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Convergence analysis of an iterative algorithm for monotone operators
Journal of Inequalities and Applications volume 2013, Article number: 199 (2013)
Abstract
In this paper, an iterative algorithm is proposed to study some nonlinear operators which are inversestrongly monotone, maximal monotone, and strictly pseudocontractive. Strong convergence of the proposed iterative algorithm is obtained in the framework of Hilbert spaces.
MSC:47H05, 47H09.
1 Introduction
Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation, and this operator is decomposed as the sum of two nonlinear operators. Study of fixed (zero) point approximation algorithms for computing fixed (zero) points constitutes now a topic of intensive research efforts. Many wellknown problems can be studied by using algorithms which are iterative in their nature. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set in which the required solution lies. The problem of finding a point in the intersection of these convex sets is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such a point. The wellknown convex feasibility problem which captures applications in various disciplines such as image restoration and radiation therapy treatment planning is to find a point in the intersection of common fixed (zero) point sets of a family of nonlinear mappings; see, for example, [1–16].
In this paper, we will investigate the problem of finding a common solution to inclusion problems and fixed point problems based on an iterative algorithm. Strong convergence of the proposed iterative algorithm has been obtained in the framework of Hilbert spaces.
The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, an iterative algorithm is proposed and analyzed. Some subresults of the main results are also discussed in this section.
2 Preliminaries
From now on, we always assume that H is a real Hilbert space with the inner product $\u3008\cdot ,\cdot \u3009$ and the norm $\parallel \cdot \parallel $, respectively. Let C be a nonempty closed convex subset of H.
Let $S:C\to C$ be a mapping. $F(S)$ stands for the fixed point set of S; that is, $F(S):=\{x\in C:x=Sx\}$.
Recall that S is said to be nonexpansive iff
S is said to be asymptotically nonexpansive iff there exists a sequence $\{{k}_{n}\}\subset [1,\mathrm{\infty})$ with ${lim}_{n\to \mathrm{\infty}}{k}_{n}=1$ such that
Recall that S is said to be strictly pseudocontractive iff there exits a positive constant κ such that
S is said to be asymptotically strictly pseudocontractive iff there exits a positive constant κ and a sequence $\{{k}_{n}\}\subset [1,\mathrm{\infty})$ with ${lim}_{n\to \mathrm{\infty}}{k}_{n}=1$ such that
Let $A:C\to H$ be a mapping. Recall that A is said to be monotone iff
A is said to be inversestrongly monotone iff there exists a constant $\alpha >0$ such that
For such a case, A is also said to be αinversestrongly monotone. It is not hard to see that inversestrongly monotone mappings are Lipschitz continuous.
A multivalued operator $T:H\to {2}^{H}$ with the domain $D(T)=\{x\in H:Tx\ne \mathrm{\varnothing}\}$ and the range $R(T)=\{Tx:x\in D(T)\}$ is said to be monotone if for ${x}_{1}\in D(T)$, ${x}_{2}\in D(T)$, ${y}_{1}\in T{x}_{1}$ and ${y}_{2}\in T{x}_{2}$, we have $\u3008{x}_{1}{x}_{2},{y}_{1}{y}_{2}\u3009\ge 0$. A monotone operator T is said to be maximal if its graph $G(T)=\{(x,y):y\in Tx\}$ is not properly contained in the graph of any other monotone operator. Let I denote the identity operator on H and $T:H\to {2}^{H}$ be a maximal monotone operator. Then we can define, for each $\lambda >0$, a nonexpansive singlevalued mapping ${J}_{\lambda}:H\to H$ by ${J}_{\lambda}={(I+\lambda T)}^{1}$. It is called the resolvent of T. We know that ${T}^{1}0=F({J}_{\lambda})$ for all $\lambda >0$ and ${J}_{\lambda}$ is firmly nonexpansive; see [17–23] and the references therein.
Recently, many authors have investigated the solution problems of nonlinear operator equations or inequalities based on iterative methods; see, for instance, [24–33] and the references therein. In [19], Kamimura and Takahashi investigated the problem of finding zero points of a maximal monotone operator via the following iterative algorithm:
where $\{{\alpha}_{n}\}$ is a sequence in $(0,1)$, $\{{\lambda}_{n}\}$ is a positive sequence, $T:H\to {2}^{H}$ is a maximal monotone and ${J}_{{\lambda}_{n}}={(I+{\lambda}_{n}T)}^{1}$. They showed that the sequence $\{{x}_{n}\}$ generated in (2.1) converges weakly to some $z\in {T}^{1}(0)$ provided that the control sequence satisfies some restrictions.
Recall that the classical variational inequality is to find an $x\in C$ such that
In this paper, we use $VI(C,A)$ to denote the solution set of (2.2). It is known that $x\in C$ is a solution to (2.1) iff x is a fixed point of the mapping ${P}_{C}(I\lambda A)$, where $\lambda >0$ is a constant, I stands for the identity mapping, and ${P}_{C}$ stands for the metric projection from H onto C. If A is αinversestrongly monotone and $\lambda \in (0,2\alpha ]$, then the mapping ${P}_{C}(IrA)$ is nonexpansive; see [28] for more details. It follows that $VI(C,A)$ is closed and convex.
In [28], Takahashi an Toyoda investigated the problem of finding a common solution of variational inequality problem (2.1) and a fixed point problem involving nonexpansive mappings by considering the following iterative algorithm:
where $\{{\alpha}_{n}\}$ is a sequence in $(0,1)$, $\{{\lambda}_{n}\}$ is a positive sequence, $S:C\to C$ is a nonexpansive mapping and $A:C\to H$ is an inversestrongly monotone mapping. They proved that the sequence $\{{x}_{n}\}$ generated in (2.3) converges weakly to some $z\in VI(C,A)\cap F(S)$ provided that the control sequence satisfies some restrictions.
In [29], Tada and Takahashi investigated the problem of finding a common solution of an equilibrium problem and a fixed point problem involving nonexpansive mappings by considering the following iterative algorithm:
for each $n\ge 1$, where $\{{\alpha}_{n}\}$ is a sequence in $(0,1)$, $\{{r}_{n}\}$ is a positive sequence, $S:C\to C$ is a nonexpansive mapping and $F:C\times C\to R$ is a bifunction. They showed that the sequence $\{{x}_{n}\}$ generated in (2.4) converges weakly to some $z\in EP(F)\cap F(S)$, where $EP(F)$ stands for the solution set of the equilibrium problem, provided that the control sequence satisfies some restrictions.
In [30], Manaka and Takahashi introduced the following iteration:
where $\{{\alpha}_{n}\}$ is a sequence in $(0,1)$, $\{{\lambda}_{n}\}$ is a positive sequence, $S:C\to C$ is a nonexpansive mapping, $A:C\to H$ is an inverselystrongly monotone mapping, $B:D(B)\subset C\to {2}^{H}$ is a maximal monotone operator, ${J}_{{\lambda}_{n}}={(I+{\lambda}_{n}B)}^{1}$ is the resolvent of B. They showed that the sequence $\{{x}_{n}\}$ generated in (2.5) converges weakly to some $z\in {(A+B)}^{1}(0)\cap F(S)$ provided that the control sequence satisfies some restrictions.
In this paper, motivated by the above results, we consider the problem of finding a common solution to the zero point problems involving two monotone operators and fixed point problems involving asymptotically strictly pseudocontractive mappings based on a onestep iterative method. Weak convergence theorems are established in the framework of Hilbert spaces.
In order to obtain our main results in this paper, we need the following lemmas.
Recall that a space is said to satisfy Opial’s property [34] if, for any sequence $\{{x}_{n}\}\subset H$ with ${x}_{n}\rightharpoonup x$, where ⇀ denotes the weak convergence, the inequality
holds for every $y\in H$ with $y\ne x$. Indeed, the above inequality is equivalent to the following:
Lemma 2.1 [20]
Let C be a nonempty, closed, and convex subset of H, $A:C\to H$ be a mapping, and $B:H\rightrightarrows H$ be a maximal monotone operator. Then $F({J}_{r}(I\lambda A))={(A+B)}^{1}(0)$.
Lemma 2.2 Let H be a real Hilbert space. For any $a\in (0,1)$ and $x,y\in H$, the following holds:
Lemma 2.3 [35]
Let $\{{a}_{n}\}$, $\{{b}_{n}\}$, and $\{{c}_{n}\}$ be three nonnegative sequences satisfying the following condition:
where ${n}_{0}$ is some nonnegative integer, ${\sum}_{n=1}^{\mathrm{\infty}}{b}_{n}<\mathrm{\infty}$ and ${\sum}_{n=1}^{\mathrm{\infty}}{c}_{n}<\mathrm{\infty}$. Then the limit ${lim}_{n\to \mathrm{\infty}}{a}_{n}$ exists.
Lemma 2.4 [36]
Let C be a nonempty closed convex subset of H and S be an asymptotically κstrictly pseudocontractive mapping. Then we have

(a)
S is uniformly Lipschitz continuous;

(b)
$IS$ is demiclosed at zero, that is, if $\{{x}_{n}\}$ is a sequence in C with ${x}_{n}\rightharpoonup x$ and ${x}_{n}S{x}_{n}\to 0$, then $x\in F(S)$.
The following lemma can be obtained from [37] immediately.
Lemma 2.5 Let H be a real Hilbert space. The following holds:
where $N\ge 2$ denotes some positive integer, ${a}_{1},{a}_{2},\dots ,{a}_{N}$ are real numbers with ${\sum}_{i=1}^{N}{a}_{i}=1$ in $(0,1)$ and ${x}_{1},{x}_{2},\dots ,{x}_{N}\in H$.
3 Main results
Theorem 3.1 Let C be a nonempty closed convex subset of H. Let $N\ge 2$ be some positive integer and $S:C\to C$ be an asymptotically strictly pseudocontractive mapping with the constant κ and the sequence $\{{k}_{n}\}$. Let ${A}_{m}:C\to H$ be an inversestrongly monotone mapping with the constant ${\alpha}_{m}$ and ${B}_{m}$ be a maximal monotone operator on H such that the domain of ${B}_{m}$ is included in C for each $m\in \{2,3,\dots ,N\}$. Assume $\mathcal{F}={\bigcap}_{m=2}^{N}{({A}_{m}+{B}_{m})}^{1}(0)\cap F(S)\ne \mathrm{\varnothing}$. Let $\{{\alpha}_{n,1}\},\{{\alpha}_{n,2}\},\dots ,\{{\alpha}_{n,N}\}$ and $\{{\beta}_{n}\}$ are real number sequences in $(0,1)$. Let $\{{r}_{n,2}\},\dots $ , and $\{{r}_{n,N}\}$ be positive real number sequences. Let $\{{x}_{n}\}$ be a sequence in C generated in the following iterative process:
where ${J}_{{r}_{n,m}}={(I+{r}_{n,m}{B}_{m})}^{1}$ is the resolvent of ${B}_{m}$. Assume that the sequences $\{{\alpha}_{n,1}\},\{{\alpha}_{n,2}\},\dots ,\{{\alpha}_{n,N}\}$, $\{{\beta}_{n}\}$, $\{{r}_{n,2}\},\dots ,\{{r}_{n,N}\}$, and $\{{k}_{n}\}$ satisfy the following restrictions:

(a)
${\sum}_{m=1}^{N}{\alpha}_{n,m}=1$ and $0<a\le {\alpha}_{n,m}<1$, $\mathrm{\forall}m\in \{2,\dots ,N\}$;

(b)
$0\le \kappa \le {\beta}_{n}\le b<1$;

(c)
$0<c\le {r}_{n,m}\le d<2{\alpha}_{m}$, $\mathrm{\forall}m\in \{2,\dots ,N\}$;

(d)
${\sum}_{n=1}^{\mathrm{\infty}}({k}_{n}1)<\mathrm{\infty}$,
where a, b, c, and d are positive real numbers. Then the sequence $\{{x}_{n}\}$ generated in (3.1) converges weakly to some point in ℱ.
Proof First, we show $I{r}_{n,m}{A}_{m}$ is nonexpansive. In view of the restriction (c), we find that
This proves that $I{r}_{n,m}{A}_{m}$ is nonexpansive. Let $p\in \mathcal{F}$. In view of Lemma 2.1, we find that
Putting ${u}_{n,m}={J}_{{r}_{n,m}}({x}_{n}{r}_{n,m}{A}_{m}{x}_{n})$, we find that
In view of Lemma 2.2, we find from the restriction (b) that
From (3.2) and (3.3), we have
We draw the conclusion that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}p\parallel $ exists with the aid of Lemma 2.3. This implies that the sequence $\{{x}_{n}\}$ is bounded. In view of Lemma 2.5, we find that
which yields
In view of the restriction (a), we find that
On the other hand, we have
It follows that
This in turn implies that
It follows from the restrictions (b) and (d) that
Notice that
It follows that
This implies that
which finds that
In view of the restriction (a), we find from (3.8) that
Notice that
From (3.6) and (3.10), we obtain that
On the other hand, we have
which yields
This implies from the restriction (c) and (3.11) that
Notice that
This implies from (3.10) and (3.11) that
On the other hand, we have
Since S is uniformly continuous, we obtain from (3.12) and (3.13) that
Since $\{{x}_{n}\}$ is bounded, there exists a subsequence $\{{x}_{{n}_{i}}\}$ of $\{{x}_{n}\}$ such that ${x}_{{n}_{i}}\rightharpoonup \omega \in C$. We find that $\omega \in F(S)$ with the aid of Lemma 2.4.
Next, we show $\omega \in {({A}_{m}+{B}_{m})}^{1}0$ for every $m\in \{1,2,\dots ,N\}$. In view of (3.10), we can choose a subsequence $\{{u}_{{n}_{i},m}\}$ of $\{{u}_{n,m}\}$ such that ${u}_{{n}_{i},m}\rightharpoonup \omega $. Notice that
This implies that
That is,
Since ${B}_{m}$ is monotone, we get for any $({u}_{m},{v}_{m})\in G({B}_{m})$ that
Replacing n by ${n}_{i}$ and letting $i\to \mathrm{\infty}$, we obtain from (3.10) that
This means ${A}_{m}{\omega}_{m}\in {B}_{m}\omega $, that is, $0\in ({A}_{m}+{B}_{m})(\omega )$. Hence we get $\omega \in {({A}_{m}+{B}_{m})}^{1}(0)$ for every $m\in \{1,2,\dots ,N\}$. This completes the proof that $\omega \in \mathcal{F}$.
Suppose there is another subsequence $\{{x}_{{n}_{j}}\}$ of $\{{x}_{n}\}$ such that ${x}_{{n}_{j}}\rightharpoonup {\omega}^{\prime}$. Then we can show that ${\omega}^{\prime}\in \mathcal{F}$ in the same way. Assume $\omega \ne {\omega}^{\prime}$. Since ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}p\parallel $ exits for any $p\in \mathcal{F}$. Put ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}\omega \parallel =d$. Since the space satisfies Opial’s condition, we see that
This is a contradiction. This shows that $\omega ={\omega}^{\prime}$. This proves that the sequence $\{{x}_{n}\}$ converges weakly to $\omega \in \mathcal{F}$. This completes the proof. □
If $N=2$, then we have the following.
Corollary 3.2 Let C be a nonempty closed convex subset of H. Let $S:C\to C$ be an asymptotically strictly pseudocontractive mapping with the constant κ and the sequence $\{{k}_{n}\}$. Let $A:C\to H$ be an inversestrongly monotone mapping with the constant α, and B be a maximal monotone operator on H such that the domain of B is included in C. Assume $\mathcal{F}={(A+B)}^{1}(0)\cap F(S)\ne \mathrm{\varnothing}$. Let $\{{\alpha}_{n,1}\}$, $\{{\alpha}_{n,2}\}$, and $\{{\beta}_{n}\}$ be real number sequences in $(0,1)$. Let $\{{r}_{n}\}$ be a positive real number sequence. Let $\{{x}_{n}\}$ be a sequence in C generated in the following iterative process:
where ${J}_{{r}_{n}}={(I+{r}_{n}B)}^{1}$ is the resolvent of B. Assume that the sequences $\{{\alpha}_{n,1}\}$, $\{{\alpha}_{n,2}\}$, $\{{\beta}_{n}\}$, $\{{r}_{n}\}$, and $\{{k}_{n}\}$ satisfy the following restrictions:

(a)
${\sum}_{m=1}^{2}{\alpha}_{n,m}=1$ and $0<a\le {\alpha}_{n,m}<1$, $\mathrm{\forall}m\in \{1,2\}$;

(b)
$0\le \kappa \le {\beta}_{n}\le b<1$;

(c)
$0<c\le {r}_{n}\le d<2\alpha $;

(d)
${\sum}_{n=1}^{\mathrm{\infty}}({k}_{n}1)<\mathrm{\infty}$,
where a, b, c, and d are positive real numbers. Then the sequence $\{{x}_{n}\}$ converges weakly to some point in ℱ.
If S is asymptotically nonexpansive, then we find from Theorem 3.1 the following by letting ${\beta}_{n}=0$.
Corollary 3.3 Let C be a nonempty closed convex subset of H. Let $N\ge 2$ be some positive integer and $S:C\to C$ be an asymptotically nonexpansive mapping with the sequence $\{{k}_{n}\}$. Let ${A}_{m}:C\to H$ be an inversestrongly monotone mapping with the constant ${\alpha}_{m}$ and let ${B}_{m}$ be a maximal monotone operator on H such that the domain of ${B}_{m}$ is included in C for each $m\in \{2,3,\dots ,N\}$. Assume $\mathcal{F}={\bigcap}_{m=2}^{N}{({A}_{m}+{B}_{m})}^{1}(0)\cap F(S)\ne \mathrm{\varnothing}$. Let $\{{\alpha}_{n,1}\},\{{\alpha}_{n,2}\},\dots ,\{{\alpha}_{n,N}\}$, and $\{{\beta}_{n}\}$ be real number sequences in $(0,1)$. Let $\{{r}_{n,2}\},\dots $ , and $\{{r}_{n,N}\}$ be positive real number sequences. Let $\{{x}_{n}\}$ be a sequence in C generated in the following iterative process:
where ${J}_{{r}_{n,m}}={(I+{r}_{n,m}{B}_{m})}^{1}$ is the resolvent of ${B}_{m}$. Assume that the sequences $\{{\alpha}_{n,1}\},\{{\alpha}_{n,2}\},\dots ,\{{\alpha}_{n,N}\}$, $\{{\beta}_{n}\}$, $\{{r}_{n,2}\},\dots ,\{{r}_{n,N}\}$, and $\{{k}_{n}\}$ satisfy the following restrictions:

(a)
${\sum}_{m=1}^{N}{\alpha}_{n,m}=1$ and $0<a\le {\alpha}_{n,m}<1$, $\mathrm{\forall}m\in \{2,\dots ,N\}$;

(b)
$0<b\le {r}_{n,m}\le c<2{\alpha}_{m}$, $\mathrm{\forall}m\in \{2,\dots ,N\}$;

(c)
${\sum}_{n=1}^{\mathrm{\infty}}({k}_{n}1)<\mathrm{\infty}$,
where a, b and c are positive real numbers. Then the sequence $\{{x}_{n}\}$ converges weakly to some point in ℱ.
If S is the identity mapping, then we draw from Theorem 3.1 the following.
Corollary 3.4 Let C be a nonempty closed convex subset of H. Let $N\ge 2$ be some positive integer. Let ${A}_{m}:C\to H$ be an inversestrongly monotone mapping with the constant ${\alpha}_{m}$ and let ${B}_{m}$ be a maximal monotone operator on H such that the domain of ${B}_{m}$ is included in C for each $m\in \{2,3,\dots ,N\}$. Assume $\mathcal{F}={\bigcap}_{m=2}^{N}{({A}_{m}+{B}_{m})}^{1}(0)\ne \mathrm{\varnothing}$. Let $\{{\alpha}_{n,1}\},\{{\alpha}_{n,2}\},\dots $ , and $\{{\alpha}_{n,N}\}$ be real number sequences in $(0,1)$. Let $\{{r}_{n,2}\},\dots $ , and $\{{r}_{n,N}\}$ be positive real number sequences. Let $\{{x}_{n}\}$ be a sequence in C generated in the following iterative process:
where ${J}_{{r}_{n,m}}={(I+{r}_{n,m}{B}_{m})}^{1}$ is the resolvent of ${B}_{m}$. Assume that the sequences $\{{\alpha}_{n,1}\},\{{\alpha}_{n,2}\},\dots ,\{{\alpha}_{n,N}\}$, $\{{r}_{n,2}\},\dots $ , and $\{{r}_{n,N}\}$ satisfy the following restrictions:

(a)
${\sum}_{m=1}^{N}{\alpha}_{n,m}=1$ and $0<a\le {\alpha}_{n,m}<1$, $\mathrm{\forall}m\in \{2,\dots ,N\}$;

(b)
$0<b\le {r}_{n,m}\le c<2{\alpha}_{m}$, $\mathrm{\forall}m\in \{2,\dots ,N\}$,
where a, b, and c are positive real numbers. Then the sequence $\{{x}_{n}\}$ converges weakly to some point in ℱ.
Let $f:H\to (\mathrm{\infty},\mathrm{\infty}]$ be a proper lower semicontinuous convex function. Define the subdifferential
for all $x\in H$. Then ∂f is a maximal monotone operator of H into itself; see [38] for more details. Let C be a nonempty closed convex subset of H and ${i}_{C}$ be the indicator function of C, that is,
Furthermore, we define the normal cone ${N}_{C}(v)$ of C at v as follows:
for any $v\in C$. Then ${i}_{C}:H\to (\mathrm{\infty},\mathrm{\infty}]$ is a proper lower semicontinuous convex function on H and $\partial {i}_{C}$ is a maximal monotone operator. Let ${J}_{r}x={(I+r\partial {i}_{C})}^{1}x$ for any $r>0$ and $x\in H$. From $\partial {i}_{C}x={N}_{C}x$ and $x\in C$, we get
where ${P}_{C}$ is the metric projection from H into C. Similarly, we can get that $x\in {(A+\partial {i}_{C})}^{1}(0)\iff x\in VI(A,C)$.
Corollary 3.5 Let C be a nonempty closed convex subset of H. Let $N\ge 2$ be some positive integer and $S:C\to C$ be an asymptotically strictly pseudocontractive mapping with the constant κ and the sequence $\{{k}_{n}\}$. Let ${A}_{m}:C\to H$ be an inversestrongly monotone mapping with the constant ${\alpha}_{m}$ for each $m\in \{2,3,\dots ,N\}$. Assume $\mathcal{F}={\bigcap}_{m=2}^{N}VI(C,{A}_{m})\cap F(S)\ne \mathrm{\varnothing}$. Let $\{{\alpha}_{n,1}\},\{{\alpha}_{n,2}\},\dots ,\{{\alpha}_{n,N}\}$, and $\{{\beta}_{n}\}$ be real number sequences in $(0,1)$. Let $\{{r}_{n,2}\},\dots $ , and $\{{r}_{n,N}\}$ be positive real number sequences. Let $\{{x}_{n}\}$ be a sequence in C generated in the following iterative process:
Assume that the sequences $\{{\alpha}_{n,1}\},\{{\alpha}_{n,2}\},\dots ,\{{\alpha}_{n,N}\}$, $\{{\beta}_{n}\}$, $\{{r}_{n,2}\},\dots ,\{{r}_{n,N}\}$, and $\{{k}_{n}\}$ satisfy the following restrictions:

(a)
${\sum}_{m=1}^{N}{\alpha}_{n,m}=1$ and $0<a\le {\alpha}_{n,m}<1$, $\mathrm{\forall}m\in \{2,\dots ,N\}$;

(b)
$0\le \kappa \le {\beta}_{n}\le b<1$;

(c)
$0<c\le {r}_{n,m}\le d<2{\alpha}_{m}$, $\mathrm{\forall}m\in \{2,\dots ,N\}$;

(d)
${\sum}_{n=1}^{\mathrm{\infty}}({k}_{n}1)<\mathrm{\infty}$,
where a, b, c, and d are positive real numbers. Then the sequence $\{{x}_{n}\}$ converges weakly to some point in ℱ.
Proof Putting ${B}_{m}=\partial {i}_{C}$ for every $m\in \{2,3,\dots ,N\}$, we see ${J}_{{r}_{n,m}}={P}_{C}$. We can immediately draw from Theorem 3.1 the desired conclusion. □
If S is the identity mapping, then we find from Corollary 3.5 the following.
Corollary 3.6 Let C be a nonempty closed convex subset of H. Let $N\ge 2$ be some positive integer. Let ${A}_{m}:C\to H$ be an inversestrongly monotone mapping with the constant ${\alpha}_{m}$ for each $m\in \{2,3,\dots ,N\}$. Assume $\mathcal{F}={\bigcap}_{m=2}^{N}VI(C,{A}_{m})\ne \mathrm{\varnothing}$. Let $\{{\alpha}_{n,1}\},\{{\alpha}_{n,2}\},\dots $ , and $\{{\alpha}_{n,N}\}$ be real number sequences in $(0,1)$. Let $\{{r}_{n,2}\},\dots $ , and $\{{r}_{n,N}\}$ be positive real number sequences. Let $\{{x}_{n}\}$ be a sequence in C generated in the following iterative process:
Assume that the sequences $\{{\alpha}_{n,1}\},\{{\alpha}_{n,2}\},\dots ,\{{\alpha}_{n,N}\}$, $\{{\beta}_{n}\}$, $\{{r}_{n,2}\},\dots ,\{{r}_{n,N}\}$, and $\{{k}_{n}\}$ satisfy the following restrictions:

(a)
${\sum}_{m=1}^{N}{\alpha}_{n,m}=1$ and $0<a\le {\alpha}_{n,m}<1$, $\mathrm{\forall}m\in \{2,\dots ,N\}$;

(b)
$0<b\le {r}_{n,m}\le c<2{\alpha}_{m}$, $\mathrm{\forall}m\in \{2,\dots ,N\}$;

(c)
${\sum}_{n=1}^{\mathrm{\infty}}({k}_{n}1)<\mathrm{\infty}$,
where a, b, and c are positive real numbers. Then the sequence $\{{x}_{n}\}$ converges weakly to some point in ℱ.
References
 1.
Manro S, Kumar S, Bhatia SS: Common fixed point theorems in intuitionistic fuzzy metric spaces using occasionally weakly compatible maps. J. Math. Comput. Sci. 2012, 2: 73–81.
 2.
Yang S, Li W: Iterative solutions of a system of equilibrium problems in Hilbert spaces. Adv. Fixed Point Theory 2011, 1: 15–26.
 3.
Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi ϕ nonexpansive mappings. Appl. Math. Comput. 2010, 215: 3874–3883. 10.1016/j.amc.2009.11.031
 4.
Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2012, 2: 374–397.
 5.
Cho SY, Qin X, Kang SM: Hybrid projection algorithms for treating common fixed points of a family of demicontinuous pseudocontractions. Appl. Math. Lett. 2012, 25: 854–857. 10.1016/j.aml.2011.10.031
 6.
Luo H, Wang Y: Iterative approximation for the common solutions of a infinite variational inequality system for inversestrongly accretive mappings. J. Math. Comput. Sci. 2012, 2: 1660–1670.
 7.
Cho SY, Qin X, Kang SM: Hybrid projection algorithms for treating common fixed points of a family of demicontinuous pseudocontractions. Appl. Math. Lett. 2012, 25: 854–857. 10.1016/j.aml.2011.10.031
 8.
Byrne C: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 2008, 20: 103–120.
 9.
Tiwari R, Shukla DP: Coincidence points and common fixed points in cone Banach spaces. J. Math. Comput. Sci. 2012, 2: 1464–1474.
 10.
Qin X, Shang M, Su Y: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Comput. Model. 2008, 48: 1033–1046. 10.1016/j.mcm.2007.12.008
 11.
Lv S, Wu C: Convergence of iterative algorithms for a generalized variational inequality and a nonexpansive mapping. Eng. Math. Lett. 2012, 1: 44–57.
 12.
Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618.
 13.
Qin X, Cho YJ, Kang SM, Zhou H: Convergence of a modified Halperntype iteration algorithm for quasi ϕ nonexpansive mappings. Appl. Math. Lett. 2009, 22: 1051–1055. 10.1016/j.aml.2009.01.015
 14.
Cho YJ, Petrot N: Regularization and iterative method for general variational inequality problem in Hilbert spaces. J. Inequal. Appl. 2011., 2011: Article ID 21
 15.
Qin X, Cho SY, Kang SM: Strong convergence of shrinking projection methods for quasi ϕ nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 2010, 234: 750–760. 10.1016/j.cam.2010.01.015
 16.
Lin LJ, Huang YJ: Generalized vector quasiequilibrium problems with applications to common fixed point theorems and optimization problems. Nonlinear Anal. 2007, 66: 1275–1289. 10.1016/j.na.2006.01.025
 17.
Cho YJ, Kang SM, Zhou H: Approximate proximal point algorithms for finding zeroes of maximal monotone operators in Hilbert spaces. J. Inequal. Appl. 2008., 2008: Article ID 598191
 18.
Qin X, Cho YJ, Kang SM: Approximating zeros of monotone operators by proximal point algorithms. J. Glob. Optim. 2010, 46: 75–87. 10.1007/s1089800994106
 19.
Kammura S, Takahashi W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 2000, 106: 226–240. 10.1006/jath.2000.3493
 20.
Aoyama K, Kimura Y, Takahashi W, Toyoda M: On a strongly nonexpansive sequence in Hilbert spaces. J. Nonlinear Convex Anal. 2007, 8: 471–489.
 21.
Takahashi S, Takahashi W, Toyoda M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 2010, 147: 27–41. 10.1007/s1095701097132
 22.
Qin X, Su Y: Approximation of a zero point of accretive operator in Banach spaces. J. Math. Anal. Appl. 2007, 329: 415–424. 10.1016/j.jmaa.2006.06.067
 23.
Eshita K, Takahashi W: Approximating zero points of accretive operators in general Banach spaces. Fixed Point Theory Appl. 2007, 2: 105–116.
 24.
Qin X, Cho YJ, Kang SM: Approximating zeros of monotone operators by proximal point algorithms. J. Glob. Optim. 2010, 46: 75–87. 10.1007/s1089800994106
 25.
Hao Y: On variational inclusion and common fixed point problems in Hilbert spaces with applications. Appl. Math. Comput. 2010, 217: 3000–3010. 10.1016/j.amc.2010.08.033
 26.
Min L, Shisheng Z: A new iterative method for finding common solutions of generalized equilibrium problem, fixed point problem of infinite kstrict pseudocontractive mappings, and quasivariational inclusion problem. Acta Math. Sci. 2012, 32: 499–519.
 27.
Zhang SS, Lee JHW, Chan CK: Algorithms of common solutions to quasi variational inclusion and fixed point problems. Appl. Math. Mech. 2008, 29: 571–581. 10.1007/s104830080502y
 28.
Takahahsi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2003, 118: 417–428. 10.1023/A:1025407607560
 29.
Tada A, Takahashi W: Weak and strong convergence theorems for a nonexpansive mappings and an equilibrium problem. J. Optim. Theory Appl. 2007, 133: 359–370. 10.1007/s109570079187z
 30.
Manaka H, Takahashi W: Weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilberts space. CUBO 2011, 13: 11–24. 10.4067/S071906462011000100002
 31.
Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011
 32.
Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017
 33.
Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi ϕ nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 10
 34.
Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591–597. 10.1090/S000299041967117610
 35.
Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 1993, 178: 301–308. 10.1006/jmaa.1993.1309
 36.
Sahu DR, Xu HK, Yao JC: Asymptotically strict pseudocontractive mappings in the intermediate sense. Nonlinear Anal. 2009, 70: 3502–3511. 10.1016/j.na.2008.07.007
 37.
Hao Y, Cho SY, Qin X: Some weak convergence theorems for a family of asymptotically nonexpansive nonself mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 218573
 38.
Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056
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Keywords
 inversestrongly monotone mapping
 maximal monotone operator
 resolvent
 strictly pseudocontractive mapping
 fixed point