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Viscosity approximation process for a sequence of quasinonexpansive mappings
Fixed Point Theory and Applications volume 2014, Article number: 17 (2014)
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
for $n\in \mathbb{N}$, where ${\alpha}_{n}$ is a real number in $[0,1]$, ${f}_{n}$ is a contractionlike 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, $\u3008\cdot ,\cdot \u3009$ 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 Txp\parallel \le \parallel xp\parallel $ for all $x\in C$ and $p\in Fix(T)$; T is said to be nonexpansive if $\parallel TxTy\parallel \le \parallel xy\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
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 θcontraction with respect to F if $\parallel f(x)f(z)\parallel \le \theta \parallel xz\parallel $ for all $x\in C$ and $z\in F$; f is said to be a θcontraction if f is a θcontraction 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 θcontraction with respect to F, where $0\le \theta <1$. If F is closed and convex, then ${P}_{F}\circ f$ is a θcontraction on F, where ${P}_{F}$ is the metric projection of H onto F.
Lemma 2.2 Let $f:C\to C$ be a θcontraction, where $0\le \theta <1$ and $T:C\to C$ a quasinonexpansive mapping. Then $f\circ T$ is a θcontraction 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
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 θcontraction 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
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 θcontraction 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
and
for $n\in \mathbb{N}$.
Lemma 3.2 $\{{x}_{n}\}$, $\{{S}_{n}{x}_{n}\}$, and $\{{f}_{n}({x}_{n})\}$ are bounded, and moreover,
and
hold for every $n\in \mathbb{N}$.
Proof Since ${f}_{n}$ is a θcontraction 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
for every $n\in \mathbb{N}$. Thus, by induction on n, we have
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
and thus
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 $1<12\theta \le 1$, we know that
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
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 $12\theta \ge 0$. 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 $12\theta <0$. 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
for every $n\in \mathbb{N}$ and
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
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
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
On the other hand, from (3.3) and (3.6), we know that
for every $n\in \mathbb{N}$. Thus, by ${\beta}_{\tau (n)}>0$, this shows that
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
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
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
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
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 θcontraction with respect to $F=Fix(S)$, where $0\le \theta <1$. Let $\{{x}_{n}\}$ be a sequence defined by ${x}_{1}\in C$ and
for $n\in \mathbb{N}$. Suppose that $IS$ 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 κstrongly monotone and ηLipschitz continuous mapping, that is, we assume that $\u3008xy,AxAy\u3009\ge \kappa {\parallel xy\parallel}^{2}$ and $\parallel AxAy\parallel \le \eta \parallel xy\parallel $ for all $x,y\in H$. Then find $z\in F$ such that
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 $, $0\le 12\kappa +{\eta}^{2}<1$ and $IA$ is a θcontraction, where $\theta =\sqrt{12\kappa +{\eta}^{2}}$;

Problem 4.1 has a unique solution and $VI(F,A)=Fix({P}_{F}(IA))$.
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 κstrongly monotone and ηLipschitz 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
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 ${f}_{n}=(IA){S}_{n}$ for $n\in \mathbb{N}$ and $\theta =\sqrt{12\kappa +{\eta}^{2}}$. Since $IA$ is a θcontraction and ${S}_{n}$ is quasinonexpansive, Lemma 2.2 implies that each ${f}_{n}$ is a θcontraction with respect to F. It is obvious that $\{{f}_{n}\}$ is stable on F. Moreover, it follows from (4.1) that
for $n\in \mathbb{N}$. Thus Theorem 3.1 implies that $\{{x}_{n}\}$ converges strongly to $w=({P}_{F}\circ {f}_{1})(w)={P}_{F}(IA)w$, 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
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
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
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) $\u3008z{P}_{g,h}x,x{P}_{g,h}x\u3009\le 0$ 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 xy\parallel $ for all $x\in H$ and
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 $g(x)=max\{x,2x1\}$ 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
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
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
It follows from (S2) that
On the other hand, we also know that
By (4.2), (4.3), and (4.4), each ${S}_{n}$ satisfies
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
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
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). □
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Acknowledgements
This paper is dedicated to Professor Wataru Takahashi on the occasion of his 70th birthday.
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Aoyama, K., Kohsaka, F. Viscosity approximation process for a sequence of quasinonexpansive mappings. Fixed Point Theory Appl 2014, 17 (2014) doi:10.1186/16871812201417
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Keywords
 viscosity approximation method
 quasinonexpansive mapping
 fixed point
 hybrid steepest descent method