A regularization algorithm for a common solution of generalized equilibrium problem, fixed point problem and the zero points of the sum of two operators

Abstract

For finding a common solution of generalized equilibrium problem, fixed point problem and the zero points of the sum of two operators, a regularization algorithm is established in the framework of real Hilbert spaces. And the strong convergence theorem is obtained under certain assumptions. The main results presented in this paper are useful in nonlinear analysis and optimization. Moreover, the results and corollaries extend the corresponding conclusions proposed by many authors.

1 Introduction

In this paper, assume that H is a real Hilbert space with the inner product $\langle\cdot,\cdot\rangle$ and the norm $\|\cdot\|$, and let C be a nonempty, closed and convex subset of H. Let $\mathbb{N}$ and $\mathbb{R}$ be the sets of positive integers and real numbers, respectively. In the following, we recall some mappings which will often be used in this paper.

• $f:C\rightarrow C$ is said to be k-contractive iff there exists a constant $k\in(0,1)$ such that $\|f(x)-f(y)\|\leq k\|x-y\|$ for all $x,y\in C$.

• $S:C\rightarrow C$ is said to be nonexpansive iff $\|Sx-Sy\|\leq\|x-y\|$ for all $x,y\in C$.

• $T:H\rightarrow H$ is said to be firmly nonexpansive iff $\|Tx-Ty\|^{2}\leq\langle Tx-Ty,x-y\rangle$ for all $x,y\in H$.

• $P_{C}: H\rightarrow C$ is said to be metric projection iff $\|x-P_{C}x\|\leq\|x-y\|$ for all $x\in H$ and $y\in C$.

• $A:H\rightarrow H$ is said to be monotone iff $\langle x-y, Ax-Ay\rangle\geq0$ for all $x,y\in H$.

• Given a number $\eta>0$, $A:H\rightarrow H$ is said to be η-strongly monotone iff $\langle x-y, Ax-Ay\rangle\geq\eta\|x-y\|^{2}$ for all $x,y\in H$.

• Given a number $\alpha>0$, $A:C\rightarrow H$ is said to be α-inverse-strongly monotone (α-ism) iff $\langle x-y, Ax-Ay\rangle\geq\alpha\|Ax-Ay\|^{2}$ for all $x,y\in C$.

• $Y:C\rightarrow H$ is a strict pseudo-contraction [1] if there exists $t\in\mathbb{R}$ with $0\leq t<1$ such that $\|Yx-Yy\|^{2}\leq\|x-y\|^{2}+t\|(I-Y)x-(I-Y)y\|^{2}$ for all $x,y\in C$.

First, we introduce the following generalized equilibrium problem.

Find $x\in C$ such that

$$ F(x,y)+\langle Tx, y-x\rangle\geq0, \quad \forall y\in C, $$

(1.1)

where $T: C\rightarrow H$ is a monotone mapping and $F: C\times C\rightarrow\mathbb{R}$ is a bifunction.

In this paper, we use $\operatorname{GEP}(F,T)$ to denote the set of such $x\in C$, i.e., $\operatorname{GEP}(F,T)=\{x\in C: F(x,y)+\langle Tx,y-x\rangle\geq0, \forall y\in C\}$.

In the case of $T\equiv0$, problem (1.1) is reduced to the following equilibrium problem [2].

Find $x\in C$ such that

$$ F(x,y)\geq0, \quad \forall y\in C. $$

(1.2)

In this paper, we use $\operatorname{EP}(F)$ to denote the set of such $x\in C$, i.e., $\operatorname{EP}(F)=\{x\in C: F(x,y)\geq0, \forall y\in C\}$.

In the case of $F\equiv0$, problem (1.1) is reduced to the classical variational inequality.

To study equilibrium problems (1.1) and (1.2), we may assume that F satisfies the following conditions:

• (A1) $F(x,x)$=0 for all $x\in C$;

• (A2) F is monotone, i.e., $F(x,y)+F(y,x)\leq0$ for all $x,y\in C$;

• (A3) for each $x, y, z\in C$,

  $$ \lim_{t\downarrow0}\sup F\bigl(tz+(1-t)x,y\bigr)\leq F(x,y); $$

• (A4) for each $x\in C$, $y\mapsto F(x,y)$ is convex and weakly lower semicontinuous.

Many problems can be transformed into finding solutions of equilibrium problems (1.1) and (1.2), for instance, image recovery, network allocation, inverse problems, transportation problems and optimization problems; see [2–7] and the references therein. Recently, many regularization methods have been extensively studied for solving solutions of equilibrium problems (1.1) and (1.2); see [7–13] and the references therein.

Second, we introduce the following fixed point problem for a family of infinitely nonexpansive mappings. Consider the fixed point problem

$$ \operatorname{Fix}(S):=\{x\in C: x=Sx\}, $$

where $S:C\rightarrow C$ is a mapping, we use $\operatorname{Fix}(S)$ to denote the fixed point set of S. If C is a bounded, closed and convex subset of H, then $\operatorname{Fix}(S)$ is not empty; see [1].

Let $\{S_{i}: C\rightarrow C\}$ be a family of infinitely nonexpansive mappings and $\{\gamma_{i}\}$ be a nonnegative real sequence with $0\leq\gamma_{i}<1$, $\forall i\geq1$. For $n\geq1$, define a mapping $W_{n}:C\rightarrow C$ as follows:

$$\begin{aligned}& U_{n,n+1}=I , \\& U_{n,n}=\gamma_{n}S_{n}U_{n,n+1}+(1- \gamma_{n})I , \\& U_{n,n-1}=\gamma_{n-1}S_{n-1}U_{n,n}+(1- \gamma_{n-1})I , \\& \ldots, \\& U_{n,k}=\gamma_{k}S_{k}U_{n,k+1}+(1- \gamma_{k})I , \\& U_{n,k-1}=\gamma_{k-1}S_{k-1}U_{n,k}+(1- \gamma_{k-1})I , \\& \ldots, \\& U_{n,2}=\gamma_{2}S_{2}U_{n,3}+(1- \gamma_{2})I , \\& W_{n}=U_{n,1}=\gamma_{1}S_{1}U_{n,2}+(1- \gamma_{1})I . \end{aligned}$$

Such a mapping $W_{n}$ is nonexpansive from C to C and it is called a W-mapping generated by $S_{n},S_{n-1},\ldots,S_{1}$ and $\gamma_{n},\gamma_{n-1},\ldots,\gamma_{1}$; see [14] and the references therein.

Third, we introduce the problem of zero points of a maximal monotone mapping

$$ M^{-1}0=\{x\in H: 0\in Mx\}, $$

(1.3)

where M is a mapping of H into $2^{H}$, the effective domain of M is denoted by domM or $D(M)$, that is, $\operatorname{dom}M=\{x\in H: Mx\neq\emptyset\}$. A multi-valued mapping M is said to be a monotone mapping on H if $\langle x-y, u-v\rangle\geq0$ for all $x,y\in \operatorname{dom}M$, $u\in Mx$, $v\in My$. A monotone mapping M on H is said to be maximal if its graph is not properly contained in the graph of any other monotone mapping on H. It is well known that a monotone mapping M is maximal if and only if, for any $(x,u)\in H\times H$, $\langle x-y,u-v\rangle\geq0$ for all $(y,v)\in \operatorname{Graph}(M)$ implies $u\in Mx$.

For a maximal monotone mapping M on H and $r>0$, we may define a single-valued mapping $J_{r}=(I+rM)^{-1}: H\rightarrow \operatorname{dom}M$, which is called the resolvent of M for r. It is easy to see that $M^{-1}0=\operatorname{Fix}(J_{r})$ for all $r>0$, and the resolvent $J_{r}$ is firmly nonexpansive, i.e.,

$$ \|J_{r}x-J_{r}y\|^{2}\leq\langle x-y, J_{r}x-J_{r}y\rangle,\quad \forall x,y\in H. $$

Finally, we introduce the subdifferential of a lower semicontinuous convex function and an indicator function.

Let h be a proper lower semicontinuous convex function on a Hilbert space H into $(-\infty,\infty]$. Then the subdifferential ∂h of h is defined as follows:

$$ \partial h(x)=\bigl\{ z\in H: h(x)+\langle z, y-x\rangle\leq h(y), \forall y\in H\bigr\} $$

(1.4)

for all $x\in H$. From Rockafellar [15] we know that ∂h is a maximal monotone operator. Let $i_{C}$ be the indicator function of C (C is a nonempty closed convex subset of H), i.e.,

$$ i_{C}(x)=\left \{ \textstyle\begin{array}{l@{\quad}l} 0, & x\in C, \\ \infty, & x\notin C. \end{array}\displaystyle \right . $$

(1.5)

Then $i_{C}$ is a proper lower semicontinuous convex function on H and the subdifferential $\partial i_{C}$ of $i_{C}$ is a maximal monotone mapping. So we can define the resolvent $J_{r}$ of $\partial i_{C}$ for $r>0$, i.e.,

$$ J_{r}x=(I+r\partial i_{C})^{-1}x $$

for all $x\in H$. We have that for any $x\in H$ and $q\in C$,

$$\begin{aligned} q=J_{r}x \quad \Longleftrightarrow\quad & x\in q+r\partial i_{C}(q) \\ \quad \Longleftrightarrow\quad & x\in q+rN_{C}(q) \\ \quad \Longleftrightarrow\quad & x-q\in rN_{C}(q) \\ \quad \Longleftrightarrow\quad & \frac{1}{r}\langle x-q,p-q\rangle\leq0, \quad \forall p\in C \\ \quad \Longleftrightarrow\quad & \langle x-q,p-q\rangle\leq0,\quad \forall p\in C \\ \quad \Longleftrightarrow\quad & q=P_{C}x, \end{aligned}$$

where $N_{C}(q)$ is the normal cone to C at q, i.e.,

$$ N_{C}(q)=\bigl\{ z\in H: \langle z,p-q\rangle\leq0, \forall p\in C \bigr\} . $$

In the present paper, we study the equilibrium problems (1.1) and (1.2), the fixed point problem for a family of infinitely nonexpansive mappings, and the problem of zero points of maximal monotone mapping (1.3). Motivated and inspired by the research going on in this direction, we propose a new regularization algorithm, and it is proved that the sequence generated by this algorithm converges strongly to a common solution of the above three problems. The results presented in this paper improve and extend the corresponding results in Chang et al. [7], Takahashi and Takahashi [8], Hao [16] and Yuan and Zhang [17].

The structure of this paper is set as follows. In Section 2, we introduce some lemmas which will be used in the proof of theorems. The main result, that is, the strong convergence of the regularization algorithm, is proved in Section 3. Corollaries to generalized equilibrium problem and the zero points of the sum of two operators are presented in Section 4. And the conclusion of this paper is given in the final section, i.e., Section 5.

2 Preliminaries

In the following, we give some useful lemmas, which will often be used in the proof of the main results and their corollaries.

Let H be a real Hilbert space with the inner product $\langle\cdot,\cdot\rangle$ and the norm $\|\cdot\|$. When $\{x_{n}\}$ is a sequence in H, we define that the strong convergence of $\{x_{n}\}$ is a sequence in H, we denote the strong convergence of $\{x_{n}\}$ to $x\in H$ by $x_{n}\rightarrow x$ and the weak convergence by $x_{n}\rightharpoonup x$. Let C be a nonempty closed convex subset of a Hilbert space H, and let $T: C\rightarrow H$ be a mapping. We denote by $\operatorname{Fix}(T)$ the set of fixed points for T. If $T: C\rightarrow H$ is a nonexpansive mapping, then $\operatorname{Fix}(T)$ is closed and convex; see [18].

Firstly, we recall the metric (nearest point) projection from H onto C is the mapping $P_{C}: H\rightarrow C$ which is defined as follows: given $x\in H$, $P_{C}x$ is the unique point in C with the property

$$ \|x-P_{C}x\|=\inf_{y\in C}\|x-y\|=: d(x,C). $$

$P_{C}$ is characterized as follows.

Lemma 2.1

Given $x\in H$ and $y\in C$. Then $y=P_{C}x$ if and only if the following inequality holds:

$$ \langle x-y,y-z\rangle\geq0, \quad \forall z\in C. $$

Then we introduce the lemma below, which shows the uniqueness of solution of the variational inequality.

Lemma 2.2

[19]

Let H be a Hilbert space, C be a closed convex subset of H, and $f:C\rightarrow C$ be a contraction with coefficient $\alpha<1$. Then

$$ \bigl\langle x-y, (I-f)x-(I-f)y\bigr\rangle \geq(1-\alpha)\|x-y\|^{2}, \quad x,y\in C. $$

That is, $I-f$ is strongly monotone with coefficient $1-\alpha$.

Lemma 2.3

[2]

Let $F:C\times C\rightarrow\mathbb{R}$ be a bifunction satisfying (A1)-(A4). Then, for any $r>0$ and $x\in H$, there exists $z\in C$ such that

$$ F(z,y)+\frac{1}{r}\langle y-z, z-x\rangle\geq0,\quad \forall y\in C. $$

Define a mapping $T_{r}: H\rightarrow C$ as follows:

$$ T_{r}x=\biggl\{ z\in C: F(z,y)+\frac{1}{r}\langle y-z,z-x\rangle \geq0, \forall y\in C\biggr\} , \quad x\in H, $$

then the following conclusions hold:

• (1) $T_{r}$ is single-valued;

• (2) $T_{r}$ is firmly nonexpansive, i.e., for any $x,y\in H$,

  $$ \|T_{r}x-T_{r}y\|^{2}\leq\langle T_{r}x-T_{r}y, x-y\rangle; $$

• (3) $F(T_{r})=\operatorname{EP}(F)$;

• (4) $\operatorname{EP}(F)$ is closed and convex.

In the following, we introduce the property of W-mapping generated by a family of infinitely nonexpansive mappings.

Lemma 2.4

[14]

Let $\{S_{i}: C\rightarrow C\}$ be a family of infinitely nonexpansive mappings with a nonempty common fixed point set, and let $\{\gamma_{i}\}$ be a real sequence such that $0<\gamma_{i}\leq l<1$, where l is some real number, $\forall i>1$. Then

• (1) $W_{n}$ is nonexpansive and $F(W_{n})=\bigcap_{i=1}^{n}F(S_{i})$ for each $n\geq1$;

• (2) for each $x\in C$ and for each positive integer k, the limit $\lim_{n\rightarrow\infty}U_{n,k}$ exists;

• (3) the mapping $W: C\rightarrow C$ defined by

  $$ Wx:= \lim_{n\rightarrow\infty}W_{n}x=\lim _{n\rightarrow\infty}U_{n,1}x,\quad x\in C, $$

  (2.1)

  is a nonexpansive mapping satisfying $F(W)=\bigcap_{i=1}^{\infty}F(S_{i})$ and it is called the W-mapping generated by $S_{1},S_{2},\ldots$ and $\gamma_{1},\gamma_{2},\ldots$ .

Lemma 2.5

[7]

Let $\{S_{i}: C\rightarrow C\}$ be a family of infinitely nonexpansive mappings with a nonempty common fixed point set, and let $\{\gamma_{i}\}$ be a real sequence such that $0<\gamma_{i}\leq l<1$, $\forall i>1$. If K is any bounded subset of C, then

$$ \limsup_{n\rightarrow\infty, x\in K}\|Wx-W_{n}x\|=0. $$

The next lemma which we introduce is about the resolvent of the maximal monotone operator.

Lemma 2.6

(see [20–22])

Let H be a real Hilbert space, and let B be a maximal monotone operator on H. For $r>0$ and $x\in H$, define the resolvent $J_{r}x$. Then the following holds:

$$ \frac{s-t}{s}\langle J_{s}x-J_{t}x, J_{s}x-x\rangle\geq\|J_{s}x-J_{t}x \|^{2} $$

for all $s,t>0$ and $x\in H$. In particular,

$$ \|J_{s}x-J_{t}x\|\leq\bigl(\vert s-t\vert /s\bigr) \|x-J_{s}x\| $$

for all $s,t>0$ and $x\in H$.

The following lemma will be used in the proof of the main results.

Lemma 2.7

[23]

Let $\{x_{n}\}$ and $\{y_{n}\}$ be bounded sequences in H, and let $\{\beta_{n}\}$ be a sequence in $(0,1)$ with $0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty }\beta_{n}<1$. Suppose that $x_{n+1}=(1-\beta_{n})y_{n}+\beta_{n}x_{n}$ for all $n\geq0$ and

$$ \limsup_{n\rightarrow\infty}\bigl(\Vert y_{n+1}-y_{n} \Vert -\|x_{n+1}-x_{n}\|\bigr)\leq0. $$

Then $\lim_{n\rightarrow\infty}\|y_{n}-x_{n}\|=0$.

Besides, the following two lemmas are extremely important in the proof of theorems. One is called a demiclosed principle for nonexpansive mappings, the other is called an important lemma.

Lemma 2.8

(Demiclosed principle [24])

Let $T : C\rightarrow C$ be a nonexpansive mapping with $F(T)\neq\emptyset$. If $\{x_{n}\}_{n=1}^{\infty}$ is a sequence in C weekly converging to x and if $\{(I-T)x_{n}\}_{n=1}^{\infty}$ converges strongly to y, then $(I-T)x = y$. In particular, if $y = 0$, then $x\in F(T)$.

Lemma 2.9

[25]

Assume that $\{a_{n}\}_{n=0}^{\infty}$ is a sequence of nonnegative real numbers such that

$$a_{n+1} \leq(1-\gamma_{n})a_{n} + \gamma_{n}\delta_{n} +\beta _{n}, \quad n \geq0, $$

where $\{\gamma_{n}\}_{n=0}^{\infty}$ and $\{\beta_{n}\}_{n=0}^{\infty}$ are sequences in $(0,1)$ and $\{\delta_{n}\}_{n=0}^{\infty}$ is a sequence in $\mathbb{R}$ such that

• (i) $\sum_{n=0}^{\infty}\gamma_{n} = \infty$;

• (ii) either $\limsup_{n\rightarrow\infty}\delta_{n} \leq0$ or $\sum_{n=0}^{\infty}\gamma_{n}|\delta_{n}| < \infty$;

• (iii) $\sum_{n=0}^{\infty}\beta_{n} < \infty$.

Then $\lim_{n\rightarrow\infty}a_{n} = 0$.

3 Main results

Theorem 3.1

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let $F: C\times C\rightarrow\mathbb{R}$ be a bifunction which satisfies (A1)-(A4), and let $f:C\rightarrow C$ be a k-contraction with the constant $k\in(0,1)$. Let $A:C\rightarrow H$ be an α-inverse-strongly monotone (α-ism) mapping with $\alpha>0$, let $B:C\rightarrow H$ be a β-inverse-strongly monotone (β-ism) mapping with $\beta>0$, and let $T: C\rightarrow H$ be a τ-inverse-strongly monotone (τ-ism) mapping with $\tau>0$. Let M be a maximal monotone operator on H such that the domain of M is included in C, and let $J_{s}=(I+sM)^{-1}$ be the resolvent of M for $s>0$. Let N be a maximal monotone operator on H such that the domain of N is included in C, and let $J_{r}=(I+rN)^{-1}$ be the resolvent of N for $r>0$. Let $\{S_{i}: C\rightarrow C\}$ be a family of infinitely nonexpansive mappings. Assume that $\Omega=\operatorname{GEP}(F,T)\cap\bigcap_{i=1}^{\infty}F(S_{i})\cap (A+N)^{-1}0\cap(B+M)^{-1}0\neq\emptyset$. Let $\{\alpha_{n}\}$, $\{\beta_{n}\}$ and $\{\gamma_{n}\}$ be sequences in $(0,1)$ such that $\alpha_{n}+\beta_{n}+\gamma_{n}=1$. Let $\{r_{n}\}$, $\{\lambda_{n}\}$ and $\{s_{n}\}$ be positive number sequences. Let $x_{1}\in C$ and let $\{x_{n}\}$ be a sequence generated by

$$ \left \{ \textstyle\begin{array}{l} y_{n}=J_{r_{n}}(u_{n}-r_{n}Au_{n}), \\ x_{n+1}=\alpha_{n}f(y_{n})+\beta _{n}W_{n}J_{s_{n}}(y_{n}-s_{n}By_{n})+\gamma_{n}x_{n}, \quad \forall n\geq1, \end{array}\displaystyle \right . $$

(3.1)

where $\{u_{n}\}$ is such that

$$ F(u_{n},y)+\langle Tx_{n},y-u_{n}\rangle+ \frac{1}{\lambda_{n}}\langle y-u_{n},u_{n}-x_{n}\rangle \geq0, \quad \forall y\in C, $$

and $\{W_{n}\}$ is the sequence generated in (2.1). Assume that the following conditions hold:

• (i) $0< a\leq r_{n}\leq b<2\alpha$ and $\lim_{n\rightarrow\infty}|r_{n+1}-r_{n}|=0$;

• (ii) $0< c\leq\lambda_{n}\leq d<2\tau$ and $\lim_{n\rightarrow\infty}|\lambda_{n+1}-\lambda_{n}|=0$;

• (iii) $0< e\leq s_{n}\leq g<2\beta$ and $\lim_{n\rightarrow\infty}|s_{n+1}-s_{n}|=0$;

• (iv) $\lim_{n\rightarrow\infty}\alpha_{n}=0$, $\sum_{n=1}^{\infty}\alpha_{n}=\infty$;

• (v) $0<\liminf_{n\rightarrow\infty}\gamma_{n}\leq\limsup_{n\rightarrow\infty }\gamma_{n}<1$,

where a, b, c, d, e, and g are real constants. Then $\{x_{n}\}$ converges strongly to a point $q\in\Omega$, which solves uniquely the following variational inequality:

$$ \bigl\langle q-f(q), q-x\bigr\rangle \leq0,\quad \forall x\in \Omega. $$

(3.2)

Equivalently, $q=P_{\Omega}f(q)$.

Proof

We divide the proof into several steps.

Step 1. We prove that the sequence $\{x_{n}\}$ is bounded.

Since A is an α-ism mapping, we see from restriction (i) that $\forall x,y\in C$,

$$\begin{aligned} \bigl\Vert (I-r_{n}A)x-(I-r_{n}A)y\bigr\Vert ^{2} =&\|x-y\|^{2}-2r_{n}\langle x-y, Ax-Ay \rangle+r_{n}^{2}\|Ax-Ay\|^{2} \\ \leq&\|x-y\|^{2}-2r_{n}\alpha\|Ax-Ay\|^{2}+r_{n}^{2} \|Ax-Ay\|^{2} \\ =&\|x-y\|^{2}+r_{n}(r_{n}-2\alpha)\|Ax-Ay \|^{2} \\ \leq&\|x-y\|^{2}. \end{aligned}$$

This implies that $I-r_{n}A$ is nonexpansive.

In the same way, we find that $I-s_{n}B$ and $I-\lambda_{n}T$ are nonexpansive. Note that $u_{n}=T_{\lambda_{n}}(I-\lambda_{n}T)x_{n}$. Let $p\in\Omega$, it follows that

$$ \|u_{n}-p\|\leq\bigl\Vert (I-\lambda_{n}T)x_{n}-(I- \lambda_{n}T)p\bigr\Vert \leq\| x_{n}-p\|. $$

Putting $z_{n}=J_{s_{n}}(y_{n}-s_{n}By_{n})$, we see that

$$ \|z_{n}-p\|\leq\|y_{n}-p\|\leq \|u_{n}-p\|\leq\|x_{n}-p\|. $$

(3.3)

From (3.3), we find that

$$\begin{aligned} \|x_{n+1}-p\| \leq&\alpha_{n}\bigl\| f(y_{n})-p\bigr\| + \beta_{n}\|W_{n}z_{n}-p\| +\gamma_{n} \|x_{n}-p\| \\ \leq&\alpha_{n}k\|y_{n}-p\|+\alpha_{n}\bigl\Vert f(p)-p\bigr\Vert +\beta_{n}\|z_{n}-p\| + \gamma_{n}\|x_{n}-p\| \\ \leq&\bigl(1-\alpha_{n}(1-k)\bigr)\|x_{n}-p\|+ \alpha_{n}(1-k)\frac{\|f(p)-p\|}{1-k}. \end{aligned}$$

By induction, we derive that

$$ \|x_{n}-p\|\leq\max\biggl\{ \|x_{1}-p\|, \frac{\|f(p)-p\|}{1-k} \biggr\} , \quad \forall n\geq1. $$

Therefore, it turns out that $\{x_{n}\}$ is bounded, and so are $\{y_{n}\}$, $\{z_{n}\}$ and $\{u_{n}\}$.

Step 2. We show that $\lim_{n\rightarrow\infty}\|x_{n+1}-x_{n}\|=0$.

Without loss of generality, we can assume that there exists a bounded set $K\subset C$ such that $x_{n}, y_{n}, z_{n}, u_{n}\in K$. Since $u_{n}=T_{\lambda_{n}}(I-\lambda_{n}T)x_{n}$, we find that

$$ F(u_{n+1},y)+\frac{1}{\lambda_{n+1}}\bigl\langle y-u_{n+1}, u_{n+1}-(I-\lambda_{n+1}T)x_{n+1} \bigr\rangle \geq0,\quad \forall y\in C $$

(3.4)

and

$$ F(u_{n},y)+\frac{1}{\lambda_{n}}\bigl\langle y-u_{n}, u_{n}-(I-\lambda_{n}T)x_{n}\bigr\rangle \geq0,\quad \forall y\in C. $$

(3.5)

Let $y=u_{n}$ in (3.4) and $y=u_{n+1}$ in (3.5). Then we add up (3.4) and (3.5) to derive that

$$ \biggl\langle u_{n+1}-u_{n}, u_{n}-u_{n+1}+u_{n+1}-(I- \lambda_{n}T)x_{n}-\frac{\lambda_{n}}{\lambda _{n+1}}\bigl(u_{n+1}-(I- \lambda_{n+1}T)x_{n+1}\bigr)\biggr\rangle \geq0. $$

This implies that

$$\begin{aligned} \|u_{n+1}-u_{n}\|^{2} \leq&\biggl\langle u_{n+1}-u_{n}, (I-\lambda_{n+1}T)x_{n+1}-(I- \lambda_{n}T)x_{n} \\ &{}+\biggl(1-\frac{\lambda_{n}}{\lambda_{n+1}}\biggr)\cdot\bigl(u_{n+1}-(I-\lambda _{n+1}T)x_{n+1}\bigr)\biggr\rangle \\ \leq&\|u_{n+1}-u_{n}\|\cdot\biggl(\bigl\Vert (I- \lambda_{n+1}T)x_{n+1}-(I-\lambda _{n}T)x_{n} \bigr\Vert \\ &{}+\biggl\vert 1-\frac{\lambda_{n}}{\lambda_{n+1}}\biggr\vert \cdot\bigl\Vert u_{n+1}-(I-\lambda _{n+1}T)x_{n+1}\bigr\Vert \biggr). \end{aligned}$$

From $I-\lambda_{n}T$ is nonexpansive and condition (ii), we obtain that

$$\begin{aligned} \|u_{n+1}-u_{n}\| \leq&\bigl\Vert (I- \lambda_{n+1}T)x_{n+1}-(I-\lambda _{n}T)x_{n} \bigr\Vert \\ &{}+\frac{|\lambda_{n+1}-\lambda_{n}|}{\lambda_{n+1}}\bigl\Vert u_{n+1}-(I-\lambda_{n+1}T)x_{n+1} \bigr\Vert \\ \leq&\bigl\Vert (I-\lambda_{n+1}T)x_{n+1}-(I- \lambda_{n+1}T)x_{n}\bigr\Vert +|\lambda _{n+1}- \lambda_{n}|\cdot\|Tx_{n}\| \\ &{}+\frac{|\lambda_{n+1}-\lambda_{n}|}{c}\bigl\Vert u_{n+1}-(I-\lambda _{n+1}T)x_{n+1}\bigr\Vert \\ \leq&\|x_{n+1}-x_{n}\|+|\lambda_{n+1}- \lambda_{n}|L_{1}, \end{aligned}$$

(3.6)

where $L_{1}$ is an appropriate constant such that

$$ L_{1}=\sup_{n\geq1}\biggl\{ \|Tx_{n}\|+ \frac{1}{c}\bigl\Vert u_{n+1}-(I-\lambda _{n+1}T)x_{n+1} \bigr\Vert \biggr\} . $$

Since both $J_{r_{n}}$ and $I-r_{n}A$ are nonexpansive, it follows from Lemma 2.6, condition (i) and (3.6) that

$$\begin{aligned} \|y_{n+1}-y_{n}\| =&\bigl\Vert J_{r_{n+1}}(I-r_{n+1}A)u_{n+1}-J_{r_{n}}(I-r_{n}A)u_{n} \bigr\Vert \\ \leq&\bigl\Vert J_{r_{n+1}}(I-r_{n+1}A)u_{n+1}-J_{r_{n+1}}(I-r_{n+1}A)u_{n} \bigr\Vert \\ &{}+\bigl\Vert J_{r_{n+1}}(I-r_{n+1}A)u_{n}-J_{r_{n+1}}(I-r_{n}A)u_{n} \bigr\Vert \\ &{}+\bigl\Vert J_{r_{n+1}}(I-r_{n}A)u_{n}-J_{r_{n}}(I-r_{n}A)u_{n} \bigr\Vert \\ \leq&\|u_{n+1}-u_{n}\|+\bigl\Vert (I-r_{n+1}A)u_{n}-(I-r_{n}A)u_{n} \bigr\Vert \\ &{}+\frac{|r_{n+1}-r_{n}|}{r_{n+1}}\bigl\Vert J_{r_{n+1}}(I-r_{n}A)u_{n}-(I-r_{n}A)u_{n} \bigr\Vert \\ \leq&\|u_{n+1}-u_{n}\|+|r_{n+1}-r_{n}| \cdot\|Au_{n}\| \\ &{}+\frac{|r_{n+1}-r_{n}|}{a}\bigl\Vert J_{r_{n+1}}(I-r_{n}A)u_{n}-(I-r_{n}A)u_{n} \bigr\Vert \\ \leq&\|x_{n+1}-x_{n}\|+|\lambda_{n+1}- \lambda_{n}|L_{1}+|r_{n+1}-r_{n}|L_{2}, \end{aligned}$$

(3.7)

where $L_{2}$ is an appropriate constant such that

$$ L_{2}=\sup_{n\geq1}\biggl\{ \|Au_{n}\|+ \frac{1}{a}\bigl\Vert J_{r_{n+1}}(I-r_{n}A)u_{n}-(I-r_{n}A)u_{n} \bigr\Vert \biggr\} . $$

Thus, from both $J_{s_{n}}$ and $I-s_{n}B$ are nonexpansive, we have from Lemma 2.6, condition (iii) and (3.7) that

$$\begin{aligned} \Vert z_{n+1}-z_{n}\Vert =&\bigl\Vert J_{s_{n+1}}(I-s_{n+1}B)y_{n+1}-J_{s_{n}}(I-s_{n}B)y_{n} \bigr\Vert \\ \leq&\bigl\Vert J_{s_{n+1}}(I-s_{n+1}B)y_{n+1}-J_{s_{n+1}}(I-s_{n+1}B)y_{n} \bigr\Vert \\ &{}+\bigl\Vert J_{s_{n+1}}(I-s_{n+1}B)y_{n}-J_{s_{n+1}}(I-s_{n}B)y_{n} \bigr\Vert \\ &{}+\bigl\Vert J_{s_{n+1}}(I-s_{n}B)y_{n}-J_{s_{n}}(I-s_{n}B)y_{n} \bigr\Vert \\ \leq&\Vert y_{n+1}-y_{n}\Vert +\bigl\Vert (I-s_{n+1}B)y_{n}-(I-s_{n}B)y_{n}\bigr\Vert \\ &{}+\frac{|s_{n+1}-s_{n}|}{s_{n+1}}\bigl\Vert J_{s_{n+1}}(I-s_{n}B)y_{n}-(I-s_{n}B)y_{n} \bigr\Vert \\ \leq&\Vert y_{n+1}-y_{n}\Vert +|s_{n+1}-s_{n}| \cdot \Vert By_{n}\Vert \\ &{}+\frac{|s_{n+1}-s_{n}|}{e}\bigl\Vert J_{s_{n+1}}(I-s_{n}B)y_{n}-(I-s_{n}B)y_{n} \bigr\Vert \\ \leq&\Vert x_{n+1}-x_{n}\Vert \\ &{}+L_{3}\bigl(|\lambda_{n+1}-\lambda_{n}|+|r_{n+1}-r_{n}|+|s_{n+1}-s_{n}|\bigr), \end{aligned}$$

(3.8)

where $L_{3}=\max\{L_{1}, L_{2}, \sup_{n\geq1}\{\|By_{n}\|+\frac{1}{e}\| J_{s_{n+1}}(I-s_{n}B)y_{n}-(I-s_{n}B)y_{n}\|\}\}$.

It yields from (3.8) that

$$\begin{aligned} \|W_{n+1}z_{n+1}-W_{n}z_{n} \| \leq&\|W_{n+1}z_{n+1}-Wz_{n+1}\|+\| Wz_{n+1}-Wz_{n}\| \\ &{}+\|Wz_{n}-W_{n}z_{n}\| \\ \leq&\sup_{x\in K}\bigl\{ \Vert W_{n+1}x-Wx\Vert + \|Wx-W_{n}x\|\bigr\} +\|x_{n+1}-x_{n}\| \\ &{}+L_{3}\bigl(\vert \lambda_{n+1}-\lambda_{n} \vert +|r_{n+1}-r_{n}|+|s_{n+1}-s_{n}| \bigr), \end{aligned}$$

(3.9)

where K is the bounded subset of C defined above. Let $x_{n+1}=(1-\gamma_{n})t_{n}+\gamma_{n}x_{n}$.

It follows that

$$\begin{aligned} t_{n+1}-t_{n} =&\frac{\alpha_{n+1}f(y_{n+1})+\beta _{n+1}W_{n+1}z_{n+1}}{1-\gamma_{n+1}}-\frac{\alpha_{n}f(y_{n})+\beta _{n}W_{n}z_{n}}{1-\gamma_{n}} \\ =&\frac{\alpha_{n+1}}{1-\gamma_{n+1}}f(y_{n+1})+\frac{1-\alpha _{n+1}-\gamma_{n+1}}{1-\gamma_{n+1}}W_{n+1}z_{n+1} \\ &{}-\biggl(\frac{\alpha_{n}}{1-\gamma_{n}}f(y_{n})+\frac{1-\alpha _{n}-\gamma_{n}}{1-\gamma_{n}}W_{n}z_{n} \biggr) \\ =&\frac{\alpha_{n+1}}{1-\gamma_{n+1}}\bigl(f(y_{n+1})-W_{n+1}z_{n+1} \bigr)-\frac {\alpha_{n}}{1-\gamma_{n}}\bigl(f(y_{n})-W_{n}z_{n} \bigr) \\ &{}+W_{n+1}z_{n+1}-W_{n}z_{n}. \end{aligned}$$

From (3.9), we derive that

$$\begin{aligned} \Vert t_{n+1}-t_{n}\Vert \leq&\frac{\alpha_{n+1}}{1-\gamma_{n+1}}\bigl\Vert f(y_{n+1})-W_{n+1}z_{n+1}\bigr\Vert + \frac{\alpha_{n}}{1-\gamma_{n}}\bigl\Vert f(y_{n})-W_{n}z_{n} \bigr\Vert \\ &{}+\Vert W_{n+1}z_{n+1}-W_{n}z_{n} \Vert \\ \leq&\frac{\alpha_{n+1}}{1-\gamma_{n+1}}\bigl\Vert f(y_{n+1})-W_{n+1}z_{n+1} \bigr\Vert +\frac{\alpha_{n}}{1-\gamma_{n}}\bigl\Vert f(y_{n})-W_{n}z_{n} \bigr\Vert \\ &{}+\sup_{x\in K}\bigl\{ \Vert W_{n+1}x-Wx\Vert + \Vert Wx-W_{n}x\Vert \bigr\} +\Vert x_{n+1}-x_{n} \Vert \\ &{}+L_{3}\bigl(\vert \lambda_{n+1}-\lambda_{n} \vert +|r_{n+1}-r_{n}|+|s_{n+1}-s_{n}| \bigr), \end{aligned}$$

which implies that

$$\begin{aligned} \Vert t_{n+1}-t_{n}\Vert -\Vert x_{n+1}-x_{n} \Vert \leq&\frac{\alpha_{n+1}}{1-\gamma _{n+1}}\bigl\Vert f(y_{n+1})-W_{n+1}z_{n+1} \bigr\Vert \\ &{}+\frac{\alpha_{n}}{1-\gamma_{n}}\bigl\Vert f(y_{n})-W_{n}z_{n} \bigr\Vert \\ &{}+\sup_{x\in K}\bigl\{ \Vert W_{n+1}x-Wx\Vert + \Vert Wx-W_{n}x\Vert \bigr\} \\ &{}+L_{3}\bigl(\vert \lambda_{n+1}-\lambda_{n} \vert +|r_{n+1}-r_{n}|+|s_{n+1}-s_{n}| \bigr). \end{aligned}$$

It follows from conditions (i)-(v) that

$$ \lim_{n\rightarrow\infty}\sup\bigl(\Vert t_{n+1}-t_{n} \Vert -\|x_{n+1}-x_{n}\|\bigr)\leq0. $$

From Lemma 2.7, we derive that $\lim_{n\rightarrow\infty}\|t_{n}-x_{n}\|=0$.

Since $t_{n}-x_{n}=\frac{1}{1-\gamma_{n}}(x_{n+1}-x_{n})$, it follows that

$$ \lim_{n\rightarrow\infty}\|x_{n+1}-x_{n} \|=0. $$

(3.10)

Step 3. We prove that $\lim_{n\rightarrow\infty}\|x_{n}-u_{n}\|=0$.

Since T is τ-ism, we find from (3.3) that

$$\begin{aligned} \Vert x_{n+1}-p\Vert ^{2} \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+\beta_{n}\Vert u_{n}-p\Vert ^{2}+\gamma_{n}\Vert x_{n}-p\Vert ^{2} \\ \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+\beta_{n}\bigl\Vert x_{n}-p-\lambda _{n}(Tx_{n}-Tp)\bigr\Vert ^{2} +\gamma_{n}\Vert x_{n}-p\Vert ^{2} \\ \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+\Vert x_{n}-p\Vert ^{2}- \lambda_{n}\beta _{n}(2\tau-\lambda_{n})\Vert Tx_{n}-Tp\Vert ^{2}, \end{aligned}$$

it turns out that

$$\begin{aligned}& \lambda_{n}\beta_{n}(2\tau-\lambda_{n})\Vert Tx_{n}-Tp\Vert ^{2} \\& \quad \leq\alpha _{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2} +\bigl(\Vert x_{n}-p\Vert +\Vert x_{n+1}-p\Vert \bigr)\Vert x_{n+1}-x_{n}\Vert . \end{aligned}$$

By virtue of conditions (ii), (iv), (v), we derive from (3.10) that

$$ \lim_{n\rightarrow\infty}\|Tx_{n}-Tp\|=0. $$

(3.11)

Since $T_{\lambda_{n}}$ is firmly nonexpansive, we find from Lemma 2.3 that

$$\begin{aligned} \|u_{n}-p\|^{2} =&\bigl\Vert T_{\lambda_{n}}(I- \lambda_{n}T)x_{n}-T_{\lambda _{n}}(I-\lambda_{n}T)p \bigr\Vert ^{2} \\ \leq&\bigl\langle (I-\lambda_{n}T)x_{n}-(I- \lambda_{n}T)p,u_{n}-p\bigr\rangle \\ \leq&\frac{1}{2}\bigl(\|x_{n}-p\|^{2}+ \|u_{n}-p\|^{2}-\|x_{n}-u_{n} \|^{2} \\ &{}+2\lambda_{n}\|Tx_{n}-Tp\|\cdot\|x_{n}-u_{n} \|\bigr), \end{aligned}$$

it implies that

$$ \|u_{n}-p\|^{2}\leq\|x_{n}-p\|^{2}- \|x_{n}-u_{n}\|^{2}+2\lambda_{n}\| Tx_{n}-Tp\|\cdot\|x_{n}-u_{n}\|. $$

Therefore, we derive that

$$\begin{aligned} \begin{aligned} \Vert x_{n+1}-p\Vert ^{2}\leq{}&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+\beta_{n}\Vert u_{n}-p\Vert ^{2}+\gamma_{n}\Vert x_{n}-p\Vert ^{2} \\ \leq{}&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}-\beta_{n}\Vert x_{n}-u_{n}\Vert ^{2} \\ & {}+2\lambda_{n}\beta_{n}\Vert Tx_{n}-Tp \Vert \cdot \Vert x_{n}-u_{n}\Vert +\Vert x_{n}-p\Vert ^{2}, \end{aligned} \end{aligned}$$

which yields that

$$\begin{aligned} \beta_{n}\Vert x_{n}-u_{n}\Vert ^{2} \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+2\lambda_{n}\beta_{n}\Vert Tx_{n}-Tp\Vert \cdot \Vert x_{n}-u_{n}\Vert \\ &{}+\bigl(\Vert x_{n}-p\Vert +\Vert x_{n+1}-p\Vert \bigr)\Vert x_{n+1}-x_{n}\Vert . \end{aligned}$$

By use of conditions (ii), (iv), (v), we obtain from (3.10) and (3.11) that

$$ \lim_{n\rightarrow\infty}\|x_{n}-u_{n}\|=0. $$

(3.12)

Step 4. We prove that $\lim_{n\rightarrow\infty}\|u_{n}-y_{n}\|=0$.

Since $J_{r_{n}}$ is nonexpansive, A is α-ism, we find from (3.3) that

$$\begin{aligned} \|y_{n}-p\|^{2} =&\bigl\Vert J_{r_{n}}(I-r_{n}A)u_{n}-J_{r_{n}}(I-r_{n}A)p \bigr\Vert ^{2} \\ \leq&\bigl\Vert (I-r_{n}A)u_{n}-(I-r_{n}A)p \bigr\Vert ^{2} \\ \leq&\|u_{n}-p\|^{2}-2r_{n}\alpha \|Au_{n}-Ap\|^{2}+r_{n}^{2}\| Au_{n}-Ap\|^{2} \\ \leq&\|x_{n}-p\|^{2}+r_{n}(r_{n}-2 \alpha)\|Au_{n}-Ap\|^{2}. \end{aligned}$$

Therefore, we derive that

$$\begin{aligned} \|x_{n+1}-p\|^{2} \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+\beta_{n} \|W_{n}z_{n}-p\|^{2}+\gamma _{n} \|x_{n}-p\|^{2} \\ \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+\beta_{n}\|y_{n}-p\|^{2}+ \gamma_{n}\| x_{n}-p\|^{2} \\ \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+r_{n}(r_{n}-2\alpha)\beta_{n}\| Au_{n}-Ap\|^{2}+\|x_{n}-p\|^{2}, \end{aligned}$$

it turns out that

$$\begin{aligned}& r_{n}(2\alpha-r_{n})\beta_{n}\|Au_{n}-Ap \|^{2} \\& \quad \leq\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2} +\bigl(\Vert x_{n}-p\Vert +\|x_{n+1}-p\|\bigr) \|x_{n+1}-x_{n}\|. \end{aligned}$$

In view of conditions (i), (iv), (v), we obtain from (3.10) that

$$ \lim_{n\rightarrow\infty}\|Au_{n}-Ap\|=0. $$

(3.13)

Since $J_{r_{n}}$ is firmly nonexpansive, $I-r_{n}A$ is nonexpansive, we find from (3.3) that

$$\begin{aligned} \begin{aligned} \|y_{n}-p\|^{2}={}&\bigl\Vert J_{r_{n}}(I-r_{n}A)u_{n}-J_{r_{n}}(I-r_{n}A)p \bigr\Vert ^{2} \\ \leq{}&\bigl\langle (I-r_{n}A)u_{n}-(I-r_{n}A)p, y_{n}-p\bigr\rangle \\ ={}&\frac{1}{2}\bigl\{ \bigl\Vert (I-r_{n}A)u_{n}-(I-r_{n}A)p \bigr\Vert ^{2}+\|y_{n}-p\|^{2} \\ & {}-\bigl\Vert (I-r_{n}A)u_{n}-(I-r_{n}A)p-(y_{n}-p) \bigr\Vert ^{2}\bigr\} \\ \leq{}&\frac{1}{2}\bigl\{ \|u_{n}-p\|^{2}+ \|y_{n}-p\|^{2}-\bigl\Vert u_{n}-y_{n}-r_{n}(Au_{n}-Ap) \bigr\Vert ^{2}\bigr\} \\ \leq{}&\frac{1}{2}\bigl\{ \|x_{n}-p\|^{2}+ \|y_{n}-p\|^{2}-\|u_{n}-y_{n} \|^{2} \\ & {}+2r_{n}\|u_{n}-y_{n}\|\cdot \|Au_{n}-Ap\|\bigr\} , \end{aligned} \end{aligned}$$

which implies that

$$ \|y_{n}-p\|^{2}\leq\|x_{n}-p\|^{2}- \|u_{n}-y_{n}\|^{2}+2r_{n}\| u_{n}-y_{n}\|\cdot\|Au_{n}-Ap\|. $$

From (3.3), this further implies that

$$\begin{aligned} \Vert x_{n+1}-p\Vert ^{2} \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+\beta_{n}\Vert W_{n}z_{n}-p\Vert ^{2}+\gamma_{n} \Vert x_{n}-p\Vert ^{2} \\ \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+\beta_{n}\Vert y_{n}-p\Vert ^{2}+\gamma_{n}\Vert x_{n}-p\Vert ^{2} \\ \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}-\beta_{n}\Vert u_{n}-y_{n}\Vert ^{2} \\ &{}+2r_{n}\beta_{n}\Vert u_{n}-y_{n} \Vert \cdot \Vert Au_{n}-Ap\Vert +\Vert x_{n}-p\Vert ^{2}, \end{aligned}$$

which yields that

$$\begin{aligned} \beta_{n}\|u_{n}-y_{n}\|^{2} \leq& \alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+2r_{n}\beta_{n}\|u_{n}-y_{n} \|\cdot\|Au_{n}-Ap\| \\ &{}+\bigl(\Vert x_{n}-p\Vert +\|x_{n+1}-p\|\bigr) \|x_{n+1}-x_{n}\|. \end{aligned}$$

By use of conditions (i), (iv), (v), we derive from (3.10) and (3.13) that

$$ \lim_{n\rightarrow\infty}\|u_{n}-y_{n}\|=0. $$

(3.14)

Step 5. We prove that $\lim_{n\rightarrow\infty}\|y_{n}-z_{n}\|=0$.

Since $J_{s_{n}}$ is nonexpansive, B is β-ism, we find from (3.3) that

$$\begin{aligned} \|z_{n}-p\|^{2} =&\bigl\Vert J_{s_{n}}(I-s_{n}B)y_{n}-J_{s_{n}}(I-s_{n}B)p \bigr\Vert ^{2} \\ \leq&\bigl\Vert (I-s_{n}B)y_{n}-(I-s_{n}B)p \bigr\Vert ^{2} \\ \leq&\|y_{n}-p\|^{2}-2s_{n}\beta \|By_{n}-Bp\|^{2}+s_{n}^{2}\| By_{n}-Bp\|^{2} \\ \leq&\|x_{n}-p\|^{2}+s_{n}(s_{n}-2 \beta)\|By_{n}-Bp\|^{2}. \end{aligned}$$

Hence, we derive that

$$\begin{aligned} \|x_{n+1}-p\|^{2} =&\bigl\Vert \alpha_{n} \bigl(f(y_{n})-p\bigr)+\beta _{n}(W_{n}z_{n}-p)+ \gamma_{n}(x_{n}-p)\bigr\Vert ^{2} \\ \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+\beta_{n}\|z_{n}-p\|^{2}+ \gamma_{n}\| x_{n}-p\|^{2} \\ \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+s_{n}(s_{n}-2\beta)\beta_{n}\| By_{n}-Bp\|^{2}+\|x_{n}-p\|^{2}, \end{aligned}$$

which implies that

$$\begin{aligned}& s_{n}(2\beta-s_{n})\beta_{n}\|By_{n}-Bp \|^{2} \\& \quad \leq\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2} +\bigl(\Vert x_{n}-p\Vert + \|x_{n+1}-p\|\bigr)\|x_{n+1}-x_{n}\|. \end{aligned}$$

By virtue of conditions (iii), (iv), (v), we derive from (3.10) that

$$ \lim_{n\rightarrow\infty}\|By_{n}-Bp\|=0. $$

(3.15)

Since $J_{s_{n}}$ is firmly nonexpansive, $I-s_{n}B$ is nonexpansive, we find from (3.3) that

$$\begin{aligned} \Vert z_{n}-p\Vert ^{2} =&\bigl\Vert J_{s_{n}}(I-s_{n}B)y_{n}-J_{s_{n}}(I-s_{n}B)p \bigr\Vert ^{2} \\ \leq&\bigl\langle (I-s_{n}B)y_{n}-(I-s_{n}B)p, z_{n}-p\bigr\rangle \\ =&\frac{1}{2}\bigl\{ \bigl\Vert (I-s_{n}B)y_{n}-(I-s_{n}B)p \bigr\Vert ^{2}+\Vert z_{n}-p\Vert ^{2} \\ &{}-\bigl\Vert (I-s_{n}B)y_{n}-(I-s_{n}B)p-(z_{n}-p) \bigr\Vert ^{2}\bigr\} \\ \leq&\frac{1}{2}\bigl\{ \Vert y_{n}-p\Vert ^{2}+ \Vert z_{n}-p\Vert ^{2}-\bigl\Vert y_{n}-z_{n}-s_{n}(By_{n}-Bp) \bigr\Vert ^{2}\bigr\} \\ \leq&\frac{1}{2}\bigl\{ \Vert x_{n}-p\Vert ^{2}+ \Vert z_{n}-p\Vert ^{2}-\Vert y_{n}-z_{n} \Vert ^{2} \\ &{}+2s_{n}\Vert y_{n}-z_{n}\Vert \cdot \Vert By_{n}-Bp\Vert \bigr\} . \end{aligned}$$

It turns out that

$$ \|z_{n}-p\|^{2}\leq\|x_{n}-p\|^{2}- \|y_{n}-z_{n}\|^{2}+2s_{n}\| y_{n}-z_{n}\|\cdot\|By_{n}-Bp\|. $$

Hence, we obtain that

$$\begin{aligned} \Vert x_{n+1}-p\Vert ^{2} \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+\beta_{n}\Vert W_{n}z_{n}-p\Vert ^{2}+\gamma_{n} \Vert x_{n}-p\Vert ^{2} \\ \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+\beta_{n}\Vert z_{n}-p\Vert ^{2}+\gamma_{n}\Vert x_{n}-p\Vert ^{2} \\ \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}-\beta_{n}\Vert y_{n}-z_{n}\Vert ^{2} \\ &{}+2s_{n}\beta_{n}\Vert y_{n}-z_{n} \Vert \cdot \Vert By_{n}-Bp\Vert +\Vert x_{n}-p\Vert ^{2}, \end{aligned}$$

it follows that

$$\begin{aligned} \beta_{n}\Vert y_{n}-z_{n}\Vert ^{2} \leq&\alpha_{n}\bigl\Vert f(y_{n})-p\bigr\Vert ^{2}+2s_{n}\beta_{n}\Vert y_{n}-z_{n} \Vert \cdot \Vert By_{n}-Bp\Vert \\ &{}+\bigl(\Vert x_{n}-p\Vert +\Vert x_{n+1}-p\Vert \bigr)\Vert x_{n+1}-x_{n}\Vert . \end{aligned}$$

By use of conditions (iii), (iv), (v), we obtain from (3.10) and (3.15) that

$$ \lim_{n\rightarrow\infty}\|y_{n}-z_{n}\|=0. $$

(3.16)

Step 6. We show that

$$ \lim_{n\rightarrow\infty}\sup\bigl\langle f(q)-q, x_{n}-q\bigr\rangle \leq0, \quad \forall x_{n}\in\Omega, $$

where $q=P_{\Omega}f(q)$. It is equivalent to show that $q\in\Omega= \operatorname{GEP}(F,T)\cap\bigcap_{i=1}^{\infty}F(S_{i})\cap (A+N)^{-1}0\cap(B+M)^{-1}0$.

First, we show that $q\in \operatorname{GEP}(F,T)$. From Lemma 2.3, we get $u_{n}=T_{\lambda_{n}}(I-\lambda_{n}T)x_{n}$, for any $y\in C$, we find from (A2) that

$$ \langle Tx_{n_{i}}, y-u_{n_{i}}\rangle+\biggl\langle y-u_{n_{i}}, \frac{u_{n_{i}}-x_{n_{i}}}{\lambda_{n_{i}}}\biggr\rangle \geq F(y, u_{n_{i}}), \quad \forall y\in C. $$

(3.17)

Putting $y_{h}=hy+(1-h)q$ for any $h\in(0,1]$ and $y\in C$, we see that $y_{h}\in C$. From (3.17), we derive that

$$\begin{aligned} \langle y_{h}-u_{n_{i}}, Ty_{h}\rangle \geq&\langle y_{h}-u_{n_{i}}, Ty_{h}\rangle-\langle Tx_{n_{i}}, y_{h}-u_{n_{i}}\rangle \\ &{}-\biggl\langle y_{h}-u_{n_{i}}, \frac{u_{n_{i}}-x_{n_{i}}}{\lambda_{n_{i}}}\biggr\rangle +F(y_{h},u_{n_{i}}) \\ =&\langle y_{h}-u_{n_{i}}, Ty_{h}-Tu_{n_{i}} \rangle+\langle y_{h}-u_{n_{i}}, Tu_{n_{i}}-Tx_{n_{i}} \rangle \\ &{}-\biggl\langle y_{h}-u_{n_{i}}, \frac{u_{n_{i}}-x_{n_{i}}}{\lambda_{n_{i}}}\biggr\rangle +F(y_{h},u_{n_{i}}). \end{aligned}$$

By virtue of the monotonicity of T and condition (ii), we obtain from (A4) that

$$ \langle y_{h}-q, Ty_{h}\rangle\geq F(y_{h},q). $$

(3.18)

From (A1) and (A4), we see that

$$\begin{aligned} 0 =&F(y_{h},y_{h})\leq hF(y_{h},y)+(1-h)F(y_{h},q) \\ \leq&hF(y_{h},y)+(1-h)\langle y_{h}-q, Ty_{h} \rangle \\ =&hF(y_{h},y)+(1-h)h\langle y-q, Ty_{h}\rangle. \end{aligned}$$

It turns out from (A3) that $q\in \operatorname{GEP}(F,T)$.

Then we show that $q\in\bigcap_{i=1}^{\infty}F(S_{i})$. Indeed, choose a subsequence $\{x_{n_{i}}\}$ of $\{x_{n}\}$ such that

$$ \lim_{n\rightarrow\infty}\sup\bigl\langle (f-I)q,x_{n}-q\bigr\rangle =\lim_{i\rightarrow\infty}\bigl\langle (f-I)q, x_{n_{i}}-q\bigr\rangle . $$

Since $\{x_{n_{i}}\}$ is bounded, there exists a subsequence $\{x_{n_{i_{j}}}\}$ of $\{x_{n_{i}}\}$ such that $\{x_{n_{i_{j}}}\}$ converges weakly to q. Without loss of generality, we may assume that $x_{n_{i}}\rightharpoonup q$. In view of (3.12), (3.14) and (3.16), we know $x_{n_{i}}\rightharpoonup q$ is equivalent to $u_{n_{i}}\rightharpoonup q$, $y_{n_{i}}\rightharpoonup q$ and $z_{n_{i}}\rightharpoonup q$.

Since $x_{n+1}=\alpha_{n}f(y_{n})+\beta_{n}W_{n}z_{n}+\gamma_{n}x_{n}$, it implies that

$$ \beta_{n}\|W_{n}z_{n}-x_{n}\|\leq \|x_{n+1}-x_{n}\|+\alpha_{n}\bigl\Vert f(y_{n})-x_{n}\bigr\Vert . $$

By virtue of conditions (iv), (v), we derive from (3.10) that

$$ \lim_{n\rightarrow\infty}\|W_{n}z_{n}-x_{n} \|=0. $$

(3.19)

Observing that

$$ \|W_{n}z_{n}-z_{n}\|\leq\|W_{n}z_{n}-x_{n} \|+\|x_{n}-u_{n}\|+\| u_{n}-y_{n}\|+ \|y_{n}-z_{n}\|, $$

and from (3.12), (3.14), (3.16) and (3.19), we derive that

$$ \lim_{n\rightarrow\infty}\|W_{n}z_{n}-z_{n} \|=0. $$

(3.20)

It is not hard to find that

$$ \|Wz_{n}-z_{n}\|\leq\|Wz_{n}-W_{n}z_{n} \|+\|W_{n}z_{n}-z_{n}\| \leq\sup_{x\in K}\|Wx-W_{n}x\|+\|W_{n}z_{n}-z_{n} \|. $$

From Lemma 2.5 and (3.20), we obtain that

$$ \lim_{n\rightarrow\infty}\|Wz_{n}-z_{n}\|=0. $$

Since $z_{n_{i}}\rightharpoonup q$, $W: C\rightarrow C$ is a nonexpansive mapping, we get by Lemma 2.8 that $q\in F(W)$. Then by Lemma 2.4, we know that $q\in F(W)$ is equivalent to $q\in \bigcap_{i=1}^{\infty}F(S_{i})$.

In the following, we show that $q\in(A+N)^{-1}0$.

As in [26], we have that for any $r>0$,

$$\begin{aligned} q\in(A+N)^{-1}0 \quad \Longleftrightarrow\quad &0\in Aq+Nq \\ \quad \Longleftrightarrow\quad &0\in rAq+rNq \\ \quad \Longleftrightarrow\quad &q-rAq\in q+rNq \\ \quad \Longleftrightarrow\quad &q=J_{r}(I-rA)q \\ \quad \Longleftrightarrow\quad &q\in \operatorname{Fix}\bigl(J_{r}(I-rA) \bigr). \end{aligned}$$

(3.21)

In view of condition (i), we can take $r_{0}\in[a,b]$. From Lemma 2.6 and $J_{r_{0}}$ is nonexpansive, we derive that

$$\begin{aligned} \bigl\Vert J_{r_{0}}(I-r_{0}A)u_{n}-y_{n} \bigr\Vert \leq&\bigl\Vert J_{r_{0}}(I-r_{0}A)u_{n}-J_{r_{0}}(I-r_{n}A)u_{n} \bigr\Vert \\ &{}+\bigl\Vert J_{r_{0}}(I-r_{n}A)u_{n}-y_{n} \bigr\Vert \\ \leq&\bigl\Vert (I-r_{0}A)u_{n}-(I-r_{n}A)u_{n} \bigr\Vert \\ &{}+\bigl\Vert J_{r_{0}}(I-r_{n}A)u_{n}-J_{r_{n}}(I-r_{n}A)u_{n} \bigr\Vert \\ \leq&|r_{n}-r_{0}|\cdot\bigl\Vert A(u_{n}) \bigr\Vert \\ &{}+\frac{|r_{n}-r_{0}|}{r_{0}}\bigl\Vert J_{r_{0}}(I-r_{n}A)u_{n} -(I-r_{n}A)u_{n}\bigr\Vert \rightarrow0. \end{aligned}$$

(3.22)

Observing that

$$ \bigl\Vert J_{r_{0}}(I-r_{0}A)u_{n}-u_{n} \bigr\Vert \leq\bigl\Vert J_{r_{0}}(I-r_{0}A)u_{n}-y_{n} \bigr\Vert +\|y_{n}-u_{n}\|, $$

and from (3.14) and (3.22), we have that

$$ \bigl\Vert J_{r_{0}}(I-r_{0}A)u_{n}-u_{n} \bigr\Vert \rightarrow0. $$

(3.23)

From the boundedness of $\{x_{n}\}$, we may assume that there exists a subsequence $\{x_{n_{i}}\}$ of $\{x_{n}\}$ such that $x_{n_{i}}\rightharpoonup q$, $q\in C$. By (3.12), we also have that $u_{n_{i}}\rightharpoonup q$, $q\in C$. On the other hand, from $r_{n}\rightarrow r_{0}\in[a,b]$, we have that $r_{n_{i}}\rightarrow r_{0}\in[a,b]$.

By use of (3.23), we have that

$$ \bigl\Vert J_{r_{0}}(I-r_{0}A)u_{n_{i}}-u_{n_{i}} \bigr\Vert \rightarrow0. $$

Since $J_{r_{0}}(I-r_{0}A)$ is nonexpansive, we have from Lemma 2.8 that $q=J_{r_{0}}(I-r_{0}A)q$. In virtue of (3.21), this means that $q\in(A+N)^{-1}0$.

By use of the similar proof method, we can also derive that

$$ \bigl\Vert J_{s_{0}}(I-s_{0}B)y_{n}-z_{n} \bigr\Vert \rightarrow0, $$

(3.24)

and we also have from (3.16) and (3.24) that

$$ \bigl\Vert J_{s_{0}}(I-s_{0}B)y_{n}-y_{n} \bigr\Vert \leq\bigl\Vert J_{s_{0}}(I-s_{0}B)y_{n}-z_{n} \bigr\Vert +\|y_{n}-z_{n}\|\rightarrow0. $$

From the above proof processing, $x_{n_{i}}\rightharpoonup q$ and $\|x_{n}-y_{n}\|\rightarrow0$, we have that $y_{n_{i}}\rightharpoonup q$, where $q\in C$. On the other hand, from $s_{n}\rightarrow s_{0}\in[e,g]$, we have that $s_{n_{i}}\rightarrow s_{0}\in[e,g]$.

In view of the above inequality, we have that

$$ \bigl\Vert J_{s_{0}}(I-s_{0}B)y_{n_{i}}-y_{n_{i}} \bigr\Vert \rightarrow0. $$

Since $J_{s_{0}}(I-s_{0}B)$ is nonexpansive, we also have from Lemma 2.8 that $q=J_{s_{0}}(I-s_{0}B)q$. By (3.21), we obtain that $q\in(B+M)^{-1}0$.

Step 7. We finally prove that $x_{n}\rightarrow q$ in norm.

Indeed, we derive from (3.1) and (3.3) that

$$\begin{aligned} \|x_{n+1}-q\|^{2} =&\alpha_{n}\bigl\langle f(y_{n})-q,x_{n+1}-q\bigr\rangle +\beta_{n}\langle W_{n}z_{n}-q,x_{n+1}-q\rangle \\ &{}+\gamma_{n}\langle x_{n}-q,x_{n+1}-q\rangle \\ \leq&\alpha_{n}k\|y_{n}-q\|\cdot\|x_{n+1}-q\|+ \alpha_{n}\bigl\langle f(q)-q,x_{n+1}-q\bigr\rangle \\ &{}+\beta_{n}\|z_{n}-q\|\cdot\|x_{n+1}-q\|+ \gamma_{n}\|x_{n}-q\| \cdot\|x_{n+1}-q\| \\ \leq&\bigl(1-\alpha_{n}(1-k)\bigr)\|x_{n}-q\|\cdot \|x_{n+1}-q\|+\alpha _{n}\bigl\langle f(q)-q,x_{n+1}-q \bigr\rangle \\ \leq&\frac{1}{2}\bigl(1-\alpha_{n}(1-k)\bigr) \bigl( \|x_{n}-q\|^{2}+\|x_{n+1}-q\| ^{2}\bigr) \\ &{}+\alpha_{n}\bigl\langle f(q)-q,x_{n+1}-q\bigr\rangle , \end{aligned}$$

it turns out that

$$ \|x_{n+1}-q\|^{2} \leq\bigl(1-\alpha_{n}(1-k)\bigr) \|x_{n}-q\|^{2}+2\alpha_{n}\bigl\langle f(q)-q,x_{n+1}-q\bigr\rangle . $$

By use of condition (iv), we obtain from Lemma 2.9 that

$$ \lim_{n\rightarrow\infty}\|x_{n}-q\|^{2}=0, $$

i.e., $x_{n}\rightarrow q$ as $n\rightarrow\infty$.

It is easy to see that the variational inequality (3.2) can be rewritten as

$$ \bigl\langle f(q)-q,q-x\bigr\rangle \geq0,\quad \forall x\in\Omega. $$

From Lemma 2.1, it is equivalent to the following fixed point equation:

$$ P_{\Omega}f(q)=q. $$

This completes the proof. □

4 Theorems

Theorem 4.1

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let $F: C\times C\rightarrow\mathbb{R}$ be a bifunction which satisfies (A1)-(A4), and let $f:C\rightarrow C$ be a k-contraction with the constant $k\in(0,1)$. Let $T: C\rightarrow H$ be a τ-ism mapping with $\tau>0$. Let $\{S_{i}: C\rightarrow C\}$ be a family of infinitely nonexpansive mappings. Assume that $\Omega=\operatorname{GEP}(F,T)\cap\bigcap_{i=1}^{\infty}F(S_{i}) \neq\emptyset$. Let $\{\alpha_{n}\}$, $\{\beta_{n}\}$ and $\{\gamma_{n}\}$ be sequences in $(0,1)$ such that $\alpha_{n}+\beta_{n}+\gamma_{n}=1$. Let $\{\lambda_{n}\}$ be a positive number sequence. Let $x_{1}\in C$ and let $\{x_{n}\}$ be a sequence generated by

$$ x_{n+1}=\alpha_{n}f(y_{n})+ \beta_{n}W_{n}u_{n}+\gamma_{n}x_{n}, \quad \forall n\in\mathbb{N}, $$

(4.1)

where $\{u_{n}\}$ is such that

$$ F(u_{n},y)+\langle Tx_{n},y-u_{n}\rangle+ \frac{1}{\lambda_{n}}\langle y-u_{n},u_{n}-x_{n}\rangle \geq0,\quad \forall y\in C, $$

and $\{W_{n}\}$ is the sequence generated in (2.1). Assume that the following conditions hold:

• (1) $0< c\leq\lambda_{n}\leq d<2\tau$ and $\lim_{n\rightarrow\infty}|\lambda_{n+1}-\lambda_{n}|=0$;

• (2) $\lim_{n\rightarrow\infty}\alpha_{n}=0$, $\sum_{n=1}^{\infty}\alpha_{n}=\infty$;

• (3) $0<\liminf_{n\rightarrow\infty}\gamma_{n}\leq\limsup_{n\rightarrow\infty }\gamma_{n}<1$,

where c and d are real constants. Then $\{x_{n}\}$ converges strongly to a point $q\in\Omega$, which solves uniquely the variational inequality (3.2).

Theorem 4.2

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let $F: C\times C\rightarrow\mathbb{R}$ be a bifunction which satisfies (A1)-(A4), and let $f:C\rightarrow C$ be a k-contraction with the constant $k\in(0,1)$. Let $A:C\rightarrow H$ be an α-ism mapping with $\alpha>0$. Let $T: C\rightarrow H$ be a τ-ism mapping with $\tau>0$. Let N be a maximal monotone operator on H such that the domain of N is included in C. Let $J_{r}=(I+rN)^{-1}$ be the resolvent of N for $r>0$. Let $\{S_{i}: C\rightarrow C\}$ be a family of infinitely nonexpansive mappings. Assume that $\Omega=\operatorname{GEP}(F,T)\cap\bigcap_{i=1}^{\infty}F(S_{i})\cap (A+N)^{-1}0\neq\emptyset$. Let $\{\alpha_{n}\}$, $\{\beta_{n}\}$ and $\{\gamma_{n}\}$ be sequences in $(0,1)$ such that $\alpha_{n}+\beta_{n}+\gamma_{n}=1$. Let $\{r_{n}\}$ and $\{\lambda_{n}\}$ be positive number sequences. Let $x_{1}\in C$ and let $\{x_{n}\}$ be a sequence generated by

$$ \left \{ \textstyle\begin{array}{l} y_{n}=J_{r_{n}}(u_{n}-r_{n}Au_{n}), \\ x_{n+1}=\alpha_{n}f(y_{n})+\beta_{n}W_{n}y_{n}+\gamma_{n}x_{n},\quad \forall n\in\mathbb{N}, \end{array}\displaystyle \right . $$

(4.2)

where $\{u_{n}\}$ is such that

$$ F(u_{n},y)+\langle Tx_{n},y-u_{n}\rangle+ \frac{1}{\lambda_{n}}\langle y-u_{n},u_{n}-x_{n}\rangle \geq0, \quad \forall y\in C, $$

and $\{W_{n}\}$ is the sequence generated in (2.1). Assume that the following conditions hold:

• (1) $0< a\leq r_{n}\leq b<2\alpha$ and $\lim_{n\rightarrow\infty}|r_{n+1}-r_{n}|=0$;

• (2) $0< c\leq\lambda_{n}\leq d<2\tau$ and $\lim_{n\rightarrow\infty}|\lambda_{n+1}-\lambda_{n}|=0$;

• (3) $\lim_{n\rightarrow\infty}\alpha_{n}=0$, $\sum_{n=1}^{\infty}\alpha_{n}=\infty$;

• (4) $0<\liminf_{n\rightarrow\infty}\gamma_{n}\leq\limsup_{n\rightarrow\infty }\gamma_{n}<1$,

where a, b, c, and d are real constants. Then $\{x_{n}\}$ converges strongly to a point $q\in\Omega$, which solves uniquely the variational inequality (3.2).

Theorem 4.3

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let $F: C\times C\rightarrow\mathbb{R}$ be a bifunction which satisfies (A1)-(A4), and let $f:C\rightarrow C$ be a k-contraction with the constant $k\in(0,1)$. Let $A:C\rightarrow H$ be an α-ism mapping with $\alpha>0$, let $B:C\rightarrow H$ be a β-ism mapping with $\beta>0$, and let $T: C\rightarrow H$ be a τ-ism mapping with $\tau>0$. Let $\{S_{i}: C\rightarrow C\}$ be a family of infinitely nonexpansive mappings. Assume that $\Omega=\operatorname{GEP}(F,T)\cap \bigcap_{i=1}^{\infty}F(S_{i})\cap \operatorname{VI}(C,A)\cap \operatorname{VI}(C,B)\neq\emptyset$. Let $\{\alpha_{n}\}$, $\{\beta_{n}\}$ and $\{\gamma_{n}\}$ be sequences in $(0,1)$ such that $\alpha_{n}+\beta_{n}+\gamma_{n}=1$. Let $\{r_{n}\}$, $\{\lambda_{n}\}$ and $\{s_{n}\}$ be positive number sequences. Let $x_{1}\in C$ and let $\{x_{n}\}$ be a sequence generated by

$$ \left \{ \textstyle\begin{array}{l} y_{n}=P_{C}(u_{n}-r_{n}Au_{n}), \\ x_{n+1}=\alpha_{n}f(y_{n})+\beta_{n}W_{n}P_{C}(y_{n}-s_{n}By_{n})+\gamma _{n}x_{n},\quad \forall n\in\mathbb{N}, \end{array}\displaystyle \right . $$

(4.3)

where $\{u_{n}\}$ is such that

$$ F(u_{n},y)+\langle Tx_{n},y-u_{n}\rangle+ \frac{1}{\lambda_{n}}\langle y-u_{n},u_{n}-x_{n}\rangle \geq0,\quad \forall y\in C, $$

and $\{W_{n}\}$ is the sequence generated in (2.1). Assume that the following conditions hold:

• (1) $0< a\leq r_{n}\leq b<2\alpha$ and $\lim_{n\rightarrow\infty}|r_{n+1}-r_{n}|=0$;

• (2) $0< c\leq\lambda_{n}\leq d<2\tau$ and $\lim_{n\rightarrow\infty}|\lambda_{n+1}-\lambda_{n}|=0$;

• (3) $0< e\leq s_{n}\leq g<2\beta$ and $\lim_{n\rightarrow\infty}|s_{n+1}-s_{n}|=0$;

• (4) $\lim_{n\rightarrow\infty}\alpha_{n}=0$, $\sum_{n=1}^{\infty}\alpha_{n}=\infty$;

• (5) $0<\liminf_{n\rightarrow\infty}\gamma_{n}\leq\limsup_{n\rightarrow\infty }\gamma_{n}<1$,

where a, b, c, d, e, and g are real constants. Then $\{x_{n}\}$ converges strongly to a point $q\in\Omega$, which solves uniquely the variational inequality (3.2).

Proof

Put $N=\partial i_{C}$ and $M=\partial i_{C}$ in Theorem 3.1. Then, for $r_{n}>0$ and $s_{n}>0$, we have that $J_{r_{n}}=P_{C}$ and $J_{s_{n}}=P_{C}$. Furthermore, we have $(A+\partial i_{C})^{-1}0=\operatorname{VI}(C,A)$ and $(B+\partial i_{C})^{-1}0=\operatorname{VI}(C,B)$. Indeed, for $q\in C$, we have

$$\begin{aligned} q\in(A+\partial i_{C})^{-1}0 \quad \Longleftrightarrow\quad &0\in Aq+\partial i_{C}(q) \\ \quad \Longleftrightarrow\quad &0\in Aq+N_{C}q \\ \quad \Longleftrightarrow\quad &-Aq\in N_{C}q \\ \quad \Longleftrightarrow\quad &\langle-Aq, p-q\rangle\leq0, \quad \forall p\in C \\ \quad \Longleftrightarrow\quad &\langle Aq, p-q\rangle\geq0,\quad \forall p\in C \\ \quad \Longleftrightarrow\quad &q\in \operatorname{VI}(C,A). \end{aligned}$$

Similarly, for $q\in C$, we also have

$$ q\in(B+\partial i_{C})^{-1}0\quad \Longleftrightarrow\quad q\in \operatorname{VI}(C,B). $$

Thus, we obtain the desired result by Theorem 3.1. □

Theorem 4.4

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let $F: C\times C\rightarrow\mathbb{R}$ be a bifunction which satisfies (A1)-(A4), and let $f:C\rightarrow C$ be a k-contraction with the constant $k\in(0,1)$. Let $A:C\rightarrow H$ be an α-ism mapping with $\alpha>0$, and let $B:C\rightarrow H$ be a β-ism mapping with $\beta>0$. Let $\{S_{i}: C\rightarrow C\}$ be a family of infinitely nonexpansive mappings. Assume that $\Omega=\operatorname{EP}(F)\cap\bigcap_{i=1}^{\infty}F(S_{i})\cap \operatorname{VI}(C,A)\cap \operatorname{VI}(C,B)\neq\emptyset$. Let $\{\alpha_{n}\}$, $\{\beta_{n}\}$ and $\{\gamma_{n}\}$ be sequences in $(0,1)$ such that $\alpha_{n}+\beta_{n}+\gamma_{n}=1$. Let $\{r_{n}\}$, $\{\lambda_{n}\}$ and $\{s_{n}\}$ be positive number sequences. Let $x_{1}\in C$ and let $\{x_{n}\}$ be a sequence generated by

$$ \left \{ \textstyle\begin{array}{l} y_{n}=P_{C}(u_{n}-r_{n}Au_{n}), \\ x_{n+1}=\alpha_{n}f(y_{n})+\beta_{n}W_{n}P_{C}(y_{n}-s_{n}By_{n})+\gamma _{n}x_{n},\quad \forall n\in\mathbb{N}, \end{array}\displaystyle \right . $$

(4.4)

where $\{u_{n}\}$ is such that

$$ F(u_{n},y)+\frac{1}{\lambda_{n}}\langle y-u_{n},u_{n}-x_{n} \rangle\geq0,\quad \forall y\in C, $$

and $\{W_{n}\}$ is the sequence generated in (2.1). Assume that the following conditions hold:

• (1) $0< a\leq r_{n}\leq b<2\alpha$ and $\lim_{n\rightarrow\infty}|r_{n+1}-r_{n}|=0$;

• (2) $0< c\leq\lambda_{n}\leq d<2\tau$ and $\lim_{n\rightarrow\infty}|\lambda_{n+1}-\lambda_{n}|=0$;

• (3) $0< e\leq s_{n}\leq g<2\beta$ and $\lim_{n\rightarrow\infty}|s_{n+1}-s_{n}|=0$;

• (4) $\lim_{n\rightarrow\infty}\alpha_{n}=0$, $\sum_{n=1}^{\infty}\alpha_{n}=\infty$;

• (5) $0<\liminf_{n\rightarrow\infty}\gamma_{n}\leq\limsup_{n\rightarrow\infty }\gamma_{n}<1$,

where a, b, c, d, e, and g are real constants. Then $\{x_{n}\}$ converges strongly to a point $q\in\Omega$, which solves uniquely the variational inequality (3.2).

Proof

In Theorem 4.3, put $T=0$. Then, for all $\tau\in(0,\infty)$, we have

$$ \langle x-y, Tx-Ty\rangle\geq\tau\|Tx-Ty\|^{2}, \quad \forall x,y\in C. $$

Taking $c, d\in(0,\infty)$ with $0< c< d<\infty$ and choosing a sequence $\{\lambda_{n}\}$ of real numbers with $c\leq\lambda_{n}\leq d$, we obtain the desired result by Theorem 4.3. □

Theorem 4.5

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let $P_{C}:H\rightarrow C$ be the metric projection. Let $f:C\rightarrow C$ be a k-contraction with the constant $k\in(0,1)$. Let $A:C\rightarrow H$ be an α-ism mapping with $\alpha>0$, let $B:C\rightarrow H$ be a β-ism mapping with $\beta>0$, and let $T: C\rightarrow H$ be a τ-ism mapping with $\tau>0$. Let $\{S_{i}: C\rightarrow C\}$ be a family of infinitely nonexpansive mappings. Assume that $\Omega=\operatorname{VI}(C,T)\cap\bigcap_{i=1}^{\infty}F(S_{i})\cap \operatorname{VI}(C,A)\cap \operatorname{VI}(C,B)\neq\emptyset$. Let $\{\alpha_{n}\}$, $\{\beta_{n}\}$ and $\{\gamma_{n}\}$ be sequences in $(0,1)$ such that $\alpha_{n}+\beta_{n}+\gamma_{n}=1$. Let $\{r_{n}\}$, $\{\lambda_{n}\}$ and $\{s_{n}\}$ be positive number sequences. Let $x_{1}\in C$ and let $\{x_{n}\}$ be a sequence generated by

$$ \left \{ \textstyle\begin{array}{l} u_{n}=P_{C}(x_{n}-\lambda_{n}Tx_{n}), \\ y_{n}=P_{C}(u_{n}-r_{n}Au_{n}), \\ x_{n+1}=\alpha_{n}f(y_{n})+\beta_{n}W_{n}P_{C}(y_{n}-s_{n}By_{n})+\gamma _{n}x_{n},\quad \forall n\in\mathbb{N}, \end{array}\displaystyle \right . $$

(4.5)

where $\{W_{n}\}$ is the sequence generated in (2.1). $\{\lambda_{n}\}$, $\{s_{n}\}$, $\{r_{n}\}$, $\{\alpha_{n}\}$ and $\{\gamma_{n}\}$ satisfy the following conditions:

• (1) $0< a\leq r_{n}\leq b<2\alpha$ and $\lim_{n\rightarrow\infty}|r_{n+1}-r_{n}|=0$;

• (2) $0< c\leq\lambda_{n}\leq d<2\tau$ and $\lim_{n\rightarrow\infty}|\lambda_{n+1}-\lambda_{n}|=0$;

• (3) $0< e\leq s_{n}\leq g<2\beta$ and $\lim_{n\rightarrow\infty}|s_{n+1}-s_{n}|=0$;

• (4) $\lim_{n\rightarrow\infty}\alpha_{n}=0$, $\sum_{n=1}^{\infty}\alpha_{n}=\infty$;

• (5) $0<\liminf_{n\rightarrow\infty}\gamma_{n}\leq\limsup_{n\rightarrow\infty }\gamma_{n}<1$,

where a, b, c, d, e, and g are real constants. Then $\{x_{n}\}$ converges strongly to a point $q\in\Omega$, which solves uniquely the variational inequality (3.2).

Proof

In Theorem 4.3, put $F=0$. Then we find that

$$ \langle Tx_{n}, y-u_{n}\rangle+\frac{1}{\lambda_{n}}\langle y-u_{n}, u_{n}-x_{n}\rangle\geq0,\quad \forall y\in C $$

is equivalent to

$$ \langle y-u_{n}, x_{n}-\lambda_{n}Tx_{n}-u_{n} \rangle\leq0,\quad \forall y\in C, $$

i.e., by Lemma 2.1, $u_{n}=P_{C}(x_{n}-\lambda_{n}Tx_{n})$. This completes the proof. □

Theorem 4.6

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let $P_{C}:H\rightarrow C$ be the metric projection. Let $f:C\rightarrow C$ be a k-contraction with the constant $k\in(0,1)$. Let $A:C\rightarrow H$ be an α-ism mapping with $\alpha>0$, let $B:C\rightarrow H$ be a β-ism mapping with $\beta>0$, and let $T: C\rightarrow H$ be a τ-ism mapping with $\tau>0$. Let $\{S_{i}: C\rightarrow C\}$ be a family of infinitely nonexpansive mappings. Let $Y: C\rightarrow H$ be a widely r-strict pseudo-contraction with $r<1$ ($r\in\mathbb{R}$). $Z: C\rightarrow H$ be a widely s-strict pseudo-contraction with $s<1$ ($s\in\mathbb{R}$). Assume that $\Omega=\operatorname{VI}(C,T)\cap\bigcap_{i=1}^{\infty}F(S_{i})\cap \operatorname{Fix}(Y)\cap \operatorname{Fix}(Z)\neq\emptyset$. Let $\{\alpha_{n}\}$, $\{\beta_{n}\}$ and $\{\gamma_{n}\}$ be sequences in $(0,1)$ such that $\alpha_{n}+\beta_{n}+\gamma_{n}=1$. Let $\{r_{n}\}$, $\{\lambda_{n}\}$ and $\{s_{n}\}$ be positive number sequences. Let $x_{1}\in C$ and let $\{x_{n}\}$ be a sequence generated by

$$ \left \{ \textstyle\begin{array}{l} u_{n}=P_{C}(x_{n}-\lambda_{n}Tx_{n}), \\ y_{n}=P_{C}(u_{n}-r_{n}Au_{n}), \\ x_{n+1}=\alpha_{n}f(y_{n})+\beta_{n}W_{n}P_{C}(y_{n}-s_{n}By_{n})+\gamma _{n}x_{n},\quad \forall n\in\mathbb{N}, \end{array}\displaystyle \right . $$

(4.6)

where $\{W_{n}\}$ is the sequence generated in (2.1). $\{\lambda_{n}\}$, $\{s_{n}\}$, $\{r_{n}\}$, $\{\alpha_{n}\}$ and $\{\gamma_{n}\}$ satisfy conditions (i)-(v) respectively, which appear in Theorem  3.1. $\{t_{n}\}$ satisfies

• (1) $\{t_{n}\}\subset(-\infty, 1)$;

• (2) $r\leq t_{n}\leq l<1$;

• (3) $\sum_{n=1}^{\infty}|t_{n}-t_{n+1}|<\infty$.

Then $\{x_{n}\}$ converges strongly to a point $q_{0}\in\Omega$.

Proof

Put $N=\partial i_{C}$ and $A=I-Y$ in Theorem 3.1. Furthermore, put $p=1-l$, $r_{n}=1-t_{n}$ and $2\alpha=1-r$ in Theorem 3.1. From $\{t_{n}\}\subset(-\infty, 1)$ and $r\leq t_{n}\leq l<1$, we get $\{r_{n}\}\subset(0,\infty)$ and $0< p\leq r_{n}\leq2\alpha$. We also get

$$ \sum_{n=1}^{\infty}|r_{n+1}-r_{n}|= \sum_{n=1}^{\infty }|t_{n+1}-t_{n}|< \infty $$

and

$$ I-r_{n}A=I-(1-t_{n}) (I-Y)=(1-t_{n})Y+t_{n}I. $$

Furthermore, we have $(A+\partial i_{C})^{-1}0=\operatorname{Fix}(Y)$. Indeed, for $q\in C$, we have

$$\begin{aligned} q\in(A+\partial i_{C})^{-1}0 \quad \Longleftrightarrow\quad & 0\in Aq+\partial i_{C}(q) \\ \quad \Longleftrightarrow\quad & 0\in q-Yq+N_{C}(q) \\ \quad \Longleftrightarrow\quad & Yq-q\in N_{C}(q) \\ \quad \Longleftrightarrow\quad & \langle Yq-q,p-q\rangle\leq0, \quad \forall p \in C \\ \quad \Longleftrightarrow\quad & P_{C}Y(q)=q. \end{aligned}$$

Since $\operatorname{Fix}(Y)\neq\emptyset$, we get from [27] that $\operatorname{Fix}(P_{C}Y)=\operatorname{Fix}(Y)$. Thus we obtain the desired result by Theorem 3.1.

Similarly, put $M=\partial i_{C}$ and $B=I-Z$ in Theorem 3.1. Furthermore, put $p=1-l$, $s_{n}=1-t_{n}$ and $2\beta=1-s$ in Theorem 3.1. From $\{t_{n}\}\subset(-\infty, 1)$ and $s\leq t_{n}\leq l<1$, we get $\{s_{n}\}\subset(0,\infty)$ and $0< p\leq s_{n}\leq2\beta$. We also get

$$ \sum_{n=1}^{\infty}|s_{n+1}-s_{n}|= \sum_{n=1}^{\infty }|t_{n+1}-t_{n}|< \infty $$

and

$$ I-s_{n}B=I-(1-t_{n}) (I-Z)=(1-t_{n})Z+t_{n}I. $$

Furthermore, we also obtain $(B+\partial i_{C})^{-1}0=\operatorname{Fix}(Z)$. □

Due to Section 3 and Section 4, we will give our conclusion in the next section.

5 Conclusion

Methods for solving a generalized equilibrium problem, a fixed point problem and the zero points of the sum of two operators have been studied by many authors respectively. However, in this paper, for finding a common solution of the above three problems, we proposed a new regularization algorithm, and it is proved that the sequence generated by this algorithm has the strong convergence. And then some corollaries to this strong convergence theorem are presented, which play important roles in nonlinear analysis and optimization problem.