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# A new explicit iterative algorithm for solving a class of variational inequalities over the common fixed points set of a finite family of nonexpansive mappings

Fixed Point Theory and Applications20142014:60

https://doi.org/10.1186/1687-1812-2014-60

• Received: 6 November 2013
• Accepted: 19 February 2014
• Published:

## Abstract

In this paper, we introduce a new explicit iterative algorithm for finding a solution for a class of variational inequalities over the common fixed points set of a finite family of nonexpansive mappings in Hilbert spaces. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to the unique solution of the variational inequality. Our result improves and extends the corresponding results announced by many others. At the end of the paper, we extend our result to the more broad family of λ-strictly pseudo-contractive mappings.

## Keywords

• nonexpansive mapping
• strong convergence
• variational inequalities
• common fixed points

## 1 Introduction

Let H be a real Hilbert space with inner product $〈\cdot ,\cdot 〉$ and norm $\parallel \cdot \parallel$. Throughout this paper, we always assume that T is a nonexpansive operator on H. The fixed point set of T is denoted by $Fix\left(T\right)$, i.e., $Fix\left(T\right)=\left\{x\in H:Tx=x\right\}$. The typical problem is to minimize a quadratic function on a real Hilbert space H:
$\underset{x\in C}{min}\frac{1}{2}〈Ax,x〉-〈x,u〉,$
(1.1)

where C is a nonempty closed convex subset of H, u is a given point in H and A is a strongly positive bounded linear operator on H.

In 2003, Xu  introduced the following iterative scheme:
${x}_{n+1}={\alpha }_{n}u+\left(I-{\alpha }_{n}A\right)T{x}_{n},$
(1.2)

where u is some point of H and $\left\{{\alpha }_{n}\right\}$ is a sequence in $\left(0,1\right)$. He proved that the sequence $\left\{{x}_{n}\right\}$ converges strongly to the unique solution of the minimization problem (1.1) with $C=Fix\left(T\right)$.

In 2006, Marino and Xu  considered the viscosity method on the iterative scheme (1.2), and they gave the following general iterative method:
${x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right)T{x}_{n},$
(1.3)
where f is a contraction on H. They proved the above sequence $\left\{{x}_{n}\right\}$ converges strongly to the unique solution of the variational inequality
$〈\left(A-\gamma f\right){x}^{\ast },x-{x}^{\ast }〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in Fix\left(T\right),$
which is the optimality condition for the minimization problem
$\underset{x\in Fix\left(T\right)}{min}\frac{1}{2}〈Ax,x〉-h\left(x\right),$

where h is a potential function for γf (i.e., ${h}^{\prime }\left(x\right)=\gamma f\left(x\right)$ for $x\in H$).

In 2001, Yamada  considered the following hybrid iterative method:
${x}_{n+1}=T{x}_{n}-\mu {\lambda }_{n}F\left(T{x}_{n}\right),$
(1.4)
where F is L-Lipschitzian continuous and η-strongly monotone operator with $L>0$, $\eta >0$ and $0<\mu <2\eta /{L}^{2}$. Under some appropriate conditions, the sequence $\left\{{x}_{n}\right\}$ generated by (1.4) converges strongly to the unique solution of the variational inequality
$〈F{x}^{\ast },x-{x}^{\ast }〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in Fix\left(T\right).$
Combining (1.3) and (1.4), Tian  considered the following general viscosity type iterative method:
${x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-\mu {\alpha }_{n}F\right)T{x}_{n}.$
(1.5)
Improving and extending the corresponding results given by Marino et al., he proved that the sequence $\left\{{x}_{n}\right\}$ generated by (1.5) converges strongly to the unique solution ${x}^{\ast }\in Fix\left(T\right)$ of the variational inequality
$〈\left(\gamma f-\mu F\right)\stackrel{˜}{x},x-\stackrel{˜}{x}〉\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in Fix\left(T\right).$
In , Tian generalized the iterative method (1.5) replacing the contraction operator f with a Lipschitzian continuous operator V to solve the following variational inequality:
$〈\left(\gamma V-\mu F\right)\stackrel{˜}{x},x-\stackrel{˜}{x}〉\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in Fix\left(T\right).$
(1.6)
On the other hand, let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of nonexpansive self-mappings of H. Assume ${\bigcap }_{i=1}^{N}Fix\left({T}_{i}\right)\ne \mathrm{\varnothing }$. In , Xu also defined the following sequence $\left\{{x}_{n}\right\}$:
${x}_{n+1}={\alpha }_{n}u+\left(I-{\alpha }_{n}A\right){T}_{n+1}{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 0,$
(1.7)
where ${T}_{n}={T}_{nmodN}$ and the mod function takes values in $\left\{1,2,\dots ,N\right\}$. He found that the sequence $\left\{{x}_{n}\right\}$ generated by (1.7) converges strongly to the unique solution of the minimization problem (1.1) with $C={\bigcap }_{i=1}^{N}Fix\left({T}_{i}\right)$ under suitable conditions on $\left\{{\alpha }_{n}\right\}$ and the following additional condition on $\left\{{T}_{n}\right\}$:
$F\left({T}_{N}\cdots {T}_{2}{T}_{1}\right)=F\left({T}_{1}{T}_{N}\cdots {T}_{3}{T}_{2}\right)=\cdots =F\left({T}_{N-1}\cdots {T}_{1}{T}_{N}\right).$
(1.8)

In fact, there are many nonexpansive mappings which do not satisfy (1.8).

In 1999, Atsushiba and Takahashi  defined the ${W}_{n}$-mappings generated by ${T}_{1},{T}_{2},\dots ,{T}_{N}$ and $\left\{{\gamma }_{n,1}\right\},\left\{{\gamma }_{n,2}\right\},\dots ,\left\{{\gamma }_{n,N}\right\}\subset \left[0,1\right]$ as follows:
$\begin{array}{c}{U}_{n,0}=I,\hfill \\ {U}_{n,1}={\gamma }_{n,1}{T}_{1}{U}_{n,0}+\left(1-{\gamma }_{n,1}\right)I,\hfill \\ {U}_{n,2}={\gamma }_{n,2}{T}_{2}{U}_{n,1}+\left(1-{\gamma }_{n,2}\right)I,\hfill \\ ⋮\hfill \\ {U}_{n,N-1}={\gamma }_{n,N-1}{T}_{N-1}{U}_{n,N-2}+\left(1-{\gamma }_{n,N-1}\right)I,\hfill \\ {W}_{n}={U}_{n,N}={\gamma }_{n,N}{T}_{N}{U}_{n,N-1}+\left(1-{\gamma }_{n,N}\right)I.\hfill \end{array}$

From [, Lemma 3.1], we know that $F\left({W}_{n}\right)={\bigcap }_{i=1}^{N}F\left({T}_{i}\right)$.

In 2006, Yao  introduced the following iterative method:
${x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\beta {x}_{n}+\left(\left(1-\beta \right)I-{\alpha }_{n}A\right){W}_{n}{x}_{n}.$
(1.9)
Without the condition (1.8), he proved that the sequence $\left\{{x}_{n}\right\}$ generated by (1.9) converges strongly to the unique solution of the following variational inequality:
$〈\left(A-\gamma f\right){x}^{\ast },{x}^{\ast }-x〉\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in \bigcap _{i=1}^{N}Fix\left({T}_{i}\right),$
(1.10)
which is the optimality condition for the minimization problem
$\underset{x\in C}{min}\frac{1}{2}〈Ax,x〉-h\left(x\right),$
(1.11)

where $C={\bigcap }_{i=1}^{N}Fix\left({T}_{i}\right)$ and h is a potential function for γf (i.e., ${h}^{\prime }\left(x\right)=\gamma f\left(x\right)$).

Shang et al.  introduced the following scheme:
$\left\{\begin{array}{c}{y}_{n}={\beta }_{n}{x}_{n}+\left(1-{\beta }_{n}\right){W}_{n}{x}_{n},\hfill \\ {x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){y}_{n}.\hfill \end{array}$
(1.12)

Under certain appropriate conditions, without (1.8), they proved that $\left\{{x}_{n}\right\}$ defined by (1.12) converges strongly to the unique solution of (1.10) which is also the optimality condition for (1.11).

Recently, combining the Krasnoselskii-Mann type algorithm and the steepest-descent method, Buong and Duong  introduced a new explicit iterative algorithm:
${x}_{k+1}=\left(1-{\beta }_{k}^{0}\right){x}_{k}+{\beta }_{k}^{0}{T}_{0}^{k}{T}_{N}^{k}\cdots {T}_{1}^{k}{x}_{k},$
(1.13)
where ${T}_{i}^{k}=\left(1-{\beta }_{k}^{i}\right)I+{\beta }_{k}^{i}{T}_{i}$ for $i=1,2,\dots ,N$, ${T}_{0}^{k}=I-{\lambda }_{k}\mu F$, and F is an L-Lipschitz continuous and η-strongly monotone mapping. Under some appropriate assumptions, they proved that the sequence $\left\{{x}_{k}\right\}$ converges strongly to the unique solution of the following variational inequality:
$〈F\left({x}^{\ast }\right),x-{x}^{\ast }〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in \bigcap _{i=1}^{N}Fix\left({T}_{i}\right).$
(1.14)
Very recently, Zhou and Wang  proposed a simpler iterative algorithm than the iterative algorithm (1.13) given by Buong and Duong:
${x}_{k+1}=\left(I-{\lambda }_{k}\mu F\right){T}_{N}^{k}\cdots {T}_{1}^{k}{x}_{k}.$
(1.15)

They proved that the sequence $\left\{{x}_{k}\right\}$ defined by (1.15) converges strongly to the unique solution of the variational inequality (1.14) in a faster rate of convergence.

Motivated and inspired by the results of Zhou et al., in this paper, we consider a new iterative algorithm to solve the class of variational inequalities (1.6). The iterative algorithm improves and extends the results of Yao et al., and the corresponding results announced by many others. At the end of this paper, we extend our iterative algorithm to the more broad family of λ-strictly pseudo-contractive mappings.

## 2 Preliminaries

Throughout this paper, we write ${x}_{n}⇀x$ and ${x}_{n}\to x$ to indicate that $\left\{{x}_{n}\right\}$ converges weakly to x and converges strongly to x, respectively.

An operator $T:H\to H$ is said to be nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for all $x,y\in H$. It is well known that $Fix\left(T\right)$ is closed and convex. A is called strongly positive if there exists a constant $\gamma >0$ such that $〈Ax,x〉\ge \gamma {\parallel x\parallel }^{2}$ for all $x\in H$. The operator F is called η-strongly monotone if there exists a constant $\eta >0$ such that
$〈x-y,Fx-Fy〉\ge \eta {\parallel x-y\parallel }^{2}$

for all $x,y\in H$.

In order to prove our results, we collect some necessary conceptions and lemmas in this section.

Definition 2.1 A mapping $T:H\to H$ is said to be an averaged mapping if there exists some number $\alpha \in \left(0,1\right)$ such that
$T=\left(1-\alpha \right)I+\alpha S,$
(2.1)

where $I:H\to H$ is the identity mapping and $S:H\to H$ is nonexpansive. More precisely, when (2.1) holds, we say that T is α-averaged.

Lemma 2.1 ()
1. (i)

The composite of finitely many averaged mappings is averaged. That is, if each of the mappings ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ is averaged, then so is the composite ${T}_{1}\cdots {T}_{N}$. In particular, if ${T}_{1}$ is ${\alpha }_{1}$-averaged and ${T}_{2}$ is ${\alpha }_{2}$-averaged, where ${\alpha }_{1},{\alpha }_{2}\in \left(0,1\right)$, then both ${T}_{1}{T}_{2}$ and ${T}_{2}{T}_{1}$ are α-averaged, where $\alpha ={\alpha }_{1}+{\alpha }_{2}-{\alpha }_{1}{\alpha }_{2}$.

2. (ii)
If the mappings ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ are averaged and have a common fixed point, then
$\bigcap _{i=1}^{N}Fix\left({T}_{i}\right)=Fix\left({T}_{1}\cdots {T}_{N}\right).$

In particular, if $N=2$, we have $Fix\left({T}_{1}\right)\cap Fix\left({T}_{2}\right)=Fix\left({T}_{1}{T}_{2}\right)=Fix\left({T}_{2}{T}_{1}\right)$.

Lemma 2.2 ()

Let C be a closed convex subset of a real Hilbert space H. Given $x\in H$ and $y\in C$. Then $y={P}_{C}x$ if and only if the following inequality holds:
$〈x-y,z-y〉\le 0$

for every $z\in C$.

Lemma 2.3 ()

Assume V is a contraction on a Hilbert space H with coefficient $\alpha >0$, and $F:H\to H$ is an L-Lipschitzian continuous and η-strongly monotone operator with $L>0$, $\eta >0$. Then, for $0<\gamma <\frac{\mu \eta }{\alpha }$, $\mu F-\gamma V$ is strongly monotone with coefficient $\mu \eta -\gamma \alpha$.

Lemma 2.4 ()

Let H be a Hilbert space, C a closed convex subset of H, and $T:C\to C$ a nonexpansive mapping with $Fix\left(T\right)\ne \mathrm{\varnothing }$. If $\left\{{x}_{n}\right\}$ is a sequence in C weakly converging to $x\in C$ and $\left\{\left(I-T\right){x}_{n}\right\}$ converges strongly to $y\in C$, then $\left(I-T\right)x=y$. In particular, if $y=0$, then $x\in Fix\left(T\right)$.

Lemma 2.5 ()

Let $\left\{{x}_{n}\right\}$ and $\left\{{z}_{n}\right\}$ be bounded sequences in a Banach space X and $\left\{{\beta }_{n}\right\}$ be a sequence in $\left[0,1\right]$ which satisfies the following condition:
$0<\underset{n\to \mathrm{\infty }}{lim inf}{\beta }_{n}\le \underset{n\to \mathrm{\infty }}{lim sup}{\beta }_{n}<1.$
Suppose ${x}_{n+1}=\left(1-{\beta }_{n}\right){z}_{n}+{\beta }_{n}{x}_{n}$ for all integers $n\ge 0$ and
$\underset{n\to \mathrm{\infty }}{lim sup}\left(\parallel {z}_{n+1}-{z}_{n}\parallel -\parallel {x}_{n+1}-{x}_{n}\parallel \right)\le 0.$

Then ${lim}_{n\to \mathrm{\infty }}\parallel {z}_{n}-{x}_{n}\parallel =0$.

Lemma 2.6 ()

Assume $\left\{{a}_{n}\right\}$ is a sequence of nonnegative real numbers such that
${a}_{n+1}\le \left(1-{\gamma }_{n}\right){a}_{n}+{\delta }_{n},\phantom{\rule{1em}{0ex}}n\ge 0,$
where $\left\{{\gamma }_{n}\right\}$ is a sequence in $\left(0,1\right)$ and $\left\{{\delta }_{n}\right\}$ is a sequence such that
1. (i)

${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_{n}=\mathrm{\infty }$,

2. (ii)

${lim sup}_{n\to \mathrm{\infty }}\frac{{\delta }_{n}}{{\gamma }_{n}}\le 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_{n}|<\mathrm{\infty }$.

Then ${lim}_{n\to \mathrm{\infty }}{a}_{n}=0$.

Lemma 2.7 ()

Assume S is a λ-strictly pseudo-contractive mapping on a Hilbert space H. Define a mapping T by $Tx=\alpha x+\left(1-\alpha \right)Sx$ for all $x\in H$ and $\alpha \in \left[\lambda ,1\right)$. Then T is a nonexpansive mapping such that $Fix\left(T\right)=Fix\left(S\right)$.

## 3 Main results

Now we state and prove our main results in this paper.

Theorem 3.1 Let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be N nonexpansive mappings of a real Hilbert space H such that $C={\bigcap }_{i=1}^{N}Fix\left({T}_{i}\right)\ne \mathrm{\varnothing }$, F be an L-Lipschitzian continuous and η-strongly monotone operator on H with $L>0$ and $\eta >0$, V be an α-Lipschitzian on H with $\alpha >0$. Suppose ${x}_{1}\in H$ and $0<\mu <\frac{2\eta }{{L}^{2}}$. Define a sequence $\left\{{x}_{k}\right\}$ as follows:
${x}_{k+1}={\alpha }_{k}\gamma V\left({x}_{k}\right)+\left(I-\mu {\alpha }_{k}F\right){T}_{N}^{k}{T}_{N-1}^{k}\cdots {T}_{1}^{k}{x}_{k},\phantom{\rule{1em}{0ex}}k\ge 0,$
(3.1)
where $0<\gamma <\frac{\tau }{\alpha }$ with $\tau =\mu \left(\eta -\frac{1}{2}\mu {L}^{2}\right)$ and ${T}_{i}^{k}=\left(1-{\beta }_{k}^{i}\right)I+{\beta }_{k}^{i}{T}_{i}$ for $i=1,2,\dots ,N$. Suppose ${\alpha }_{k}\in \left(0,1\right)$ and ${\beta }_{k}^{i}\in \left(\xi ,\zeta \right)$ for some $\xi ,\zeta \in \left(0,1\right)$. If the following conditions are satisfied:
1. (i)

${lim}_{k\to \mathrm{\infty }}{\alpha }_{k}=0$;

2. (ii)

${\sum }_{k=1}^{\mathrm{\infty }}{\alpha }_{k}=\mathrm{\infty }$;

3. (iii)

${lim}_{k\to \mathrm{\infty }}|{\beta }_{k+1}^{i}-{\beta }_{k}^{i}|=0$ for $i=1,2,\dots ,N$.

Then the sequence $\left\{{x}_{k}\right\}$ converges strongly to the unique solution ${x}^{\ast }$ of the variational inequality:
$〈\left(\mu F-\gamma V\right){x}^{\ast },x-{x}^{\ast }〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in \bigcap _{i=1}^{N}Fix\left({T}_{i}\right).$
(3.2)

Equivalently, we have ${P}_{C}\left(I-\mu F+\gamma V\right){x}^{\ast }={x}^{\ast }$.

Proof Since our methods easily deduce the general case, we prove Theorem 3.1 for $N=2$.

First, we show $\left\{{x}_{k}\right\}$ is bounded. In fact, for some point $p\in C$, by (3.1) we have
$\begin{array}{rcl}\parallel {x}_{k+1}-p\parallel & =& \parallel {\alpha }_{k}\gamma V{x}_{k}+\left(I-\mu {\alpha }_{k}F\right){T}_{2}^{k}{T}_{1}^{k}{x}_{k}-p\parallel \\ =& \parallel \left(I-\mu {\alpha }_{k}F\right){T}_{2}^{k}{T}_{1}^{k}{x}_{k}-\left(I-\mu {\alpha }_{k}F\right)p+{\alpha }_{k}\left(\gamma V{x}_{k}-\mu Fp\right)\parallel \\ \le & \left(1-{\alpha }_{k}\tau \right)\parallel {T}_{2}^{k}{T}_{1}^{k}{x}_{k}-{T}_{2}^{k}{T}_{1}^{k}p\parallel +{\alpha }_{k}\left(\parallel \gamma V{x}_{k}-\gamma Vp\parallel +\parallel \gamma Vp-\mu Fp\parallel \right)\\ \le & \left(1-{\alpha }_{k}\tau \right)\parallel {x}_{k}-p\parallel +{\alpha }_{k}\gamma \alpha \parallel {x}_{k}-p\parallel +{\alpha }_{k}\parallel \gamma Vp-\mu Fp\parallel \\ =& \left(1-{\alpha }_{k}\left(\tau -\gamma \alpha \right)\right)\parallel {x}_{k}-p\parallel +{\alpha }_{k}\left(\tau -\gamma \alpha \right)\frac{\parallel \gamma Vp-\mu Fp\parallel }{\tau -\gamma \alpha }\\ \le & max\left\{\parallel {x}_{k}-p\parallel ,\frac{1}{\tau -\gamma \alpha }\parallel \gamma Vp-\mu Fp\parallel \right\}\\ \le & \cdots \le max\left\{\parallel {x}_{0}-p\parallel ,\frac{1}{\tau -\gamma \alpha }\parallel \gamma Vp-\mu Fp\parallel \right\}.\end{array}$
Therefore, $\left\{{x}_{k}\right\}$ is bounded. Hence we also see that $\left\{{T}_{2}^{k}{T}_{1}^{k}{x}_{k}\right\}$, $\left\{F{T}_{2}^{k}{T}_{1}^{k}{x}_{k}\right\}$, and $\left\{V{x}_{k}\right\}$ are all bounded. From (3.1), it follows that
$\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{k+1}-{T}_{2}^{k}{T}_{1}^{k}{x}_{k}\parallel =0.$
(3.3)
We next show that ${lim}_{k\to \mathrm{\infty }}\parallel {x}_{k+1}-{x}_{k}\parallel =0$. Noting that ${T}_{1}^{k}$ and ${T}_{2}^{k}$ are ${\beta }_{k}^{1}$-averaged and ${\beta }_{k}^{2}$-averaged, respectively, by Lemma 2.1, we find that ${T}_{2}^{k}{T}_{1}^{k}$ is ${t}_{k}$-averaged for every k, where ${t}_{k}={\beta }_{k}^{1}+{\beta }_{k}^{2}-{\beta }_{k}^{1}{\beta }_{k}^{2}$. Set ${\xi }^{\ast }=2\xi -{\xi }^{2}$ and ${\zeta }^{\ast }=2\zeta -{\zeta }^{2}$. It is easy to deduce that $0<{\xi }^{\ast }\le {t}_{k}\le {\zeta }^{\ast }<1$ for all k and
$\underset{k\to \mathrm{\infty }}{lim}\parallel {t}_{k+1}-{t}_{k}\parallel =0.$
(3.4)
Since for every k, ${T}_{2}^{k}{T}_{1}^{k}$ is ${t}_{k}$-averaged, we can find a family of nonexpansive mappings ${\left\{{S}_{k}\right\}}_{k\ge 0}$ on H such that
${T}_{2}^{k}{T}_{1}^{k}=\left(1-{t}_{k}\right)I+{t}_{k}{S}_{k},\phantom{\rule{1em}{0ex}}k\ge 0.$
(3.5)
Substituting (3.4) into (3.1) yields
$\begin{array}{rcl}{x}_{k+1}& =& {\alpha }_{k}\gamma V{x}_{k}+\left(I-\mu {\alpha }_{k}F\right)\left[\left(1-{t}_{k}\right){x}_{k}+{t}_{k}{S}_{k}{x}_{k}\right]\\ =& \left(1-{t}_{k}\right){x}_{k}+{t}_{k}\left[{S}_{k}{x}_{k}+\frac{{\alpha }_{k}}{{t}_{k}}\left(\gamma V{x}_{k}-\mu F{T}_{2}^{k}{T}_{1}^{k}{x}_{k}\right)\right].\end{array}$
Define a sequence $\left\{{z}_{k}\right\}$ by ${z}_{k}={S}_{k}{x}_{k}+\frac{{\alpha }_{k}}{{t}_{k}}\left(\gamma V{x}_{k}-\mu F{T}_{2}^{k}{T}_{1}^{k}{x}_{k}\right)$, so
${x}_{k+1}=\left(1-{t}_{k}\right){x}_{k}+{t}_{k}{z}_{k}.$
(3.6)
Now, we claim that
$\underset{k\to \mathrm{\infty }}{lim sup}\left(\parallel {z}_{k+1}-{z}_{k}\parallel -\parallel {x}_{k+1}-{x}_{k}\parallel \right)\le 0.$
To this end, we observe that
$\begin{array}{rcl}\parallel {z}_{k+1}-{z}_{k}\parallel & \le & \parallel {S}_{k+1}{x}_{k+1}-{S}_{k}{x}_{k}\parallel +\frac{{\alpha }_{k+1}}{{t}_{k+1}}\parallel \gamma V{x}_{k+1}-\mu F{T}_{2}^{k+1}{T}_{1}^{k+1}{x}_{k+1}\parallel \\ +\frac{{\alpha }_{k}}{{t}_{k}}\parallel \gamma V{x}_{k}-\mu F{T}_{2}^{k}{T}_{1}^{k}{x}_{k}\parallel \\ \le & \parallel {S}_{k+1}{x}_{k+1}-{S}_{k+1}{x}_{k}\parallel +\parallel {S}_{k+1}{x}_{k}-{S}_{k}{x}_{k}\parallel \\ +\frac{{\alpha }_{k+1}}{{t}_{k+1}}\left(\parallel \gamma V{x}_{k+1}\parallel +\parallel \mu F{T}_{2}^{k+1}{T}_{1}^{k+1}{x}_{k+1}\parallel \right)\\ +\frac{{\alpha }_{k}}{{t}_{k}}\left(\parallel \gamma V{x}_{k}\parallel +\parallel \mu F{T}_{2}^{k}{T}_{1}^{k}{x}_{k}\parallel \right)\\ \le & \parallel {x}_{k+1}-{x}_{k}\parallel +\parallel {S}_{k+1}{x}_{k}-{S}_{k}{x}_{k}\parallel \\ +\frac{{\alpha }_{k+1}}{{t}_{k+1}}\left(\parallel \gamma V{x}_{k+1}\parallel +\parallel \mu F{T}_{2}^{k+1}{T}_{1}^{k+1}{x}_{k+1}\parallel \right)\\ +\frac{{\alpha }_{k}}{{t}_{k}}\left(\parallel \gamma V{x}_{k}\parallel +\parallel \mu F{T}_{2}^{k}{T}_{1}^{k}{x}_{k}\parallel \right)\end{array}$
(3.7)
and
$\begin{array}{rcl}\parallel {S}_{k+1}{x}_{k}-{S}_{k}{x}_{k}\parallel & =& \parallel \frac{1}{{t}_{k+1}}{T}_{2}^{k+1}{T}_{1}^{k+1}{x}_{k}-\frac{1}{{t}_{k}}{T}_{2}^{k}{T}_{1}^{k}{x}_{k}-\frac{1-{t}_{k+1}}{{t}_{k+1}}{x}_{k}+\frac{1-{t}_{k}}{{t}_{k}}{x}_{k}\parallel \\ \le & |\frac{{t}_{k+1}-{t}_{k}}{{t}_{k+1}{t}_{k}}|\left(\parallel {T}_{2}^{k+1}{T}_{1}^{k+1}{x}_{k}\parallel +\parallel {x}_{k}\parallel \right)+\frac{1}{{t}_{k}}\parallel {T}_{2}^{k+1}{T}_{1}^{k+1}{x}_{k}-{T}_{2}^{k}{T}_{1}^{k}{x}_{k}\parallel \\ \le & |\frac{{t}_{k+1}-{t}_{k}}{{t}_{k+1}{t}_{k}}|M+\frac{1}{{t}_{k}}\left(\parallel {T}_{2}^{k+1}{T}_{1}^{k+1}{x}_{k}-{T}_{2}^{k+1}{T}_{1}^{k}{x}_{k}\parallel \\ +\parallel {T}_{2}^{k+1}{T}_{1}^{k}{x}_{k}-{T}_{2}^{k}{T}_{1}^{k}{x}_{k}\parallel \right)\\ \le & |\frac{{t}_{k+1}-{t}_{k}}{{t}_{k+1}{t}_{k}}|M+\frac{1}{{\xi }^{\ast }}\left(\parallel {T}_{1}^{k+1}{x}_{k}-{T}_{1}^{k}{x}_{k}\parallel \\ +\parallel {T}_{2}^{k+1}{T}_{1}^{k}{x}_{k}-{T}_{2}^{k}{T}_{1}^{k}{x}_{k}\parallel \right),\end{array}$
(3.8)
where M is a fixed constant satisfying
$M\ge \underset{k\ge 0}{sup}\left\{\parallel {T}_{2}^{k+1}{T}_{1}^{k+1}{x}_{k}\parallel +\parallel {x}_{k}\parallel \right\}.$
Note that
$\begin{array}{rcl}\parallel {T}_{1}^{k+1}{x}_{k}-{T}_{1}^{k}{x}_{k}\parallel & =& \parallel \left(1-{\beta }_{k+1}^{1}\right){x}_{k}+{\beta }_{k+1}^{1}{T}_{1}{x}_{k}-\left(1-{\beta }_{k}^{1}\right){x}_{k}-{\beta }_{k}^{1}{T}_{1}{x}_{k}\parallel \\ \le & |{\beta }_{k+1}^{1}-{\beta }_{k}^{1}|\left(\parallel {x}_{k}\parallel +\parallel {T}_{1}{x}_{k}\parallel \right).\end{array}$
Since ${lim}_{k\to \mathrm{\infty }}|{\beta }_{k+1}^{i}-{\beta }_{k}^{i}|=0$ for $i=1,2$, and $\left\{{x}_{k}\right\}$ and $\left\{{T}_{1}{x}_{k}\right\}$ are bounded, we easily obtain
$\underset{k\to \mathrm{\infty }}{lim}\parallel {T}_{1}^{k+1}{x}_{k}-{T}_{1}^{k}{x}_{k}\parallel =0.$
(3.9)
Similarly,
$\parallel {T}_{2}^{k+1}{T}_{1}^{k}{x}_{k}-{T}_{2}^{k}{T}_{1}^{k}{x}_{k}\parallel \le |{\beta }_{k+1}^{2}-{\beta }_{k}^{2}|\left(\parallel {T}_{1}^{k}{x}_{k}\parallel +\parallel {T}_{2}{T}_{1}^{k}{x}_{k}\parallel \right),$
from which it follows that
$\underset{k\to \mathrm{\infty }}{lim}\parallel {T}_{2}^{k+1}{T}_{1}^{k}{x}_{k}-{T}_{2}^{k}{T}_{1}^{k}{x}_{k}\parallel =0.$
(3.10)
Using (3.4), (3.9), and (3.10), from (3.8) we have
$\underset{k\to \mathrm{\infty }}{lim}\parallel {S}_{k+1}{x}_{k}-{S}_{k}{x}_{k}\parallel =0.$
(3.11)
Since ${lim}_{k\to \mathrm{\infty }}{\alpha }_{k}=0$ and $0<{\xi }^{\ast }<{t}_{k}<{\zeta }^{\ast }<1$, combining (3.7) and (3.11) we get
$\underset{k\to \mathrm{\infty }}{lim sup}\left(\parallel {z}_{k+1}-{z}_{k}\parallel -\parallel {x}_{k+1}-{x}_{k}\parallel \right)\le 0.$
By Lemma 2.5, we conclude that ${lim}_{k\to \mathrm{\infty }}\parallel {z}_{k}-{x}_{k}\parallel =0$, which implies that ${lim}_{k\to \mathrm{\infty }}\parallel {x}_{k+1}-{x}_{k}\parallel =0$ by (3.6). Thus from (3.3), it is true that
$\underset{k\to \mathrm{\infty }}{lim}\parallel {x}_{k}-{T}_{2}^{k}{T}_{1}^{k}{x}_{k}\parallel =0.$
(3.12)
From [, Theorem 3.2], we know that the solution of the variational inequality (3.2) is unique. We use ${x}^{\ast }$ to denote the unique solution of (3.2). Since ${\left\{{x}_{k}\right\}}_{k\ge 0}$ is bounded, there exists a subsequence ${\left\{{x}_{{k}_{j}}\right\}}_{j\ge 1}$ of ${\left\{{x}_{k}\right\}}_{k\ge 0}$ such that ${x}_{{k}_{j}}⇀\stackrel{ˆ}{x}$ as $j\to \mathrm{\infty }$ and
$\underset{k\to \mathrm{\infty }}{lim sup}〈\left(\mu F-\gamma V\right){x}^{\ast },{x}^{\ast }-{x}_{k}〉=\underset{j\to \mathrm{\infty }}{lim}〈\left(\mu F-\gamma V\right){x}^{\ast },{x}^{\ast }-{x}_{{k}_{j}}〉.$
Since $\left\{{\beta }_{k}^{i}\right\}$ is bounded for $i=1,2$, we can assume that ${\beta }_{{k}_{j}}^{i}\to {\beta }_{\mathrm{\infty }}^{i}$ as $j\to \mathrm{\infty }$, where $0<\xi \le {\beta }_{\mathrm{\infty }}^{i}\le \zeta <1$ for $i=1,2$. Define ${T}_{i}^{\mathrm{\infty }}=\left(1-{\beta }_{\mathrm{\infty }}^{i}\right)I+{\beta }_{\mathrm{\infty }}^{i}{T}_{i}$ ($i=1,2$). Then we have $Fix\left({T}_{i}^{\mathrm{\infty }}\right)=Fix\left({T}_{i}\right)$ for $i=1,2$. Note that
$\parallel {T}_{i}^{{k}_{j}}x-{T}_{i}^{\mathrm{\infty }}x\parallel \le |{\beta }_{{k}_{j}}^{i}-{\beta }_{\mathrm{\infty }}^{i}|\left(\parallel x\parallel +\parallel {T}_{i}x\parallel \right).$
Hence, we deduce that
$\underset{j\to \mathrm{\infty }}{lim}\underset{x\in D}{sup}\parallel {T}_{i}^{{k}_{j}}x-{T}_{i}^{\mathrm{\infty }}x\parallel =0,$
(3.13)

where D is an arbitrary bounded subset of H.

Since $Fix\left({T}_{1}^{\mathrm{\infty }}\right)\cap Fix\left({T}_{2}^{\mathrm{\infty }}\right)=Fix\left({T}_{1}\right)\cap Fix\left({T}_{2}\right)=C\ne \mathrm{\varnothing }$ and ${T}_{i}^{\mathrm{\infty }}$ is ${\beta }_{\mathrm{\infty }}^{i}$-averaged for $i=1,2$, by Lemma 2.1, we know that $Fix\left({T}_{2}^{\mathrm{\infty }}{T}_{1}^{\mathrm{\infty }}\right)=Fix\left({T}_{2}^{\mathrm{\infty }}\right)\cap Fix\left({T}_{1}^{\mathrm{\infty }}\right)=C$. Combining (3.12) and (3.13), we obtain
$\begin{array}{rcl}\parallel {x}_{{k}_{j}}-{T}_{2}^{\mathrm{\infty }}{T}_{1}^{\mathrm{\infty }}{x}_{{k}_{j}}\parallel & \le & \parallel {x}_{{k}_{j}}-{T}_{2}^{{k}_{j}}{T}_{1}^{{k}_{j}}{x}_{{k}_{j}}\parallel +\parallel {T}_{2}^{{k}_{j}}{T}_{1}^{{k}_{j}}{x}_{{k}_{j}}-{T}_{2}^{\mathrm{\infty }}{T}_{1}^{{k}_{j}}{x}_{{k}_{j}}\parallel \\ +\parallel {T}_{2}^{\mathrm{\infty }}{T}_{1}^{{k}_{j}}{x}_{{k}_{j}}-{T}_{2}^{\mathrm{\infty }}{T}_{1}^{\mathrm{\infty }}{x}_{{k}_{j}}\parallel \\ \le & \parallel {x}_{{k}_{j}}-{T}_{2}^{{k}_{j}}{T}_{1}^{{k}_{j}}{x}_{{k}_{j}}\parallel +\parallel {T}_{2}^{{k}_{j}}{T}_{1}^{{k}_{j}}{x}_{{k}_{j}}-{T}_{2}^{\mathrm{\infty }}{T}_{1}^{{k}_{j}}{x}_{{k}_{j}}\parallel \\ +\parallel {T}_{1}^{{k}_{j}}{x}_{{k}_{j}}-{T}_{1}^{\mathrm{\infty }}{x}_{{k}_{j}}\parallel \\ \le & \parallel {x}_{{k}_{j}}-{T}_{2}^{{k}_{j}}{T}_{1}^{{k}_{j}}{x}_{{k}_{j}}\parallel +\underset{x\in {D}^{\prime }}{sup}\parallel {T}_{2}^{{k}_{j}}x-{T}_{2}^{\mathrm{\infty }}x\parallel \\ +\underset{x\in {D}^{″}}{sup}\parallel {T}_{1}^{{k}_{j}}x-{T}_{1}^{\mathrm{\infty }}x\parallel ,\end{array}$
where ${D}^{\prime }$ is a bounded subset including $\left\{{T}_{1}^{{k}_{j}}{x}_{{k}_{j}}\right\}$ and ${D}^{″}$ is a bounded subset including $\left\{{x}_{{k}_{j}}\right\}$. Hence ${lim}_{j\to \mathrm{\infty }}\parallel {x}_{{k}_{j}}-{T}_{2}^{\mathrm{\infty }}{T}_{1}^{\mathrm{\infty }}{x}_{{k}_{j}}\parallel =0$. From Lemma 2.4, we have $\stackrel{ˆ}{x}\in Fix\left({T}_{2}^{\mathrm{\infty }}{T}_{1}^{\mathrm{\infty }}\right)=C$. It follows that
$\begin{array}{rcl}\underset{k\to \mathrm{\infty }}{lim sup}〈\left(\mu F-\gamma V\right){x}^{\ast },{x}^{\ast }-{T}_{2}^{k}{T}_{1}^{k}{x}_{k}〉& =& \underset{k\to \mathrm{\infty }}{lim sup}〈\left(\mu F-\gamma V\right){x}^{\ast },{x}^{\ast }-{x}_{k}〉\\ =& \underset{j\to \mathrm{\infty }}{lim}〈\left(\mu F-\gamma V\right){x}^{\ast },{x}^{\ast }-{x}_{{k}_{j}}〉\\ =& 〈\left(\mu F-\gamma V\right){x}^{\ast },{x}^{\ast }-\stackrel{ˆ}{x}〉\le 0.\end{array}$
(3.14)
Finally, we show that ${x}_{k}\to {x}^{\ast }$ as $k\to \mathrm{\infty }$. From (3.1), we have
$\begin{array}{rcl}{\parallel {x}_{k+1}-{x}^{\ast }\parallel }^{2}& =& {\parallel {\alpha }_{k}\gamma V{x}_{k}+\left(I-\mu {\alpha }_{k}F\right){T}_{2}^{k}{T}_{1}^{k}{x}_{k}-{x}^{\ast }\parallel }^{2}\\ =& {\parallel \left(I-\mu {\alpha }_{k}F\right){T}_{2}^{k}{T}_{1}^{k}{x}_{k}-\left(I-\mu {\alpha }_{k}F\right){x}^{\ast }+{\alpha }_{k}\left(\gamma V{x}_{k}-\mu F{x}^{\ast }\right)\parallel }^{2}\\ =& {\parallel \left(I-\mu {\alpha }_{k}F\right){T}_{2}^{k}{T}_{1}^{k}{x}_{k}-\left(I-\mu {\alpha }_{k}F\right){x}^{\ast }\parallel }^{2}+{\alpha }_{k}^{2}{\parallel \gamma V{x}_{k}-\mu F{x}^{\ast }\parallel }^{2}\\ +2{\alpha }_{k}〈\left(I-\mu {\alpha }_{k}F\right){T}_{2}^{k}{T}_{1}^{k}{x}_{k}-\left(I-\mu {\alpha }_{k}F\right){x}^{\ast },\gamma V{x}_{k}-\mu F{x}^{\ast }〉\\ \le & {\left(1-{\alpha }_{k}\tau \right)}^{2}{\parallel {x}_{k}-{x}^{\ast }\parallel }^{2}+{\alpha }_{k}^{2}{\parallel \gamma V{x}_{k}-\mu F{x}^{\ast }\parallel }^{2}\\ +2{\alpha }_{k}〈{T}_{2}^{k}{T}_{1}^{k}{x}_{k}-{x}^{\ast },\gamma V{x}_{k}-\mu F{x}^{\ast }〉\\ -2\mu {\alpha }_{k}^{2}〈F{T}_{2}^{k}{T}_{1}^{k}{x}_{k}-F{x}^{\ast },\gamma V{x}_{k}-\mu F{x}^{\ast }〉\\ \le & {\left(1-{\alpha }_{k}\tau \right)}^{2}{\parallel {x}_{k}-{x}^{\ast }\parallel }^{2}+{\alpha }_{k}^{2}{\parallel \gamma V{x}_{k}-\mu F{x}^{\ast }\parallel }^{2}\\ +2{\alpha }_{k}\gamma 〈{T}_{2}^{k}{T}_{1}^{k}{x}_{k}-{x}^{\ast },V{x}_{k}-V{x}^{\ast }〉\\ +2{\alpha }_{k}〈{T}_{2}^{k}{T}_{1}^{k}{x}_{k}-{x}^{\ast },\gamma V{x}^{\ast }-\mu F{x}^{\ast }〉\\ -2\mu {\alpha }_{k}^{2}〈F{T}_{2}^{k}{T}_{1}^{k}{x}_{k}-F{x}^{\ast },\gamma V{x}_{k}-\mu F{x}^{\ast }〉\\ \le & \left[{\left(1-{\alpha }_{k}\tau \right)}^{2}+2\alpha {\alpha }_{k}\gamma \right]{\parallel {x}_{k}-{x}^{\ast }\parallel }^{2}+{\alpha }_{k}\left[2〈{T}_{2}^{k}{T}_{1}^{k}{x}_{k}-{x}^{\ast },\left(\gamma V-\mu F\right){x}^{\ast }〉\\ +{\alpha }_{k}{\parallel \gamma V{x}_{k}-\mu F{x}^{\ast }\parallel }^{2}+2{\alpha }_{k}L\parallel {T}_{2}^{k}{T}_{1}^{k}{x}_{k}-{x}^{\ast }\parallel \parallel \gamma V{x}_{k}-\mu F{x}^{\ast }\parallel \right]\\ =& \left[1-2{\alpha }_{k}\left(\tau -\alpha \gamma \right)\right]{\parallel {x}_{k}-{x}^{\ast }\parallel }^{2}+{\alpha }_{k}\left[2〈{T}_{2}^{k}{T}_{1}^{k}{x}_{k}-{x}^{\ast },\left(\gamma V-\mu F\right){x}^{\ast }〉\\ +{\alpha }_{k}\left({\parallel \gamma V{x}_{k}-\mu F{x}^{\ast }\parallel }^{2}+2L\parallel {x}_{k}-{x}^{\ast }\parallel \parallel \gamma V{x}_{k}-\mu F{x}^{\ast }\parallel +{\tau }^{2}{\parallel {x}_{k}-{x}^{\ast }\parallel }^{2}\right)\right]\\ \le & \left[1-2{\alpha }_{k}\left(\tau -\alpha \gamma \right)\right]{\parallel {x}_{k}-{x}^{\ast }\parallel }^{2}\\ +{\alpha }_{k}\left[2〈{T}_{2}^{k}{T}_{1}^{k}{x}_{k}-{x}^{\ast },\left(\gamma V-\mu F\right){x}^{\ast }〉+{\alpha }_{k}{M}^{\prime }\right],\end{array}$
where ${M}^{\prime }$ is a constant satisfying
${M}^{\prime }\ge \underset{k\ge 0}{sup}\left\{{\parallel \gamma V{x}_{k}-\mu F{x}^{\ast }\parallel }^{2}+2L\parallel {T}_{2}^{k}{T}_{1}^{k}{x}_{k}-{x}^{\ast }\parallel \parallel \gamma V{x}_{k}-\mu F{x}^{\ast }\parallel +{\tau }^{2}{\parallel {x}_{k}-{x}^{\ast }\parallel }^{2}\right\}.$

Consequently, according to the conditions (i) and (ii), (3.14), and Lemma 2.6, we conclude that ${x}_{k}\to {x}^{\ast }$ as $k\to \mathrm{\infty }$. This completes the proof. □

## 4 An extension of our result

In this section, we extend our result to the more broad family of λ-strictly pseudo-contractive mappings. Now let us recall that a mapping $S:H\to H$ is said to be λ-strictly pseudo-contractive if there exists a constant $\lambda \in \left[0,1\right)$ such that
${\parallel Sx-Sy\parallel }^{2}\le {\parallel x-y\parallel }^{2}+\lambda {\parallel \left(I-S\right)x-\left(I-S\right)y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in H.$
Let ${\left\{{S}_{i}\right\}}_{i=1}^{N}$ be a family of ${\lambda }_{i}$-strictly pseudo-contractive self-mappings of H with $0\le {\lambda }_{i}<1$. For $i=1,2,\dots ,N$, define
${\stackrel{ˆ}{T}}_{i}={\omega }_{i}I+\left(1-{\omega }_{i}\right){S}_{i},$
(4.1)

where $0\le {\lambda }_{i}\le {\omega }_{i}<1$. By virtue of Lemma 2.7, we know that ${\left\{{\stackrel{ˆ}{T}}_{i}\right\}}_{i=1}^{N}$ is a family of nonexpansive mappings. Thus we extend Theorem 3.1 to the family of ${\lambda }_{i}$-strictly pseudo-contractions.

Theorem 4.1 Let H be a real Hilbert space, $F:H\to H$ be an L-Lipschitizian continuous and η-strongly monotone operator on H with $L>0$ and $\eta >0$, V be an α-Lipschitzian continuous on H with $\alpha >0$. Let ${\left\{{S}_{i}\right\}}_{i=1}^{N}$ be N ${\lambda }_{i}$-strictly pseudo-contractive mappings on H such that $C={\bigcap }_{i=1}^{N}Fix\left({S}_{i}\right)\ne \mathrm{\varnothing }$. Suppose $0<\mu <\frac{\tau }{\alpha }$, $0<\gamma <\frac{\tau }{\alpha }$ with $\tau =\mu \left(\eta -\frac{1}{2}\mu {L}^{2}\right)$, ${\alpha }_{k}\in \left(0,1\right)$, ${\beta }_{k}^{i}\in \left(\xi ,\zeta \right)$ for some $\xi ,\zeta \in \left(0,1\right)$ and $0\le {\lambda }_{i}\le {\omega }_{i}<1$ for $i=1,2,\dots ,N$. If the conditions (i)-(iii) of Theorem  3.1 are satisfied, the sequence ${\left\{{x}_{k}\right\}}_{k\ge 0}$ defined by (3.1) with ${T}_{i}$ replaced by (4.1), converges strongly to the unique solution ${x}^{\ast }$ of the following variational inequality:
$〈\left(\mu F-\gamma V\right){x}^{\ast },x-{x}^{\ast }〉\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in \bigcap _{i=1}^{N}Fix\left({S}_{i}\right).$

## Declarations

### Acknowledgements

This research is supported by the Fundamental Science Research Funds for the Central Universities (Program No. 3122013k004).

## Authors’ Affiliations

(1)
College of Science, Civil Aviation University of China, Tianjin, 300300, P.R. China

## References

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