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

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.

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 $Fix(T)$, i.e., $Fix(T)=\{x\in H:Tx=x\}$. 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 [1] introduced the following iterative scheme:

$$x_{n+1}={\alpha }_{n}u+(I-{\alpha }_{n}A)T{x}_n,$$

(1.2)

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

In 2006, Marino and Xu [2] considered the viscosity method on the iterative scheme (1.2), and they gave the following general iterative method:

$$x_{n+1}={\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)T{x}_n,$$

(1.3)

where f is a contraction on H. They proved the above sequence $\{x_n\}$ converges strongly to the unique solution of the variational inequality

$$〈(A-\gamma f)x^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in Fix(T),$$

which is the optimality condition for the minimization problem

$$\underset{x\in Fix(T)}{min}\frac{1}{2}〈Ax,x〉-h(x),$$

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

In 2001, Yamada [3] considered the following hybrid iterative method:

$$x_{n+1}=T{x}_n-\mu {\lambda }_{n}F(T{x}_n),$$

(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 $\{x_n\}$ generated by (1.4) converges strongly to the unique solution of the variational inequality

$$〈F{x}^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in Fix(T).$$

Combining (1.3) and (1.4), Tian [4] considered the following general viscosity type iterative method:

$$x_{n+1}={\alpha }_n\gamma f(x_n)+(I-\mu {\alpha }_{n}F)T{x}_n.$$

(1.5)

Improving and extending the corresponding results given by Marino et al., he proved that the sequence $\{x_n\}$ generated by (1.5) converges strongly to the unique solution $x^{\ast }\in Fix(T)$ of the variational inequality

$$〈(\gamma f-\mu F)\tilde{x},x-\tilde{x}〉\le 0,\mathrm{\forall }x\in Fix(T).$$

In [5], Tian generalized the iterative method (1.5) replacing the contraction operator f with a Lipschitzian continuous operator V to solve the following variational inequality:

$$〈(\gamma V-\mu F)\tilde{x},x-\tilde{x}〉\le 0,\mathrm{\forall }x\in Fix(T).$$

(1.6)

On the other hand, let ${\{T_i\}}_{i=1}^N$ be a finite family of nonexpansive self-mappings of H. Assume ${\bigcap }_{i=1}^{N}Fix(T_i)\ne \mathrm{\varnothing }$. In [1], Xu also defined the following sequence $\{x_n\}$:

$$x_{n+1}={\alpha }_{n}u+(I-{\alpha }_{n}A)T_{n+1}{x}_n,n\ge 0,$$

(1.7)

where $T_n=T_{nmodN}$ and the mod function takes values in $\{1,2,\dots ,N\}$. He found that the sequence $\{x_n\}$ generated by (1.7) converges strongly to the unique solution of the minimization problem (1.1) with $C={\bigcap }_{i=1}^{N}Fix(T_i)$ under suitable conditions on $\{{\alpha }_n\}$ and the following additional condition on $\{T_n\}$:

$$F(T_N\cdots T_2{T}_1)=F(T_1{T}_N\cdots T_3{T}_2)=\cdots =F(T_{N-1}\cdots T_1{T}_N).$$

(1.8)

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

In 1999, Atsushiba and Takahashi [6] defined the $W_n$-mappings generated by $T_1,T_2,\dots ,T_N$ and $\{{\gamma }_{n,1}\},\{{\gamma }_{n,2}\},\dots ,\{{\gamma }_{n,N}\}\subset [0,1]$ as follows:

$$\begin{array}{c}{U}_{n,0}=I,\hfill \\ U_{n,1}={\gamma }_{n,1}{T}_1{U}_{n,0}+(1-{\gamma }_{n,1})I,\hfill \\ U_{n,2}={\gamma }_{n,2}{T}_2{U}_{n,1}+(1-{\gamma }_{n,2})I,\hfill \\ ⋮\hfill \\ U_{n,N-1}={\gamma }_{n,N-1}{T}_{N-1}{U}_{n,N-2}+(1-{\gamma }_{n,N-1})I,\hfill \\ W_n=U_{n,N}={\gamma }_{n,N}{T}_N{U}_{n,N-1}+(1-{\gamma }_{n,N})I.\hfill \end{array}$$

From [[6], Lemma 3.1], we know that $F(W_n)={\bigcap }_{i=1}^{N}F(T_i)$.

In 2006, Yao [7] introduced the following iterative method:

$$x_{n+1}={\alpha }_n\gamma f(x_n)+\beta x_n+((1-\beta )I-{\alpha }_{n}A)W_n{x}_n.$$

(1.9)

Without the condition (1.8), he proved that the sequence $\{x_n\}$ generated by (1.9) converges strongly to the unique solution of the following variational inequality:

$$〈(A-\gamma f)x^{\ast },x^{\ast }-x〉\le 0,\mathrm{\forall }x\in \bigcap _{i=1}^{N}Fix(T_i),$$

(1.10)

which is the optimality condition for the minimization problem

$$\underset{x\in C}{min}\frac{1}{2}〈Ax,x〉-h(x),$$

(1.11)

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

Shang et al. [8] introduced the following scheme:

$$\{\begin{array}{c}{y}_n={\beta }_n{x}_n+(1-{\beta }_n)W_n{x}_n,\hfill \\ x_{n+1}={\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)y_n.\hfill \end{array}$$

(1.12)

Under certain appropriate conditions, without (1.8), they proved that $\{x_n\}$ 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 [9] introduced a new explicit iterative algorithm:

$$x_{k+1}=(1-{\beta }_k^0)x_k+{\beta }_k^0{T}_0^k{T}_N^k\cdots T_1^k{x}_k,$$

(1.13)

where $T_i^k=(1-{\beta }_k^i)I+{\beta }_k^i{T}_i$ 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 $\{x_k\}$ converges strongly to the unique solution of the following variational inequality:

$$〈F\left(x^{\ast }\right),x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in \bigcap _{i=1}^{N}Fix(T_i).$$

(1.14)

Very recently, Zhou and Wang [10] proposed a simpler iterative algorithm than the iterative algorithm (1.13) given by Buong and Duong:

$$x_{k+1}=(I-{\lambda }_k\mu F)T_N^k\cdots T_1^k{x}_k.$$

(1.15)

They proved that the sequence $\{x_k\}$ 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 $x_n\rightharpoonup x$ and $x_n\to x$ to indicate that $\{x_n\}$ 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 $Fix(T)$ 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 $\alpha \in (0,1)$ such that

$$T=(1-\alpha )I+\alpha S,$$

(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 ([11])

• (i) The composite of finitely many averaged mappings is averaged. That is, if each of the mappings ${\{T_i\}}_{i=1}^N$ 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 ${\alpha }_1,{\alpha }_2\in (0,1)$, then both $T_1{T}_2$ and $T_2{T}_1$ are α-averaged, where $\alpha ={\alpha }_1+{\alpha }_2-{\alpha }_1{\alpha }_2$.

• (ii) If the mappings ${\{T_i\}}_{i=1}^N$ are averaged and have a common fixed point, then

  $$\bigcap _{i=1}^{N}Fix(T_i)=Fix(T_1\cdots T_N).$$

In particular, if $N=2$, we have $Fix(T_1)\cap Fix(T_2)=Fix(T_1{T}_2)=Fix(T_2{T}_1)$.

Lemma 2.2 ([12])

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 ([5])

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 ([13])

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

Lemma 2.5 ([14])

Let $\{x_n\}$ and $\{z_n\}$ be bounded sequences in a Banach space X and $\{{\beta }_n\}$ be a sequence in $[0,1]$ which satisfies the following condition:

$$0<\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\beta }_n\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\beta }_n<1.$$

Suppose $x_{n+1}=(1-{\beta }_n)z_n+{\beta }_n{x}_n$ for all integers $n\ge 0$ and

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}(\parallel z_{n+1}-z_n\parallel -\parallel x_{n+1}-x_n\parallel )\le 0.$$

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

Lemma 2.6 ([1])

Assume $\{a_n\}$ is a sequence of nonnegative real numbers such that

$$a_{n+1}\le (1-{\gamma }_n)a_n+{\delta }_n,n\ge 0,$$

where $\{{\gamma }_n\}$ is a sequence in $(0,1)$ and $\{{\delta }_n\}$ is a sequence such that

• (i) ${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_n=\mathrm{\infty }$,

• (ii) ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\frac{{\delta }_n}{{\gamma }_n}\le 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}|{\delta }_n|<\mathrm{\infty }$.

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

Lemma 2.7 ([15])

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

3 Main results

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

Theorem 3.1 Let ${\{T_i\}}_{i=1}^N$ be N nonexpansive mappings of a real Hilbert space H such that $C={\bigcap }_{i=1}^{N}Fix(T_i)\ne \mathrm{\varnothing }$, 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 $\{x_k\}$ as follows:

$$x_{k+1}={\alpha }_k\gamma V(x_k)+(I-\mu {\alpha }_{k}F)T_N^k{T}_{N-1}^k\cdots T_1^k{x}_k,k\ge 0,$$

(3.1)

where $0<\gamma <\frac{\tau }{\alpha }$ with $\tau =\mu (\eta -\frac{1}{2}\mu L^2)$ and $T_i^k=(1-{\beta }_k^i)I+{\beta }_k^i{T}_i$ for $i=1,2,\dots ,N$. Suppose ${\alpha }_k\in (0,1)$ and ${\beta }_k^i\in (\xi ,\zeta )$ for some $\xi ,\zeta \in (0,1)$. If the following conditions are satisfied:

• (i) ${lim}_{k\to \mathrm{\infty }}{\alpha }_k=0$;

• (ii) ${\sum }_{k=1}^{\mathrm{\infty }}{\alpha }_k=\mathrm{\infty }$;

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

Then the sequence $\{x_k\}$ converges strongly to the unique solution $x^{\ast }$ of the variational inequality:

$$〈(\mu F-\gamma V)x^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in \bigcap _{i=1}^{N}Fix(T_i).$$

(3.2)

Equivalently, we have $P_C(I-\mu F+\gamma V)x^{\ast }=x^{\ast }$.

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

First, we show $\{x_k\}$ is bounded. In fact, for some point $p\in C$, by (3.1) we have

$$\begin{array}{rcl}\parallel x_{k+1}-p\parallel & =& \parallel {\alpha }_k\gamma V{x}_k+(I-\mu {\alpha }_{k}F)T_2^k{T}_1^k{x}_k-p\parallel \\ =& \parallel (I-\mu {\alpha }_{k}F)T_2^k{T}_1^k{x}_k-(I-\mu {\alpha }_{k}F)p+{\alpha }_k(\gamma V{x}_k-\mu Fp)\parallel \\ \le & (1-{\alpha }_k\tau )\parallel T_2^k{T}_1^k{x}_k-T_2^k{T}_1^{k}p\parallel +{\alpha }_k(\parallel \gamma V{x}_k-\gamma Vp\parallel +\parallel \gamma Vp-\mu Fp\parallel )\\ \le & (1-{\alpha }_k\tau )\parallel x_k-p\parallel +{\alpha }_k\gamma \alpha \parallel x_k-p\parallel +{\alpha }_k\parallel \gamma Vp-\mu Fp\parallel \\ =& (1-{\alpha }_k(\tau -\gamma \alpha ))\parallel x_k-p\parallel +{\alpha }_k(\tau -\gamma \alpha )\frac{\parallel \gamma Vp-\mu Fp\parallel }{\tau -\gamma \alpha }\\ \le & max\{\parallel x_k-p\parallel ,\frac{1}{\tau -\gamma \alpha }\parallel \gamma Vp-\mu Fp\parallel \}\\ \le & \cdots \le max\{\parallel x_0-p\parallel ,\frac{1}{\tau -\gamma \alpha }\parallel \gamma Vp-\mu Fp\parallel \}.\end{array}$$

Therefore, $\{x_k\}$ is bounded. Hence we also see that $\{T_2^k{T}_1^k{x}_k\}$, $\{F{T}_2^k{T}_1^k{x}_k\}$, and $\{V{x}_k\}$ 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 ${\{S_k\}}_{k\ge 0}$ on H such that

$$T_2^k{T}_1^k=(1-t_k)I+t_k{S}_k,k\ge 0.$$

(3.5)

Substituting (3.4) into (3.1) yields

$$\begin{array}{rcl}{x}_{k+1}& =& {\alpha }_k\gamma V{x}_k+(I-\mu {\alpha }_{k}F)[(1-t_k)x_k+t_k{S}_k{x}_k]\\ =& (1-t_k)x_k+t_k[S_k{x}_k+\frac{{\alpha }_k}{t_k}(\gamma V{x}_k-\mu F{T}_2^k{T}_1^k{x}_k)].\end{array}$$

Define a sequence $\{z_k\}$ by $z_k=S_k{x}_k+\frac{{\alpha }_k}{t_k}(\gamma V{x}_k-\mu F{T}_2^k{T}_1^k{x}_k)$, so

$$x_{k+1}=(1-t_k)x_k+t_k{z}_k.$$

(3.6)

Now, we claim that

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}(\parallel z_{k+1}-z_k\parallel -\parallel x_{k+1}-x_k\parallel )\le 0.$$

To this end, we observe that

$$\begin{array}{rcl}\parallel z_{k+1}-z_k\parallel & \le & \parallel S_{k+1}{x}_{k+1}-S_k{x}_k\parallel +\frac{{\alpha }_{k+1}}{t_{k+1}}\parallel \gamma V{x}_{k+1}-\mu F{T}_2^{k+1}{T}_1^{k+1}{x}_{k+1}\parallel \\ +\frac{{\alpha }_k}{t_k}\parallel \gamma V{x}_k-\mu F{T}_2^k{T}_1^k{x}_k\parallel \\ \le & \parallel S_{k+1}{x}_{k+1}-S_{k+1}{x}_k\parallel +\parallel S_{k+1}{x}_k-S_k{x}_k\parallel \\ +\frac{{\alpha }_{k+1}}{t_{k+1}}(\parallel \gamma V{x}_{k+1}\parallel +\parallel \mu F{T}_2^{k+1}{T}_1^{k+1}{x}_{k+1}\parallel )\\ +\frac{{\alpha }_k}{t_k}(\parallel \gamma V{x}_k\parallel +\parallel \mu F{T}_2^k{T}_1^k{x}_k\parallel )\\ \le & \parallel x_{k+1}-x_k\parallel +\parallel S_{k+1}{x}_k-S_k{x}_k\parallel \\ +\frac{{\alpha }_{k+1}}{t_{k+1}}(\parallel \gamma V{x}_{k+1}\parallel +\parallel \mu F{T}_2^{k+1}{T}_1^{k+1}{x}_{k+1}\parallel )\\ +\frac{{\alpha }_k}{t_k}(\parallel \gamma V{x}_k\parallel +\parallel \mu F{T}_2^k{T}_1^k{x}_k\parallel )\end{array}$$

(3.7)

and

$$\begin{array}{rcl}\parallel S_{k+1}{x}_k-S_k{x}_k\parallel & =& \parallel \frac{1}{t_{k+1}}{T}_2^{k+1}{T}_1^{k+1}{x}_k-\frac{1}{t_k}{T}_2^k{T}_1^k{x}_k-\frac{1-t_{k+1}}{t_{k+1}}{x}_k+\frac{1-t_k}{t_k}{x}_k\parallel \\ \le & \left|\frac{t_{k+1}-t_k}{t_{k+1}{t}_k}\right|(\parallel T_2^{k+1}{T}_1^{k+1}{x}_k\parallel +\parallel x_k\parallel )+\frac{1}{t_k}\parallel T_2^{k+1}{T}_1^{k+1}{x}_k-T_2^k{T}_1^k{x}_k\parallel \\ \le & \left|\frac{t_{k+1}-t_k}{t_{k+1}{t}_k}\right|M+\frac{1}{t_k}(\parallel T_2^{k+1}{T}_1^{k+1}{x}_k-T_2^{k+1}{T}_1^k{x}_k\parallel \\ +\parallel T_2^{k+1}{T}_1^k{x}_k-T_2^k{T}_1^k{x}_k\parallel )\\ \le & \left|\frac{t_{k+1}-t_k}{t_{k+1}{t}_k}\right|M+\frac{1}{{\xi }^{\ast }}(\parallel T_1^{k+1}{x}_k-T_1^k{x}_k\parallel \\ +\parallel T_2^{k+1}{T}_1^k{x}_k-T_2^k{T}_1^k{x}_k\parallel ),\end{array}$$

(3.8)

where M is a fixed constant satisfying

$$M\ge \underset{k\ge 0}{sup}\{\parallel T_2^{k+1}{T}_1^{k+1}{x}_k\parallel +\parallel x_k\parallel \}.$$

Note that

$$\begin{array}{rcl}\parallel T_1^{k+1}{x}_k-T_1^k{x}_k\parallel & =& \parallel (1-{\beta }_{k+1}^1)x_k+{\beta }_{k+1}^1{T}_1{x}_k-(1-{\beta }_k^1)x_k-{\beta }_k^1{T}_1{x}_k\parallel \\ \le & |{\beta }_{k+1}^1-{\beta }_k^1|(\parallel x_k\parallel +\parallel T_1{x}_k\parallel ).\end{array}$$

Since ${lim}_{k\to \mathrm{\infty }}|{\beta }_{k+1}^i-{\beta }_k^i|=0$ for $i=1,2$, and $\{x_k\}$ and $\{T_1{x}_k\}$ 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,

$$\parallel T_2^{k+1}{T}_1^k{x}_k-T_2^k{T}_1^k{x}_k\parallel \le |{\beta }_{k+1}^2-{\beta }_k^2|(\parallel T_1^k{x}_k\parallel +\parallel T_2{T}_1^k{x}_k\parallel ),$$

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

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}(\parallel z_{k+1}-z_k\parallel -\parallel x_{k+1}-x_k\parallel )\le 0.$$

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 [[8], 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 ${\{x_k\}}_{k\ge 0}$ is bounded, there exists a subsequence ${\{x_{k_j}\}}_{j\ge 1}$ of ${\{x_k\}}_{k\ge 0}$ such that $x_{k_j}\rightharpoonup \hat{x}$ as $j\to \mathrm{\infty }$ and

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈(\mu F-\gamma V)x^{\ast },x^{\ast }-x_k〉=\underset{j\to \mathrm{\infty }}{lim}〈(\mu F-\gamma V)x^{\ast },x^{\ast }-x_{k_j}〉.$$

Since $\{{\beta }_k^i\}$ 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 $T_i^{\mathrm{\infty }}=(1-{\beta }_{\mathrm{\infty }}^i)I+{\beta }_{\mathrm{\infty }}^i{T}_i$ ($i=1,2$). Then we have $Fix(T_i^{\mathrm{\infty }})=Fix(T_i)$ for $i=1,2$. Note that

$$\parallel T_i^{k_j}x-T_i^{\mathrm{\infty }}x\parallel \le |{\beta }_{k_j}^i-{\beta }_{\mathrm{\infty }}^i|(\parallel x\parallel +\parallel T_{i}x\parallel ).$$

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 $Fix(T_1^{\mathrm{\infty }})\cap Fix(T_2^{\mathrm{\infty }})=Fix(T_1)\cap Fix(T_2)=C\ne \mathrm{\varnothing }$ and $T_i^{\mathrm{\infty }}$ is ${\beta }_{\mathrm{\infty }}^i$-averaged for $i=1,2$, by Lemma 2.1, we know that $Fix(T_2^{\mathrm{\infty }}{T}_1^{\mathrm{\infty }})=Fix(T_2^{\mathrm{\infty }})\cap Fix(T_1^{\mathrm{\infty }})=C$. 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 $\{T_1^{k_j}{x}_{k_j}\}$ and $D^{″}$ is a bounded subset including $\{x_{k_j}\}$. 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 $\hat{x}\in Fix(T_2^{\mathrm{\infty }}{T}_1^{\mathrm{\infty }})=C$. It follows that

$$\begin{array}{rcl}\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈(\mu F-\gamma V)x^{\ast },x^{\ast }-T_2^k{T}_1^k{x}_k〉& =& \underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈(\mu F-\gamma V)x^{\ast },x^{\ast }-x_k〉\\ =& \underset{j\to \mathrm{\infty }}{lim}〈(\mu F-\gamma V)x^{\ast },x^{\ast }-x_{k_j}〉\\ =& 〈(\mu F-\gamma V)x^{\ast },x^{\ast }-\hat{x}〉\le 0.\end{array}$$

(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+(I-\mu {\alpha }_{k}F)T_2^k{T}_1^k{x}_k-x^{\ast }\parallel }^2\\ =& {\parallel (I-\mu {\alpha }_{k}F)T_2^k{T}_1^k{x}_k-(I-\mu {\alpha }_{k}F)x^{\ast }+{\alpha }_k(\gamma V{x}_k-\mu F{x}^{\ast })\parallel }^2\\ =& {\parallel (I-\mu {\alpha }_{k}F)T_2^k{T}_1^k{x}_k-(I-\mu {\alpha }_{k}F)x^{\ast }\parallel }^2+{\alpha }_k^2{\parallel \gamma V{x}_k-\mu F{x}^{\ast }\parallel }^2\\ +2{\alpha }_k〈(I-\mu {\alpha }_{k}F)T_2^k{T}_1^k{x}_k-(I-\mu {\alpha }_{k}F)x^{\ast },\gamma V{x}_k-\mu F{x}^{\ast }〉\\ \le & {(1-{\alpha }_k\tau )}^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 & {(1-{\alpha }_k\tau )}^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 & [{(1-{\alpha }_k\tau )}^2+2\alpha {\alpha }_k\gamma ]{\parallel x_k-x^{\ast }\parallel }^2+{\alpha }_k[2〈T_2^k{T}_1^k{x}_k-x^{\ast },(\gamma V-\mu F)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 ]\\ =& [1-2{\alpha }_k(\tau -\alpha \gamma )]{\parallel x_k-x^{\ast }\parallel }^2+{\alpha }_k[2〈T_2^k{T}_1^k{x}_k-x^{\ast },(\gamma V-\mu F)x^{\ast }〉\\ +{\alpha }_k({\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)]\\ \le & [1-2{\alpha }_k(\tau -\alpha \gamma )]{\parallel x_k-x^{\ast }\parallel }^2\\ +{\alpha }_k[2〈T_2^k{T}_1^k{x}_k-x^{\ast },(\gamma V-\mu F)x^{\ast }〉+{\alpha }_k{M}^{\prime }],\end{array}$$

where $M^{\prime }$ is a constant satisfying

$$M^{\prime }\ge \underset{k\ge 0}{sup}\{{\parallel \gamma V{x}_k-\mu F{x}^{\ast }\parallel }^2+2L\parallel T_2^k{T}_1^k{x}_k-x^{\ast }\parallel \parallel \gamma V{x}_k-\mu F{x}^{\ast }\parallel +{\tau }^2{\parallel x_k-x^{\ast }\parallel }^2\}.$$

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 $\lambda \in [0,1)$ such that

$${\parallel Sx-Sy\parallel }^2\le {\parallel x-y\parallel }^2+\lambda {\parallel (I-S)x-(I-S)y\parallel }^2,\mathrm{\forall }x,y\in H.$$

Let ${\{S_i\}}_{i=1}^N$ 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

$${\hat{T}}_i={\omega }_{i}I+(1-{\omega }_i)S_i,$$

(4.1)

where $0\le {\lambda }_i\le {\omega }_i<1$. By virtue of Lemma 2.7, we know that ${\{{\hat{T}}_i\}}_{i=1}^N$ 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 ${\{S_i\}}_{i=1}^N$ be N ${\lambda }_i$-strictly pseudo-contractive mappings on H such that $C={\bigcap }_{i=1}^{N}Fix(S_i)\ne \mathrm{\varnothing }$. Suppose $0<\mu <\frac{\tau }{\alpha }$, $0<\gamma <\frac{\tau }{\alpha }$ with $\tau =\mu (\eta -\frac{1}{2}\mu L^2)$, ${\alpha }_k\in (0,1)$, ${\beta }_k^i\in (\xi ,\zeta )$ for some $\xi ,\zeta \in (0,1)$ 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 ${\{x_k\}}_{k\ge 0}$ defined by (3.1) with $T_i$ replaced by (4.1), converges strongly to the unique solution $x^{\ast }$ of the following variational inequality:

$$〈(\mu F-\gamma V)x^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in \bigcap _{i=1}^{N}Fix(S_i).$$