\times(X\times X)\to\mathbb{R}\) defined by

$$\langle\overrightarrow{ab}, \overrightarrow{cd}\rangle=\frac {1}{2} \bigl( \rho^{2}(a, d) + \rho^{2}(b, c) -\rho^{2}(a, c)- \rho^{2}(b, d) \bigr) \quad\mbox{for all } a, b, c, d \in X. $$

It is easy to see that \(\langle\overrightarrow{ab}, \overrightarrow{cd}\rangle=\langle\overrightarrow{cd}, \overrightarrow{ab}\rangle\), \(\langle\overrightarrow{ab}, \overrightarrow{cd}\rangle=- \langle\overrightarrow{ba}, \overrightarrow{cd}\rangle\), and \(\langle\overrightarrow{ax}, \overrightarrow{cd}\rangle+\langle\overrightarrow{xb}, \overrightarrow{cd}\rangle=\langle\overrightarrow{ab}, \overrightarrow{cd}\rangle\) for all \(a, b, c, d, x \in X\). We say that \((X,\rho)\) satisfies the Cauchy-Schwarz inequality if

$$\bigl\vert \langle\overrightarrow{ab}, \overrightarrow{cd}\rangle\bigr\vert \leq\rho(a, b)\rho(c, d) \quad \mbox{for all } a, b, c, d \in X. $$

It is known from [3], Corollary 3, that a geodesic space X is a CAT(0) space if and only if X satisfies the Cauchy-Schwarz inequality. Some other properties of quasi-linearization are included as follows.

Lemma 2.8

([4], Lemma 2.9)

Let X be a CAT(0) space. Then

$$\rho^{2}(x,u) \leq\rho^{2}(y,u) + 2 \langle \overrightarrow{xy}, \overrightarrow{xu} \rangle $$

for all \(u, x, y \in X\).

Lemma 2.9

([4], Lemma 2.10)

Let u and v be two points in a CAT(0) space X. For each \(t\in[0, 1]\), we set \(u_{t} = t u\oplus (1-t)v\). Then, for each \(x, y\in X\), we have

  1. (i)

    \(\langle\overrightarrow{u_{t} x}, \overrightarrow{u_{t} y}\rangle\leq t\langle\overrightarrow{ux}, \overrightarrow{u_{t} y}\rangle+(1-t)\langle\overrightarrow{vx}, \overrightarrow{u_{t} y}\rangle\);

  2. (ii)

    \(\langle\overrightarrow{u_{t} x}, \overrightarrow{uy}\rangle\leq t\langle\overrightarrow{ux}, \overrightarrow{uy}\rangle+(1-t)\langle\overrightarrow{vx}, \overrightarrow{uy}\rangle\) and \(\langle\overrightarrow{u_{t} x}, \overrightarrow{vy}\rangle\leq t\langle\overrightarrow{ux}, \overrightarrow{vy}\rangle+(1-t)\langle\overrightarrow{vx}, \overrightarrow{vy}\rangle\).

The following fact, which can be found in [15], is an immediate consequence of Lemma 2.4.

Lemma 2.10

Let X be a CAT(0) space. Then

$$\rho^{2}\bigl(t x\oplus(1-t)y, z\bigr)\leq t^{2} \rho^{2}(x,z)+(1-t)^{2} \rho ^{2}(y,z)+2t(1-t) \langle\overrightarrow{xz}, \overrightarrow{yz}\rangle $$

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

Main theorem

Before proving our main theorem, we need one more lemma, which is proved by Wangkeeree and Preechasilp (see [4], Theorem 3.1).

Lemma 3.1

Let C be a nonempty, closed, and convex subset of a complete CAT(0) space X, \(T : C \to C\) be a nonexpansive mapping with \(\operatorname{Fix}(T)\neq\emptyset\), and \(f:C\to C\) be a contraction with \(k\in[0,1)\). For each \(t \in (0, 1)\), let \(\{z_{t}\}\) be given by

$$z_{t}=t f(z_{t})\oplus(1-t) T(z_{t}). $$

Then \(\{z_{t}\}\) converges strongly to as \(t\to0\). Moreover, \(\tilde{x}=P_{\operatorname{Fix}(T)}(f(\tilde{x}))\) and also satisfies the following variational inequality:

$$ \bigl\langle \overrightarrow{\tilde{x}f(\tilde{x})}, \overrightarrow{x \tilde{x}}\bigr\rangle \geq0, \quad x\in \operatorname{Fix}(T). $$
(4)

Now, we are ready to prove our main theorem.

Theorem 3.2

Let C be a nonempty, closed, and convex subset of a complete CAT(0) space X, \(T : C \to C\) be a nonexpansive mapping with \(\operatorname{Fix}(T)\neq\emptyset\), and \(f:C\to C\) be a contraction with \(k\in [0,\frac{1}{2} )\). For the arbitrary initial point \(u\in C\), let \(\{x_{n}\}\) be generated by

$$\begin{aligned}[b] &x_{1}=u, \\ &y_{n}=\alpha_{n} f(x_{n})\oplus(1- \alpha_{n}) T(x_{n}), \\ &x_{n+1}=\beta_{n} x_{n}\oplus(1- \beta_{n}) y_{n}, \quad \forall n\geq1, \end{aligned} $$

where \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are sequences in \((0,1)\) satisfying the following conditions:

  1. (i)

    \(\lim_{n\to\infty} \alpha_{n} =0\);

  2. (ii)

    \(\sum^{\infty}_{n=1} \alpha_{n} = \infty\);

  3. (iii)

    \(0<\liminf_{n} \beta_{n} \leq\limsup_{n} \beta_{n} <1\).

Then \(\{x_{n}\}\) converges strongly to such that \(\tilde{x}=P_{\operatorname{Fix}(T)}(f(\tilde{x}))\) and also satisfies

$$\bigl\langle \overrightarrow{\tilde{x}f(\tilde{x})}, \overrightarrow{x \tilde {x}}\bigr\rangle \geq0, \quad x\in \operatorname{Fix}(T). $$

Proof

We divide the proof into three steps.

Step 1. We show that \(\{x_{n}\}\), \(\{y_{n}\}\), \(\{T(x_{n})\}\), and \(\{f(x_{n})\}\) are bounded sequences. Let \(p\in \operatorname{Fix}(T)\). By Lemma 2.3, we have

$$\begin{aligned} \rho(x_{n+1}, p) \leq& \beta_{n} \rho(x_{n}, p)+(1-\beta _{n}) \rho(y_{n}, p) \\ \leq& \beta_{n} \rho(x_{n}, p)+(1-\beta_{n}) \bigl[\alpha_{n} \rho\bigl(f(x_{n}), p\bigr)+(1- \alpha_{n})\rho\bigl(T(x_{n}), p\bigr)\bigr] \\ \leq& \bigl[\beta_{n}+(1-\beta_{n}) (1- \alpha_{n})\bigr]\rho(x_{n}, p)+(1-\beta_{n})\alpha _{n} \rho\bigl(f(x_{n}), f(p)\bigr)\\ &{}+(1-\beta_{n}) \alpha_{n} \rho\bigl(f(p), p\bigr) \\ \leq& \bigl[1-(1-k)\alpha_{n}+(1-k)\alpha_{n} \beta_{n}\bigr]\rho(x_{n}, p)+(1-\beta _{n}) \alpha_{n} \rho\bigl(f(p), p\bigr) \\ \leq& \max \biggl\{ \rho(x_{n},p), \frac{\rho(f(p), p)}{1-k} \biggr\} . \end{aligned}$$

By induction, we also have

$$\rho(x_{n},p)\leq\max \biggl\{ \rho(x_{1},p), \frac{\rho(f(p), p)}{1-k} \biggr\} . $$

Hence, \(\{x_{n}\}\) is bounded and so are \(\{y_{n}\}\), \(\{f(x_{n})\}\), and \(\{T(x_{n})\}\).

Step 2. We show that \(\lim_{n\to\infty} \rho(x_{n},T(x_{n}))=0\). By applying Lemma 2.5 twice for geodesic triangles \(\triangle(f(x_{n}), T(x_{n}), T(x_{n+1}))\) and \(\triangle(f(x_{n}), f(x_{n+1}), T(x_{n+1}))\), respectively, we obtain

$$\begin{aligned} \rho(y_{n}, y_{n+1}) \leq& (1-\alpha_{n}) \rho \bigl(T(x_{n}), T(x_{n+1})\bigr) + |\alpha_{n}- \alpha_{n+1}| \rho\bigl(f(x_{n}), T(x_{n+1})\bigr) \\ &{}+\alpha_{n+1}\rho\bigl(f(x_{n}), f(x_{n+1}) \bigr) \\ \leq& (1-\alpha_{n}) \rho(x_{n}, x_{n+1}) + | \alpha_{n}-\alpha_{n+1}| \rho\bigl(f(x_{n}), T(x_{n+1})\bigr) \\ &{} +\alpha_{n+1} k \rho(x_{n}, x_{n+1}), \end{aligned}$$

which implies

$$\rho(y_{n}, y_{n+1})-\rho(x_{n}, x_{n+1}) \leq(\alpha_{n+1} k-\alpha_{n}) \rho (x_{n}, x_{n+1})+|\alpha_{n}-\alpha_{n+1}| \rho \bigl(f(x_{n}), T(x_{n+1})\bigr). $$

Since \(\lim_{n\to\infty} \alpha_{n} =0\), \(\limsup_{n\to\infty} (\rho(y_{n+1},y_{n})-\rho(x_{n+1},x_{n}) )\leq0\). By Lemma 2.6 we have \(\lim_{n\to\infty} \rho(x_{n}, y_{n})=0\). Thus,

$$\begin{aligned} \rho\bigl(x_{n}, T(x_{n})\bigr)&\leq\rho(x_{n}, y_{n}) +\rho\bigl(y_{n}, T(x_{n})\bigr)\\ &= \rho (x_{n},y_{n})+\alpha_{n} \rho\bigl(f(x_{n}), T(x_{n})\bigr)\to0 \quad \mbox{as } n\to\infty. \end{aligned}$$

Step 3. We show that \(\{x_{n}\}\) converges to , which satisfies \(\tilde{x}=P_{\operatorname{Fix}(T)}(f(\tilde{x}))\) and

$$\bigl\langle \overrightarrow{\tilde{x}f(\tilde{x})}, \overrightarrow{x \tilde {x}}\bigr\rangle \geq0, \quad x\in \operatorname{Fix}(T). $$

Let \(\{z_{m}\}\) be a sequence in C defined by

$$z_{m}=\alpha_{m} f(z_{m})\oplus(1- \alpha_{m}) T(z_{m}), \quad \forall m\in\mathbb{N}. $$

By Lemma 3.1, \(\{z_{m}\}\) converges strongly as \(m\to \infty\) to which satisfies (4) and \(\tilde{x}=P_{\operatorname{Fix}(T)}(f(\tilde{x}))\). We claim that

$$\limsup_{n\to\infty} \bigl\langle \overrightarrow{f(\tilde{x}) \tilde{x}}, \overrightarrow{x_{n}\tilde{x}}\bigr\rangle \leq0. $$

It follows from Lemma 2.9(i) that

$$\begin{aligned} \rho^{2}(z_{m},x_{n}) = & \langle \overrightarrow{z_{m} x_{n}}, \overrightarrow{z_{m} x_{n}}\rangle \\ \leq& \alpha_{m}\bigl\langle \overrightarrow{f(z_{m}) x_{n}}, \overrightarrow{z_{m} x_{n}}\bigr\rangle +(1-\alpha_{m})\bigl\langle \overrightarrow{T(z_{m}) x_{n}}, \overrightarrow{z_{m} x_{n}}\bigr\rangle \\ = & \alpha_{m}\bigl\langle \overrightarrow{f(z_{m}) f( \tilde{x})}, \overrightarrow{z_{m} x_{n}}\bigr\rangle + \alpha_{m}\bigl\langle \overrightarrow{f(\tilde{x}) \tilde{x}}, \overrightarrow{z_{m} x_{n}}\bigr\rangle + \alpha_{m}\langle\overrightarrow{\tilde{x} z_{m}}, \overrightarrow{z_{m} x_{n}}\rangle+\alpha_{m} \langle\overrightarrow{z_{m} x_{n}}, \overrightarrow{z_{m} x_{n}}\rangle \\ &{}+(1-\alpha_{m})\bigl\langle \overrightarrow{T(z_{m}) T(x_{n})}, \overrightarrow{z_{m} x_{n}}\bigr\rangle +(1-\alpha_{m})\bigl\langle \overrightarrow{T(x_{n}) x_{n}}, \overrightarrow{z_{m} x_{n}}\bigr\rangle \\ \leq& \alpha_{m} k \rho(z_{m},\tilde{x}) \rho(z_{m},x_{n})+ \alpha_{m}\bigl\langle \overrightarrow{f(\tilde{x}) \tilde{x}}, \overrightarrow{z_{m} x_{n}}\bigr\rangle +\alpha_{m}\rho(\tilde{x},z_{m}) \rho(z_{m},x_{n}) \\ &{}+\alpha_{m}\rho ^{2}(z_{m},x_{n})+(1-\alpha_{m})\rho^{2}(z_{m},x_{n})+(1- \alpha_{m})\rho\bigl(T(x_{n}),x_{n}\bigr)\rho (z_{m},x_{n}) \\ \leq& \alpha_{m} (k+1) \rho(z_{m},\tilde{x})M+\rho \bigl(T(x_{n}),x_{n}\bigr)M+\rho^{2}(z_{m},x_{n})+ \alpha_{m}\bigl\langle \overrightarrow{f(\tilde{x}) \tilde{x}}, \overrightarrow{z_{m} x_{n}}\bigr\rangle , \end{aligned}$$

for some \(M>0\). This implies

$$ \bigl\langle \overrightarrow{f(\tilde{x}) \tilde{x}}, \overrightarrow{x_{n} z_{m}}\bigr\rangle \leq (k+1) \rho(z_{m},\tilde{x})M+\frac{\rho(x_{n},T(x_{n}))}{\alpha_{m}}M. $$
(5)

Taking the upper limit as \(n\to\infty\) first and then \(m\to\infty\), the inequality (5) yields

$$ \limsup_{m\to\infty}\limsup_{n\to\infty } \bigl\langle \overrightarrow{f(\tilde{x}) \tilde{x}}, \overrightarrow{x_{n} z_{m}}\bigr\rangle \leq0. $$
(6)

Notice also that

$$\begin{aligned} \bigl\langle \overrightarrow{f(\tilde{x}) \tilde{x}}, \overrightarrow{x_{n} \tilde{x}}\bigr\rangle = \bigl\langle \overrightarrow{f(\tilde{x}) \tilde{x}}, \overrightarrow{x_{n} z_{m}}\bigr\rangle + \bigl\langle \overrightarrow{f(\tilde{x}) \tilde{x}}, \overrightarrow{z_{m} \tilde{x}}\bigr\rangle \leq \bigl\langle \overrightarrow{f(\tilde{x}) \tilde{x}}, \overrightarrow{x_{n} z_{m}}\bigr\rangle + \rho\bigl(f( \tilde{x}), \tilde{x}\bigr) \rho(z_{m}, \tilde{x}). \end{aligned}$$

This, together with (6), implies that

$$\limsup_{n\to\infty} \bigl\langle \overrightarrow{f(\tilde{x}) \tilde{x}}, \overrightarrow{x_{n}\tilde{x}}\bigr\rangle \leq0. $$

Finally, we show that \(x_{n}\to\tilde{x}\) as \(n\to\infty\). It follows from Lemmas 2.4, 2.8, 2.9, and 2.10 that

$$\begin{aligned} \rho^{2}(x_{n+1},\tilde{x}) \leq& \beta_{n} \rho^{2}(x_{n},\tilde{x})+(1-\beta_{n}) \rho^{2}(y_{n},\tilde{x}) \\ \leq& \beta_{n} \rho^{2}(x_{n},\tilde{x})+(1- \beta_{n}) \bigl[\alpha_{n}^{2}\rho^{2} \bigl(f(x_{n}),\tilde {x}\bigr)+(1-\alpha_{n})^{2} \rho^{2}\bigl(T(x_{n}),\tilde{x}\bigr) \bigr] \\ &{}+2\alpha_{n}(1-\alpha_{n}) (1-\beta_{n})\bigl\langle \overrightarrow{f(x_{n})\tilde {x}},\overrightarrow{T(x_{n}) \tilde{x}}\bigr\rangle \\ \leq& \beta_{n}\rho^{2}(x_{n},\tilde{x})+(1- \beta_{n}) (1-\alpha_{n})^{2}\rho ^{2}(x_{n},\tilde{x}) \\ &{}+\alpha^{2}_{n}(1-\beta_{n}) \bigl[ \rho^{2}\bigl(x_{n+1},f(x_{n})\bigr)+2\bigl\langle \overrightarrow{\tilde{x}x_{n+1}},\overrightarrow{\tilde {x}f(x_{n})}\bigr\rangle \bigr] \\ &{}+2\alpha_{n}(1-\alpha_{n}) (1-\beta_{n}) \bigl[\bigl\langle \overrightarrow {f(x_{n})\tilde{x}}, \overrightarrow{T(x_{n})x_{n}}\bigr\rangle +\bigl\langle \overrightarrow{f(x_{n})\tilde{x}},\overrightarrow{x_{n} \tilde{x}}\bigr\rangle \bigr] \\ \leq& \bigl[\beta_{n}+(1-\beta_{n}) (1- \alpha_{n}) \bigr]\rho^{2}(x_{n},\tilde {x})+ \alpha^{2}_{n}(1-\beta_{n})\rho^{2} \bigl(x_{n+1},f(x_{n})\bigr) \\ &{}+2\alpha^{2}_{n}(1-\beta_{n}) \bigl[\bigl\langle \overrightarrow{f(x_{n})f(\tilde {x})},\overrightarrow{x_{n+1} \tilde{x}}\bigr\rangle +\bigl\langle \overrightarrow {f(\tilde{x})\tilde{x}}, \overrightarrow{x_{n+1}\tilde{x}}\bigr\rangle \bigr] \\ &{}+2\alpha_{n}(1-\alpha_{n}) (1-\beta_{n})\bigl\langle \overrightarrow{f(x_{n})\tilde {x}},\overrightarrow{T(x_{n})x_{n}} \bigr\rangle \\ &{}+2\alpha_{n}(1-\alpha_{n}) (1-\beta_{n}) \bigl[\bigl\langle \overrightarrow {f(x_{n})f(\tilde{x})}, \overrightarrow{x_{n}\tilde{x}}\bigr\rangle +\bigl\langle \overrightarrow{f(\tilde{x})\tilde{x}},\overrightarrow{x_{n}\tilde {x}} \bigr\rangle \bigr] \\ \leq& \bigl[\beta_{n}+(1-\beta_{n}) (1- \alpha_{n}) \bigr]\rho^{2}(x_{n},\tilde {x})+ \alpha^{2}_{n}(1-\beta_{n})\rho^{2} \bigl(x_{n+1},f(x_{n})\bigr) \\ &{}+2\alpha^{2}_{n}(1-\beta_{n})\rho \bigl(f(x_{n}),f(\tilde{x})\bigr)\rho(x_{n+1},\tilde {x})+2 \alpha^{2}_{n}(1-\beta_{n})\bigl\langle \overrightarrow{f(\tilde{x})\tilde {x}},\overrightarrow{x_{n+1}\tilde{x}} \bigr\rangle \\ &{}+2\alpha_{n}(1-\alpha_{n}) (1-\beta_{n})\rho \bigl(f(x_{n}),\tilde{x}\bigr)\rho \bigl(T(x_{n}),x_{n} \bigr) \\ &{}+2\alpha_{n}(1-\alpha_{n}) (1-\beta_{n})\rho \bigl(f(x_{n}),f(\tilde{x})\bigr)\rho (x_{n},\tilde{x})\\ &{}+2 \alpha_{n}(1-\alpha_{n}) (1-\beta_{n})\bigl\langle \overrightarrow {f(\tilde{x})\tilde{x}},\overrightarrow{x_{n}\tilde{x}} \bigr\rangle \\ \leq& \bigl[\beta_{n}+(1-\beta_{n}) (1- \alpha_{n}) \bigr]\rho^{2}(x_{n},\tilde {x})+ \alpha^{2}_{n}(1-\beta_{n})\rho^{2} \bigl(x_{n+1},f(x_{n})\bigr) \\ &{}+2k\alpha^{2}_{n}(1-\beta_{n}) \rho(x_{n},\tilde{x})\rho(x_{n+1},\tilde {x})+2 \alpha^{2}_{n}(1-\beta_{n})\bigl\langle \overrightarrow{f(\tilde{x})\tilde {x}},\overrightarrow{x_{n+1}\tilde{x}} \bigr\rangle \\ &{}+2\alpha_{n}(1-\alpha_{n}) (1-\beta_{n})\rho \bigl(f(x_{n}),\tilde{x}\bigr)\rho \bigl(x_{n},T(x_{n}) \bigr) \\ &{}+2k\alpha_{n}(1-\alpha_{n}) (1-\beta_{n}) \rho^{2}(x_{n},\tilde{x})+2\alpha _{n}(1- \alpha_{n}) (1-\beta_{n})\bigl\langle \overrightarrow{f( \tilde{x})\tilde {x}},\overrightarrow{x_{n}\tilde{x}}\bigr\rangle \\ \leq& \bigl[\beta_{n}+(1-\beta_{n}) (1- \alpha_{n}) \bigr]\rho^{2}(x_{n},\tilde {x})+ \alpha^{2}_{n}(1-\beta_{n})\rho^{2} \bigl(x_{n+1},f(x_{n})\bigr) \\ &{}+k\alpha^{2}_{n}(1-\beta_{n}) \bigl[ \rho^{2}(x_{n},\tilde{x})+\rho ^{2}(x_{n+1}, \tilde{x}) \bigr]+2\alpha^{2}_{n}(1-\beta_{n})\bigl\langle \overrightarrow{f(\tilde{x})\tilde{x}},\overrightarrow{x_{n+1} \tilde {x}}\bigr\rangle \\ &{}+2\alpha_{n}(1-\alpha_{n}) (1-\beta_{n})\rho \bigl(f(x_{n}),\tilde{x}\bigr)\rho \bigl(x_{n},T(x_{n}) \bigr) \\ &{}+2k\alpha_{n}(1-\alpha_{n}) (1-\beta_{n}) \rho^{2}(x_{n},\tilde{x})+2\alpha _{n}(1- \alpha_{n}) (1-\beta_{n})\bigl\langle \overrightarrow{f( \tilde{x})\tilde {x}},\overrightarrow{x_{n}\tilde{x}}\bigr\rangle . \end{aligned}$$

This implies that

$$\begin{aligned} \rho^{2}(x_{n+1},\tilde{x}) \leq& \biggl[\frac{\beta_{n}+(1-\beta_{n})(1-\alpha_{n})+2k\alpha_{n}(1-\alpha _{n})(1-\beta_{n})}{1-k\alpha^{2}_{n}(1-\beta_{n})} \biggr]\rho^{2}(x_{n},\tilde{x}) \\ &{}+\frac{k\alpha^{2}_{n}(1-\beta_{n})}{1-k\alpha^{2}_{n}(1-\beta_{n})}\rho ^{2}(x_{n},\tilde{x})+ \frac{\alpha^{2}_{n}(1-\beta_{n})}{1-k\alpha^{2}_{n}(1-\beta _{n})}\rho^{2}\bigl(x_{n+1},f(x_{n})\bigr) \\ &{}+\frac{2\alpha_{n}(1-\alpha_{n})(1-\beta_{n})}{1-k\alpha^{2}_{n}(1-\beta_{n})}\rho \bigl(f(x_{n}),\tilde{x}\bigr)\rho \bigl(x_{n},T(x_{n})\bigr) \\ &{}+\frac{2\alpha^{2}_{n}(1-\beta_{n})}{1-k\alpha^{2}_{n}(1-\beta_{n})}\bigl\langle \overrightarrow{f(\tilde{x})\tilde{x}}, \overrightarrow{x_{n+1}\tilde {x}}\bigr\rangle +\frac{2\alpha_{n}(1-\alpha_{n})(1-\beta_{n})}{1-k\alpha^{2}_{n}(1-\beta _{n})}\bigl\langle \overrightarrow{f(\tilde{x})\tilde{x}},\overrightarrow {x_{n} \tilde{x}}\bigr\rangle . \end{aligned}$$

Thus,

$$ \rho^{2}(x_{n+1},\tilde{x})\leq \bigl(1- \alpha_{n}'\bigr)\rho^{2}(x_{n}, \tilde{x})+\alpha_{n}'\beta_{n}', $$
(7)

where \(\alpha_{n}'=\frac{\alpha_{n}(1-\beta_{n})(1-k(2-\alpha_{n}))}{1-k\alpha ^{2}_{n}(1-\beta_{n})}\) and

$$\begin{aligned} \beta_{n}' = & \frac{k\alpha_{n}}{1-k(2-\alpha_{n})}\rho ^{2}(x_{n},\tilde{x})+\frac{\alpha_{n}}{1-k(2-\alpha_{n})}\rho ^{2} \bigl(x_{n+1},f(x_{n})\bigr) \\ &{}+\frac{2(1-\alpha_{n})}{1-k(2-\alpha_{n})}\rho\bigl(f(x_{n}),\tilde{x}\bigr)\rho \bigl(x_{n},T(x_{n})\bigr) \\ &{}+\frac{2\alpha_{n}}{1-k(2-\alpha_{n})}\bigl\langle \overrightarrow{f(\tilde {x})\tilde{x}}, \overrightarrow{x_{n+1}\tilde{x}}\bigr\rangle +\frac{2(1-\alpha_{n})}{1-k(2-\alpha_{n})}\bigl\langle \overrightarrow{f(\tilde {x})\tilde{x}},\overrightarrow{x_{n} \tilde{x}}\bigr\rangle . \end{aligned}$$

Since \(k\in [0,\frac{1}{2} )\), \(\alpha_{n}'\in(0,1)\). Applying Lemma 2.7 to the inequality (7), we can conclude that \(x_{n}\to\tilde{x}\) as \(n\to\infty\). This completes the proof. □

Concluding remarks and open problems

  1. (1)

    Our main theorem can be applied to \(\operatorname{CAT}(\kappa)\) spaces with \(\kappa\leq0\) since any \(\operatorname{CAT}(\kappa)\) space is a \(\operatorname{CAT}(\kappa')\) space for \(\kappa' \geq\kappa\) (see [6]). However, the result for \(\kappa> 0\) is still unknown (see [5], p.1264).

  2. (2)

    Our main theorem can be viewed as an extension of Corollary 8 in [16] for a contraction f with \(k\in [0,\frac{1}{2} )\). It remains an open problem whether Theorem 3.2 holds for \(k\in [\frac{1}{2}, 1 )\).

References

  1. 1.

    Moudafi, A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46-55 (2000)

  2. 2.

    Shi, LY, Chen, RD: Strong convergence of viscosity approximation methods for nonexpansive mappings in CAT(0) spaces. J. Appl. Math. 2012, Article ID 421050 (2012)

  3. 3.

    Berg, ID, Nikolaev, IG: Quasilinearization and curvature of Alexandrov spaces. Geom. Dedic. 133, 195-218 (2008)

  4. 4.

    Wangkeeree, R, Preechasilp, P: Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces. J. Inequal. Appl. 2013, Article ID 93 (2013)

  5. 5.

    Piatek, B: Viscosity iteration in \(\operatorname{CAT}(\kappa)\) spaces. Numer. Funct. Anal. Optim. 34, 1245-1264 (2013)

  6. 6.

    Bridson, M, Haefliger, A: Metric Spaces of Non-Positive Curvature. Springer, Berlin (1999)

  7. 7.

    Burago, D, Burago, Y, Ivanov, S: A Course in Metric Geometry. Graduate Studies in Math., vol. 33. Am. Math. Soc., Providence (2001)

  8. 8.

    Brown, KS (ed.): Buildings. Springer, New York (1989)

  9. 9.

    Espinola, R, Fernandez-Leon, A: \(\operatorname{CAT}(k)\)-Spaces, weak convergence and fixed points. J. Math. Anal. Appl. 353, 410-427 (2009)

  10. 10.

    Dhompongsa, S, Panyanak, B: On Δ-convergence theorems in CAT(0) spaces. Comput. Math. Appl. 56, 2572-2579 (2008)

  11. 11.

    Kirk, WA: Geodesic geometry and fixed point theory II. In: International Conference on Fixed Point Theory and Applications, pp. 113-142. Yokohama Publishers, Yokohama (2004)

  12. 12.

    Laowang, W, Panyanak, B: Strong and Δ convergence theorems for multivalued mappings in CAT(0) spaces. J. Inequal. Appl. 2009, Article ID 730132 (2009)

  13. 13.

    Suzuki, T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 103-123 (2005)

  14. 14.

    Xu, HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659-678 (2003)

  15. 15.

    Dehghan, H, Rooin, J: Metric projection and convergence theorems for nonexpansive mappings in Hadamard spaces. J. Nonlinear Convex Anal. (to appear)

  16. 16.

    Nilsrakoo, W, Saejung, S: Equilibrium problems and Moudafi’s viscosity approximation methods in Hilbert spaces. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 17, 195-213 (2010)

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Acknowledgements

This research was supported by Chiang Mai University and Thailand Research Fund under Grant RTA5780007.

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Kaewkhao, A., Panyanak, B. & Suantai, S. Viscosity iteration method in CAT(0) spaces without the nice projection property. J Inequal Appl 2015, 278 (2015). https://doi.org/10.1186/s13660-015-0801-6

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Keywords

  • viscosity iteration method
  • fixed point
  • strong convergence
  • the nice projection property
  • CAT(0) space