Viscosity approximation process for a sequence of quasinonexpansive mappings

Abstract

We study the viscosity approximation method due to Moudafi for a fixed point problem of quasinonexpansive mappings in a Hilbert space. First, we establish a strong convergence theorem for a sequence of quasinonexpansive mappings. Then we employ our result to approximate a solution of the variational inequality problem over the common fixed point set of the sequence of quasinonexpansive mappings.

MSC:47H09, 47H10, 41A65.

1 Introduction

Let C be a nonempty closed convex subset of a Hilbert space. This paper is devoted to the study of strong convergence of a sequence $\{x_n\}$ in C defined by an arbitrary point $x_1\in C$ and

$$x_{n+1}={\alpha }_n{f}_n(x_n)+(1-{\alpha }_n)T_n{x}_n$$

(1.1)

for $n\in \mathbb{N}$, where ${\alpha }_n$ is a real number in $[0,1]$, $f_n$ is a contraction-like mapping on C, and $T_n$ is a quasinonexpansive mapping on C. This iterative method (1.1) is called the viscosity approximation method [1]. In Section 3, we establish that, under some appropriate assumptions, the sequence $\{x_n\}$ converges strongly to a certain common fixed point of $\{T_n\}$ by using the technique developed in [2]. Then, in Section 4, we apply our result to approximate a solution of a variational inequality problem over the common fixed point set of $\{T_n\}$.

The viscosity approximation method (1.1) is based on the study of Moudafi [1], who considered a fixed point problem of a single nonexpansive mapping and proved strong convergence of sequences generated by the method. After that, Xu [3] extended Moudafi’s results [1] in the framework of Hilbert spaces and Banach spaces; Suzuki [4] gave simple proofs of Xu’s results [3]; Aoyama and Kimura [5] investigated a relationship between viscosity approximation methods and Halpern [6] type iterative methods for a sequence of nonexpansive mappings.

On the other hand, Maingé [7] adopted the viscosity approximation method for a fixed point problem of a single quasinonexpansive mapping; Wongchan and Saejung [8] extended Maingé’s result [7]. Our main result (Theorem 3.1) is a generalization of Wongchan and Saejung’s result [8] and is closely related to the study in [5]. Moreover, it is also applicable to an approximation method, which is called the hybrid steepest descent method [9, 10], for a variational inequality problem over the common fixed point set of a sequence of quasinonexpansive mappings.

2 Preliminaries

Throughout the present paper, H denotes a real Hilbert space, $〈\cdot ,\cdot 〉$ the inner product of H, $\parallel \cdot \parallel$ the norm of H, C a nonempty closed convex subset of H, I the identity mapping on H, ℝ the set of real numbers, and ℕ the set of positive integers. Strong convergence of a sequence $\{x_n\}$ in H to $x\in H$ is denoted by $x_n\to x$ and weak convergence by $x_n\rightharpoonup x$.

Let $T:C\to H$ be a mapping. The set of fixed points of T is denoted by $Fix(T)$. A mapping T is said to be quasinonexpansive if $Fix(T)\ne \mathrm{\varnothing }$ and $\parallel Tx-p\parallel \le \parallel x-p\parallel$ for all $x\in C$ and $p\in Fix(T)$; T is said to be nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for all $x,y\in C$; T is said to be strongly quasinonexpansive if T is quasinonexpansive and $T{x}_n-x_n\to 0$ whenever $\{x_n\}$ is a bounded sequence in C and $\parallel x_n-p\parallel -\parallel T{x}_n-p\parallel \to 0$ for some point $p\in Fix(T)$; T is demiclosed at 0 if $Tp=0$ whenever $\{x_n\}$ is a sequence in C such that $x_n\rightharpoonup p$ and $T{x}_n\to 0$. We know that if $T:C\to H$ is quasinonexpansive, then $Fix(T)$ is closed and convex; see [[11], Theorem 1].

It is known that, for each $x\in H$, there exists a unique point $x_0\in C$ such that

$$\parallel x-x_0\parallel =min\{\parallel x-y\parallel :y\in C\}.$$

Such a point $x_0$ is denoted by $P_C(x)$ and $P_C$ is called the metric projection of H onto C. It is known that the metric projection $P_C$ is nonexpansive; see [12].

Let $f:C\to C$ be a mapping, F a nonempty subset of C, and θ a real number in $[0,1)$. A mapping f is said to be a θ-contraction with respect to F if $\parallel f(x)-f(z)\parallel \le \theta \parallel x-z\parallel$ for all $x\in C$ and $z\in F$; f is said to be a θ-contraction if f is a θ-contraction with respect to C. By definition, it is easy to check the following results.

Lemma 2.1 Let F be a nonempty subset of C and $f:C\to C$ a θ-contraction with respect to F, where $0\le \theta <1$. If F is closed and convex, then $P_F\circ f$ is a θ-contraction on F, where $P_F$ is the metric projection of H onto F.

Lemma 2.2 Let $f:C\to C$ be a θ-contraction, where $0\le \theta <1$ and $T:C\to C$ a quasinonexpansive mapping. Then $f\circ T$ is a θ-contraction with respect to $Fix(T)$.

Let D be a nonempty subset of C. A sequence $\{f_n\}$ of mappings of C into H is said to be stable on D [5] if $\{f_n(z):n\in \mathbb{N}\}$ is a singleton for every $z\in D$. It is clear that if $\{f_n\}$ is stable on D, then $f_n(z)=f_1(z)$ for all $n\in \mathbb{N}$ and $z\in D$.

A function $\tau :\mathbb{N}\to \mathbb{N}$ is said to be eventually increasing [2] if ${lim}_{n\to \mathrm{\infty }}\tau (n)=\mathrm{\infty }$ and $\tau (n)\le \tau (n+1)$ for all $n\in \mathbb{N}$. By definition, we easily obtain the following.

Lemma 2.3 Let $\tau :\mathbb{N}\to \mathbb{N}$ be an eventually increasing function and $\{{\xi }_n\}$ a sequence of real numbers such that ${\xi }_n\to \xi$. Then ${\xi }_{\tau (n)}\to \xi$.

The following is a direct consequence of [[13], Lemma 3.1].

Lemma 2.4 ([[2], Lemma 3.4])

Let $\{{\xi }_n\}$ be a sequence of nonnegative real numbers which is not convergent. Then there exist $N\in \mathbb{N}$ and an eventually increasing function $\tau :\mathbb{N}\to \mathbb{N}$ such that ${\xi }_{\tau (n)}\le {\xi }_{\tau (n)+1}$ for all $n\in \mathbb{N}$ and ${\xi }_n\le {\xi }_{\tau (n)+1}$ for all $n\ge N$.

Under the assumptions of Lemma 2.4, we cannot choose a strictly increasing function τ; see [[2], Example 3.3].

Let $\{T_n\}$ be a sequence of mappings of C into H such that $F={\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n)$ is nonempty. Then

• $\{T_n\}$ is said to be strongly quasinonexpansive type if each $T_n$ is quasinonexpansive and $T_n{x}_n-x_n\to 0$ whenever $\{x_n\}$ is a bounded sequence in C and

  $$\parallel x_n-p\parallel -\parallel T_n{x}_n-p\parallel \to 0$$

for some point $p\in F$;

• $\{T_n\}$ is said to satisfy the condition (Z) [2, 14–16] if every weak cluster point of $\{x_n\}$ belongs to F whenever $\{x_n\}$ is a bounded sequence in C such that $T_n{x}_n-x_n\to 0$.

Remark 2.5 Since ${\beta }_n-{\alpha }_n\to 0$ if and only if ${\beta }_n^2-{\alpha }_n^2\to 0$ for all bounded sequences $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ in $[0,\mathrm{\infty })$, $\{T_n\}$ is strongly quasinonexpansive type if and only if it is a strongly relatively nonexpansive sequence in the sense of [2, 17]. See also [18, 19].

We know several examples of strongly quasinonexpansive type sequences satisfying the condition (Z); see [17] and Example 4.5 in Section 4.

The following lemma follows from [[2], Lemma 3.5] and Remark 2.5.

Lemma 2.6 Let $\{T_n\}$ be a sequence of mappings of C into H such that $F={\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n)$ is nonempty, $\tau :\mathbb{N}\to \mathbb{N}$ an eventually increasing function, and $\{z_n\}$ a bounded sequence in C such that $\parallel z_n-p\parallel -\parallel T_{\tau (n)}{z}_n-p\parallel \to 0$ for some $p\in F$. If $\{T_n\}$ is strongly quasinonexpansive type, then $T_{\tau (n)}{z}_n-z_n\to 0$.

In order to prove our main result in Section 3, we need the following lemmas.

Lemma 2.7 ([[2], Lemma 3.6])

Let $\{T_n\}$ be a sequence of mappings of C into H such that $F={\bigcap }_{n=1}^{\mathrm{\infty }}Fix(T_n)$ is nonempty, $\tau :\mathbb{N}\to \mathbb{N}$ an eventually increasing function, and $\{z_n\}$ a bounded sequence in C such that $T_{\tau (n)}{z}_n-z_n\to 0$. Suppose that $\{T_n\}$ satisfies the condition (Z). Then every weak cluster point of $\{z_n\}$ belongs to F.

Lemma 2.8 ([[2], Lemma 3.7])

Let $\{T_n\}$ be a sequence of mappings of C into H, F a nonempty closed convex subset of H, $\{z_n\}$ a bounded sequence in C such that $T_n{z}_n-z_n\to 0$, and $u\in H$. Suppose that every weak cluster point of $\{z_n\}$ belongs to F. Then

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈T_n{z}_n-w,u-w〉\le 0,$$

where $w=P_F(u)$.

The following lemma is well known; see [20, 21].

Lemma 2.9 Let $\{{\xi }_n\}$ be a sequence of nonnegative real numbers, $\{{\delta }_n\}$ a sequence of real numbers, and $\{{\beta }_n\}$ a sequence in $[0,1]$. Suppose that ${\xi }_{n+1}\le (1-{\beta }_n){\xi }_n+{\beta }_n{\delta }_n$ for every $n\in \mathbb{N}$, ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\delta }_n\le 0$, and ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$. Then ${\xi }_n\to 0$.

3 Strong convergence of a viscosity approximation process

In this section, we prove the following strong convergence theorem.

Theorem 3.1 Let H be a real Hilbert space, C a nonempty closed convex subset of H, $\{S_n\}$ a sequence of mappings of C into C such that $F={\bigcap }_{n=1}^{\mathrm{\infty }}Fix(S_n)$ is nonempty, $\{{\alpha }_n\}$ a sequence in $(0,1]$ such that ${\alpha }_n\to 0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, and $\{f_n\}$ a sequence of mappings of C into C such that each $f_n$ is a θ-contraction with respect to F and $\{f_n\}$ is stable on F, where $0\le \theta <1$. Let $\{x_n\}$ be a sequence defined by $x_1\in C$ and

$$x_{n+1}={\alpha }_n{f}_n(x_n)+(1-{\alpha }_n)S_n{x}_n$$

(3.1)

for $n\in \mathbb{N}$. Suppose that $\{S_n\}$ is strongly quasinonexpansive type and satisfies the condition (Z). Then $\{x_n\}$ converges strongly to $w\in F$, where w is the unique fixed point of a contraction $P_F\circ f_1$.

Note that Lemma 2.1 implies that $P_F\circ f_1$ is a θ-contraction on F and hence it has a unique fixed point on F.

First, we show some lemmas; then we prove Theorem 3.1. In the rest of this section, we set

$${\beta }_n={\alpha }_n(1+(1-2\theta )(1-{\alpha }_n))$$

and

$${\gamma }_n={\alpha }_n^2{\parallel f_n(x_n)-w\parallel }^2+2{\alpha }_n(1-{\alpha }_n)〈S_n{x}_n-w,f_1(w)-w〉$$

for $n\in \mathbb{N}$.

Lemma 3.2 $\{x_n\}$, $\{S_n{x}_n\}$, and $\{f_n(x_n)\}$ are bounded, and moreover,

$$\parallel x_{n+1}-w\parallel \le {\alpha }_n\parallel f_n(x_n)-w\parallel +\parallel S_n{x}_n-w\parallel$$

(3.2)

and

$${\parallel x_{n+1}-w\parallel }^2\le (1-{\beta }_n){\parallel x_n-w\parallel }^2+{\gamma }_n$$

(3.3)

hold for every $n\in \mathbb{N}$.

Proof Since $f_n$ is a θ-contraction with respect to F, $S_n$ is quasinonexpansive, $w\in F\subset Fix(S_n)$, and $\{f_n\}$ is stable on F, it follows that

$$\begin{array}{c}\parallel x_{n+1}-w\parallel \hfill \\ \le {\alpha }_n\parallel f_n(x_n)-w\parallel +(1-{\alpha }_n)\parallel S_n{x}_n-w\parallel \hfill \\ \le {\alpha }_n(\parallel f_n(x_n)-f_n(w)\parallel +\parallel f_n(w)-w\parallel )+(1-{\alpha }_n)\parallel S_n{x}_n-w\parallel \hfill \\ \le (1-{\alpha }_n(1-\theta ))\parallel x_n-w\parallel +{\alpha }_n(1-\theta )\frac{\parallel f_1(w)-w\parallel }{1-\theta }\hfill \end{array}$$

(3.4)

for every $n\in \mathbb{N}$. Thus, by induction on n, we have

$$\parallel S_n{x}_n-w\parallel \le \parallel x_n-w\parallel \le max\{\parallel x_1-w\parallel ,\parallel f_1(w)-w\parallel /(1-\theta )\}.$$

Therefore, it turns out that $\{x_n\}$ and $\{S_n{x}_n\}$ are bounded, and moreover, $\{f_n(x_n)\}$ is also bounded.

Equation (3.2) follows from (3.4).

Next, we show (3.3). By assumption, it follows that

$$\begin{array}{rl}〈S_n{x}_n-w,f_n(x_n)-w〉& \le \parallel S_n{x}_n-w\parallel \parallel f_n(x_n)-f_n(w)\parallel +〈S_n{x}_n-w,f_n(w)-w〉\\ \le \theta {\parallel x_n-w\parallel }^2+〈S_n{x}_n-w,f_1(w)-w〉,\end{array}$$

and thus

$$\begin{array}{rcl}{\parallel x_{n+1}-w\parallel }^2& =& {\alpha }_n^2{\parallel f_n(x_n)-w\parallel }^2+{(1-{\alpha }_n)}^2{\parallel S_n{x}_n-w\parallel }^2\\ +2{\alpha }_n(1-{\alpha }_n)〈S_n{x}_n-w,f_n(x_n)-w〉\\ \le & {\alpha }_n^2{\parallel f_n(x_n)-w\parallel }^2+({(1-{\alpha }_n)}^2+2{\alpha }_n(1-{\alpha }_n)\theta ){\parallel x_n-w\parallel }^2\\ +2{\alpha }_n(1-{\alpha }_n)〈S_n{x}_n-w,f_1(w)-w〉\\ =& (1-{\beta }_n){\parallel x_n-w\parallel }^2+{\gamma }_n\end{array}$$

(3.5)

for every $n\in \mathbb{N}$. Therefore, (3.3) holds. □

Lemma 3.3 The following hold:

• $0<{\beta }_n\le 1$ for every $n\in \mathbb{N}$;

• $2{\alpha }_n(1-{\alpha }_n)/{\beta }_n\to 1/(1-\theta )$;

• ${\alpha }_n^2{\parallel f_n(x_n)-w\parallel }^2/{\beta }_n\to 0$;

• ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$.

Proof Since $0<{\alpha }_n\le 1$ and $-1<1-2\theta \le 1$, we know that

$$0<{\alpha }_n^2={\alpha }_n(1+(-1)(1-{\alpha }_n))\le {\beta }_n\le {\alpha }_n(1+(1-{\alpha }_n))={\alpha }_n(2-{\alpha }_n)\le 1.$$

It follows from ${\alpha }_n\to 0$ that $2{\alpha }_n(1-{\alpha }_n)/{\beta }_n\to 1/(1-\theta )$.

Since $\{f_n(x_n)\}$ is bounded by Lemma 3.2 and

$$\frac{{\alpha }_n^2}{{\beta }_n}=\frac{{\alpha }_n}{1+(1-2\theta )(1-{\alpha }_n)}\to 0,$$

it follows that ${\alpha }_n^2{\parallel f_n(x_n)-w\parallel }^2/{\beta }_n\to 0$.

Finally, we prove ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$. Suppose that $1-2\theta \ge 0$. Then it is clear that ${\beta }_n\ge {\alpha }_n$ for every $n\in \mathbb{N}$. Thus, ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n\ge {\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$. Next, we suppose that $1-2\theta <0$. Then it is clear that ${\beta }_n>2(1-\theta ){\alpha }_n$ for every $n\in \mathbb{N}$. Thus, ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n\ge 2(1-\theta ){\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$. This completes the proof. □

Lemma 3.4 $\{\parallel x_n-w\parallel \}$ is convergent.

Proof We assume, in order to obtain a contraction, that $\{\parallel x_n-w\parallel \}$ is not convergent. Then Lemma 2.4 implies that there exist $N\in \mathbb{N}$ and an eventually increasing function $\tau :\mathbb{N}\to \mathbb{N}$ such that

$$\parallel x_{\tau (n)}-w\parallel \le \parallel x_{\tau (n)+1}-w\parallel$$

(3.6)

for every $n\in \mathbb{N}$ and

$$\parallel x_n-w\parallel \le \parallel x_{\tau (n)+1}-w\parallel$$

(3.7)

for every $n\ge N$.

We show that $S_{\tau (n)}{x}_{\tau (n)}-x_{\tau (n)}\to 0$. Since $S_{\tau (n)}$ is quasinonexpansive and $w\in F\subset Fix(S_{\tau (n)})$, it follows from (3.6), (3.2), and Lemmas 2.3 and 3.2 that

$$\begin{array}{rl}0& \le \parallel x_{\tau (n)}-w\parallel -\parallel S_{\tau (n)}{x}_{\tau (n)}-w\parallel \\ \le \parallel x_{\tau (n)+1}-w\parallel -\parallel S_{\tau (n)}{x}_{\tau (n)}-w\parallel \\ \le {\alpha }_{\tau (n)}\parallel f_{\tau (n)}(x_{\tau (n)})-w\parallel \to 0\end{array}$$

as $n\to \mathrm{\infty }$. Since $\{x_{\tau (n)}\}$ is bounded and $\{S_n\}$ is strongly quasinonexpansive type, Lemma 2.6 implies that $S_{\tau (n)}{x}_{\tau (n)}-x_{\tau (n)}\to 0$.

Since $\{S_n\}$ satisfies the condition (Z), it follows from Lemma 2.7 that every weak cluster point of $\{x_{\tau (n)}\}$ belongs to F. Thus Lemma 2.8 shows that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈S_{\tau (n)}{x}_{\tau (n)}-w,f_1(w)-w〉\le 0.$$

Moreover, Lemmas 2.3 and 3.3 imply that ${\alpha }_{\tau (n)}^2{\parallel f_{\tau (n)}(x_{\tau (n)})-w\parallel }^2/{\beta }_{\tau (n)}\to 0$ and $2{\alpha }_{\tau (n)}(1-{\alpha }_{\tau (n)})/{\beta }_{\tau (n)}\to 1/(1-\theta )$. Therefore, we obtain

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{{\gamma }_{\tau (n)}}{{\beta }_{\tau (n)}}\le 0.$$

(3.8)

On the other hand, from (3.3) and (3.6), we know that

$$\begin{array}{rl}{\parallel x_{\tau (n)+1}-w\parallel }^2& \le (1-{\beta }_{\tau (n)}){\parallel x_{\tau (n)}-w\parallel }^2+{\gamma }_{\tau (n)}\\ \le (1-{\beta }_{\tau (n)}){\parallel x_{\tau (n)+1}-w\parallel }^2+{\gamma }_{\tau (n)}\end{array}$$

for every $n\in \mathbb{N}$. Thus, by ${\beta }_{\tau (n)}>0$, this shows that

$${\parallel x_{\tau (n)+1}-w\parallel }^2\le \frac{{\gamma }_{\tau (n)}}{{\beta }_{\tau (n)}}$$

(3.9)

for every $n\in \mathbb{N}$.

Finally, we obtain a contradiction that $\parallel x_n-w\parallel \to 0$. Using (3.7), (3.9), and (3.8), we conclude that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel x_n-w\parallel }^2\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\parallel x_{\tau (n)+1}-w\parallel }^2\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{{\gamma }_{\tau (n)}}{{\beta }_{\tau (n)}}\le 0,$$

and hence $\parallel x_n-w\parallel \to 0$, which is a contradiction. □

Proof of Theorem 3.1 We first show that $S_n{x}_n-x_n\to 0$. Since $S_n$ is quasinonexpansive, it follows from (3.2) that

$$0\le \parallel x_n-w\parallel -\parallel S_n{x}_n-w\parallel \le \parallel x_n-w\parallel -\parallel x_{n+1}-w\parallel +{\alpha }_n\parallel f_n(x_n)-w\parallel$$

for every $n\in \mathbb{N}$, so that $\parallel x_n-w\parallel -\parallel S_n{x}_n-w\parallel \to 0$ by Lemma 3.4, ${\alpha }_n\to 0$, and Lemma 3.2. Since $\{S_n\}$ is strongly quasinonexpansive type and $\{x_n\}$ is bounded, we conclude that $S_n{x}_n-x_n\to 0$.

Since $\{S_n\}$ satisfies the condition (Z), Lemma 2.8 implies that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈S_n{x}_n-w,f_1(w)-w〉\le 0.$$

This shows that ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\gamma }_n/{\beta }_n\le 0$ by using Lemmas 3.2 and 3.3. On the other hand, it follows from (3.3) that

$${\parallel x_{n+1}-w\parallel }^2\le (1-{\beta }_n){\parallel x_n-w\parallel }^2+{\beta }_n\frac{{\gamma }_n}{{\beta }_n}$$

for every $n\in \mathbb{N}$. Therefore, noting that ${\sum }_{n=1}^{\mathrm{\infty }}{\beta }_n=\mathrm{\infty }$ and using Lemma 2.9, we conclude that $x_n-w\to 0$. □

A direct consequence of Theorem 3.1 is the following corollary, which is a slight generalization of [[8], Theorem 2.3].

Corollary 3.5 Let H be a real Hilbert space, C a nonempty closed convex subset of H, $S:C\to C$ a strongly quasinonexpansive mapping, $\{{\alpha }_n\}$ a sequence in $(0,1]$ such that ${\alpha }_n\to 0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, and $f:C\to C$ a θ-contraction with respect to $F=Fix(S)$, where $0\le \theta <1$. Let $\{x_n\}$ be a sequence defined by $x_1\in C$ and

$$x_{n+1}={\alpha }_{n}f(x_n)+(1-{\alpha }_n)S{x}_n$$

(3.10)

for $n\in \mathbb{N}$. Suppose that $I-S$ is demiclosed at 0. Then $\{x_n\}$ converges strongly to $w\in F$, where w is the unique fixed point of a contraction $P_F\circ f$.

Proof Set $S_n=S$ and $f_n=f$ for $n\in \mathbb{N}$. Then it is clear that ${\bigcap }_{n=1}^{\mathrm{\infty }}Fix(S_n)=Fix(S)$, $\{S_n\}$ is strongly quasinonexpansive type, $\{S_n\}$ satisfies the condition (Z), and $\{f_n\}$ is stable on C. Thus Theorem 3.1 implies the conclusion. □

4 Application to a variational inequality problem

In this section, applying Theorem 3.1, we study an approximation method for the following variational inequality problem.

Problem 4.1 Let κ and η be positive real numbers such that ${\eta }^2<2\kappa$. Let F be a nonempty closed convex subset of H and $A:H\to H$ a κ-strongly monotone and η-Lipschitz continuous mapping, that is, we assume that $〈x-y,Ax-Ay〉\ge \kappa {\parallel x-y\parallel }^2$ and $\parallel Ax-Ay\parallel \le \eta \parallel x-y\parallel$ for all $x,y\in H$. Then find $z\in F$ such that

$$〈y-z,Az〉\ge 0\text{for all }y\in F.$$

The solution set of Problem 4.1 is denoted by $VI(F,A)$. Under the assumptions of Problem 4.1, it is known that the following hold; see, for example, [22].

• $\kappa \le \eta$, $0\le 1-2\kappa +{\eta }^2<1$ and $I-A$ is a θ-contraction, where $\theta =\sqrt{1-2\kappa +{\eta }^2}$;

• Problem 4.1 has a unique solution and $VI(F,A)=Fix(P_F(I-A))$.

Remark 4.2 The assumption that ${\eta }^2<2\kappa$ in Problem 4.1 is not restrictive. Indeed, let F be a nonempty closed convex subset of H and $\tilde{A}$ a $\tilde{\kappa }$-strongly monotone and $\tilde{\eta }$-Lipschitz continuous mapping, where $\tilde{\kappa }>0$ and $\tilde{\eta }>0$. Set $A=\mu \tilde{A}$, $\kappa =\mu \tilde{\kappa }$, and $\eta =\mu \tilde{\eta }$, where μ is a positive constant such that $\mu {\tilde{\eta }}^2<2\tilde{\kappa }$. Then it is easy to verify that A is κ-strongly monotone and η-Lipschitz continuous, ${\eta }^2<2\kappa$, and moreover, $VI(F,A)=VI(F,\tilde{A})$.

Using Theorem 3.1, we obtain the following convergence theorem for Problem 4.1.

Theorem 4.3 Let H, κ, η, and A be the same as in Problem  4.1. Let $\{S_n\}$ be a sequence of mappings of H into H such that $F={\bigcap }_{n=1}^{\mathrm{\infty }}Fix(S_n)$ is nonempty, and $\{{\alpha }_n\}$ the same as in Theorem  3.1. Let $\{x_n\}$ be a sequence defined by $x_1\in H$ and

$$x_{n+1}=S_n{x}_n-{\alpha }_{n}A{S}_n{x}_n$$

(4.1)

for $n\in \mathbb{N}$. Suppose that $\{S_n\}$ is strongly quasinonexpansive type and $\{S_n\}$ satisfies the condition (Z). Then $\{x_n\}$ converges strongly to the unique solution of Problem  4.1.

Proof Set $f_n=(I-A)S_n$ for $n\in \mathbb{N}$ and $\theta =\sqrt{1-2\kappa +{\eta }^2}$. Since $I-A$ is a θ-contraction and $S_n$ is quasinonexpansive, Lemma 2.2 implies that each $f_n$ is a θ-contraction with respect to F. It is obvious that $\{f_n\}$ is stable on F. Moreover, it follows from (4.1) that

$$x_{n+1}={\alpha }_n{f}_n(x_n)+(1-{\alpha }_n)S_n{x}_n$$

for $n\in \mathbb{N}$. Thus Theorem 3.1 implies that $\{x_n\}$ converges strongly to $w=(P_F\circ f_1)(w)=P_F(I-A)w$, which is the unique solution of Problem 4.1. □

Remark 4.4 The iteration (4.1) is called the hybrid steepest descent method; see [9, 10] for more details.

We finally construct an example of $\{S_n\}$ in Theorem 4.3 by using the notion of a subgradient projection.

Let $g:H\to \mathbb{R}$ be a continuous and convex function such that

$$C=\{x\in H:g(x)\le 0\}$$

is nonempty and $h:H\to H$ a mapping such that $h(x)\in \partial g(x)$ for all $x\in H$, where ∂g denotes the subdifferential mapping of g defined by

$$\partial g(x)=\{z\in H:g(x)+〈y-x,z〉\le g(y)(\mathrm{\forall }y\in H)\}$$

for all $x\in H$. Then the subgradient projection $P_{g,h}:H\to H$ with respect to g and h is defined by $P_{g,h}x=P_{L(x)}x$ for all $x\in H$, where $P_{L(x)}$ denotes the metric projection of H onto the set $L(x)$ defined by

$$L(x)=\{y\in H:g(x)+〈y-x,h(x)〉\le 0\}$$

for all $x\in H$. Note that C is a subset of $L(x)$ for all $x\in H$ and that $L(x)$ is a closed half space for all $x\in H\setminus C$. According to [[23], Section 7], [[24], Proposition 2.3], and [[25], Proposition 1.1.11], we know the following:

(S1) $Fix(P_{g,h})=C$;

(S2) $〈z-P_{g,h}x,x-P_{g,h}x〉\le 0$ for all $z\in C$ and $x\in H$;

(S3) if $g(V)$ is bounded for each bounded subset V of H, then $I-P_{g,h}$ is demiclosed at 0.

It is known that the metric projection $P_D$ of H onto a nonempty closed convex subset D of H coincides with the subgradient projection $P_{g,h}$ with respect to g and h defined by $g(x)={inf}_{y\in D}\parallel x-y\parallel$ for all $x\in H$ and

$$h(x)=\{\begin{array}{cc}0\hfill & (x\in D);\hfill \\ (x-P_{D}x)/\parallel x-P_{D}x\parallel \hfill & (x\in H\setminus D).\hfill \end{array}$$

The subgradient projection is not necessarily nonexpansive. In fact, if $g:\mathbb{R}\to \mathbb{R}$ and $h:\mathbb{R}\to \mathbb{R}$ are defined by $g(x)=max\{x,2x-1\}$ for all $x\in \mathbb{R}$ and $h(x)=1$ if $x<1$; $h(x)=2$ if $x\ge 1$, then $P_{g,h}$ is given by

$$P_{g,h}(x)=\{\begin{array}{cc}x\hfill & (x\le 0);\hfill \\ 0\hfill & (0<x<1);\hfill \\ 1/2\hfill & (x\ge 1)\hfill \end{array}$$

and is not nonexpansive.

Using (S1), (S2), and (S3), we show the following.

Example 4.5 Let $g:H\to \mathbb{R}$ be a continuous and convex function such that $C=\{x\in H:g(x)\le 0\}$ is nonempty and $g(V)$ is bounded for each bounded subset V of H, $h:H\to H$ a mapping such that $h(x)\in \partial g(x)$ for all $x\in H$, and $\{S_n\}$ a sequence of mappings of H into H defined by

$$S_n={\beta }_{n}I+(1-{\beta }_n)P_{g,h}$$

for all $n\in \mathbb{N}$, where $\{{\beta }_n\}$ is a sequence of real numbers such that $-1<{inf}_n{\beta }_n$ and ${sup}_n{\beta }_n<1$. Then the following hold:

• (i) $Fix(S_n)=C$ for all $n\in \mathbb{N}$;

• (ii) $\{S_n\}$ is strongly quasinonexpansive type;

• (iii) $\{S_n\}$ satisfies the condition (Z).

Proof Since ${\beta }_n\ne 1$ for all $n\in \mathbb{N}$, the part (i) obviously follows from (S1).

We first show (ii). By (i), we know that ${\bigcap }_{n=1}^{\mathrm{\infty }}Fix(S_n)=C$ is nonempty. Let $n\in \mathbb{N}$, $p\in C$, and $x\in H$ be given. Then we have

$$\begin{array}{rl}{\parallel S_{n}x-p\parallel }^2+{\parallel x-S_{n}x\parallel }^2-{\parallel x-p\parallel }^2& =2〈S_{n}x-x,S_{n}x-p〉\\ =2(1-{\beta }_n)〈p-S_{n}x,x-P_{g,h}x〉.\end{array}$$

(4.2)

It follows from (S2) that

$$〈p-S_{n}x,x-P_{g,h}x〉\le 〈P_{g,h}x-S_{n}x,x-P_{g,h}x〉.$$

(4.3)

On the other hand, we also know that

$$\begin{array}{c}〈P_{g,h}x-S_{n}x,x-P_{g,h}x〉\hfill \\ =-{\parallel P_{g,h}x-x\parallel }^2+〈x-S_{n}x,x-P_{g,h}x〉\hfill \\ \le -{(\parallel P_{g,h}x-x\parallel -\frac{1}{2}\parallel x-S_{n}x\parallel )}^2+\frac{1}{4}{\parallel x-S_{n}x\parallel }^2\le \frac{1}{4}{\parallel x-S_{n}x\parallel }^2.\hfill \end{array}$$

(4.4)

By (4.2), (4.3), and (4.4), each $S_n$ satisfies

$${\parallel S_{n}x-p\parallel }^2+\frac{1}{2}(1+{\beta }_n){\parallel x-S_{n}x\parallel }^2\le {\parallel x-p\parallel }^2$$

(4.5)

for all $p\in C$ and $x\in H$. Since $(1+{\beta }_n)/2>0$, we know that each $S_n$ is quasinonexpansive.

Let $\{x_n\}$ be a bounded sequence in H such that $\parallel x_n-p\parallel -\parallel S_n{x}_n-p\parallel \to 0$ for some $p\in C$. Since $\{S_n{x}_n\}$ is bounded, it follows from (4.5) that

$$\frac{1}{2}(1+{\beta }_n){\parallel x_n-S_n{x}_n\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel S_n{x}_n-p\parallel }^2\to 0$$

and hence $S_n{x}_n-x_n\to 0$ by ${inf}_n(1+{\beta }_n)>0$. Thus $\{S_n\}$ is strongly quasinonexpansive type.

We finally show (iii). Let $\{y_n\}$ be a bounded sequence in H such that $S_n{y}_n-y_n\to 0$. By the definition of $S_n$, we have

$$\parallel P_{g,h}{y}_n-y_n\parallel =\frac{1}{1-{\beta }_n}\parallel S_n{y}_n-y_n\parallel$$

for all $n\in \mathbb{N}$. Since ${inf}_n(1-{\beta }_n)>0$, we obtain $P_{g,h}{y}_n-y_n\to 0$. Consequently, by (S1) and (S3), we know that $\{S_n\}$ satisfies the condition (Z). □