 Research
 Open Access
 Published:
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). □
References
 1.
Moudafi A: Viscosity approximation methods for fixedpoints problems. J. Math. Anal. Appl. 2000, 241: 46–55. 10.1006/jmaa.1999.6615
 2.
Aoyama K, Kimura Y, Kohsaka F: Strong convergence theorems for strongly relatively nonexpansive sequences and applications. J. Nonlinear Anal. Optim. 2012, 3: 67–77.
 3.
Xu HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059
 4.
Suzuki T: Moudafi’s viscosity approximations with MeirKeeler contractions. J. Math. Anal. Appl. 2007, 325: 342–352. 10.1016/j.jmaa.2006.01.080
 5.
Aoyama, K, Kimura, Y: Viscosity approximation methods with a sequence of contractions. CUBO (to appear)
 6.
Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S000299041967118640
 7.
Maingé PE: The viscosity approximation process for quasinonexpansive mappings in Hilbert spaces. Comput. Math. Appl. 2010, 59: 74–79. 10.1016/j.camwa.2009.09.003
 8.
Wongchan K, Saejung S: On the strong convergence of viscosity approximation process for quasinonexpansive mappings in Hilbert spaces. Abstr. Appl. Anal. 2011., 2011: Article ID 385843
 9.
Yamada I, Ogura N: Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasinonexpansive mappings. Numer. Funct. Anal. Optim. 2004, 25: 619–655.
 10.
Cegielski A, Zalas R: Methods for variational inequality problem over the intersection of fixed point sets of quasinonexpansive operators. Numer. Funct. Anal. Optim. 2013, 34: 255–283. 10.1080/01630563.2012.716807
 11.
Dotson WG Jr.: Fixed points of quasinonexpansive mappings. J. Aust. Math. Soc. 1972, 13: 167–170. 10.1017/S144678870001123X
 12.
Takahashi W: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama; 2009.
 13.
Maingé PE: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. SetValued Anal. 2008, 16: 899–912. 10.1007/s112280080102z
 14.
Aoyama K: Asymptotic fixed points of sequences of quasinonexpansive type mappings. In Banach and Function Spaces III. Yokohama Publishers, Yokohama; 2011:343–350.
 15.
Aoyama K, Kimura Y: Strong convergence theorems for strongly nonexpansive sequences. Appl. Math. Comput. 2011, 217: 7537–7545. 10.1016/j.amc.2011.01.092
 16.
Aoyama K, Kohsaka F, Takahashi W: Strong convergence theorems by shrinking and hybrid projection methods for relatively nonexpansive mappings in Banach spaces. In Nonlinear Analysis and Convex Analysis. Yokohama Publishers, Yokohama; 2009:7–26.
 17.
Aoyama K, Kohsaka F, Takahashi W: Strongly relatively nonexpansive sequences in Banach spaces and applications. J. Fixed Point Theory Appl. 2009, 5: 201–224. 10.1007/s1178400901087
 18.
Aoyama K, Kimura Y, Takahashi W, Toyoda M: On a strongly nonexpansive sequence in Hilbert spaces. J. Nonlinear Convex Anal. 2007, 8: 471–489.
 19.
Aoyama K, Kimura Y, Takahashi W, Toyoda M: Strongly nonexpansive sequences and their applications in Banach spaces. In Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2008:1–18.
 20.
Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. (2) 2002, 66: 240–256. 10.1112/S0024610702003332
 21.
Aoyama K, Kimura Y, Takahashi W, Toyoda M: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 2007, 67: 2350–2360. 10.1016/j.na.2006.08.032
 22.
 23.
Bauschke HH, Borwein JM: On projection algorithms for solving convex feasibility problems. SIAM Rev. 1996, 38: 367–426. 10.1137/S0036144593251710
 24.
Bauschke HH, Combettes PL: A weaktostrong convergence principle for Fejérmonotone methods in Hilbert spaces. Math. Oper. Res. 2001, 26: 248–264. 10.1287/moor.26.2.248.10558
 25.
Butnariu D, Iusem AN Applied Optimization 40. In Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Kluwer Academic, Dordrecht; 2000.
Acknowledgements
This paper is dedicated to Professor Wataru Takahashi on the occasion of his 70th birthday.
Author information
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Aoyama, K., Kohsaka, F. Viscosity approximation process for a sequence of quasinonexpansive mappings. Fixed Point Theory Appl 2014, 17 (2014). https://doi.org/10.1186/16871812201417
Received:
Accepted:
Published:
Keywords
 viscosity approximation method
 quasinonexpansive mapping
 fixed point
 hybrid steepest descent method