Wiener-Hopf equation technique for solving equilibrium problems and variational inequalities and fixed points of a nonexpansive mapping

Abstract

In this paper, we introduce some new iterative schemes based on the Wiener-Hopf equation technique and auxiliary principle for finding common elements of the set of solutions of equilibrium problems, the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality. Several strong convergence results for the sequences generated by these iterative schemes are established in Hilbert spaces. As the generation, we also consider two generalized variational inequalities, and obtain some iterative schemes and the proposed strong convergence theorems for solving these generalized variational inequalities, equilibrium problems, and a nonexpansive mapping. Our results and proof are new, and they extend the corresponding results of Verma (Appl. Math. Lett. 10:107-109, 1997), Wu and Li (4th International Congress on Image and Signal Processing, pp. 2802-2805, 2011), and Noor and Huang (Appl. Math. Comput. 191:504-510, 2007).

1 Introduction

Let H be a real Hilbert space, whose inner product and norm are denoted by $〈\cdot ,\cdot 〉$ and $\parallel \cdot \parallel$, respectively. Let K be a nonempty closed convex subset of H. Let $T,S:K\to K$ be nonlinear mappings. Let $M:H\to 2^H$ be a multi-valued operator and let $f:K\times K\to R$ be a bifunction, where R is the set of real numbers. Let $P_K$ be the projection of H onto the closed convex set K and $Q_K=I-P_K$, where the I is the identity operator.

In 1994, Blum and Oettli (see [1]) introduced the equilibrium problem (EP) which is to find $\overline{x}\in K$, such that

$$f(\overline{x},y)\ge 0,\mathrm{\forall }y\in K.$$

(1.1)

The set of solutions of (1.1) is denoted by $EP(f)$. Let $f(x,y)=〈Tx,y-x〉$ for all $x,y\in K$, then the EP reduces to the variational inequality problem (VIP) which is to find $\overline{x}\in K$ such that

$$〈T\overline{x},y-x〉\ge 0,\mathrm{\forall }y\in K.$$

(1.2)

This problem was introduced by Stampacchia (see [2]) in 1964. Related to EP and VIP, fixed point problems (FP) of a nonexpansive mapping are also considered by many authors. Recall that a mapping S is nonexpansive if $\parallel Sx-Sy\parallel \le \parallel x-y\parallel$ for all $x,y\in K$. Let us denote the set of fixed points of S and set of solutions of problem (1.2) by $F(S)$, $VI(K,T)$, respectively.

Recently, for solving the EP, VIP, and FP, many authors have introduced and extended lots of iterative schemes.

For solving variational inequality problems:

In 1991, Shi (see [3]) demonstrated the equivalence between the VIP: $〈Tu-f,v-u〉\ge 0$, $\mathrm{\forall }v\in K$ and Wiener-Hopf equation: $(T{P}_K+Q_K)v=f$, where $f\in H$. Noor (see [4]) established the equivalence between the generalized VIP: $〈Tu,g(v)-g(u)〉\ge 0$, $\mathrm{\forall }g(v)\in K$ and the generalized Wiener-Hopf equation: $T{g}^{-1}{P}_{K}z+{\rho }^{-1}{Q}_{K}z=0$ and introduced an iterative scheme based on this equivalence,

$$\{\begin{array}{c}g(u_n)=P_K{z}_n,\hfill \\ z_{n+1}=g(u_n)-\rho T{u}_n,\hfill \end{array}$$

(1.3)

where $g^{-1}$ exists.

Afterwards, by using different generalized Wiener-Hopf equation techniques, Verma and Al-Shemas et al. introduced several algorithms for solving generalized VIPs, respectively (see, for example, [5–7]). It has been shown that the Wiener-Hopf equation techniques are more flexible and general than projection methods.

On the other hand, for getting the unified approach to solve EP, VIP and FP, many authors also suggest and analyze lots of iterative schemes for common elements of $F(S)$, $EP(f)$, $VI(K,T)$.

For solving $EP(f)\cap F(S)$:

Takahashi and Takahashi (see [8]) introduced the following iterative schemes based on the viscosity approximation method:

$$\{\begin{array}{c}F(u_n,y)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in K,\hfill \\ x_{n+1}={\alpha }_{n}f(x_n)+(1-{\alpha }_n)S{u}_n.\hfill \end{array}$$

For solving $EP(f)\cap VI(K,T)$:

Li and Su (see [9]) introduced the following iterative schemes:

$$\{\begin{array}{c}{y}_n={\mathrm{\Pi }}_C{J}^{-1}(J{x}_n-r_{n}A{x}_n),\hfill \\ u_n\in C:f(u_n,y)+\frac{1}{r_n}〈y-u_n,J{u}_n-J{y}_n〉\ge 0,\mathrm{\forall }y\in K,\hfill \\ C_{n+1}=\{z\in C_n:\phi (z,u_n)\le \phi (z,x_n)\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0.\hfill \end{array}$$

For solving $EP(f)\cap VIP(K,T)\cap F(S)$:

Plubtieng and Punpaeng (see [10]) introduced the following iterative schemes:

$$\{\begin{array}{c}f(u_n,y)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in K,\hfill \\ y_n=P_C(u_n-{\lambda }_{n}A{u}_n),\hfill \\ x_{n+1}={\alpha }_n{x}_1+{\beta }_n{x}_n+{\gamma }_{n}S{P}_C(y_n-{\lambda }_{n}A{y}_n).\hfill \end{array}$$

For more algorithms, please see, for instance, [1–33] and the references therein.

Summarizing the above algorithms, we know that these papers have mainly used the auxiliary principle, the projection technique and iterative schemes of fixed points. However, the Wiener-Hopf equation technique which is more flexible and general than projection methods has not been used for solving $EP(f)\cap F(S)$, $EP(f)\cap VI(K,T)$, and $EP(f)\cap VI(K,T)\cap F(S)$.

Remark 1.1 It is worth mentioning that although there are some papers which have applied the Wiener-Hopf equation technique to solve VIP and $VI(K,T)\cap F(S)$, their research does not include $EP(f)$. However, our idea is just to apply the Wiener-Hopf equation technique to study the common element problem which is related to $EP(f)$.

In this paper, motivated and inspired by the above analysis and ongoing research in this field, we combine the Wiener-Hopf equation technique and auxiliary principle to introduce some iterative schemes for solving the common element problem which is related to $EP(f)$. This paper is organized as follows: In Section 2, some preliminaries are presented. Section 3 is devoted to solving the $EP(f)\cap VI(K,T)$. In Section 4, we consider a nonexpansive mapping S and obtain some iterative schemes and strong convergent results for solving $EP(f)\cap VI(K,T)\cap F(S)$. In Section 5, we extend the VIPs in Section 3 and Section 4, and we get some iterative schemes and strong convergent theorems for solving $GVI(K,T)\cap EP(f)$ and $GVI(K,T)\cap EP(f)\cap F(S)$, respectively. Our results extend the corresponding results of Verma (see [6]), Wu and Li (see [29]), and Noor and Huang (see [33]).

2 Preliminaries

In the rest of this paper, let H be a real Hilbert space, whose inner product and norm are denoted by $〈\cdot ,\cdot 〉$ and $\parallel \cdot \parallel$, respectively. Let K be a nonempty closed convex subset of H. Let $T,S:K\to K$ be nonlinear mapping. Let $M:H\to 2^H$ be a multi-valued operator and let $f:K\times K\to R$ be a bifunction, where R is the set of real numbers. Let $P_K$ be the projection of H onto the closed convex set K and $Q_K=I-P_K$, where I is the identity operator.

Definition 2.1 The operator $T:K\to K$ is said to be:

• (i) μ-Lipschitz continuous, if there exists a constant $\mu >0$ such that $\parallel Tx-Ty\parallel \le \mu \parallel x-y\parallel$ for all $x,y\in K$;

• (ii) r-strongly monotone, if there exists a constant $r>0$ such that $〈Tx-Ty,x-y〉\ge r{\parallel x-y\parallel }^2$ for all $x,y\in K$;

• (iii) γ-co-coercive, if there exists a constant $\gamma >0$ such that $〈Tx-Ty,x-y〉\ge \gamma {\parallel Tx-Ty\parallel }^2$ for all $x,y\in K$;

• (iv) relaxed γ-co-coercive, if there exists a constant $\gamma >0$ such that $〈Tx-Ty,x-y〉\ge -\gamma {\parallel Tx-Ty\parallel }^2$ for all $x,y\in K$;

• (v) relaxed $(\gamma ,r)$-co-coercive, if there exist two constants $\gamma ,r>0$ such that $〈Tx-Ty,x-y〉\ge -\gamma {\parallel Tx-Ty\parallel }^2+r{\parallel x-y\parallel }^2$ for all $x,y\in K$.

Definition 2.2 The multi-valued operator $M:H\to 2^H$ is said to be:

• (i) a relaxed monotone operator, if there exists a constant $k>0$ such that $〈w_1-w_2,u-v〉\ge -k{\parallel u-v\parallel }^2$, $\mathrm{\forall }{w}_1\in Mu$, $\mathrm{\forall }{w}_2\in Mv$;

• (ii) Lipschitz continuous if there exists a constant $\lambda >0$ such that $\parallel w_1-w_2\parallel \le \lambda \parallel u-v\parallel$, $\mathrm{\forall }{w}_1\in Mu$, $\mathrm{\forall }{w}_2\in Mv$.

Lemma 2.1 (see [10])

Let the bifunction $f:K\times K\to R$ satisfy the following conditions:

• (i) $f(x,x)=0$ for all $x\in K$;

• (ii) f is monotone, i.e. $f(x,y)+f(y,x)\le 0$ for all $x,y\in K$;

• (iii) for each $x,y,z\in K$, ${lim}_{t\to 0}f(tz+(1-t)x,y)\le f(x,y)$;

• (iv) for each $x\in K$, $f(x,\cdot )$ is convex and lower semicontinuous.

Then $EP(f)\ne \mathrm{\varnothing }$.

Lemma 2.2 (see [10])

Let $r>0$, $x\in H$, and f satisfy the conditions (i)-(iv) in Lemma  2.1. Then there exists $z\in K$ such that $f(z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0$, $\mathrm{\forall }y\in K$.

Lemma 2.3 (see [10])

Let $r>0$, $x\in H$, and f satisfy the conditions (i)-(iv) in Lemma  2.1. Define a mapping $T_r:H\to K$ as $T_r(x)=\{z\in K:f(z,y)+\frac{1}{r}\cdot 〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in K\}$.

Then the following hold:

• (a) $T_r$ is single-valued;

• (b) $T_r$ is firmly nonexpansive, i.e. $\parallel T_{r}x-T_{r}y\parallel \le 〈T_{r}x-T_{r}y,x-y〉$ for all $x,y\in H$;

• (c) $EP(f)=F(T_r)$, where $F(T_r)$ denotes the sets of fixed point of $T_r$;

• (d) $EP(f)$ is closed and convex.

In [4], Noor introduced the following generalized Wiener-Hopf equation:

$$T{g}^{-1}{P}_{K}z+{\rho }^{-1}{Q}_{K}z=0.$$

(2.1)

Here he assumed $g^{-1}$ exists, and note that if $g=I$, the identity operator, then (2.1) reduces to

$$T{P}_{K}z+{\rho }^{-1}{Q}_{K}z=0,$$

(2.2)

which was introduced by Shi (see [3]). Denote the sets of solutions of (2.1) and (2.2) by $WHE(T,g)$ and $WHE(T)$, respectively.

Lemma 2.4 (see [3])

The variational inequality (1.2) has a solution $\overline{x}\in H$ if and only if the Wiener-Hopf equation (2.2) has a solution $\overline{z}\in H$, where $\overline{x}=P_K\overline{z}$, $\overline{z}=\overline{x}-\rho T\overline{x}$.

In 1988, Noor (see [19]) introduced the generalized variational inequality (GVIP) which is to find $\overline{x}\in H$ such that $g(\overline{x})\in H$ and

$$〈T\overline{x},g(y)-g(\overline{x})〉\ge 0,\mathrm{\forall }g(y)\in K.$$

(2.3)

Denote the set of solutions of (2.3) by $GVI(K,T)$. Clearly, if $g=I$, the identity operator, the GVIP (2.3) reduces to VIP (1.2).

Lemma 2.5 (see [4])

The variational inequality (2.3) has a solution $\tilde{x}\in H$ if and only if the Wiener-Hopf equation (2.1) has a solution $\tilde{z}\in H$, where $g(\overline{x})=P_K\tilde{z}$, $\tilde{z}=g(\overline{x})-\rho T\overline{x}$, and $\rho >0$ is a constant.

For finding the common element of the set of fixed points of a nonexpansive mapping and the set of solution of the variational inequality, Noor and Huang [33] introduced the Wiener-Hopf equation which included a nonexpansive mapping:

$$TS{P}_{K}z+{\rho }^{-1}{Q}_{K}z=0.$$

(2.4)

Furthermore the equivalence was established between the Wiener-Hopf equation (2.4) and the variational inequality (1.2) as follows.

Lemma 2.6 (see [33])

The variational inequality (1.2) has a solution $\tilde{x}$ if and only if the Wiener-Hopf equation (2.4) has a solution $\tilde{z}$, where $\tilde{z}=\tilde{x}-\rho T\tilde{x}$, $\tilde{x}=S{P}_K\tilde{z}$.

Wu and Li (see [29]) introduced the Wiener-Hopf equation which includes a nonexpansive mapping S:

$$TS{P}_{K}z+w+{\rho }^{-1}{Q}_{K}z=0,\mathrm{\forall }w\in MS{P}_{K}z$$

(2.5)

and established the equivalence between the Wiener-Hopf equation (2.5) and the generalized variational inequality which is to find $u\in K$ such that

$$〈Tu+w,v-u〉\ge 0,\mathrm{\forall }v\in C,\mathrm{\forall }w\in Mu.$$

(2.6)

Next, denote the sets of solutions of (2.5) by $WHE(T,S)$.

Lemma 2.7 (see [29])

The variational inequality (2.6) has a solution $\tilde{c}\in H$ if and only if the Wiener-Hopf equation (2.5) has a solution $\tilde{z}\in H$, where $\tilde{z}=\tilde{c}-\rho (T\tilde{c}+w)$, $\tilde{c}=S{P}_K\tilde{z}$.

Lemma 2.8 (see [30])

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

$$a_{n+1}\le (1-{\lambda }_n)a_n+b_n,\mathrm{\forall }n\ge n_0,$$

where $n_0$ is some nonnegative integer, and $\{{\lambda }_n\}$ is a sequence in $[0,1]$ such that ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n=\mathrm{\infty }$, $b_n=o({\lambda }_n)$. Then ${lim}_{n\to \mathrm{\infty }}{a}_n=0$.

Lemma 2.9 For given $x,z\in K$, if

$$〈y-z,z-x〉\ge 0,$$

(2.7)

for any $y\in K$. Then $x=z$.

Proof Assume $x\ne z$. Put $y=x$, then (2.7) reduces to $0\le 〈x-z,z-x〉=-{\parallel x-z\parallel }^2<0$, which is a contradiction. □

3 Results for solving $EP(f)\cap VI(K,T)$

In this section, we firstly use Lemma 2.4 to introduce some iterative schemes and the convergence theorems for solving $EP(f)\cap VI(K,T)$.

Algorithm 3.1 For a given $z_0$, compute the approximate solution $z_{n+1}$ by the iterative schemes:

$$\{\begin{array}{c}{u}_n=P_K{z}_n,\hfill \\ f(v_n,y)+\frac{1}{r}〈y-v_n,v_n-u_n〉\ge 0,\mathrm{\forall }y\in K,\hfill \\ z_{n+1}=v_n-\rho T{v}_n.\hfill \end{array}$$

By an appropriate rearrangement, Algorithm 3.1 can be written in the following form.

Algorithm 3.2 For a given $z_0$, compute the approximate solution $z_{n+1}$ by the iterative schemes:

$$\{\begin{array}{c}f(v_n,y)+\frac{1}{r}〈y-v_n,v_n-P_K{z}_n〉\ge 0,\mathrm{\forall }y\in K,\hfill \\ z_{n+1}=v_n-\rho T{v}_n.\hfill \end{array}$$

If $f(x,y)=0$ for all $x,y\in K$, Algorithm 3.1 collapses to the following iterative method for solving variational inequalities (1.2), which is mainly due to Shi [3].

Algorithm 3.3 For a given $z_0$, compute the approximate solution $z_{n+1}$ by the iterative schemes:

$$\{\begin{array}{c}{u}_n=P_K{z}_n,\hfill \\ z_{n+1}=u_n-\rho T{u}_n.\hfill \end{array}$$

Theorem 3.1 Let K be the nonempty closed convex subset of H. The bifunction $f:K\times K\to R$ satisfies the conditions (i)-(iv) of Lemma  2.1. Let $T:K\to K$ be a α-strongly monotone and β-Lipschitz continuous operator such that $EP(f)\cap VI(K,T)\ne \mathrm{\varnothing }$. Let $\{z_n\}$, $\{u_n\}$, $\{v_n\}$ be the sequences generated by Algorithm 3.1, where $\rho >0$ is a constant and

$$0<1-2\alpha \rho +{\beta }^2{\rho }^2<1.$$

Then $\{u_n\}$, $\{v_n\}$ generated by Algorithm 3.1 converge to $s\in EP(f)\cap VI(K,T)$, and $\{z_n\}$ generated by Algorithm 3.1 converges to $\tilde{z}\in WHE(T)$.

Proof Let $\tilde{z}\in WHE(T)$ and $s\in EP(f)\cap VI(K,T)$.

Step 1. Estimate $\parallel z_{n+1}-\tilde{z}\parallel$.

From Lemma 2.4, we have

$$\begin{array}{c}\tilde{z}=s-\rho Ts,\hfill \\ s=P_K\tilde{z}.\hfill \end{array}$$

Hence

$$\parallel z_{n+1}-\tilde{z}\parallel =\parallel v_n-\rho T{v}_n-s+\rho Ts\parallel$$

(3.1)

$$=\sqrt{{\parallel v_n-s-(\rho T{v}_n-\rho Ts)\parallel }^2}$$

(3.2)

$$\le \sqrt{{\parallel v_n-s\parallel }^2-2\rho 〈T{v}_n-Ts,v_n-s〉+{\parallel T{v}_n-Ts\parallel }^2}$$

(3.3)

$$\le \sqrt{1-2\alpha \rho +{\beta }^2{\rho }^2}\parallel v_n-s\parallel .$$

(3.4)

Here, we have used the α-strong monotonicity and μ-Lipschitz continuity of T in (3.3) and (3.4).

Step 2. Estimate $\parallel v_n-s\parallel$.

Since $s\in EP(f)\cap VI(K,T)$, we have

$$f(s,y)\ge 0,\mathrm{\forall }y\in K.$$

(3.5)

Putting $y=v_n$ in (3.5) and $y=s$ in Algorithm 3.1, we have

$$f(s,v_n)\ge 0\text{and}f(v_n,s)+\frac{1}{r}〈s-v_n,v_n-u_n〉\ge 0.$$

(3.6)

From the monotonicity of f, we get

$$f(s,v_n)\ge 0\Rightarrow f(v_n,s)\le 0.$$

(3.7)

Combining (3.6) and (3.7), we obtain

$$〈s-v_n,v_n-u_n〉\ge 0.$$

It follows that

$$\begin{array}{c}〈s-v_n,v_n-s+s-u_n〉\ge 0\hfill \\ \Rightarrow 〈s-v_n,v_n-s〉+〈s-v_n,s-u_n〉\ge 0\hfill \\ \Rightarrow {\parallel s-v_n\parallel }^2\le 〈s-v_n,s-u_n〉\le \parallel s-v_n\parallel \cdot \parallel s-u_n\parallel \hfill \\ \Rightarrow \parallel s-v_n\parallel \le \parallel s-u_n\parallel \hfill \\ \Rightarrow \parallel v_n-s\parallel \le \parallel u_n-s\parallel .\hfill \end{array}$$

Step 3. Estimate $\parallel u_n-s\parallel$ and prove the strong convergence of sequences generated by Algorithm 3.1.

Due to the nonexpansivity of $P_K$, we find

$$\parallel u_n-s\parallel =\parallel P_K{z}_n-P_K\tilde{z}\parallel \le \parallel z_n-\tilde{z}\parallel .$$

By the above three steps, we have

$$\begin{array}{rcl}\parallel z_{n+1}-\tilde{z}\parallel & \le & \sqrt{(1-2\alpha \rho +{\beta }^2{\rho }^2)}\parallel v_n-s\parallel \\ \le & \sqrt{(1-2\alpha \rho +{\beta }^2{\rho }^2)}\parallel u_n-s\parallel \\ \le & \sqrt{(1-2\alpha \rho +{\beta }^2{\rho }^2)}\parallel z_n-\tilde{z}\parallel ,\end{array}$$

which implies that

$$\begin{array}{rcl}\parallel z_{n+1}-\tilde{z}\parallel & \le & \sqrt{(1-2\alpha \rho +{\beta }^2{\rho }^2)}\parallel z_n-\tilde{z}\parallel \le \cdots \\ \le & {\left(\sqrt{(1-2\alpha \rho +{\beta }^2{\rho }^2)}\right)}^{n+1}\parallel z_0-\tilde{z}\parallel .\end{array}$$

Since

$$\sqrt{(1-2\alpha \rho +{\beta }^2{\rho }^2)}<1,$$

we conclude

$$\parallel z_{n+1}-\tilde{z}\parallel \to 0(n\to \mathrm{\infty }).$$

Therefore, from

$$\parallel v_n-s\parallel \le \parallel u_n-s\parallel =\parallel P_K{z}_n-P_{K}z\parallel \le \parallel z_n-z\parallel ,$$

we have

$$\begin{array}{c}\parallel v_n-s\parallel \to 0(n\to \mathrm{\infty }),\hfill \\ \parallel u_n-s\parallel \to 0(n\to \mathrm{\infty }).\hfill \end{array}$$

 □

4 Results for solving $EP(f)\cap VI(K,T)\cap F(S)$

In this section, applying Lemma 2.6, we consider a nonexpansive mapping and analyze several iterative schemes and convergence theorems for solving $EP(f)\cap VI(K,T)\cap F(S)$.

Algorithm 4.1 For a given $z_0\in H$, compute the approximate solution $z_{n+1}$ by the iterative schemes

$$\{\begin{array}{c}{u}_n={\alpha }_n{P}_C{z}_n+(1-{\alpha }_n)S{P}_C{z}_n,\hfill \\ f(v_n,y)+\frac{1}{r}〈y-v_n,v_n-u_n〉\ge 0,\mathrm{\forall }y\in K,\hfill \\ z_{n+1}=(1-{\alpha }_n)z_n+{\alpha }_n(v_n-\rho T{v}_n).\hfill \end{array}$$

For $S=I$, the identity operator, Algorithm 4.1 collapses to the following iterative method.

Algorithm 4.2 For a given $z_0\in H$, compute the approximate solution $z_{n+1}$ by the iterative schemes

$$\{\begin{array}{c}{u}_n=P_C{z}_n,\hfill \\ f(v_n,y)+\frac{1}{r}〈y-v_n,v_n-u_n〉\ge 0,\mathrm{\forall }y\in K,\hfill \\ z_{n+1}=(1-{\alpha }_n)z_n+{\alpha }_n(v_n-\rho T{v}_n).\hfill \end{array}$$

For ${\alpha }_n=1$, $S=I$, the identity operator, Algorithm 4.1 collapses to the following iterative method.

Algorithm 4.3 For a given $z_0\in H$, compute the approximate solution $z_{n+1}$ by the iterative schemes

$$\{\begin{array}{c}{u}_n=P_C{z}_n,\hfill \\ f(v_n,y)+\frac{1}{r}〈y-v_n,v_n-u_n〉\ge 0,\mathrm{\forall }y\in K,\hfill \\ z_{n+1}=v_n-\rho T{v}_n.\hfill \end{array}$$

For $f=0$ and via Lemma 2.9, Algorithm 4.1 reduces to the following iterative method for solving $VI(K,T)\cap F(S)$.

Algorithm 4.4 For a given $z_0\in H$, compute the approximate solution $z_{n+1}$ by the iterative schemes

$$\{\begin{array}{c}{u}_n={\alpha }_n{P}_C{z}_n+(1-{\alpha }_n)S{P}_C{z}_n,\hfill \\ z_{n+1}=(1-{\alpha }_n)z_n+{\alpha }_n(u_n-\rho T{u}_n).\hfill \end{array}$$

For $f=0$, $S=I$, ${\alpha }_n=1$, and via Lemma 2.9, Algorithm 4.1 reduces to the following iterative method for solving VIP.

Algorithm 4.5 For a given $z_0\in H$, compute the approximate solution $z_{n+1}$ by the iterative schemes

$$\{\begin{array}{c}{u}_n=P_C{z}_n,\hfill \\ z_{n+1}=u_n-\rho T{u}_n.\hfill \end{array}$$

Theorem 4.1 Let K be the nonempty closed convex subset of H. The bifunction f satisfies the conditions (i)-(iv) of Lemma  2.1. Let $T:K\to K$ be relaxed $(\gamma ,r)$-co-coercive and μ-Lipschitz continuous, and let $S:K\to K$ be a k-strictly pseudocontractive mapping such that $EP(f)\cap VIP(K,T)\cap F(S)\ne \mathrm{\varnothing }$, where

$$\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty },\alpha \in [k,1),0<\rho <\frac{2(r-\gamma \mu -k)}{{(\mu +m)}^2},r>\gamma \mu +k.$$

Then the sequences $\{u_n\}$, $\{v_n\}$ generated by Algorithm 4.1 converge to $\tilde{c}\in EP(f)\cap VI(K,T)\cap F(S)$ and the sequence $\{z_n\}$ generated by Algorithm 4.1 converges to $\tilde{z}\in WHE(T)$.

Proof Let $\tilde{z}\in WHE(T)$ and $\tilde{c}\in EP(f)\cap VI(K,T)\cap F(S)$.

Step 1. Estimate $\parallel z_{n+1}-\tilde{z}\parallel$.

From Lemma 2.6, we have

$$\begin{array}{c}\tilde{z}=\tilde{c}-\rho T\tilde{c},\hfill \\ \tilde{c}=S{P}_K\tilde{z}.\hfill \end{array}$$

This implies that

$$\begin{array}{r}\tilde{c}={\alpha }_n{P}_C\tilde{z}+(1-{\alpha }_n)S{P}_C\tilde{z},\\ \tilde{z}=(1-{\alpha }_n)\tilde{z}+{\alpha }_n(\tilde{c}-\rho T\tilde{c}).\end{array}$$

Hence

$$\begin{array}{c}\parallel z_{n+1}-\tilde{z}\parallel \hfill \\ =\parallel (1-{\alpha }_n)z_n+{\alpha }_n(v_n-\rho T{v}_n)-\tilde{z}\parallel \hfill \end{array}$$

(4.1)

$$=\parallel (1-{\alpha }_n)z_n+{\alpha }_n(v_n-\rho T{v}_n)-(1-{\alpha }_n)\tilde{z}-{\alpha }_n(\tilde{c}-\rho T\tilde{c})\parallel$$

(4.2)

$$=\parallel (1-{\alpha }_n)z_n-(1-{\alpha }_n)\tilde{z}+{\alpha }_n(v_n-\rho T{v}_n)-{\alpha }_n(\tilde{c}-\rho T\tilde{c})\parallel$$

(4.3)

$$\le (1-{\alpha }_n)\parallel z_n-\tilde{z}\parallel +{\alpha }_n\parallel v_n-\tilde{c}-\rho (T{v}_n-T\tilde{c})\parallel$$

(4.4)

$$=(1-{\alpha }_n)\parallel z_n-\tilde{z}\parallel +{\alpha }_n\sqrt{{\parallel v_n-\tilde{c}-\rho (T{v}_n-T\tilde{c})\parallel }^2}$$

(4.5)

$$=(1-{\alpha }_n)\parallel z_n-\tilde{z}\parallel +{\alpha }_n\sqrt{{\parallel v_n-\tilde{c}\parallel }^2-2\rho 〈T{v}_n-\rho T\tilde{c},v_n-\tilde{c}〉+{\rho }^2{\parallel T{v}_n-T\tilde{c}\parallel }^2}$$

(4.6)

$$\le (1-{\alpha }_n)\parallel z_n-\tilde{z}\parallel$$

(4.7)

$$+{\alpha }_n\sqrt{{\parallel v_n-\tilde{c}\parallel }^2-2\rho [-\gamma {\parallel T{v}_n-\rho T\tilde{c}\parallel }^2+r{\parallel v_n-\tilde{c}\parallel }^2]+{\rho }^2{\parallel T{v}_n-\rho T\tilde{c}\parallel }^2}$$

(4.8)

$$\le (1-{\alpha }_n)\parallel z_n-\tilde{z}\parallel$$

(4.9)

$$+{\alpha }_n\sqrt{{\parallel v_n-\tilde{c}\parallel }^2-2\rho [-\gamma {\mu }^2{\parallel v_n-\tilde{c}\parallel }^2+r{\parallel v_n-\tilde{c}\parallel }^2]+{\rho }^2{\mu }^2{\parallel v_n-\tilde{c}\parallel }^2}$$

(4.10)

$$=(1-{\alpha }_n)\parallel z_n-\tilde{z}\parallel +{\alpha }_n\sqrt{(1+2\rho \gamma {\mu }^2-2\rho r+{\rho }^2{\mu }^2)}\parallel v_n-\tilde{c}\parallel .$$

(4.11)

In (4.8), (4.10), (4.11) of the above induction, we have used the $(\gamma ,r)$-co-coercivity and μ-Lipschitz continuity of the operator T.

Step 2. Estimate $\parallel v_n-\tilde{c}\parallel$.

Since $\tilde{c}\in EP(f)\cap VI(K,T)\cap F(S)$, we get

$$f(\tilde{c},y)\ge 0,\mathrm{\forall }y\in K.$$

(4.12)

Putting $y=v_n$ in (4.12) and $y=\tilde{c}$ in Algorithm 4.1, respectively, we have

$$f(\tilde{c},v_n)\ge 0\text{and}f(v_n,\tilde{c})+\frac{1}{r}〈\tilde{c}-v_n,v_n-u_n〉\ge 0.$$

(4.13)

From the monotonicity of f, we obtain

$$f(\tilde{c},v_n)\ge 0\Rightarrow f(v_n,\tilde{c})\le 0.$$

(4.14)

Combining (4.13) and (4.14), we know

$$〈\tilde{c}-v_n,v_n-u_n〉\ge 0.$$

It follows that

$$\begin{array}{c}〈\tilde{c}-v_n,v_n-\tilde{c}+\tilde{c}-u_n〉\hfill \\ \Rightarrow 〈\tilde{c}-v_n,v_n-\tilde{c}〉+〈\tilde{c}-v_n,\tilde{c}-u_n〉\hfill \\ \Rightarrow {\parallel \tilde{c}-v_n\parallel }^2\le 〈\tilde{c}-v_n,\tilde{c}-u_n〉\le \parallel \tilde{c}-v_n\parallel \cdot \parallel \tilde{c}-u_n\parallel \hfill \\ \Rightarrow \parallel \tilde{c}-v_n\parallel \le \parallel \tilde{c}-u_n\parallel \hfill \\ \Rightarrow \parallel v_n-\tilde{c}\parallel \le \parallel u_n-\tilde{c}\parallel .\hfill \end{array}$$

Step 3. Estimate $\parallel u_n-\tilde{c}\parallel$ and prove the strong convergence of sequences generated by Algorithm 4.1.

Since

$$\begin{array}{r}{u}_n={\alpha }_n{P}_C{z}_n+(1-{\alpha }_n)S{P}_C{z}_n,\\ \tilde{c}={\alpha }_n{P}_C\tilde{z}+(1-{\alpha }_n)S{P}_C\tilde{z},\end{array}$$

we have

$$\begin{array}{rcl}\parallel u_n-\tilde{c}\parallel & =& \parallel {\alpha }_n{P}_C{z}_n+(1-{\alpha }_n)S{P}_C{z}_n-{\alpha }_n{P}_C\tilde{z}-(1-{\alpha }_n)S{P}_C\tilde{z}\parallel \\ =& \parallel {\alpha }_n{P}_C{z}_n-{\alpha }_n{P}_C\tilde{z}+(1-{\alpha }_n)S{P}_C{z}_n-(1-{\alpha }_n)S{P}_C\tilde{z}\parallel \\ \le & {\alpha }_n\parallel P_C{z}_n-P_C\tilde{z}\parallel +(1-{\alpha }_n)\parallel S{P}_C{z}_n-S{P}_C\tilde{z}\parallel \\ \le & \parallel z_n-\tilde{z}\parallel .\end{array}$$

By the above three steps, we get

$$\begin{array}{rcl}\parallel z_{n+1}-\tilde{z}\parallel & \le & (1-{\alpha }_n)\parallel z_n-\tilde{z}\parallel +{\alpha }_n\sqrt{(1+2\rho \gamma {\mu }^2-2\rho r+{\rho }^2{\mu }^2)}\parallel v_n-\tilde{c}\parallel \\ \le & (1-{\alpha }_n)\parallel z_n-\tilde{z}\parallel +{\alpha }_n\sqrt{(1+2\rho \gamma {\mu }^2-2\rho r+{\rho }^2{\mu }^2)}\parallel u_n-\tilde{c}\parallel \\ \le & (1-{\alpha }_n)\parallel z_n-\tilde{z}\parallel +{\alpha }_n\sqrt{(1+2\rho \gamma {\mu }^2-2\rho r+{\rho }^2{\mu }^2)}\parallel z_n-\tilde{c}\parallel \\ =& [1-{\alpha }_n(1-\theta )]\parallel z_n-\tilde{c}\parallel ,\end{array}$$

where $\theta =\sqrt{(1+2\rho \gamma {\mu }^2-2\rho r+{\rho }^2{\mu }^2)}$.

From Lemma 2.8, it is easy to see that

$$\parallel z_n-\tilde{z}\parallel \to 0(n\to \mathrm{\infty }),$$

and from

$$\parallel v_n-s\parallel \le \parallel u_n-s\parallel =\parallel P_K{z}_n-P_{K}z\parallel \le \parallel z_n-z\parallel$$

we have

$$\begin{array}{c}\parallel u_n-s\parallel \to 0(n\to \mathrm{\infty }),\hfill \\ \parallel v_n-s\parallel \to 0(n\to \mathrm{\infty }).\hfill \end{array}$$

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5 Generation

Ever since the classical variational inequality was introduced, it has been extended to many forms in different directions, such as the nonconvex variational inequality, the multi-valued variational inequality, the extended general quasi-variational inequalities and so on. Related to the variational inequality, the Wiener-Hopf equation has also been extended, and the equivalence between generalized variational inequalities and generalized Wiener-Hopf equations has been studied.

In this section, we consider the two generalized variational inequalities (2.3) and (2.6), respectively. Furthermore, applying Lemma 2.5, we firstly suggest and give some iterative schemes for solving the variational inequality (2.3) and the equilibrium problem; secondly, via Lemma 2.7, we also analyze and introduce several iterative schemes for solving the variational inequality (2.6), the equilibrium problem, and a nonexpansive mapping.

5.1 Results for solving variational inequality (2.3) and equilibrium problem

In this subsection, we use a similar technique to Section 3, and the proposed results in this subsection can be considered as the generation of Section 3.

Algorithm 5.1 For a given $z_0\in H$, compute the approximate solution $z_{n+1}$ by the iterative schemes

$$\{\begin{array}{c}g(u_n)=P_K{z}_n,\hfill \\ f(v_n,y)+\frac{1}{r}〈y-v_n,v_n-u_n〉\ge 0,\mathrm{\forall }y\in K,\hfill \\ z_{n+1}=g(v_n)-\rho T{v}_n.\hfill \end{array}$$

Theorem 5.1 Let K be the nonempty closed convex subset of H and the bifunction $f:K\times K\to R$ satisfy the conditions (i)-(iv) of Lemma  2.1. Let $T:K\to K$ be a α-strongly monotone and β-Lipschitz continuous single-valued operator and let $g:K\to K$ be a σ-strongly monotone and δ-Lipschitz continuous single-valued operator such that $g^{-1}$ exists and $EP(f)\cap GVI(K,T)\ne \mathrm{\varnothing }$, where

$$k=\sqrt{1-2\sigma +{\delta }^2}\ne 1,t=\sqrt{1-2\rho \alpha +{\beta }^2{\rho }^2},0<\frac{k+t}{1-k}<1,\rho >0.$$

(5.1)

Then the sequences $\{u_n\}$, $\{v_n\}$ generated by Algorithm 5.1 converge strongly to $s\in EP(f)\cap GVI(K,T)$ and $\{z_n\}$ generated by Algorithm 5.1 converges strongly to $\tilde{z}\in WHE(T,g)$.

Proof Let $s\in EP(f)\cap GVI(K,T)$, $\tilde{z}\in WHE(T,g)$.

Step 1. Estimate $\parallel z_{n+1}-\tilde{z}\parallel$.

From the Lemma 2.5, we have

$$g(s)=P_K\tilde{z},$$

(5.2)

$$\tilde{z}=g(s)-\rho Ts.$$

(5.3)

This implies that

$$\parallel z_{n+1}-\tilde{z}\parallel =\parallel g(v_n)-g(s)-\rho (T{v}_n-Ts)\parallel$$

(5.4)

$$\le \parallel v_n-s-(g(v_n)-g(s))\parallel +\parallel v_n-s-\rho (T{v}_n-Ts)\parallel$$

(5.5)

$$=\sqrt{{\parallel v_n-s\parallel }^2-2〈g(v_n)-g(s),v_n-s〉+{\parallel g(v_n)-g(s)\parallel }^2}$$

(5.6)

$$+\sqrt{{\parallel v_n-s\parallel }^2-2\rho 〈T{v}_n-Ts,v_n-s〉+{\rho }^2{\parallel T{v}_n-Ts\parallel }^2}$$

(5.7)

$$\le \sqrt{1-2\sigma +{\delta }^2}\parallel v_n-s\parallel +\sqrt{1-2\rho \alpha +{\beta }^2{\rho }^2}\parallel v_n-s\parallel$$

(5.8)

$$\le (k+t){\parallel v_n-s\parallel }^2.$$

(5.9)

Note that we use the strong monotonicity and Lipschitz continuity of T, g in (5.8), where $k=\sqrt{1-2\sigma +{\delta }^2}\ne 1$, $t=\sqrt{1-2\rho \alpha +{\beta }^2{\rho }^2}$.

Step 2. Estimate $\parallel v_n-s\parallel$.

Employing the technique that we use in the proof of Theorem 3.1, we get

$$\parallel v_n-s\parallel \le \parallel u_n-s\parallel .$$

Step 3. Estimate $\parallel u_n-s\parallel$ and prove the strong convergence of sequences generated by Algorithm 5.1.

From (5.3), we obtain

$$\begin{array}{rcl}\parallel u_n-s\parallel & \le & \parallel u_n-s-(g(u_n)-g(s))+P_K{z}_n-P_{K}z\parallel \\ \le & k\parallel u_n-s\parallel +\parallel z_n-z\parallel .\end{array}$$

It follows that

$$\parallel u_n-s\parallel \le \left(\frac{1}{1-k}\right)\parallel z_n-z\parallel (k\ne 1).$$

Combining the above three steps, we have

$$\begin{array}{rcl}\parallel z_{n+1}-z\parallel & \le & (k+t)\parallel v_n-s\parallel \le (k+t)\parallel u_n-s\parallel \\ \le & \left(\frac{k+t}{1-k}\right)\parallel z_n-z\parallel .\end{array}$$

Since

$$0<\frac{k+t}{1-k}<1,$$

we get

$$\begin{array}{c}\parallel z_n-\tilde{z}\parallel \to 0(n\to \mathrm{\infty }),\hfill \\ \parallel u_n-s\parallel \to 0(n\to \mathrm{\infty }),\hfill \\ \parallel v_n-s\parallel \to 0(n\to \mathrm{\infty }).\hfill \end{array}$$

 □

Remark 5.1 Clearly, if g is the identity mapping, Theorem 5.1 reduces to Theorem 3.1.

Remark 5.2 Via Lemma 2.9, it is easy to see that if the bifunction $f(x,y)=0$, $\mathrm{\forall }x,y\in K$, Algorithm 5.1 reduces to

$$\{\begin{array}{c}g(u_n)=P_K{z}_n,\hfill \\ z_{n+1}=g(u_n)-\rho T{u}_n.\hfill \end{array}$$

This algorithm was introduced by Noor in [4], which implies that Algorithm 5.1 in this paper extends the results in [4].

Remark 5.3 It is easy to show the condition (5.1) can be satisfied, for instance

$$\alpha =\frac{7}{4},\beta =\sqrt{2},\delta =\frac{\sqrt{2}}{10},\sigma =\frac{49}{100},\rho =\frac{1}{4}.$$

5.2 Results for solving variational inequality (2.6), an equilibrium problem, and a nonexpansive mapping

In this subsection, we use a similar technique to Section 4, and the proposed results in this subsection can be considered as the generation of Section 4.

Algorithm 5.2 For a given $z_0\in H$, compute the approximate solution $z_{n+1}$ by the iterative schemes

$$\{\begin{array}{c}{u}_n=\alpha P_C{z}_n+(1-\alpha )S{P}_C{z}_n,\hfill \\ f(v_n,y)+\frac{1}{r}〈y-v_n,v_n-u_n〉\ge 0,\mathrm{\forall }y\in K,\hfill \\ z_{n+1}=(1-{\alpha }_n)z_n+{\alpha }_n[v_n-\rho (T{v}_n+w_n)].\hfill \end{array}$$

If $S=I$, Algorithm 5.2 reduces to the following.

Algorithm 5.3 For a given $z_0\in H$, compute the approximate solution $z_{n+1}$ by the iterative schemes

$$\{\begin{array}{c}{u}_n=P_C{z}_n,\hfill \\ f(v_n,y)+\frac{1}{r}〈y-v_n,v_n-u_n〉\ge 0,\mathrm{\forall }y\in K,\hfill \\ z_{n+1}=(1-{\alpha }_n)z_n+{\alpha }_n[v_n-\rho (T{v}_n+w_n)].\hfill \end{array}$$

If $S=I$ and ${\alpha }_n=1$, Algorithm 5.2 reduces to the following.

Algorithm 5.4 For a given $z_0\in H$, compute the approximate solution $z_{n+1}$ by the iterative schemes

$$\{\begin{array}{c}{u}_n=P_C{z}_n,\hfill \\ f(v_n,y)+\frac{1}{r}〈y-v_n,v_n-u_n〉\ge 0,\mathrm{\forall }y\in K,\hfill \\ z_{n+1}=v_n-\rho (T{v}_n+w_n).\hfill \end{array}$$

If $f=0$, Algorithm 5.2 reduces to the following.

Algorithm 5.5 For a given $z_0\in H$, compute the approximate solution $z_{n+1}$ by the iterative schemes

$$\{\begin{array}{c}{u}_n=\alpha P_C{z}_n+(1-\alpha )S{P}_C{z}_n,\hfill \\ z_{n+1}=(1-{\alpha }_n)z_n+{\alpha }_n[v_n-\rho (T{v}_n+w_n)],\hfill \end{array}$$

which was introduced in [8].

If $f=0$, $S=I$, Algorithm 5.2 reduces to the following.

Algorithm 5.6 For a given $z_0\in H$, compute the approximate solution $z_{n+1}$ by the iterative schemes

$$\{\begin{array}{c}{u}_n=S{P}_C{z}_n,\hfill \\ z_{n+1}=(1-{\alpha }_n)z_n+{\alpha }_n[v_n-\rho (T{v}_n+w_n)],\hfill \end{array}$$

which was introduced in [8].

If $f=0$, $S=I$, and ${\alpha }_n=1$, Algorithm 5.2 reduces to the following.

Algorithm 5.7 For a given $z_0\in H$, compute the approximate solution $z_{n+1}$ by the iterative schemes

$$\{\begin{array}{c}{u}_n=S{P}_C{z}_n,\hfill \\ z_{n+1}=u_n-\rho (T{u}_n+w_n),\hfill \end{array}$$

which was introduced in [6].

Theorem 5.2 Let K be the nonempty closed convex subset of H and the bifunction f satisfy the conditions (i)-(iv) of Lemma  2.1. Let $T:K\to K$ be a relaxed $(\gamma ,r)$-co-coercive and μ-Lipschitz continuous operator, and $S:K\to K$ be k-strictly such that $EP(f)\cap VI(K,T)\cap F(S)\ne \mathrm{\varnothing }$. Let $M:H\to 2^H$ be a multi-valued relaxed monotone and Lipschitz continuous operator with the corresponding constant $k>0$, $m>0$, where

$$\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty },\alpha \in [k,1),0<\rho <\frac{2(r-\gamma \mu -k)}{{(\mu +m)}^2},r>\gamma \mu +k.$$

Then the sequences $\{u_n\}$, $\{v_n\}$ generated by Algorithm 5.2 converge strongly to $\tilde{c}\in EP(f)\cap VI(K,T)\cap F(S)$, and $\{z_n\}$ generated by Algorithm 5.2 converges strongly to $\tilde{z}\in WHE(T,S)$.

Proof Let $\tilde{c}\in EP(f)\cap VI(K,T)\cap F(S)$ and $\tilde{z}\in WHE(T,S)$.

Step 1. Estimate $\parallel z_{n+1}-\tilde{z}\parallel$ and $\parallel u_n-\tilde{c}\parallel$.

By the same technique in [29], we have

$$\begin{array}{c}\parallel z_{n+1}-\tilde{z}\parallel \le (1-{\alpha }_n)\parallel z_n-\tilde{z}\parallel +{\alpha }_n\theta \parallel v_n-\tilde{c}\parallel ,\hfill \\ \parallel u_n-\tilde{c}\parallel \le \parallel z_n-\tilde{z}\parallel ,\hfill \end{array}$$

where $\theta =\sqrt{1+2\rho (\gamma \mu -r+k)+{\rho }^2{(\mu +m)}^2}$.

Step 2. Estimate $\parallel v_n-\tilde{c}\parallel$.

Applying the technique in Theorem 3.2 of this paper, we get

$$\parallel v_n-\tilde{c}\parallel \le \parallel u_n-\tilde{c}\parallel .$$

Combining the above two steps, we can obtain

$$\begin{array}{rl}\parallel z_{n+1}-\tilde{z}\parallel & \le (1-{\alpha }_n)\parallel z_n-\tilde{z}\parallel +\alpha \theta \parallel u_n-\tilde{c}\parallel \\ \le (1-{\alpha }_n)\parallel z_n-\tilde{z}\parallel +\alpha \theta \parallel z_n-\tilde{c}\parallel \\ =[1-{\alpha }_n(1-\theta )]\parallel z_n-\tilde{z}\parallel .\end{array}$$

From Lemma 2.8, it follows that

$$\parallel z_n-\tilde{z}\parallel \to 0(n\to \mathrm{\infty }).$$

Note that

$$\parallel v_n-s\parallel \le \parallel u_n-s\parallel \le \parallel z_n-z\parallel .$$

We conclude that

$$\begin{array}{c}\parallel u_n-s\parallel \to 0(n\to \mathrm{\infty }),\hfill \\ \parallel v_n-s\parallel \to 0(n\to \mathrm{\infty }).\hfill \end{array}$$

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