A New Viscosity Implicit Approximation Method for Solving Variational Inequalities over the Common Fixed Points of Nonexpansive Mappings in Symmetric Hilbert Space

Abstract

In this paper, based on the viscosity approximation method and the hybrid steepest-descent iterative method, a new implicit iterative algorithm is presented for finding the common fixed points set of a finite family of nonexpansive mappings in a reflexive Hilbert space, which is called a symmetric space. We prove that the sequence generated by this new implicit rule strongly converges to the unique solution of a class of variational inequalities under certain appropriate conditions of the parameters. Moreover, we also study the applications to a broader family of strictly pseudo-contractive mappings and generalized equilibrium problems that involve several variational inequality problems, optimization problems, and fixed-point problems. Finally, numerical results are provided to clarify the stability and effectiveness of the algorithm and to compare with some existing iterative algorithms.

1. Introduction

The problem of variational inequality originally appeared in mathematical equations. Hartman and Stampacchia [1] proposed and established the initial theory of variational inequality in 1964. Since then, scholars have carried out extensive research on variational inequality that covers a wide range of disciplines, including optimization, optimal control, mechanics, and finance (see, e.g., [2,3,4,5]). In the theory of variational inequalities, an important and interesting problem is determination of the approximate solutions of variational inequalities by creating a feasible and effective iterative algorithm. In combination with general iterative methods, many scholars have constructed compound iterative schemes. Below, we list their main conclusions.

Let H be a real symmetric Hilbert space possessed of the inner product $⟨·,·⟩$ and the induced norm $\parallel ·\parallel$, and let C be a nonempty closed and convex subset of H. Recall that a mapping $T:C\to C$ is nonexpansive if

$$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\forall x,y\in C.$$

(1)

We define $Fix\left(T\right)=\{x\in C:Tx=x\}$ to express the set of all fixed points of T. Additionally, a $f:C\to C$ is a contraction on H if there exists $\alpha \in (0,1)$ such that

$$\parallel f\left(x\right)-f\left(y\right)\parallel \le \alpha \parallel x-y\parallel ,\forall x,y\in C.$$

(2)

In 2003, Xu [6] created the iterative scheme by means of the following recurrence relation:

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

(3)

where ${\alpha }_n\in (0,1)$, $u\in C$ and A is a strongly positive linear bounded operator. He not only proved the strong convergence from $\left\{x_n\right\}$ to a fixed point of T, but also showed that the solution of sequence $\left\{x_n\right\}$ is equivalent to the unique solution of the following minimization problem

$$\underset{\begin{array}{c}x\in C\end{array}}{min}\frac{1}{2}⟨Ax,x⟩-(x,u).$$

In the following way, for the nonexpansive mapping T, Moudafi [7] established the viscosity approximation method. The sequence $\left\{x_n\right\}$ can be created by

$$x_{n+1}={\alpha }_{n}f\left(x_n\right)+(1-{\alpha }_n)T{x}_n,n\ge 0.$$

(4)

where $\left\{{\alpha }_n\right\}$ is a sequence in $(0,1)$ and f is a contraction on H. It was proven that the sequence $\left\{x_n\right\}$ constructed by Equation (4) strongly converges to the unique solution of the following form of variational inequality

$$⟨(I-f)x^{*},x-x^{*}⟩\ge 0,\forall x\in Fix\left(T\right).$$

(5)

In 2006, combining the iterative methods of Equations (3) and (4), Marino and Xu [8] created the viscosity iterative method below:

$$x_{n+1}={\alpha }_n\gamma f\left(x_n\right)+(I-{\alpha }_{n}A)T{x}_n,n\ge 0,$$

(6)

where f is a contraction and A is a strongly positive linear bounded operator. Under some appropriate conditions, they proved that the solution of the sequence $\left\{x_n\right\}$ constructed by Equation (6) is equal to the union solution of the following form of variational inequality

$$⟨(A-\gamma f)x^{*},x-x^{*}⟩\ge 0,\forall x\in Fix\left(T\right),$$

which also becomes the optimal solution 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 $\gamma f$ (i.e., $h^{\prime }\left(x\right)=\gamma f\left(x\right)$ for $x\in H$).

In 2001, Yamada [9] et al. created the hybrid steepest-descent iterative method:

$$x_{n+1}=T{x}_n-\mu {\lambda }_{n}F\left(T{x}_n\right),n\ge 0,$$

(7)

where F is Lipschitzian continuous and strongly monotone operator and $0<\mu <\frac{2\eta }{{\kappa }^2}$. They certified that the solution of the sequence $\left\{x_n\right\}$ constructed by Equation (7) is equivalent to the unique solution of the following form of variational inequality

$$⟨F\left(x^{*}\right),x-x^{*}⟩\ge 0,\forall x\in Fix\left(T\right).$$

In 2010, combining with the work of previous scholars, Tian [10] proposed a generalized viscosity iterative algorithm:

$$x_{n+1}={\alpha }_n\gamma V\left(x_n\right)+(I-\mu {\alpha }_{n}F)T{x}_n,n\ge 0,$$

(8)

where F is Lipschitzian continuous and strongly monotone operator, and V is a Lipschitzian continuous operator. It was proven that the solution of the sequence $\left\{x_n\right\}$ produced by Equation (8) is equivalent to the unique solution of the following form of variational inequality

$$⟨(\mu F-\gamma V)x^{*},x-x^{*}⟩\ge 0,\forall x\in Fix\left(T\right).$$

In 2013, Zhou and Wang [11] created a new iterative scheme:

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

(9)

They showed that the sequence $\left\{x_n\right\}$ proposed by Equation (9) converges faster and, at the same time, can solve the following type of variational inequality:

$$⟨F\left(x^{*}\right),x-x^{*}⟩\ge 0,\forall x\in {\cap }_{i=1}^{N}Fix\left(T_i\right),$$

where $T_i^n=(1-{\beta }_n^i)I+{\beta }_n^i{T}_i,\forall i=1,2,\cdots ,N,T_0^k=I-{\lambda }_k\mu F$, F is Lipschitzian continuous and strongly monotone operator.

In 2014, combining the iterative methods of Equations (8) and (9), Zhang and Yang [12] explored the following explicit iterative algorithm based on the viscosity method:

$$x_{n+1}={\alpha }_n\gamma V\left(x_n\right)+(I-\mu {\alpha }_{n}F)T_N^n{T}_{N-1}^n\cdots T_1^n{x}_n,n\ge 0,$$

(10)

where V is Lipschitzian, $T_i^n=(1-{\beta }_n^i)I+{\beta }_n^i{T}_i,\forall i=1,2,\cdots ,N,$ and ${\beta }_n^i\in (0,1)$. The following variational inequality was proven by them:

$$⟨(\mu F-\gamma V)x^{*},x-x^{*}⟩\ge 0,\forall x\in {\cap }_{i=1}^{N}Fix\left(T_i\right).$$

(11)

The implicit midpoint rules have played an important role in settling the ordinary differential equations in the development of the research pursuing a solution to fixed point problems of nonexpansive mappings (see the detailed references [13,14,15,16,17]). As a consequence, this method has recently aroused the interest of some scholars and is gradually being studied more. In 2015, using the viscosity approximation method, Xu [18] et al. built the iterative sequence of implicit midpoint rules for nonexpansive mappings:

$$x_{n+1}={\alpha }_{n}f\left(x_n\right)+(1-{\alpha }_n)T\left(\frac{x_n+x_{n+1}}{2}\right),n\ge 0,$$

(12)

where $\left\{{\alpha }_n\right\}$ is sequence of numbers in $(0,1)$, and f is a compressed mapping on H. It was proven that the solution of sequence $\left\{x_n\right\}$ produced by Equation (12) is equivalent to the unique solution of variational inequality Equation (5).

In the same year, the implicit rule of generalized viscosity was established by Ke and Ma [19]:

$$x_{n+1}={\alpha }_{n}f\left(x_n\right)+(1-{\alpha }_n)T(s_n{x}_n+(1-s_n)x_{n+1}),n\ge 0.$$

(13)

where $\left\{{\alpha }_n\right\}$ and $\left\{s_n\right\}$ are the real sequence for $(0,1)$. They verified that the solution of sequence $\left\{x_n\right\}$ produced by Equation (13) is equal to the unique solution of the above variational inequality Equation (5).

In 2017, He and Mao [20] showed the following new iterative method of implicit rules combined with the viscosity approximation method:

$$x_{n+1}={\alpha }_{n}f\left(x_n\right)+(1-{\alpha }_n)T^n(s_n{x}_n+(1-s_n)x_{n+1}),n\ge 0.$$

(14)

It was proven that the solution of sequence $\left\{x_n\right\}$ produced by Equation (14) is equivalent to the unique solution of the above variational inequality Equation (5).

Recently, Cai and Yekini [21] studied a modified viscosity implicit rule of the nonexpansive mapping

$$x_{n+1}=P_C[{\alpha }_{n}f\left(x_n\right)+(I-\mu {\alpha }_{n}F)T(s_n{x}_n+(1-s_n)x_{n+1})],n\ge 0,$$

(15)

where $\left\{{\alpha }_n\right\}$ and $\left\{s_n\right\}$ are two sequences in $(0,1]$, and F is a Lipschitzian continuous and strongly monotone operator.

They proved that the solution of sequence $\left\{x_n\right\}$ is equal to the union solution of the variational inequality

$$⟨(\mu F-f)x^{*},x-x^{*}⟩\ge 0,\forall x\in Fix\left(T\right).$$

After studying the results of the above scholars, we realize that viscosity approximation methods can be used to solve the fixed-point problem, that is to say, it is an efficient method which amounts to choosing a particular fixed point for a given nonexpansive self-mapping.

As a matter of fact, in Hilbert space, a variational inequality with respect to a closed convex subset is equal to a fixed-point equation involving a metric projection from any point onto the closed convex set, that is, the feasible set.

Therefore, solving the variational inequality depends on the projection mapping. However, when the closed form of the projection mapping is incomplete, it is not easy to compute. In this case, by assuming that the common fixed points set of a finite family of nonexpansive mappings becomes the new feasible set, the hybrid steepest-descent method is created, which overcomes the difficulty of estimating projection operators due to the complexity of feasible sets.

Inspired by the above methods and ideas and in combination with the viscosity approximation technique and hybrid steepest descent iterative method of nonexpansive mappings, we study a new generalized viscosity implicit iterative scheme in Hilbert space.

Let ${\left\{T_i\right\}}_{i=1}^N$ be a finite family of nonexpansive mappings, and start with an arbitrary $x_0\in C$, let V be an $\alpha$-Lipschitzian operator on H with coefficient $\alpha \in (0,1)$, let F be a $\zeta$-Lipschitzian continuous and $\kappa$-strongly monotone operator on H with constants $\zeta >0$ and $\kappa >0$; then, define the sequences $\left\{x_n\right\}$ with:

$$x_{n+1}={\alpha }_n\gamma V\left(x_n\right)+(I-\mu {\alpha }_{n}F)T_N^n{T}_{N-1}^n\cdots T_1^n(t_n{x}_n+(1-t_n)x_{n+1}),n\ge 0.$$

Under appropriate conditions, we prove that the sequence $\left\{x_n\right\}$ strongly converges to the union solution of the variational inequality in Equation (11).

The remainder of the content of this paper is as follows. In Section 2, some useful definitions and lemmas are recalled for use in the main results. In Section 3, with the help of some suitable conditions, the strong convergence of the iterative sequence is proved. In Section 4, the new iterative algorithm is applied to the broader family of $\xi$-strictly pseudo-contractive mappings and generalized equilibrium problems. In Section 5, with the purpose of supporting the main results and discussing the convergence, two numerical examples are provided. In the final section, the main work of this article is summarized.

2. Preliminaries

To prove our main results, in this section we recall some helpful definitions and lemmas. When $\left\{x_n\right\}$ is a sequence in the real Hilbert space H, we use $x_n\to x$ and $x_n\rightharpoonup x$, respectively, to denote that $\left\{x_n\right\}$ converges strongly to x and $\left\{x_n\right\}$ converges weakly to x.

A mapping $P_C$ is called a metric projection from H to C when C is a nonempty closed and convex subset of H. Then, for any $u\in H$, there exists a unique nearest point $P_C\left(u\right)\in C$:

$$\parallel u-P_C\left(u\right)\parallel \le \parallel u-v\parallel ,\forall v\in C.$$

As a matter of fact, $P_C$ is nonexpansive. Moreover, the next inequality holds:

$$⟨u-v,P_C\left(u\right)-P_C\left(v\right)⟩\ge {\parallel P_C\left(u\right)-P_C\left(v\right)\parallel }^2,\forall u,v\in H.$$

For $u\in H$ and $z\in C$, $P_C$ satisfies

$$z=P_C\left(u\right)\iff ⟨u-z,v-z⟩\le 0,\forall v\in C.$$

(16)

Furthermore,

$$\parallel P_C\left(u\right)-P_C\left(v\right)\parallel \le \parallel u-v\parallel ,\forall u,v\in H.$$

Definition 1.

An operator $G:C\to C$ is said to be

(1) 

A strongly positive bounded linear operator with coefficient ρ if there exists a constant $\rho >0$ such that

$$⟨Gu,u⟩\ge {\rho \parallel u\parallel }^2,\forall u\in C.$$

(2) 

κ-strongly monotone if there exists a positive constant κ such that

$$⟨Gu-Gv,u-v⟩\ge {\kappa \parallel u-v\parallel }^2,\forall u,v\in C.$$

(3) 

ζ-Lipschitzian if there exists a positive constant ζ such that

$$\parallel Gu-Gv\parallel \le \zeta \parallel u-v\parallel ,\forall u,v\in C.$$

(4) 

θ-inverse strongly monotone (for short, θ-ism) if there exists a $\theta >0$ such that

$$⟨Gu-Gv,u-v⟩\ge {\theta \parallel Gu-Gv\parallel }^2,\forall u,v\in C.$$

(5) 

Firmly nonexpansive if

$$⟨Gu-Gv,u-v⟩\ge {\parallel Gu-Gv\parallel }^2,\forall u,v\in C.$$

Remark 1. (1) It is not hard to find that the strongly positive bounded linear operator G turns into $\parallel G\parallel$-Lipschitzian and ρ-strongly monotone.

(2) Projection $P_C$ is an example of a firmly nonexpansive projection that is $\frac{1}{2}$-averaged.

Definition 2

([22]). An averaged mapping is defined by a mapping $T:H⟶H$ if there exists some constant $\lambda \in (0,1)$ for

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

where $I:H\to H$ is the identity mapping, and $S:H\to H$ is a nonexpansive mapping. More precisely, it can be said to be λ-averaged, which is also non-expansive and $Fix\left(S\right)=Fix\left(T\right).$

Lemma 1

([23]). The composite of the limited multiple averaged mappings is still averaged. If the mappings ${\left\{T_i\right\}}_{i=1}^N$ are averaged and they all have a common fixed point, then

$${\cap }_{i=1}^{N}Fix\left(T_i\right)=Fix(T_N{T}_{N-1}\cdots T_1).$$

Distinctively, when $N=2,Fix\left(T_1\right)\cap Fix\left(T_2\right)=Fix\left(T_1{T}_2\right)=Fix\left(T_2{T}_1\right)$.

Lemma 2

([24]). Let H be a real Hilbert space, C be a nonempty closed and convex subset of H, and $T:C\to C$ be a nonexpansive mapping with $Fix\left(T\right)\ne \varphi$. If $\left\{x_n\right\}\subset C,$ and $u,v\in C$ with $\left\{x_n\right\}\rightharpoonup u$; $\left\{(I-T)x_n\right\}\to v$, then $(I-T)u=v$. Distinctively, if $v=0$, then $u\in Fix\left(T\right)$.

Lemma 3

([6]). Assume that $\left\{{\alpha }_n\right\}$ is a sequence with nonnegative real numbers satisfying the condition

$${\alpha }_{n+1}\le (1-{\beta }_n){\alpha }_n+{\phi }_n,n\ge 0,$$

where $\left\{{\beta }_n\right\}$ is a sequence in $(0,1)$ and $\left\{{\phi }_n\right\}$ is a sequence in $\mathbb{R}$ such that

(i) 

${\sum }_{n=1}^{\infty }{\beta }_n=\infty$,

(ii) 

${lim}_{n\to \infty }sup\frac{{\phi }_n}{{\beta }_n}\le 0$ or ${\sum }_{n=1}^{\infty }\left|{\phi }_n\right|<\infty$.

Then, ${lim}_{n\to \infty }{\alpha }_n=0$.

Lemma 4.

Let $F:H⟶H$ be a ζ-Lipschitzian continuous and κ-strongly monotone operator, ${\left\{T_i\right\}}_{i=1}^N:C⟶C$ be an N nonexpansive mapping of H, $T_i^n=(1-{\omega }_n^i)I+{\omega }_n^i{T}_i$ for $i=1,2,\dots N$, and ${\omega }_n^i\in (0,1]$. For a digit λ in $(0,1]$ and a fixed $\mu \in (0,\frac{2\kappa }{{\zeta }^2})$, we define a family of nonexpansive mappings $T_N^{\lambda }\dots T_1^{\lambda }:C\to H$ by

$$T_N^{\lambda }\dots T_1^{\lambda }u:=T_N^n\dots T_1^{n}u-\lambda \mu F(T_N^n\dots T_1^{n}u),\forall u\in C.$$

Then, $T_N^{\lambda }\dots T_1^{\lambda }$ forms a family of contractions, which satisfies the inequality

$$\parallel T_N^{\lambda }\dots T_1^{\lambda }u-T_N^{\lambda }\dots T_1^{\lambda }v\parallel \le (1-\lambda \tau )\parallel u-v\parallel ,\forall u,v\in C,$$

where $\tau =1-\sqrt{1-\mu (2\kappa -\mu {\zeta }^2)}\in (0,1]$.

This lemma plays a significant role in the main results section.

Proof. 

By applying the $\zeta$-Lipschitz continuity and $\kappa$-strong monotonicity of F over $T\left(H\right)$ to $G:\mu F-I$, we can obtain

(i)

$\begin{array}{cc}& \parallel T_N^n\dots T_1^{n}u-T_N^n\dots T_1^{n}v\parallel \hfill \\ \hfill =& \parallel (1-{\omega }_n^N)(T_{N-1}^n\cdots T_1^{n}u)+{\omega }_n^N{T}_N(T_{N-1}^n\cdots T_1^{n}u)-(1-{\omega }_n^N)(T_{N-1}^n\cdots T_1^{n}v)\hfill \\ & -{\omega }_n^N{T}_N(T_{N-1}^n\cdots T_1^{n}v)\parallel \hfill \\ \hfill =& \parallel (1-{\omega }_n^N)(T_{N-1}^n\cdots T_1^{n}u-T_{N-1}^n\cdots T_1^{n}v)+{\omega }_n^N(T_N{T}_{N-1}^n\cdots T_1^{n}u-T_N{T}_{N-1}^n\cdots T_1^{n}v)\parallel \hfill \\ \hfill \le & \parallel (1-{\omega }_n^N)(T_{N-1}^n\cdots T_1^{n}u-T_{N-1}^n\cdots T_1^{n}v)\parallel +\parallel {\omega }_n^N(T_N{T}_{N-1}^n\cdots T_1^{n}u-T_N{T}_{N-1}^n\cdots T_1^{n}v)\parallel \hfill \\ \hfill \le & (1-{\omega }_n^N)\parallel T_{N-1}^n\cdots T_1^{n}u-T_{N-1}^n\cdots T_1^{n}v\parallel +{\omega }_n^N\parallel T_{N-1}^n\cdots T_1^{n}u-T_{N-1}^n\cdots T_1^{n}v\parallel \hfill \\ \hfill \le & \cdots \le \parallel u-v\parallel \hfill \end{array}$

(ii)

$\begin{array}{cc}& \parallel G(T_N^n\cdots T_1^{n}u)-G(T_N^n\cdots T_1^{n}v){\parallel }^2\hfill \\ \hfill =& \parallel (\mu F-I)(T_N^n\cdots T_1^{n}u)-(\mu F-I)(T_N^n\cdots T_1^{n}v){\parallel }^2\hfill \\ \hfill =& \parallel \mu F(T_N^n\cdots T_1^{n}u)-T_N^n\cdots T_1^{n}u-\mu F(T_N^n\cdots T_1^{n}v)+T_N^n\cdots T_1^{n}v{)\parallel }^2\hfill \\ \hfill =& \parallel \mu [F(T_N^n\cdots T_1^{n}u)-F(T_N^n\cdots T_1^{n}v)]-(T_N^n\cdots T_1^{n}u-T_N^n\cdots T_1^{n}v){\parallel }^2\hfill \\ \hfill =& {\mu }^2\parallel F(T_N^n\cdots T_1^{n}u)-F(T_N^n\cdots T_1^{n}v){\parallel }^2+{\parallel T_N^n\cdots T_1^{n}u-T_N^n\cdots T_1^{n}v\parallel }^2\hfill \\ & -2\mu ⟨F(T_N^n\cdots T_1^{n}u)-F(T_N^n\cdots T_1^{n}v),T_N^n\cdots T_1^{n}u-T_N^n\cdots T_1^{n}v⟩\hfill \\ \hfill \le & {\mu }^2{L}^2\parallel T_N^n\cdots T_1^{n}u-T_N^n\cdots T_1^n{v\parallel }^2+{\parallel T_N^n\cdots T_1^{n}u-T_N^n\cdots T_1^{n}v\parallel }^2\hfill \\ & -2\mu \eta \parallel T_N^n\cdots T_1^{n}u-T_N^n\cdots T_1^n{v\parallel }^2\hfill \\ \hfill =& [1-\mu (2\eta -\mu L^2)]{\parallel T_N^n\cdots T_1^{n}u-T_N^n\cdots T_1^{n}v\parallel }^2\hfill \end{array}$

(iii)

$\begin{array}{cc}& \parallel T_N^{\lambda }\cdots T_1^{\lambda }u-T_N^{\lambda }\cdots T_1^{\lambda }v\parallel \hfill \\ \hfill =& \parallel T_N^n\cdots T_1^{n}u-\lambda \mu F(T_N^n\cdots T_1^{n}u)-T_N^n\cdots T_1^{n}v+\lambda \mu F(T_N^n\cdots T_1^{n}v)\parallel \hfill \\ \hfill =& \parallel T_N^n\cdots T_1^{n}u-T_N^n\cdots T_1^{n}v-\lambda T_N^n\cdots T_1^{n}u+\lambda T_N^n\cdots T_1^{n}u\hfill \\ & -\lambda T_N^n\cdots T_1^{n}v+\lambda T_N^n\cdots T_1^{n}v-\lambda \mu F(T_N^n\cdots T_1^{n}u)+\lambda \mu F(T_N^n\cdots T_1^{n}v)\parallel \hfill \\ \hfill =& \parallel (1-\lambda )(T_N^n\cdots T_1^{n}u-T_N^n\cdots T_1^{n}v)-\lambda [\mu F(T_N^n\cdots T_1^{n}u)-T_N^n\cdots T_1^{n}u\hfill \\ & -\mu F(T_N^n\cdots T_1^{n}v)+T_N^n\cdots T_1^{n}v]\parallel \hfill \\ \hfill =& \parallel (1-\lambda )(T_N^n\cdots T_1^{n}u-T_N^n\cdots T_1^{n}v)-\lambda [(\mu F-I)(T_N^n\cdots T_1^{n}u)-(\mu F-I)(T_N^n\cdots T_1^{n}v)]\parallel \hfill \\ \hfill =& \parallel (1-\lambda )(T_N^n\cdots T_1^{n}u-T_N^n\cdots T_1^{n}v)-\lambda [G(T_N^n\cdots T_1^{n}u)-G(T_N^n\cdots T_1^{n}v)]\parallel \hfill \\ \hfill \le & (1-\lambda )\parallel T_N^n\cdots T_1^{n}u-T_N^n\cdots T_1^{n}v\parallel +\lambda \sqrt{1-\mu (2\kappa -\mu {\zeta }^2)}\parallel T_N^n\cdots T_1^{n}u-T_N^n\cdots T_1^{n}v\parallel \hfill \\ \hfill \le & (1-\lambda )\parallel u-v\parallel +\lambda \sqrt{1-\mu (2\kappa -\mu {\zeta }^2)}\parallel u-v\parallel \hfill \\ \hfill =& [1-\lambda +\lambda \sqrt{1-\mu (2\kappa -\mu {\zeta }^2)}]\parallel u-v\parallel \hfill \\ \hfill =& (1-\lambda \tau )\parallel u-v\parallel .\hfill \end{array}$

where $\tau =1-\sqrt{1-\mu (2\kappa -\mu {\zeta }^2)}.$ □

Lemma 5

([25]). Let H be a real Hilbert space; then, for all $u,v\in H$,

${\parallel u+v\parallel }^2\le {\parallel u\parallel }^2+2⟨v,u+v⟩.$

3. Main Results

Theorem 1.

Let C be a nonempty closed and convex subset of the real Hilbert space H, ${\left\{T_i\right\}}_{i=1}^N:C\to C$ be N nonexpansive mapping of H such that $C={\bigcap }_{i=1}^{N}Fix\left(T_i\right)\ne \varnothing$, $V:C\to C$ be an α-Lipschitzian operator on H with coefficient $\alpha \in (0,1)$, $F:C\to H$ be a ζ-Lipschitzian continuous and κ-strongly monotone operator on H with constants $\zeta >0$ and $\kappa >0$. Let a sequence $\left\{x_n\right\}$ be created by:

$$x_{n+1}={\alpha }_n\gamma V\left(x_n\right)+(I-\mu {\alpha }_{n}F)T_N^n{T}_{N-1}^n\cdots T_1^n(t_n{x}_n+(1-t_n)x_{n+1}),n\ge 0.$$

(17)

where $\left\{{\alpha }_n\right\}$ and $\left\{t_n\right\}$ both belong to $(0,1],0<\gamma <\frac{\tau }{\alpha },0<\mu <\frac{2\kappa }{{\zeta }^2},\tau =1-\sqrt{1-\mu (2\kappa -\mu {\zeta }^2)}\in (0,1],T_i^n=(1-{\omega }_n^i)I+{\omega }_n^i{T}_i$ for $i=1,2,\cdots ,N$ and ${\omega }_n^i\in (0,1]$, satisfying the following conditions:

(i) 

${lim}_{n\to \infty }{\alpha }_n=0;$

(ii) 

${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$ and ${\sum }_{n=0}^{\infty }|{\alpha }_{n+1}-{\alpha }_n|<\infty$;

(iii) 

$0<b\le t_n\le t_{n+1}<1,\forall n\ge 0.$

Then, the sequence $\left\{x_n\right\}$ converges strongly to the common fixed points set $x^{*}$ of a finite family of nonexpansive mappings, which is equivalent to the unique solution of the following variational inequality

$$⟨(\mu F-\gamma f)x^{*},x-x^{*}⟩\ge 0,\forall x\in {\cap }_{i=1}^{N}Fix\left(T_i\right).$$

Equally, $P_C(I-\mu F+\gamma V)x^{*}=x^{*}$ holds.

Proof. 

Our proof can be easily extended to the general case, and for the sake of simplicity of computation, we will show proofs of Theorem 1 in five steps for $N=2$.

Step 1. We prove that the sequence $\left\{x_n\right\}$ is bounded.

Suppose $q\in C$, we obtain

$$\begin{array}{cc}\hfill \parallel x_{n+1}-q\parallel =& \parallel {\alpha }_n\gamma V\left(x_n\right)+(I-\mu {\alpha }_{n}F)T_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})-q\parallel \hfill \\ \hfill =& \parallel {\alpha }_n[\gamma V\left(x_n\right)-\mu F\left(q\right)]+(I-\mu {\alpha }_{n}F)T_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})\hfill \\ & -(I-\mu {\alpha }_{n}F)T_2^n{T}_1^{n}q\parallel \hfill \\ \hfill \le & {\alpha }_n\parallel \gamma V\left(x_n\right)-\mu F\left(q\right)\parallel +(1-\tau {\alpha }_n)\parallel t_n{x}_n+(1-t_n)x_{n+1}-q\parallel \hfill \\ \hfill \le & {\alpha }_n\parallel \gamma V\left(x_n\right)-\gamma V\left(q\right)\parallel +{\alpha }_n\parallel \gamma V\left(q\right)-\mu F\left(q\right)\parallel +(1-\tau {\alpha }_n)t_n\parallel x_n-q\parallel \hfill \\ & +(1-\tau {\alpha }_n)(1-t_n)\parallel x_{n+1}-q\parallel \hfill \\ \hfill \le & \gamma \alpha {\alpha }_n\parallel x_n-q\parallel +{\alpha }_n\parallel \gamma V\left(q\right)-\mu F\left(q\right)\parallel +(1-\tau {\alpha }_n)t_n\parallel x_n-q\parallel \hfill \\ & +(1-\tau {\alpha }_n)(1-t_n)\parallel x_{n+1}-q\parallel \hfill \\ \hfill =& (t_n-\tau {\alpha }_n{t}_n+\gamma \alpha {\alpha }_n)\parallel x_n-q\parallel +(1-\tau {\alpha }_n)(1-t_n)\parallel x_{n+1}-q\parallel \hfill \\ \hfill & +{\alpha }_n\parallel \gamma V\left(q\right)-\mu F\left(q\right)\parallel .\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill (1-(1-\tau {\alpha }_n)(1-t_n))\parallel x_{n+1}-q\parallel \le & (t_n-\tau {\alpha }_n{t}_n+\gamma \alpha {\alpha }_n)\parallel x_n-q\parallel \hfill \\ \hfill & +{\alpha }_n\parallel \gamma V\left(q\right)-\mu F\left(q\right)\parallel ,\hfill \end{array}$$

(18)

which implies that

$\begin{array}{cc}\hfill \parallel x_{n+1}-q\parallel \le & \frac{t_n-\tau {\alpha }_n{t}_n+\gamma \alpha {\alpha }_n}{t_n+\tau {\alpha }_n(1-t_n)}\parallel x_n-q\parallel +\frac{{\alpha }_n\parallel \gamma V\left(q\right)-\mu F\left(q\right)\parallel }{t_n+\tau {\alpha }_n(1-t_n)}\hfill \\ \hfill =& [1-\frac{{\alpha }_n(\tau -\gamma \alpha )}{t_n+\tau {\alpha }_n(1-t_n)}]\parallel x_n-q\parallel +\frac{{\alpha }_n(\tau -\gamma \alpha )}{t_n+\tau {\alpha }_n(1-t_n)}\frac{1}{\tau -\gamma \alpha }\parallel \gamma V\left(q\right)-\mu F\left(q\right)\parallel \hfill \end{array}$

$\begin{array}{cc}\hfill \le & max\{\parallel x_n-q\parallel ,\frac{1}{\tau -\gamma \alpha }\parallel \gamma V\left(q\right)-\mu F\left(q\right)\parallel \}\hfill \\ \hfill \le & \cdots \le max\{\parallel x_0-q\parallel ,\frac{1}{\tau -\gamma \alpha }\parallel \gamma V\left(q\right)-\mu F\left(q\right)\parallel \}.\hfill \end{array}$

Making use of the induction rule, we obtain

$\parallel x_n-q\parallel \le max\{\parallel x_0-q\parallel ,\frac{1}{\tau -\gamma \alpha }\parallel \gamma V\left(q\right)-\mu F\left(q\right)\parallel \},\forall n\ge 0.$

Hence, $\left\{x_n\right\}$ is determined to be bounded.

Therefore, $\left\{f\left(x_n\right)\right\},\left\{T_2^n{T}_1^n[t_n{x}_n+(1-t_n)x_{n+1}]\right\}$ are both inferred as bounded.

Step 2. We show that ${lim}_{n\to \infty }\parallel x_{n+1}-x_n\parallel =0.$

In order to achieve this purpose, Equation (17) is used to realize

$$\begin{array}{cc}\hfill \parallel x_{n+2}-x_{n+1}\parallel =& \parallel {\alpha }_{n+1}\gamma V\left(x_{n+1}\right)+(I-\mu {\alpha }_{n+1}F)T_2^{n+1}{T}_1^{n+1}(t_{n+1}{x}_{n+1}+(1-t_{n+1})x_{n+2})\hfill \\ & -[{\alpha }_n\gamma V\left(x_n\right)+(I-\mu {\alpha }_{n}F)T_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})]\parallel \hfill \\ \hfill =& \parallel {\alpha }_{n+1}\gamma [V\left(x_{n+1}\right)-V\left(x_n\right)]+({\alpha }_{n+1}-{\alpha }_n)\gamma V\left(x_n\right)\hfill \\ & +T_2^{{\alpha }_{n+1}}{T}_1^{{\alpha }_{n+1}}(t_{n+1}{x}_{n+1}+(1-t_{n+1})x_{n+2})\hfill \\ & -T_2^{{\alpha }_{n+1}}{T}_1^{{\alpha }_{n+1}}(t_n{x}_n+(1-t_n)x_{n+1})+T_2^{{\alpha }_{n+1}}{T}_1^{{\alpha }_{n+1}}(t_n{x}_n+(1-t_n)x_{n+1})\hfill \\ & -T_2^{{\alpha }_n}{T}_1^{{\alpha }_n}(t_n{x}_n+(1-t_n)x_{n+1})\parallel \hfill \\ \hfill =& \parallel {\alpha }_{n+1}\gamma [V\left(x_{n+1}\right)-V\left(x_n\right)]+({\alpha }_{n+1}-{\alpha }_n)\gamma V\left(x_n\right)\hfill \\ & +T_2^{{\alpha }_{n+1}}{T}_1^{{\alpha }_{n+1}}(t_{n+1}{x}_{n+1}+(1-t_{n+1})x_{n+2})\hfill \\ & -T_2^{{\alpha }_{n+1}}{T}_1^{{\alpha }_{n+1}}(t_n{x}_n+(1-t_n)x_{n+1})+T_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})\hfill \\ & -\mu {\alpha }_{n+1}F{T}_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})-T_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})\hfill \\ & +\mu {\alpha }_{n}F{T}_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})\parallel \hfill \\ \hfill \le & \gamma \alpha {\alpha }_{n+1}\parallel x_{n+1}-x_n\parallel \hfill \\ & +(1-\tau {\alpha }_{n+1})\parallel t_{n+1}{x}_{n+1}+(1-t_{n+1})x_{n+2}-t_n{x}_n-(1-t_n)x_{n+1}\parallel \hfill \\ & +|{\alpha }_{n+1}-{\alpha }_n|\parallel \gamma V\left(x_n\right)-\mu F{T}_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})\parallel \hfill \\ \hfill \le & \gamma \alpha {\alpha }_{n+1}\parallel x_{n+1}-x_n\parallel +(1-\tau {\alpha }_{n+1})(1-t_{n+1})\parallel x_{n+2}-x_{n+1}\parallel \hfill \\ & +(1-\tau {\alpha }_{n+1})t_n\parallel (x_{n+1}-x_n)\parallel +|{\alpha }_{n+1}-{\alpha }_n|M\hfill \\ \hfill =& (t_n-\tau {\alpha }_{n+1}{t}_n+\gamma \alpha {\alpha }_{n+1})\parallel x_{n+1}-x_n\parallel \hfill \\ & +(1-\tau {\alpha }_{n+1})(1-t_{n+1})\parallel x_{n+2}-x_{n+1}\parallel +|{\alpha }_{n+1}-{\alpha }_n|M\hfill \end{array}$$

where $M=\underset{n\ge 0}{sup}\parallel \gamma V\left(x_n\right)-\mu F{T}_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})\parallel .$

It follows that

$$\begin{array}{cc}\hfill (1-(1-\tau {\alpha }_{n+1})(1-t_{n+1}))\parallel x_{n+2}-x_{n+1}\parallel \le & (t_n-\tau {\alpha }_{n+1}{t}_n+\gamma \alpha {\alpha }_{n+1})\hfill \\ & \parallel x_{n+1}-x_n\parallel +|{\alpha }_{n+1}-{\alpha }_n|M,\hfill \end{array}$$

that is,

$$\begin{array}{cc}\hfill \parallel x_{n+2}-x_{n+1}\parallel \le & \frac{t_n-\tau {\alpha }_{n+1}{t}_n+\gamma \alpha {\alpha }_{n+1}}{1-(1-\tau {\alpha }_{n+1})(1-t_{n+1})}\parallel x_{n+1}-x_n\parallel +\frac{|{\alpha }_{n+1}-{\alpha }_n|M}{1-(1-\tau {\alpha }_{n+1})(1-t_{n+1})}\hfill \\ \hfill =& [1-\frac{{\alpha }_{n+1}(\tau -\gamma \alpha )+(1-\tau {\alpha }_{n+1})(t_{n+1}-t_n)}{1-(1-\tau {\alpha }_{n+1})(1-t_{n+1})}]\parallel x_{n+1}-x_n\parallel \hfill \\ \hfill =& [1-\frac{{\alpha }_{n+1}(\tau -\gamma \alpha )+(1-\tau {\alpha }_{n+1})(t_{n+1}-t_n)}{1-(1-\tau {\alpha }_{n+1})(1-t_{n+1})}]\parallel x_{n+1}-x_n\parallel \hfill \\ & +\frac{|{\alpha }_{n+1}-{\alpha }_n|M}{1-(1-\tau {\alpha }_{n+1})(1-t_{n+1})}.\hfill \end{array}$$

Note that $\left\{{\alpha }_n\right\}$ and $\left\{t_n\right\}$ both belong to $(0,1]$ and from condition (iii), we have

$$0<b\le t_n\le t_{n+1}\le 1-(1-\tau {\alpha }_{n+1})(1-t_{n+1})<1.$$

This implies

$$\frac{{\alpha }_{n+1}(\tau -\gamma \alpha )+(1-\tau {\alpha }_{n+1})(t_{n+1}-t_n)}{1-(1-\tau {\alpha }_{n+1})(1-t_{n+1})}\ge {\alpha }_{n+1}(\tau -\gamma \alpha ).$$

Thus,

$$\parallel x_{n+2}-x_{n+1}\parallel \le [1-{\alpha }_{n+1}(\tau -\gamma \alpha )]\parallel x_{n+1}-x_n\parallel +\frac{M}{b}|{\alpha }_{n+1}-{\alpha }_n|.$$

By the virtue of condition (ii) and Lemma 3, we show that ${lim}_{n\to \infty }\parallel x_{n+1}-x_n\parallel =0.$

Step 3. We prove that ${lim}_{n\to \infty }\parallel x_n-T_2^n{T}_1^n{x}_n\parallel =0$.

In reality, we obtain

$$\begin{array}{cc}\hfill \parallel x_n-T_2^n{T}_1^n{x}_n\parallel =& \parallel x_n-x_{n+1}+x_{n+1}-T_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})\hfill \\ & +T_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})-T_2^n{T}_1^n{x}_n\parallel \hfill \\ \hfill \le & \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-T_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})\parallel \hfill \\ & +\parallel T_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})-T_2^n{T}_1^n{x}_n\parallel \hfill \\ \hfill \le & \parallel x_n-x_{n+1}\parallel \hfill \\ & +\parallel {\alpha }_n[\gamma V\left(x_n\right)-\mu F{T}_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})]\parallel \hfill \\ & +(1-\tau {\alpha }_n)\parallel t_n{x}_n+(1-t_n)x_{n+1}-x_n\parallel \hfill \\ \hfill \le & \parallel x_n-x_{n+1}\parallel +{\alpha }_{n}M+(1-t_n)\parallel x_{n+1}-x_n\parallel \hfill \\ \hfill =& (2-t_n)\parallel x_{n+1}-x_n\parallel +{\alpha }_{n}M\hfill \\ \hfill \le & 2\parallel x_{n+1}-x_n\parallel +{\alpha }_{n}M.\hfill \end{array}$$

Combining Step 2 and condition (i), we obtain ${lim}_{n\to \infty }\parallel x_n-T_2^n{T}_1^n{x}_n\parallel =0$.

Step 4. We claim that ${lim}_{n\to \infty }sup⟨(\mu F-\gamma V)x^{*},x^{*}-x_n⟩\le 0$ where $x^{*}$ is the unique solution of the variational inequality Equation (11).

Above all, we provide a proof procedure for $P_C(\gamma V+I-\mu F)$, which is a contraction. With all $u,v\in C$, by using Lemma 4, we obtain

$$\begin{array}{cc}& \parallel P_C(\gamma V+I-\mu F)\left(u\right)-P_C(\gamma V+I-\mu F)\left(v\right)\parallel \hfill \\ \hfill \le & \parallel \gamma V\left(u\right)-\gamma V\left(v\right)\parallel +\parallel (I-\mu F)\left(u\right)-(I-\mu F)\left(v\right)\parallel \hfill \\ \hfill \le & \alpha \gamma \parallel u-v\parallel +(1-\tau )\parallel u-v\parallel \hfill \\ \hfill =& [1-(\tau -\alpha \gamma \left)\right]\parallel u-v\parallel ,\hfill \end{array}$$

which explains that $P_C(\gamma V+I-\mu F)$ is a contractive mapping. Then, because of the contraction mapping principle, we obtain there exists a unique fixed point expressed as $x^{*}\in C$, which is $x^{*}=P_C(\gamma V+I-\mu F)x^{*}$. Because $\left\{x_n\right\}$ is bounded and according to the supremum and infimum principle, we know there must exist a subsequence of $\left\{x_n\right\}$ that can obtain the least upper bound. Additionally, a bounded point column in a reflexive space must have a weakly convergent subcolumn. For convenience, we take a weakly convergent subcolumn that is equal to the one that obtains the upper bound, make that $x_{n_k}\rightharpoonup \widehat{x}\in C$ as $k\to \infty$ and

$$\underset{n\to \infty }{lim}sup⟨(\mu F-\gamma V)x^{*},x^{*}-x_n⟩=\underset{n\to \infty }{lim}⟨(\mu F-\gamma V)x^{*},x^{*}-x_{n_k}⟩.$$

Due to the fact that $\left\{{\omega }_n^i\right\}$ is a bounded set for $i=1,2$, we suppose that ${\omega }_{n_k}^i\to {\omega }_{\infty }^i$ ($k\to \infty$), where $0<{\omega }_{\infty }^i<1$. Define $T_i^{\infty }=(1-{\omega }_{\infty }^i)I+{\omega }_{\infty }^i{T}_i$ for $i=1,2$. So, we obtain that $Fix\left(T_i^{\infty }\right)=Fix\left(T_i\right)$. Notice that

$$\begin{array}{cc}\hfill \parallel T_i^{n_k}x-T_i^{\infty }x\parallel & =\parallel [(1-{\omega }_{n_k}^i)x+{\omega }_{n_k}^i{T}_{i}x]-[(1-{\omega }_{\infty }^i)x+{\omega }_{\infty }^i{T}_{i}x]\parallel \hfill \\ & \le |{\omega }_{n_k}^i-{\omega }_{\infty }^i\left|\right(\parallel x\parallel +\parallel T_{i}x\parallel )\to 0\hfill \end{array}$$

Hence, we have

$$\underset{k\to \infty }{lim}\underset{x\in E}{sup}\parallel T_i^{n_k}x-T_i^{\infty }x\parallel =0,$$

(19)

where E is an arbitrary bounded subset of H.

On account of the fact that $Fix\left(T_1^{\infty }\right)\cap Fix\left(T_2^{\infty }\right)=Fix\left(T_1\right)\cap Fix\left(T_2\right)=\mathrm{C}\ne \varnothing$, $T_i^{\infty }$ is ${\omega }_{\infty }^i$-averaged for $i=1,2$, using Lemma 1, we have that $Fix\left(T_2^{\infty }{T}_1^{\infty }\right)=Fix\left(T_2\right)\cap Fix\left(T_1\right)=\mathrm{C}$. Consider

$$\begin{array}{cc}\hfill \parallel x_{n_k}-T_2^{\infty }{T}_1^{\infty }{x}_{n_k}\parallel \le & \parallel x_{n_k}-T_2^{n_k}{T}_1^{n_k}{x}_{n_k}\parallel +\parallel T_2^{n_k}{T}_1^{n_k}{x}_{n_k}-T_2^{\infty }{T}_1^{n_k}{x}_{n_k}\parallel \hfill \\ & +\parallel T_2^{\infty }{T}_1^{n_k}{x}_{n_k}-T_2^{\infty }{T}_1^{\infty }{x}_{n_k}\parallel \hfill \\ \hfill \le & \parallel x_{n_k}-T_2^{n_k}{T}_1^{n_k}{x}_{n_k}\parallel +\parallel T_2^{n_k}{x}_{n_k}-T_2^{\infty }{x}_{n_k}\parallel \hfill \\ & +\parallel T_1^{n_k}{x}_{n_k}-T_1^{\infty }{x}_{n_k}\parallel \hfill \\ \hfill \le & \parallel x_{n_k}-T_2^{n_k}{T}_1^{n_k}{x}_{n_k}\parallel +\underset{x\in E_1}{sup}\parallel T_2^{n_k}x-T_2^{\infty }x\parallel \hfill \\ & +\underset{x\in E_2}{sup}\parallel T_1^{n_k}x-T_1^{\infty }x\parallel ,\hfill \end{array}$$

where $E_1$ and $E_2$ are bounded subsets including $\left\{T_1^{n_k}{x}_{n_k}\right\}$ and $\left\{x_{n_k}\right\}$, respectively. From step 3 and Equation (19), we obtain that ${lim}_{k\to \infty }\parallel x_{n_k}-T_2^{\infty }{T}_1^{\infty }{x}_{n_k}\parallel =0$. From Lemma 2, we obtain that $\widehat{x}\in Fix\left(T_2^{\infty }{T}_1^{\infty }\right)=Fix\left(T_2^{\infty }\right)\cap Fix\left(T_1^{\infty }\right)=Fix\left(T_2\right)\cap Fix\left(T_1\right)=\mathrm{C}$.

Then, it follows from Equation (16) that

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim}sup⟨(\mu F-\gamma V)x^{*},x^{*}-x_n⟩& =\underset{k\to \infty }{lim}⟨(\mu F-\gamma V)x^{*},x^{*}-x_{n_k}⟩\hfill \\ \hfill & =⟨(\mu F-\gamma V)x^{*},x^{*}-\widehat{x}⟩\le 0.\hfill \end{array}$$

(20)

Step 5. We show that $x_n\to x^{*}$ as $n\to \infty$; here, $x^{*}=P_C(\gamma V+I-\mu F)x^{*}$.

It follows from Equation (17) and Lemma 5 that

$$\begin{array}{cc}& \parallel x_{n+1}-x^{*}{\parallel }^2\hfill \\ \hfill =& \parallel {\alpha }_n\gamma V\left(x_n\right)+(I-\mu {\alpha }_{n}F)T_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})-x^{*}{\parallel }^2\hfill \\ \hfill =& \parallel (I-\mu {\alpha }_{n}F)T_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})-(I-\mu {\alpha }_{n}F)T_2^n{T}_1^n{x}^{*}+{\alpha }_n[\gamma V\left(x_n\right)-\mu F\left(x^{*}\right)]{\parallel }^2\hfill \\ \hfill \le & \parallel (I-\mu {\alpha }_{n}F)T_2^n{T}_1^n(t_n{x}_n+(1-t_n)x_{n+1})-(I-\mu {\alpha }_{n}F)T_2^n{T}_1^n{x}^{*}{\parallel }^2\hfill \\ & +2{\alpha }_n⟨\gamma V\left(x_n\right)-\mu F\left(x^{*}\right),x_{n+1}-x^{*}⟩\hfill \\ \hfill \le & {(1-\tau {\alpha }_n)}^2{\parallel t_n{x}_n+(1-t_n)x_{n+1}-x^{*}\parallel }^2+2\gamma {\alpha }_n⟨V\left(x_n\right)-V\left(x^{*}\right),x_{n+1}-x^{*}⟩\hfill \\ & +2{\alpha }_n⟨\gamma V\left(x^{*}\right)-\mu F\left(x^{*}\right),x_{n+1}-x^{*}⟩\hfill \\ \hfill \le & {(1-\tau {\alpha }_n)}^2{t}_n{}^2\parallel x_n-x^{*}{\parallel }^2+{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2{\parallel x_{n+1}-x^{*}\parallel }^2\hfill \\ & +2t_n(1-t_n){(1-\tau {\alpha }_n)}^2\parallel (x_n-x^{*})\parallel \parallel (x_{n+1}-x^{*})\parallel \hfill \\ & +2\gamma \alpha {\alpha }_n\parallel x_n-x^{*}\parallel \parallel x_{n+1}-x^{*}\parallel +2{\alpha }_n⟨\gamma V\left(x^{*}\right)-\mu F\left(x^{*}\right),x_{n+1}-x^{*}⟩\hfill \\ \hfill =& {(1-\tau {\alpha }_n)}^2{t}_n{}^2\parallel x_n-x^{*}{\parallel }^2+{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2{\parallel x_{n+1}-x^{*}\parallel }^2\hfill \\ & +2(t_n(1-t_n){(1-\tau {\alpha }_n)}^2+\gamma \alpha {\alpha }_n)\parallel (x_n-x^{*})\parallel \parallel (x_{n+1}-x^{*})\parallel \hfill \\ & +2{\alpha }_n⟨\gamma V\left(x^{*}\right)-\mu F\left(x^{*}\right),x_{n+1}-x^{*}⟩\hfill \\ \hfill \le & {(1-\tau {\alpha }_n)}^2{t}_n{}^2\parallel x_n-x^{*}{\parallel }^2+{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2{\parallel x_{n+1}-x^{*}\parallel }^2\hfill \\ & +(t_n(1-t_n){(1-\tau {\alpha }_n)}^2+\gamma \alpha {\alpha }_n)[\parallel (x_n-x^{*}){\parallel }^2+\parallel x_{n+1}-x^{*}{\parallel }^2]\hfill \\ & +2{\alpha }_n⟨\gamma V\left(x^{*}\right)-\mu F\left(x^{*}\right),x_{n+1}-x^{*}⟩\hfill \\ \hfill \le & [{(1-\tau {\alpha }_n)}^2{t}_n^2+t_n(1-t_n){(1-\tau {\alpha }_n)}^2+\gamma \alpha {\alpha }_n]{\parallel (x_n-x^{*})\parallel }^2\hfill \\ & +[{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2+t_n(1-t_n){(1-\tau {\alpha }_n)}^2+\gamma \alpha {\alpha }_n]{\parallel x_{n+1}-x^{*}\parallel }^2+L_n,\hfill \end{array}$$

where

$$L_n:=2{\alpha }_n⟨\gamma V\left(x^{*}\right)-\mu F\left(x^{*}\right),x_{n+1}-x^{*}⟩.$$

It follows that

$$\begin{array}{l}[1-{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2-t_n(1-t_n){(1-\tau {\alpha }_n)}^2-\gamma \alpha {\alpha }_n]\parallel x_{n+1}-x^{*}{\parallel }^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le [{(1-\tau {\alpha }_n)}^2{t}_n^2+t_n(1-t_n){(1-\tau {\alpha }_n)}^2+\gamma \alpha {\alpha }_n]{\parallel x_n-x^{*}\parallel }^2+L_n.\end{array}$$

This indicates that

$$\begin{array}{cc}\hfill & \parallel x_{n+1}-x^{*}{\parallel }^2\hfill \\ \hfill \le & \frac{{(1-\tau {\alpha }_n)}^2{t}_n^2+t_n(1-t_n){(1-\tau {\alpha }_n)}^2+\gamma \alpha {\alpha }_n}{1-{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2-t_n(1-t_n){(1-\tau {\alpha }_n)}^2-\gamma \alpha {\alpha }_n}{\parallel x_n-x^{*}\parallel }^2\hfill \\ \hfill & +\frac{L_n}{1-{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2-t_n(1-t_n){(1-\tau {\alpha }_n)}^2-\gamma \alpha {\alpha }_n}.\hfill \end{array}$$

(21)

Let

$$\begin{array}{cc}\hfill {\psi }_n:& =\frac{1}{{\alpha }_n}\{1-\frac{{(1-\tau {\alpha }_n)}^2{t}_n{}^2+t_n(1-t_n){(1-\tau {\alpha }_n)}^2+\gamma \alpha {\alpha }_n}{1-{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2-t_n(1-t_n){(1-\tau {\alpha }_n)}^2-\gamma \alpha {\alpha }_n}\}\hfill \\ & =\frac{1}{{\alpha }_n}\{\frac{1-{(1-\tau {\alpha }_n)}^2-2\gamma \alpha {\alpha }_n}{1-{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2-t_n(1-t_n){(1-\tau {\alpha }_n)}^2-\gamma \alpha {\alpha }_n}\}\hfill \\ & =\frac{1}{{\alpha }_n}\{\frac{2\tau {\alpha }_n-{\tau }^2{\alpha }_n{}^2-2\gamma \alpha {\alpha }_n}{1-{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2-t_n(1-t_n){(1-\tau {\alpha }_n)}^2-\gamma \alpha {\alpha }_n}\}\hfill \\ & =\frac{2(\tau -\gamma \alpha )-{\tau }^2{\alpha }_n}{1-{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2-t_n(1-t_n){(1-\tau {\alpha }_n)}^2-\gamma \alpha {\alpha }_n}.\hfill \end{array}$$

Because $0<\epsilon <\tau -\gamma \alpha$, $\left\{t_n\right\}$ satisfies $0<b<t_n\le t_{n+1}<1$ for $\forall n>0$. If ${lim}_{n\to \infty }{t}_n$ exists, we suppose that ${lim}_{n\to \infty }{t}_n=t^{*}>0.$

Then,

$$\underset{n\to \infty }{lim}{\psi }_n=\frac{2(\tau -\gamma \alpha )}{2t^{*}-{\left(s^{*}\right)}^2}>0.$$

Let $\rho$ satisfy $0<\rho <\frac{2(\tau -\gamma \alpha )}{2t^{*}-{\left(t^{*}\right)}^2}$; then, there exists a large integer M such that ${\psi }_n>\rho$ for all $n\ge M$. Therefore, we can obtain

$$\frac{{(1-\tau {\alpha }_n)}^2{t}_n{}^2+2(t_n(1-t_n){(1-\tau {\alpha }_n)}^2+\gamma \alpha {\alpha }_n)}{1-{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2-t_n(1-t_n){(1-\tau {\alpha }_n)}^2-\gamma \alpha {\alpha }_n}<1-\rho {\alpha }_n,\forall n\ge M.$$

From Equation (21), we can obtain that

$$\begin{array}{cc}\hfill & \parallel x_{n+1}-x^{*}{\parallel }^2\hfill \\ \hfill \le & (1-\rho {\alpha }_n){\parallel x_n-x^{*}\parallel }^2+\frac{L_n}{1-{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2-t_n(1-t_n){(1-\tau {\alpha }_n)}^2\gamma \alpha {\alpha }_n}\hfill \end{array}$$

(22)

By Equation (20), we have

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}sup\frac{L_n}{\rho {\alpha }_n[1-{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2-t_n(1-t_n){(1-\tau {\alpha }_n)}^2-\gamma \alpha {\alpha }_n]}\hfill \\ \hfill =& \underset{n\to \infty }{lim}sup\frac{2⟨\gamma V\left(x^{*}\right)-\mu F\left(x^{*}\right),x_{n+1}-x^{*}⟩}{\rho [1-{(1-\tau {\alpha }_n)}^2{(1-t_n)}^2-t_n(1-t_n){(1-\tau {\alpha }_n)}^2-\gamma \alpha {\alpha }_n]}\hfill \\ \hfill \le & 0.\hfill \end{array}$$

(23)

From Equation (22), Equation (23), and Lemma 3, we can gain

$$\underset{n\to \infty }{lim}\parallel x_n-x^{*}\parallel =0.$$

The proof is completed. □

The following theorems can be easily gained from Theorem 1.

Theorem 2.

Let C be a nonempty closed and convex subset of the real Hilbert space H, ${\left\{T_i\right\}}_{i=1}^N:C\to C$ be N nonexpansive mapping of H such that $C={\bigcap }_{i=1}^{N}Fix\left(T_i\right)\ne \varphi$, $V:C\to C$ be an α-Lipschitzian operator on H with coefficient $\alpha \in (0,1)$, $F:C\to H$ be a ζ-Lipschitzian continuous and κ-strongly monotone operator on H with constants $\zeta >0$ and $\kappa >0$. Let a sequence $\left\{x_n\right\}$ be generated by:

$$x_{n+1}={\alpha }_n\gamma V\left(x_n\right)+(I-\mu {\alpha }_{n}F)T_N^n{T}_{N-1}^n\cdots T_1^n(\frac{x_n+x_{n+1}}{2}),n\ge 0,$$

(24)

where $T_i^n=(1-{\omega }_n^i)I+{\omega }_n^i{T}_i$ for $i=1,2,\dots ,N$ and $\left\{{\alpha }_n\right\},\left\{s_n\right\},\gamma ,\mu ,\tau ,{\omega }_n^i\in (0,1]$ satisfy the same conditions as Theorem 1.

Then, the sequence $\left\{x_n\right\}$ converges strongly to $x^{*}$, which settles the following variational inequality as well:

$$⟨(\mu F-\gamma V)x^{*},x-x^{*}⟩\ge 0,\forall x\in {\cap }_{i=1}^{N}Fix\left(T_i\right).$$

Theorem 3.

Let C be a nonempty closed and convex subset of the real Hilbert space H, ${\left\{T_i\right\}}_{i=1}^N:C\to C$ be N non-expansive mapping of H such that $C={\bigcap }_{i=1}^{N}Fix\left(T_i\right)\ne \varphi$, and $V:C\to C$ be an α-Lipschitzian operator on H with coefficient $\alpha \in (0,1)$. Let A be a strongly positive bounded linear operator with a constant $\kappa >0$ such that $\alpha <\tau$ and $0<\mu <\frac{2\kappa }{{\parallel A\parallel }^2}$, where $\tau =1-\sqrt{1-\mu (2\kappa -\mu \parallel A{\parallel }^2)}$. Let a sequence $\left\{x_n\right\}$ be generated by:

$$x_{n+1}={\alpha }_n\gamma V\left(x_n\right)+\left(I-\mu {\alpha }_{n}A\right)T_N^n{T}_{N-1}^n\cdots T_1^n(s_n{x}_n+(1-s_n)x_{n+1}),n\ge 0,$$

(25)

where $T_i^n=(1-{\omega }_n^i)I+{\omega }_n^i{T}_i$ for $i=1,2,\dots ,N$ and $\left\{{\alpha }_n\right\},\left\{s_n\right\},\gamma ,\mu ,\tau ,{\omega }_n^i\in (0,1]$ satisfy the same conditions as in Theorem 1.

Then, the sequence $\left\{x_n\right\}$ converges strongly to $x^{*}$, which settles the following variational inequality as well:

$$⟨(\mu F-\gamma V)x^{*},x-x^{*}⟩\ge 0,\forall x\in {\cap }_{i=1}^{N}Fix\left(T_i\right).$$

4. Application

In this section, the iterative algorithm Equation (17) is effectively applied to settle some important problems.

4.1. Strict Pseudo-Contractive Mappings

A mapping $G:C\to H$ is named to be a $\xi$-strictly pseudo-contraction if there exists a constant $\xi \in [0,1)$ such that

$${\parallel Gx-Gy\parallel }^2\le {\parallel x-y\parallel }^2+\xi {\parallel (I-G)x-(I-G)y\parallel }^2,\forall x,y\in H.$$

(26)

Lemma 6

([26]). Let $G:C\to H$ be a ξ-strictly pseudo-contractive mapping, $S:C\to H$ by $Sx=\delta x+(1-\delta )Gx$ for $\forall x\in C$. Then, as $\delta \in [\xi ,1)$, S is a nonexpansive mapping such that $Fix\left(S\right)=Fix\left(G\right)$.

Theorem 4.

Let H be a real Hilbert space, ${\left\{G_i\right\}}_{i=1}^N$ be N ${\xi }_i$-strictly pseudo-contraction mappings of H, $C={\bigcap }_{i=1}^{N}Fix\left(G_i\right)\ne \varnothing$. Let $V:C\to C$ be an α-Lipschitzian operator with coefficient $\alpha \in (0,1)$ and $F:C\to H$ be a ζ-Lipschitzian continuous and κ-strongly monotone operator with constants $\zeta >0$ and $\kappa >0$ such that $0<\mu <\frac{2\kappa }{{\zeta }^2},0<\gamma <\frac{\tau }{\alpha },\tau =1-\sqrt{1-\mu (2\kappa -\mu {\zeta }^2)}\in (0,1]$. For an arbitrarily given $x_0\in C$, $\left\{x_n\right\}$ is defined as follows:

$$\left\{\begin{array}{c}\widehat{S_i}x={\delta }_{i}x+(1-{\delta }_i)G_{i}x,i=1,\dots N\hfill \\ x_{n+1}={\alpha }_n\gamma V\left(x_n\right)+(I-\mu {\alpha }_{n}F)\widehat{S_N}\widehat{S_{N-1}}\cdots \widehat{S_1}(t_n{x}_n+(1-t_n)x_{n+1}),n\ge 0\hfill \end{array}\right.$$

(27)

where ${\delta }_i\in [\lambda ,1)$,$\left\{{\alpha }_n\right\}$ and $\left\{t_n\right\}$ both belong to $(0,1]$, satisfying the next conditions:

(i) 

${lim}_{n\to \infty }{\alpha }_n=0;$

(ii) 

${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$ and ${\sum }_{n=0}^{\infty }|{\alpha }_{n+1}-{\alpha }_n|<\infty$;

(iii) 

$0<b\le t_n\le t_{n+1}<1,\forall n\ge 0.$

Then, the sequence $\left\{x_n\right\}$ converges strongly to the common fixed points set $x^{*}$ of a finite family of nonexpansive mappings, which also settles the following type of variational inequality:

$$⟨(\mu F-\gamma V)x^{*},x-x^{*}⟩\ge 0,\forall x\in {\cap }_{i=1}^{N}Fix\left(T_i\right).$$

Proof. 

Let ${\left\{G_i\right\}}_{i=1}^N$ be a family of ${\xi }_i$-strictly pseudo-contractions of H. We define $\widehat{S_i}:C\to H$ by $\widehat{S_i}x={\delta }_{i}x+(1-{\delta }_i)G_{i}x$ for $\forall x\in C,0\le {\xi }_i\le {\delta }_i<1$, and $i=1,\dots ,N$. By virtue of Lemma 6, we can clearly obtain that ${\{\widehat{S_i}\}}_{i=1}^N$ is a family of nonexpansive mappings and $Fix(\widehat{S_i})=Fix\left(G_i\right)$. Therefore, the desired results can be easily obtained using Theorem 1. □

4.2. Generalized Equilibrium Problems

Let ${\psi }_i$ be a bifunction from $C_i\times C_i$ into $\mathbb{R}$, where ${\left\{C_i\right\}}_{i=1}^N$ are nonempty closed convex subsets of H for $i=1,\dots ,N$, and $\mathbb{R}$ is the set of real numbers. In [27], Mihai and Ashish considered the next generalized equilibrium problem to find $(\widehat{x_1},\widehat{x_2},\dots ,\widehat{x_N})$ satisfying

$$\left\{\begin{array}{c}{\psi }_1(\widehat{x_1},x_1)+⟨D_1\widehat{x_2},x_1-\widehat{x_1}⟩+\frac{1}{{\delta }_1}⟨\widehat{x_1}-\widehat{x_2},x_1-\widehat{x_1}⟩\ge 0,\forall x_1\in C_1,\hfill \\ {\psi }_2(\widehat{x_2},x_2)+⟨D_2\widehat{x_3},x_2-\widehat{x_2}⟩+\frac{1}{{\delta }_2}⟨\widehat{x_2}-\widehat{x_3},x_2-\widehat{x_2}⟩\ge 0,\forall x_2\in C_2,\hfill \\ \cdots \hfill \\ {\psi }_N\left(\widehat{x_N}{x}_N\right)+⟨D_N\widehat{x_1},x_N-\widehat{x_N}⟩+\frac{1}{{\delta }_N}⟨\widehat{x_N}-\widehat{x_1},x_N-\widehat{x_N}⟩\ge 0,\forall x_N\in C_N,\hfill \end{array}\right.$$

(28)

where $D_i:H\to H$ is a nonlinear mapping and ${\delta }_i>0$ for $i=1,2,\dots N$. Here, $EP\left(\psi \right)$ is denoted as its set of solutions, as it is well known that generalized equilibrium problems contain a number of variational inequality problems, optimization problems, and fixed-point problems.

To solve the problem generated by Equation (28), we assume the following assumptions are satisfied by the bifunction $\psi$:

(A)

$\psi (u,u)=0$ for $\forall u\in C$;

(B)

$\psi$ is monotone, i.e., $\psi (u,v)+\psi (v,u)\le 0$ for $\forall u,v\in C$;

(C)

$\psi$ is upper-hemi continuous, i.e., for each $u,v,w\in C$

$$\underset{t\to 0^{+}}{lim}sup\psi (tu+(1-t)v,w)\le \psi (v,w),$$

(D)

$\psi (u,•)$ is convex and weakly lower semicontinuous for any $u\in C$.

Lemma 7

([28]). Let $g:H\to H$ be a α-ism operator on H. Then, $I-2\alpha g$ is nonexpansive.

Lemma 8

([29]). Let $\psi :C\times C\to \mathbb{R}$ be a bifunction meeting the conditions (A)–(D). Then, for $\delta >0$ and $u\in H$, there exists $z\in C$ such that

$$\psi (z,v)+\frac{1}{\delta }⟨v-z,z-u⟩\ge 0,\forall v\in C.$$

Lemma 9

([29]). Let $\psi :C\times C\to \mathbb{R}$ satisfy (A)–(D). Define a mapping $T_{\delta }^{\psi }:H\to C$ for $\delta >0$ and $u\in H$ as follows:

$$T_{\delta }^{\psi }\left(u\right)=\{z\in C:\psi (z,v)+\frac{1}{\delta }⟨v-z,z-u⟩\ge 0,\forall v\in C\}.$$

For all $u\in H$, the next conditions hold:

(a) 

$T_{\delta }^{\psi }$ is single valued;

(b) 

$T_{\delta }^{\psi }$ is firmly non-expansive, i.e., $\parallel T_{\delta }^{\psi }u-T_{\delta }^{\psi }{v\parallel }^2\le ⟨T_{\delta }^{\psi }u-T_{\delta }^{\psi }v,u-v⟩$.

This implies that $\parallel T_{\delta }^{\psi }u-T_{\delta }^{\psi }v\parallel \le \parallel u-v\parallel$, namely, $T_{\delta }^{\psi }$ is a nonexpansive mapping;

(c) 

$Fix(T_{\delta }^{\psi })=EP\left(\psi \right);$

(d) 

$EP\left(\psi \right)$ is a closed and convex set.

Lemma 10

([27]). Let ${\left\{C_i\right\}}_{i=1}^N$ be nonempty closed convex subsets of Hilbert space H, ${\psi }_i:C_i\times C_i\to \mathbb{R}$ be a bifunction meeting the conditions (A)–(D) for $i=1,2,\dots ,N$ and $D_i:C\to C$ be a nonlinear mapping.

Then, for $\widehat{x_i}\in C_i,i=1,2,\cdots ,N$, $(\widehat{x_1},\widehat{x_2},\cdots ,\widehat{x_N})\in C_1\times C_2\times \cdots \times C_N$ is a solution of Equation (28) if and only if $\widehat{x_i}$ is a fixed point of the following mapping:

$$T=T_{{\delta }_1}^{{\psi }_1}(I-{\delta }_1{D}_1)T_{{\delta }_2}^{{\psi }_2}(I-{\delta }_2{D}_2)\cdots T_{{\delta }_N}^{{\psi }_N}(I-{\delta }_N{D}_N).$$

Theorem 5.

Let ${\left\{C_i\right\}}_{i=1}^N$ be nonempty closed convex subsets of H, ${\psi }_i:C_i\times C_i\to \mathbb{R}$ be a bifunction meeting the conditions (A)–(D) for $i=1,2,\cdots ,N$ and $D_i:C\to C$ be a ${\sigma }_i$-ism self-mapping. Let $V:C\to C$ be an α-Lipschitzian operator with coefficient $\alpha \in (0,1)$ and let $F:C\to H$ be a ζ-Lipschitzian continuous and κ-strongly monotone operator with constants $\zeta >0$ and $\kappa >0.$ Assume that $\Omega =Fix\left(T\right)\ne \varnothing ,$ where T is given in Lemma 10. Let a sequence $\left\{x_n\right\}$ be generated by:

$$\left\{\begin{array}{c}\widehat{T_i}=T_{{\delta }_i}^{{\psi }_i}(I-{\delta }_i{D}_i),i=1,\cdots ,N\hfill \\ x_{n+1}={\alpha }_n\gamma V\left(x_n\right)+(I-\mu {\alpha }_{n}F)\widehat{T_N}\cdots \widehat{T_1}(t_n{x}_n+(1-t_n)x_{n+1}),n\ge 0\hfill \end{array}\right.$$

(29)

where $\left\{{\alpha }_n\right\}$ and $\left\{s_n\right\}$ both belong to $[0,1),0<\mu <\frac{2\kappa }{{\zeta }^2},\tau =1-\sqrt{1-\mu (2\kappa -\mu {\zeta }^2)}\in (0,1],0<\gamma <\frac{\tau }{\alpha }$, and ${\delta }_i\in (0,2{\sigma }_i)$. The following conditions are satisfied:

(i) 

${lim}_{n\to \infty }{\alpha }_n=0;$

(ii) 

${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$ and ${\sum }_{n=0}^{\infty }|{\alpha }_{n+1}-{\alpha }_n|<\infty$;

(iii) 

$0<b\le t_n\le t_{n+1}<1,\forall n\ge 0.$

Then, the sequence $\left\{x_n\right\}$ converges strongly to the common fixed points of Ω.

Proof .

We need to prove that $T=T_{{\delta }_1}^{{\psi }_1}(I-{\delta }_1{D}_1)T_{{\delta }_2}^{{\psi }_2}(I-{\delta }_2{D}_2)\cdots T_{{\delta }_N}^{{\psi }_N}(I-{\delta }_N{D}_N)$ is an averaged mapping. Observe that $I-{\delta }_i{D}_i$ can be written as $I-{\delta }_i{D}_i=(1-\frac{{\delta }_i}{2{\sigma }_i})I+\frac{{\delta }_i}{2{\sigma }_i}(I-2{\sigma }_i{D}_i)$, where $\frac{{\delta }_i}{2{\sigma }_i}\in (0,1)$. From Lemma 7, we obtain that $I-2{\sigma }_i{D}_i$ is nonexpansive. Therefore, by Definition 2, $I-{\delta }_i{D}_i$ is averaged with ${\delta }_i\in (0,2{\sigma }_i)$ for $i=1,2,\cdots ,N$. Next, applying Lemma 9, we can obtain that $T_{{\delta }_i}^{{\psi }_i}$ is firmly nonexpansive, that is to say, $T_{{\delta }_i}^{{\psi }_i}$ is $\frac{1}{2}$-averaged for $i=1,2,\cdots ,N$. Then, Lemma 1 means that T is averaged on H. As a consequence, T can be expressed in the form of the identity mapping nonexpansive mapping, for example, $T=(1-\lambda )I+T^{{}^{\prime }}$ for some $\lambda \in (0,1)$. Here, $T^{{}^{\prime }}$ is a nonexpansive mapping and $Fix\left(T^{{}^{\prime }}\right)=Fix\left(T\right)$. Therefore, we can easily obtain the expected results by employing Theorem 1. □

5. Numerical Example

In this section, the first numerical example is provided to indicate the convergence of the proposed sequence. Then, the other numerical example is presented to compare the convergence rate with a number of implicit iterative sequences.

Example 1.

We establish the inner product $⟨·,·⟩:{\mathbb{R}}^3\times {\mathbb{R}}^3\to \mathbb{R}$ by

$$⟨x,y⟩=x·y=x_1·y_1+x_2·y_2+x_3·y_3$$

and the usual norm $\parallel ·\parallel :{\mathbb{R}}^3\to \mathbb{R}$ is expressed as

$$〚x〛={(x_1^2+x_2^2+x_3^2)}^{1/2},x=(x_1,x_2,x_3)\in {\mathbb{R}}^3.$$

Let ${\alpha }_n=\frac{1}{2n},s_n=\frac{1}{3},{\omega }_n^i=\frac{1}{2}$ and $T_{i}x=\frac{x}{i}$, $i=1,2,\cdots ,N$. Assume $V\left(x\right)=\frac{1}{4}x$, $F\left(x\right)=x$. Hence, V is $\frac{1}{4}$-Lipschitzian, F is 1-Lipschitzian and 1-strongly monotone, and $T_i^n=\frac{1}{2}I+\frac{1}{2}{T}_i.$ Let $\gamma =\frac{1}{4},\mu =1.$ For convenience of calculation, we choose the situation of $N=2$ in Theorem 1. Then, the sequence $\left\{x_n\right\}$ created by Equation (17) can be simplified as:

$$\begin{array}{cc}\hfill x_{n+1}& ={\alpha }_n\gamma V\left(x_n\right)+(I-\mu {\alpha }_{n}F)T_N^n{T}_{N-1}^n\cdots T_1^n(t_n{x}_n+(1-t_n)x_{n+1})\hfill \\ \hfill & =\frac{1}{32n}{x}_n+(1-\frac{1}{2n})(\frac{1}{4}{x}_n+\frac{1}{2}{x}_{n+1})\hfill \\ \hfill & =\frac{8n-3}{16n+8}{x}_n.\hfill \end{array}$$

(30)

Choosing $x_1$ in Equation (30), the numerical result is represented in the form of Figure 1 and Figure 2.

Figure 1. Convergence in two dimensions.

Figure 2. Convergence in three dimensions.

Remark 2.

As can be seen from the results in Figure 1 and Figure 2, the iterative sequence generated by Equation (17) is strongly convergent.

Example 2.

Let all the assumptions of Example 1 be satisfied except ${\omega }_n^i=\frac{3}{4},\mu =\frac{3}{2}.$ Hence, $T_i^n=\frac{1}{4}I+\frac{3}{4}{T}_i.$ In order to make the numerical result more obvious, let us consider the case where $N=3$.

Firstly, the sequence $\left\{x_n\right\}$ generated by Equation (17) can be simplified as

$$\begin{array}{cc}\hfill x_{n+1}& ={\alpha }_n\gamma V\left(x_n\right)+(I-\mu {\alpha }_{n}F)T_N^n{T}_{N-1}^n\cdots T_1^n(t_n{x}_n+(1-t_n)x_{n+1})\hfill \\ \hfill & =\frac{1}{32n}{x}_n+(1-\frac{3}{4n})(\frac{10}{96}{x}_n+\frac{10}{48}{x}_{n+1})\hfill \\ \hfill & =\frac{40n-18}{304n+60}{x}_n.\hfill \end{array}$$

(31)

Secondly, when $i=1,T_{1}x=Tx=x$, the sequence $\left\{x_n\right\}$ generated by Equation (15) can be simplified as

$$\begin{array}{cc}\hfill x_{n+1}& =P_C[{\alpha }_{n}f\left(x_n\right)+(I-\mu {\alpha }_{n}F)T(t_n{x}_n+(1-t_n)x_{n+1})]\hfill \\ \hfill & =\frac{1}{8n}{x}_n+(1-\frac{3}{4n})(\frac{1}{3}{x}_n+\frac{2}{3}{x}_{n+1})\hfill \\ \hfill & =\frac{8n-3}{8n+12}{x}_n.\hfill \end{array}$$

(32)

Thirdly, the sequence $\left\{x_n\right\}$ generated by Equation (13) can be simplified as

$$\begin{array}{cc}\hfill x_{n+1}& ={\alpha }_{n}f\left(x_n\right)+(1-{\alpha }_n)T(t_n{x}_n+(1-t_n)x_{n+1})\hfill \\ \hfill & =\frac{1}{8n}{x}_n+(1-\frac{1}{2n})(\frac{1}{3}{x}_n+\frac{2}{3}{x}_{n+1})\hfill \\ \hfill & =\frac{8n-1}{8n+8}{x}_n.\hfill \end{array}$$

(33)

Lastly, the sequence $\left\{x_n\right\}$ generated by Equation (12) can be simplified as

$$\begin{array}{cc}\hfill x_{n+1}& ={\alpha }_{n}f\left(x_n\right)+(1-{\alpha }_n)T(\frac{x_n+x_{n+1}}{2})\hfill \\ \hfill & =\frac{1}{8n}{x}_n+(1-\frac{1}{2n})(\frac{1}{2}{x}_n+\frac{1}{2}{x}_{n+1})\hfill \\ \hfill & =\frac{4n-1}{4n+2}{x}_n.\hfill \end{array}$$

(34)

Numerical comparison of Algorithms (12), (13), (15), and (17).

Table 1 and Figure 3 indicate that when $x_1=80$ and $n=20$, the sequences $\left\{x_n\right\}$ produced by Algorithms (12), (13), (15), and (17) all converge to 0. An effective comparison can be clearly seen.

Table 1. Convergence numerical comparison between Algorithms (12), (13), (15), and (17).

n	$\parallel {\mathit{x}}_{\mathit{n}}-0\parallel$ for (12)	$\parallel {\mathit{x}}_{\mathit{n}}-0\parallel$ for (13)	$\parallel {\mathit{x}}_{\mathit{n}}-0\parallel$ for (15)	$\parallel {\mathit{x}}_{\mathit{n}}-0\parallel$ for (17)
1	80.0000	80.0000	80.0000	80.0000
2	40.0000	35.0000	20.0000	4.8352
3	28.0000	21.8750	9.2857	0.4488
4	22.0000	15.7227	5.4167	0.0471
5	18.3333	12.1851	3.5701	0.0052
6	15.8333	9.9004	2.5402	0.0006
7	14.0064	8.3092	1.9052	0.0001
8	12.6058	7.1407	1.4849	0.0000
9	11.4935	6.2482	1.1918	0.0000
…	…	…	…	…
16	7.3614	3.2591	0.4069	0.0000
17	7.0268	3.0434	0.3633	0.0000
18	6.7257	2.8532	0.3265	0.0000
19	6.4530	2.6843	0.2951	0.0000
20	6.2048	2.5333	0.2681	0.0000

Figure 3. Convergence in three dimensions.

Remark 3.

Table 1 and Figure 3 show that the iterative Algorithm (17) enjoys a faster convergence rate than Algorithms (12), (13), and (15). Table 1 shows that the convergence rate of Algorithm (17) is not only faster, but also converges to zero in advance when compared with the iterations in Algorithms (12), (13) and (15), which all do not approach zero even until the twentieth term.

6. Conclusions

In this paper, we considered a combination of the viscosity approximation method and the hybrid steepest-descent iterative method into the implicit iterative algorithm, which has been proven to strongly converge to the unique solution of the variational inequality. As for its applications, we extended the main results to the case of treating the common fixed-point set of a finite family of strictly pseudo-compressed self-mappings as a feasible set and associated the fixed-point set of the nonexpansive mapping with the solution set of the generalized equilibrium problem, which includes a number of variational inequality problems, optimization problems, and fixed-point problems. In the numerical examples section, our Algorithm (17) required less iteration time and had a faster rate of convergence than the existing Algorithms (12), (13) and (15).