On the Ishikawa iteration processes for multivalued mappings in some $CAT(\kappa )$ spaces

Abstract

The purpose of this paper is to prove the strong convergence of the Ishikawa iteration processes for some generalized multivalued nonexpansive mappings in the framework of CAT(1) spaces. Our results extend the corresponding results given by Shahzad and Zegeye (Nonlinear Anal. 71:838-844, 2009), Puttasontiphot (Appl. Math. Sci. 4:3005-3018, 2010), Song and Cho (Bull. Korean Math. Soc. 48:575-584, 2011) and many others.

1 Introduction

Roughly speaking, a $CAT(\kappa )$ space is a geodesic space of bounded curvature. The precise definition is given below. Here CAT means the initials of three mathematician’s names (E Cartan, AD Alexandrov and A 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 as the curvature bound of the space.

Fixed point theory in $CAT(\kappa )$ spaces was first studied by Kirk [1, 2]. His works were followed by a series of new works by many authors (see, e.g., [3–12]) mainly focusing on CAT(0) spaces. Since any $CAT(\kappa )$ space is a $CAT({\kappa }^{\prime })$ space for ${\kappa }^{\prime }\ge \kappa$ (see [[13], p.165]), all results for CAT(0) spaces immediately apply to any $CAT(\kappa )$ space with $\kappa \le 0$. Notice also that all $CAT(\kappa )$ spaces (with appropriate sizes) are uniformly convex metric spaces in the sense of [14]. Thus, the results in [14] concerning uniformly convex metric spaces also hold in $CAT(\kappa )$ spaces as well.

In 1974, Ishikawa [15] introduced an iteration process for approximating fixed points of a single-valued mapping t on a Hilbert space H by

$$x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}t((1-{\beta }_n)x_n+{\beta }_{n}t(x_n)),n\ge 1,$$

where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are sequences in $[0,1]$ satisfying some certain restrictions. For more details and literature on the convergence of the Ishikawa iteration process for single-valued mappings, see, e.g., [16–28].

The first result concerning the convergence of an Ishikawa iteration process for multivalued mappings was proved by Sastry and Babu [29] in a Hilbert space. Panyanak [30] extended the result of Sastry and Babu to a uniformly convex Banach space. Since then the strong convergence of the Ishikawa iteration processes for multivalued mappings has been rapidly developed and many of papers have appeared (see, e.g., [31–36]). Among other things, Shahzad and Zegeye [37] defined two types of Ishikawa iteration processes as follows.

Let E be a nonempty closed convex subset of a uniformly convex Banach space X, $\{{\alpha }_n\},\{{\beta }_n\}\subset [0,1]$, and $T:E\to 2^E$ be a multivalued mapping whose values are nonempty proximinal subsets of E. For each $x\in E$, let $P_T:E\to 2^E$ be a multivalued mapping defined by

$$P_T(x):=\{u\in T(x):\parallel x-u\parallel =\underset{y\in T(x)}{inf}\parallel x-y\parallel \}.$$

(A): The sequence of Ishikawa iterates is defined by $x_1\in E$,

$$y_n={\beta }_n{z}_n+(1-{\beta }_n)x_n,n\ge 1,$$

where $z_n\in T(x_n)$, and

$$x_{n+1}={\alpha }_n{z}_n^{\prime }+(1-{\alpha }_n)x_n,n\ge 1,$$

where $z_n^{\prime }\in T(y_n)$.

(B): The sequence of Ishikawa iterates is defined by $x_1\in E$,

$$y_n={\beta }_n{z}_n+(1-{\beta }_n)x_n,n\ge 1,$$

where $z_n\in P_T(x_n)$, and

$$x_{n+1}={\alpha }_n{z}_n^{\prime }+(1-{\alpha }_n)x_n,n\ge 1,$$

where $z_n^{\prime }\in P_T(y_n)$.

They proved, under some suitable assumptions, that the sequence $\{x_n\}$ defined by (A) and (B) converges strongly to a fixed point of T. In 2010, Puttasontiphot [38] gave analogous results to those of Shahzad and Zegeye in complete CAT(0) spaces.

In this paper, we extend Puttasontiphot’s results to the setting of $CAT(\kappa )$ spaces with $\kappa \ge 0$.

2 Preliminaries

Let $(X,d)$ be a metric space, and let $x\in X$, $E\subseteq X$. The distance from x to E is defined by

$$dist(x,E)=inf\{d(x,y):y\in E\}.$$

The diameter of E is defined by

$$diam(E)=sup\{d(u,v):u,v\in E\}.$$

The set E is called proximinal if for each $x\in X$, there exists an element $y\in E$ such that $d(x,y)=dist(x,E)$. We shall denote by $2^E$ the family of nonempty subsets of E, by $\mathcal{P}(E)$ the family of nonempty proximinal subsets of E and by $\mathcal{C}(E)$ the family of nonempty closed subsets of E. Let $H(\cdot ,\cdot )$ be the Hausdorff (generalized) distance on $2^E$, i.e.,

$$H(A,B)=max\{\underset{a\in A}{sup}dist(a,B),\underset{b\in B}{sup}dist(b,A)\},A,B\in 2^E.$$

Definition 2.1 Let E be a nonempty subset of a metric space $(X,d)$ and $T:E\to 2^E$. Then T is said to

• (i) be nonexpansive if $H(T(x),T(y))\le d(x,y)$ for all $x,y\in E$;

• (ii) be quasi-nonexpansive if $Fix(T)\ne \mathrm{\varnothing }$ and

  $$H(T(x),T(p))\le d(x,p)\text{for all }x\in E\text{ and }p\in Fix(T);$$

• (iii) satisfy condition (I) if there is a nondecreasing function $f:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $f(0)=0$, $f(r)>0$ for $r\in (0,\mathrm{\infty })$ such that

  $$dist(x,T(x))\ge f(dist(x,Fix(T)))\text{for all }x\in E;$$

• (iv) be hemicompact if for any sequence $\{x_n\}$ in E such that

  $$\underset{n\to \mathrm{\infty }}{lim}dist(x_n,T(x_n))=0,$$

there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ and $q\in E$ such that ${lim}_{k\to \mathrm{\infty }}{x}_{n_k}=q$.

A point $x\in E$ is called a fixed point of T if $x\in T(x)$. We denote by $Fix(T)$ the set of all fixed points of T.

The following lemma can be found in [39]. We observe that the boundedness of the images of T is superfluous.

Lemma 2.2 Let E be a nonempty subset of a metric space $(X,d)$ and $T:E\to \mathcal{P}(E)$ be a multivalued mapping. Then

• (i) $dist(x,T(x))=dist(x,P_T(x))$ for all $x\in E$;

• (ii) $x\in Fix(T)⟺x\in Fix(P_T)⟺P_T(x)=\{x\}$;

• (iii) $Fix(T)=Fix(P_T)$.

Let $(X,d)$ be a metric space. A geodesic path joining $x\in X$ to $y\in X$ (or, more briefly, a geodesic from x to y) is a map c from a closed interval $[0,l]\subset R$ to X such that $c(0)=x$, $c(l)=y$, and $d(c(t),c(t^{\prime }))=|t-t^{\prime }|$ for all $t,t^{\prime }\in [0,l]$. In particular, c is an isometry and $d(x,y)=l$. The image $c([0,l])$ of c is called a geodesic (or metric) 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

$$d(x,z)=(1-\alpha )d(x,y)\text{and}d(y,z)=\alpha d(x,y).$$

In this case, we write $z=\alpha x\oplus (1-\alpha )y$. The space $(X,d)$ is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each $x,y\in X$. A subset E of X is said to be convex if E includes every geodesic segment joining any two of its points.

In a geodesic space $(X,d)$, the metric $d:X\times X\to R$ is convex if for any $x,y,z\in X$ and $\alpha \in [0,1]$, one has

$$d(x,\alpha y\oplus (1-\alpha )z)\le \alpha d(x,y)+(1-\alpha )d(x,z).$$

Let $D\in (0,\mathrm{\infty }]$, then $(X,d)$ is called a D-geodesic space if any two points of X with their distance smaller than D are joined by a geodesic segment. Notice that $(X,d)$ is a geodesic space if and only if it is a D-geodesic space.

Let $n\in N$, we denote by $〈\cdot |\cdot 〉$ the Euclidean scalar product in $R^n$, that is,

$$〈x|y〉=x_1{y}_1+\cdots +x_n{y}_n,\text{where }x=(x_1,\dots ,x_n),y=(y_1,\dots ,y_n).$$

Let $S^n$ denote the n-dimensional sphere defined by

$$S^n=\{x=(x_1,\dots ,x_{n+1})\in R^{n+1}:〈x|x〉=1\},$$

with metric $d(x,y)=arccos〈x|y〉$, $x,y\in S^n$ (see [[13], Proposition 2.1]).

From now on, we assume that $\kappa \ge 0$ and define

$$D_{\kappa }:=\frac{\pi }{\sqrt{\kappa }}\text{if }\kappa >0\text{and}{D}_{\kappa }:=\mathrm{\infty }\text{if }\kappa =0.$$

We denote by $M_{\kappa }^n$ the following metric spaces:

• (i) if $\kappa =0$ then $M_0^n$ is the Euclidean space $R^n$;

• (ii) if $\kappa >0$ then $M_{\kappa }^n$ is obtained from $S^n$ by multiplying the distance function by the constant $1/\sqrt{\kappa }$.

A geodesic triangle $\mathrm{△}(x,y,z)$ in the metric space $(X,d)$ consists of three points x, y, z in X (the vertices of △) and three geodesic segments between each pair of vertices (the edges of △). We write $p\in \mathrm{△}(x,y,z)$ when $p\in [x,y]\cup [y,z]\cup [z,x]$. For $\mathrm{△}(x,y,z)$ in a geodesic space X satisfying $d(x,y)+d(y,z)+d(z,x)<2D_{\kappa }$, there exist points $\overline{x},\overline{y},\overline{z}\in M_{\kappa }^2$ such that

$$d(x,y)=d_{M_{\kappa }^2}(\overline{x},\overline{y}),d(y,z)=d_{M_{\kappa }^2}(\overline{y},\overline{z}),\text{and}d(z,x)=d_{M_{\kappa }^2}(\overline{z},\overline{x})$$

(see [[13], Lemma 2.14]). We call the triangle having vertices $\overline{x}$, $\overline{y}$, $\overline{z}$ in $M_{\kappa }^2$ a comparison triangle of $\mathrm{△}(x,y,z)$. Notice that it is unique up to an isometry of $M_{\kappa }^2$, and we denote it by $\overline{\mathrm{△}}(\overline{x},\overline{y},\overline{z})$. A point $\overline{p}\in [\overline{x},\overline{y}]$ is called a comparison point for $p\in [x,y]$ if $d(x,p)=d_{M_{\kappa }^2}(\overline{x},\overline{p})$.

A geodesic triangle $\mathrm{△}(x,y,z)$ in X with $d(x,y)+d(y,z)+d(z,x)<2D_{\kappa }$ is said to satisfy the $CAT(\kappa )$ inequality if for any $p,q\in \mathrm{△}(x,y,z)$ and for their comparison points $\overline{p},\overline{q}\in \overline{\mathrm{△}}(\overline{x},\overline{y},\overline{z})$, one has

$$d(p,q)\le d_{M_{\kappa }^2}(\overline{p},\overline{q}).$$

Definition 2.3 A metric space $(X,d)$ is called a $CAT(\kappa )$ space if it is $D_{\kappa }$-geodesic and any geodesic triangle $\mathrm{△}(x,y,z)$ in X with $d(x,y)+d(y,z)+d(z,x)<2D_{\kappa }$ satisfies the $CAT(\kappa )$ inequality.

It follows from [[13], Proposition 1.4] that $CAT(\kappa )$ spaces are uniquely geodesic spaces. In this paper, we consider $CAT(\kappa )$ spaces with $\kappa \ge 0$. Since most of the results for such spaces are easily deduced from those for CAT(1) spaces, in what follows, we mainly focus on CAT(1) spaces. The following lemma is a consequence of Proposition 3.1 in [40].

Lemma 2.4 If $(X,d)$ is a CAT(1) space with $diam(X)<\pi /2$, then there is a constant $K>0$ such that

$$d^2((1-\alpha )x\oplus \alpha y,z)\le (1-\alpha )d^2(x,z)+\alpha d^2(y,z)-\frac{K}{2}\alpha (1-\alpha )d^2(x,y)$$

for any $\alpha \in [0,1]$ and any points $x,y,z\in X$.

The following lemma is also needed.

Lemma 2.5 [30]

Let $\{{\alpha }_n\}$, $\{{\beta }_n\}$ be two real sequences such that

• (i) $0\le {\alpha }_n,{\beta }_n<1$;

• (ii) ${\beta }_n\to 0$ as $n\to \mathrm{\infty }$;

• (iii) $\sum {\alpha }_n{\beta }_n=\mathrm{\infty }$.

Let $\{{\gamma }_n\}$ be a nonnegative real sequence such that ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n{\beta }_n(1-{\beta }_n){\gamma }_n$ is bounded. Then $\{{\gamma }_n\}$ has a subsequence which converges to zero.

3 Main results

We begin this section by proving a crucial lemma.

Lemma 3.1 Let $(X,d)$ be a CAT(1) space with convex metric, E be a nonempty closed convex subset of X, and $T:E\to 2^E$ be a quasi-nonexpansive mapping with $Fix(T)\ne \mathrm{\varnothing }$ and $T(p)=\{p\}$ for each $p\in Fix(T)$. Let $\{x_n\}$ be the sequence of Ishikawa iterates defined by (A) (replacing + with ⊕). Then ${lim}_{n\to \mathrm{\infty }}d(x_n,p)$ exists for each $p\in Fix(T)$.

Proof Let $p\in Fix(T)$. For each $n\ge 1$, we have

$$\begin{array}{rcl}d(y_n,p)& =& d({\beta }_n{z}_n\oplus (1-{\beta }_n)x_n,p)\\ \le & {\beta }_{n}d(z_n,p)+(1-{\beta }_n)d(x_n,p)\\ \le & {\beta }_{n}H(T(x_n),T(p))+(1-{\beta }_n)d(x_n,p)\\ \le & {\beta }_{n}d(x_n,p)+(1-{\beta }_n)d(x_n,p)\\ \le & d(x_n,p)\end{array}$$

and

$$\begin{array}{rcl}d(x_{n+1},p)& =& d({\alpha }_n{z}_n^{\prime }\oplus (1-{\alpha }_n)x_n,p)\\ \le & {\alpha }_{n}d(z_n^{\prime },p)+(1-{\alpha }_n)d(x_n,p)\\ \le & {\alpha }_{n}H(T(y_n),T(p))+(1-{\alpha }_n)d(x_n,p)\\ \le & {\alpha }_{n}d(y_n,p)+(1-{\alpha }_n)d(x_n,p)\\ \le & d(x_n,p).\end{array}$$

This shows that the sequence $\{d(x_n,p)\}$ is decreasing and bounded below. Thus ${lim}_{n\to \mathrm{\infty }}d(x_n,p)$ exists for any $p\in Fix(T)$. □

Now, we prove the strong convergence of the Ishikawa iteration process defined by (A).

Theorem 3.2 Let $(X,d)$ be a complete CAT(1) space with convex metric and $diam(X)<\pi /2$, E be a nonempty closed convex subset of X, and $T:E\to \mathcal{C}(E)$ be a quasi-nonexpansive mapping with $Fix(T)\ne \mathrm{\varnothing }$ and $T(p)=\{p\}$ for each $p\in Fix(T)$. Let ${\alpha }_n,{\beta }_n\in [a,b]\subset (0,1)$ and $\{x_n\}$ be the sequence of Ishikawa iterates defined by (A) (replacing + with ⊕). If T satisfies condition (I), then $\{x_n\}$ converges strongly to a fixed point of T.

Proof Let $p\in Fix(T)$. By using Lemma 2.4, we have

$$\begin{array}{rcl}{d}^2(x_{n+1},p)& =& d^2({\alpha }_n{z}_n^{\prime }\oplus (1-{\alpha }_n)x_n,p)\\ \le & (1-{\alpha }_n)d^2(x_n,p)+{\alpha }_n{d}^2(z_n^{\prime },p)-\frac{K}{2}{\alpha }_n(1-{\alpha }_n)d^2(x_n,z_n^{\prime })\\ \le & (1-{\alpha }_n)d^2(x_n,p)+{\alpha }_n{H}^2(T(y_n),T(p))-\frac{K}{2}{\alpha }_n(1-{\alpha }_n)d^2(x_n,z_n^{\prime })\\ \le & (1-{\alpha }_n)d^2(x_n,p)+{\alpha }_n{d}^2(y_n,p)\end{array}$$

and

$$\begin{array}{rcl}{d}^2(y_n,p)& =& d^2({\beta }_n{z}_n\oplus (1-{\beta }_n)x_n,p)\\ \le & (1-{\beta }_n)d^2(x_n,p)+{\beta }_n{d}^2(z_n,p)-\frac{K}{2}{\beta }_n(1-{\beta }_n)d^2(x_n,z_n)\\ \le & (1-{\beta }_n)d^2(x_n,p)+{\beta }_n{H}^2(T(x_n),T(p))-\frac{K}{2}{\beta }_n(1-{\beta }_n)d^2(x_n,z_n)\\ \le & (1-{\beta }_n)d^2(x_n,p)+{\beta }_n{d}^2(x_n,p)-\frac{K}{2}{\beta }_n(1-{\beta }_n)d^2(x_n,z_n)\\ \le & d^2(x_n,p)-\frac{K}{2}{\beta }_n(1-{\beta }_n)d^2(x_n,z_n).\end{array}$$

So that

$$d^2(x_{n+1},p)\le (1-{\alpha }_n)d^2(x_n,p)+{\alpha }_n{d}^2(x_n,p)-\frac{K}{2}{\alpha }_n{\beta }_n(1-{\beta }_n)d^2(x_n,z_n).$$

This implies that

$$\frac{K}{2}{a}^2(1-b)d^2(x_n,z_n)\le \frac{K}{2}{\alpha }_n{\beta }_n(1-{\beta }_n)d^2(x_n,z_n)\le d^2(x_n,p)-d^2(x_{n+1},p)$$

(1)

and so

$$\sum _{n=1}^{\mathrm{\infty }}\frac{K}{2}{a}^2(1-b)d^2(x_n,z_n)<\mathrm{\infty }.$$

Thus, ${lim}_{n\to \mathrm{\infty }}d(x_n,z_n)=0$. Also, $dist(x_n,T(x_n))\le d(x_n,z_n)\to 0$ as $n\to \mathrm{\infty }$. Since T satisfies condition (I), we have ${lim}_{n\to \mathrm{\infty }}dist(x_n,Fix(T))=0$. The proof of the remaining part follows the proof of Theorem 3.2 in [38], therefore we omit it. □

Theorem 3.3 Let $(X,d)$ be a complete CAT(1) space with convex metric and $diam(X)<\pi /2$, E be a nonempty closed convex subset of X, and $T:E\to \mathcal{C}(E)$ be a quasi-nonexpansive mapping with $Fix(T)\ne \mathrm{\varnothing }$ and $T(p)=\{p\}$ for each $p\in Fix(T)$. Assume that (i) $0\le {\alpha }_n,{\beta }_n<1$; (ii) ${\beta }_n\to 0$; (iii) $\sum {\alpha }_n{\beta }_n=\mathrm{\infty }$, and let $\{x_n\}$ be the sequence of Ishikawa iterates defined by (A) (replacing + with ⊕). If T is hemicompact and continuous, then $\{x_n\}$ converges strongly to a fixed point of T.

Proof Let $p\in Fix(T)$. By (1) we have

$$\frac{K}{2}\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n{\beta }_n(1-{\beta }_n)d^2(x_n,z_n)<\mathrm{\infty }.$$

By Lemma 2.5, there exist subsequences $\{x_{n_k}\}$ and $\{z_{n_k}\}$ of $\{x_n\}$ and $\{z_n\}$ respectively such that ${lim}_{k\to \mathrm{\infty }}d(x_{n_k},z_{n_k})=0$. Hence

$$\underset{k\to \mathrm{\infty }}{lim}dist(x_{n_k},T(x_{n_k}))\le \underset{k\to \mathrm{\infty }}{lim}d(x_{n_k},z_{n_k})=0.$$

Since T is hemicompact, by passing through a subsequence, we may assume that $x_{n_k}\to q$ for some $q\in E$. Since T is continuous,

$$dist(q,T(q))\le d(q,x_{n_k})+dist(x_{n_k},T(x_{n_k}))+H(T(x_{n_k}),T(q))\to 0\text{as }k\to \mathrm{\infty }.$$

This implies that $q\in Fix(T)$ since $T(q)$ is closed. Thus ${lim}_{n\to \mathrm{\infty }}d(x_n,q)$ exists by Lemma 3.1 and hence q is the limit of $\{x_n\}$ itself. □

To avoid the restriction of T, that is, $T(p)=\{p\}$ for $p\in Fix(T)$, we use the iteration process defined by (B).

Theorem 3.4 Let $(X,d)$ be a complete CAT(1) space with convex metric and $diam(X)<\pi /2$, E be a nonempty closed convex subset of X, and $T:E\to \mathcal{P}(E)$ be a multivalued mapping with $Fix(T)\ne \mathrm{\varnothing }$ and $P_T$ is quasi-nonexpansive. Let ${\alpha }_n,{\beta }_n\in [a,b]\subset (0,1)$ and $\{x_n\}$ be the sequence of Ishikawa iterates defined by (B) (replacing + with ⊕). If T satisfies condition (I), then $\{x_n\}$ converges strongly to a fixed point of T.

Proof It follows from Lemma 2.2 that $dist(x,P_T(x))=dist(x,T(x))$ for all $x\in E$,

$$Fix(P_T)=Fix(T)\text{and}{P}_T(p)=\{p\}\text{for each }p\in Fix(P_T).$$

Since T satisfies condition (I), for each $x\in E$ we have

$$dist(x,P_T(x))=dist(x,T(x))\ge f(dist(x,Fix(T)))=f(dist(x,Fix(P_T))).$$

That is, $P_T$ satisfies condition (I). Next, we show that $P_T(x)$ is closed for any $x\in E$. Let $\{y_n\}\subset P_T(x)$ and ${lim}_{n\to \mathrm{\infty }}{y}_n=y$ for some $y\in E$. Then

$$d(x,y_n)=dist(x,T(x))\text{and}\underset{n\to \mathrm{\infty }}{lim}d(x,y_n)=d(x,y).$$

It follows that $d(x,y)=dist(x,T(x))$ and this implies $y\in P_T(x)$. Applying Theorem 3.2 to the map $P_T$, we can conclude that the sequence $\{x_n\}$ defined by (B) converges to a point $z\in Fix(P_T)=Fix(T)$. This completes the proof. □

The following theorem is an analogue of Theorem 3 in [39].

Theorem 3.5 Let $(X,d)$ be a complete CAT(1) space with convex metric and $diam(X)<\pi /2$, E be a nonempty closed convex subset of X, and $T:E\to \mathcal{P}(E)$ be a hemicompact mapping with $Fix(T)\ne \mathrm{\varnothing }$ and $P_T$ is quasi-nonexpansive and continuous. Assume that (i) $0\le {\alpha }_n,{\beta }_n<1$; (ii) ${\beta }_n\to 0$; (iii) $\sum {\alpha }_n{\beta }_n=\mathrm{\infty }$, and let $\{x_n\}$ be the sequence of Ishikawa iterates defined by (B) (replacing + with ⊕). Then $\{x_n\}$ converges strongly to a fixed point of T.

Proof As in the proof of Theorem 3.4, we have

$$Fix(P_T)=Fix(T)\text{and}{P}_T(p)=\{p\}\text{for each }p\in Fix(P_T).$$

The hemicompactness of $P_T$ follows from that of T. The conclusion follows from Theorem 3.3. □