Strong convergence of the viscosity approximation method for the split generalized equilibrium problem

Abstract

In this paper, we consider a common solution of three problems in real Hilbert spaces including the split generalized equilibrium problem, the variational inequality problem and the fixed point problem for nonexpansive multivalued mappings. For finding the solution, we present a modified viscosity approximation method and prove a strong convergence theorem under mild conditions. Moreover, we also provide a numerical example to illustrate the convergence behavior of the proposed iterative method.

1 Introduction

Let $H_1$ and $H_2$ be real Hilbert spaces with inner product $\langle \cdot ,\cdot \rangle$ and induced norm $\Vert \cdot \Vert$. Let C and Q be nonempty closed convex subsets of $H_1$ and $H_2$, respectively. We denote the strong convergence and the weak convergence of the sequence $\{x_n\}$ to a point x in a Hilbert space by $x_n\rightarrow x$ and $x_n\rightharpoonup x$, respectively.

The classical variational inequality problem is the problem to find $u\in C$ such that

$$\begin{aligned} \left\langle {Du,v - u} \right\rangle \ge 0,\,\,\, \forall v\in C, \end{aligned}$$

(1.1)

where $D:C\rightarrow H_1$ is a bounded linear operator. The solution set of the variational inequality problem (1.1) is denoted by VI(C, D). It is well known that the variational inequality problem (1.1) has a unique solution when the operator D is a strongly monotone and Lipschitz continuous mapping on C.

The equilibrium problem for a bifunction $F:C\times C\rightarrow \mathbb {R}$ is to find a point $x^*\in C$ such that

$$\begin{aligned} F(x^*,x)\ge 0,\,\,\,\forall x\in C. \end{aligned}$$

(1.2)

The solution set of the equilibrium problem (1.2) is denoted by EP(F). It is easy to see that $EP(F)=VI(C, D)$ when $F(x,y) = \left\langle {Dx,y - x} \right\rangle$ for all $x,y\in C$. Let $\varphi :C\times C\rightarrow \mathbb {R}$ be a nonlinear bifunction, then the generalized equilibrium problem is to find $x^*\in C$ such that

$$\begin{aligned} F(x^*,x)+\varphi (x^*,x)\ge 0,\,\,\,\forall x\in C. \end{aligned}$$

(1.3)

The solution set of the generalized equilibrium problem (1.3) is denoted by $GEP(F,\varphi )$. In particular, if $\varphi =0$, this problem reduces to the equilibrium problem (1.2).

In this paper, we are interested to find the solution of the split generalized equilibrium problem which is introduced by Kazmi and Rizvi [16] in 2013 as the following problem; find $x^*\in C$ such that

$$\begin{aligned} F_1(x^*,x)+\varphi _1(x^*,x)\ge 0,\,\,\,\forall x\in C \end{aligned}$$

(1.4)

and such that

$$\begin{aligned} y^*=Ax^*\in Q\,\,\text {solves}\,\,F_2(y^*,y)+\varphi _2(y^*,y)\ge 0,\,\,\,\forall y\in Q, \end{aligned}$$

(1.5)

where $F_1,\varphi _1:C\times C\rightarrow \mathbb {R}$ and $F_2,\varphi _2:Q\times Q\rightarrow \mathbb {R}$ are nonlinear bifunctions and $A:H_1\rightarrow H_2$ is a bounded linear operator.

The solution set of the split generalized equilibrium problem (1.4)–(1.5) is denoted by

$$\begin{aligned} SGEP(F_1,\varphi _1,F_2,\varphi _2):=\{x^*\in C: x^*\in GEP(F_1,\varphi _1)\,\,\text {and } Ax^*\in GEP(F_2,\varphi _2)\}. \end{aligned}$$

If $\varphi _1=0$ and $\varphi _2=0$, the split generalized equilibrium problem reduces to the split equilibrium problem; see [27]. If $F_2=0$ and $\varphi _2=0$, the split generalized equilibrium problem reduces to the generalized equilibrium problem considered by Cianciaruso et al. [9].

The split generalized equilibrium problem generalizes multiple-sets split feasibility problem. It also includes as special case, the split variational inequality problem [3] which is the generalization of split zero problems and split feasibility problems, see for details [4,5,6, 8, 11, 12, 15, 19, 20, 25, 27, 30, 32, 33]. This formalism is also at the core of modeling of many inverse problems arising for phase retrieval and other real world problems; for instance, in sensor networks in computerized tomography and data compression; see, e.g., [1, 2, 7, 10].

A single-valued mapping $S:C\rightarrow C$ is called nonexpansive if

$$\begin{aligned} \Vert Sx-Sy\Vert \le \Vert x-y\Vert ,\,\,\,x,y\in C. \end{aligned}$$

A point $x\in C$ is called a fixed point of a mapping S if $Sx=x$ and denote by F(S) the set of all fixed points of S. A single-valued mapping $g:C\rightarrow C$ is called contraction if there exists a constant $k\in (0,1)$ such that $\Vert g(x)-g(y)\Vert \le k\Vert x-y\Vert$ for all $x,y\in C$. There are some algorithms for approximation of fixed points of a nonexpansive single-valued mapping. In 2000, Moudafi [26] introduced the following iterative algorithm, which is known as the viscosity approximation method, for finding a fixed point of a nonexpansive single-valued mapping in Hilbert spaces under some suitable conditions:

$$\begin{aligned} x_{n+1}=\alpha _ng(x_n)+(1-\alpha _n) Sx_n,\,\,\,n\in \mathbb {N}, \end{aligned}$$

(1.6)

where $\{\alpha _n\}$ is a sequence in [0, 1], g is a contraction and S is a nonexpansive single-valued mapping on C. We note that the Halpern approximation method [14],

$$\begin{aligned} x_{n+1}=\alpha _nu+(1-\alpha _n) Sx_n,\,\,\,n\in \mathbb {N}, \end{aligned}$$

(1.7)

where u is a fixed element in C, is a special case of (1.6).

Viscosity approximation methods are very important because they are applied to linear programming, convex optimization and monotone inclusions. In Hilbert spaces, many authors have studied the fixed points problems of the fixed points for the nonexpansive single-valued mappings and monotone mappings by the viscosity approximation methods, and obtained a series of good results (see [12, 22, 23, 26, 29, 34, 38]).

Recently, Kazmi and Rizvi [17] introduced the iterative process combined with Halpern approximation method (1.7) for finding a common solution of the split equilibrium problem, the variational inequality problem and the fixed point problem for nonexpansive single-valued mapping in real Hilbert spaces.

Motivated by the works of Kazmi and Rizvi [16, 17] and Moudafi [26], we introduce and study a modified viscosity approximation method for approximating a common solution of three problems in real Hilbert spaces including the split generalized equilibrium problem, the variational inequality problem for a $\tau$-inverse strongly monotone mapping and the fixed point problem for a nonexpansive multivalued mapping. We prove the strong convergence of the purposed iterative method under mild conditions. Our results extend and improve recent results announced by many others. Moreover, we give a numerical example to illustrate our main result.

2 Preliminaries

In this section, we recall some concepts and results which are needed in sequel. Let C be a nonempty closed convex subset of a real Hilbert space H. We denote by CB(C) and K(C) the collections of all nonempty closed bounded subsets and nonempty compact subsets of C, respectively. The Hausdorff metric $\mathcal {H}$ on CB(C) is defined by

$$\begin{aligned} \mathcal {H}(B_1,B_2):=\max \left\{ \sup _{x\in B_1} \text {dist} (x,B_2),\sup _{y\in B_2} \text {dist} (y,B_1)\right\} ,\,\,\,\,\forall B_1,B_2 \in CB(C), \end{aligned}$$

where $\text {dist}(x,B_2)=\inf \{d(x,y):y\in B_2\}$ is the distance from a point x to a subset $B_2$. Let $S:C\rightarrow CB(C)$ be a multivalued mapping. An element $x\in C$ is called a fixed point of a multivalued mapping S if $x\in Sx$. The set of all fixed points of S is denoted by F(S). Recall that a multivalued mapping $S:C\rightarrow CB(C)$ is called nonexpansive if

$$\begin{aligned} \mathcal {H}(Sx,Sy)\le \Vert x-y\Vert ,\,\,\forall x,y\in C. \end{aligned}$$

If S is a nonexpansive single-valued mapping on a closed convex subset of a Hilbert space, then F(S) is always closed and convex. The closedness of F(S) can be easily extended to the multivalued case. But the convexity of F(S) cannot be extended (see, e.g., [18]). However, if S is a nonexpansive multivalued mapping and $Sp=\{p\}$ for each $p\in F(S)$, then F(S) is always closed and convex.

For every point x in a real Hilbert space H, there exists a unique nearest point of C, denoted by $P_Cx$, such that $\Vert x-P_Cx\Vert \le \Vert x-y\Vert$ for all $y\in C$. Such a $P_C$ is called the metric projection from H onto C. It means that $z={P_C}x$ if and only if $\Vert x-z\Vert \le \Vert x-y\Vert$ for all $y\in C$. Moreover, it is equivalent to

$$\begin{aligned} \left\langle {x-z,y-z} \right\rangle \le 0,\,\,\forall y\in C. \end{aligned}$$

(2.1)

It is well known that $P_C$ is a nonexpansive mapping and is characterized by the following properties:

• (i) ${\left\| {{P_C}x - {P_C}y} \right\| ^2} \le \left\langle {x - y,{P_C}x - {P_C}y} \right\rangle ,\,\forall x,y\in H$;

• (ii) ${\left\| {x - {P_C}x} \right\| ^2} + {\left\| {y - {P_C}x} \right\| ^2} \le {\left\| {x - y} \right\| ^2},\,\forall x\in H,y\in C$;

• (iii) ${\left\| {x - y} \right\| ^2} - {\left\| {{P_C}x - {P_C}y} \right\| ^2} \le {\left\| {(x - y) - ({P_C}x - {P_C}y)} \right\| ^2},\,\forall x,y\in H$.

For more properties of $P_C$ can be found in [13, 21].

We now give some concepts of the monotonicity of a nonlinear mapping.

Definition 2.1

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. A mapping $D:C\rightarrow H$ is said to be:

(i):

monotone if $\left\langle {Dx-Dy,x-y} \right\rangle \ge 0,\,\forall x,y\in C$;

(ii):

$\tau$-inverse strongly monotone if there exists a constant $\tau >0$ such that

$$\begin{aligned} \left\langle {Dx-Dy,x-y} \right\rangle \ge \tau \Vert Dx-Dy\Vert ^2,\,\forall x,y\in C. \end{aligned}$$

It is easy to observe that every $\tau$-inverse strongly monotone mapping D is monotone.

Lemma 2.2

([36]) Let H be a real Hilbert space, C be a nonempty closed convex subset of H, and D be a mapping of C into H. Let $u\in C$. Then, for $\lambda >0$, $u=P_C (I-\lambda D)u$ if and only if $u\in VI(C,D).$

Lemma 2.3

([28]) Let $\{x_n\}$ be any sequence in a Hilbert space H. Then, we have $\{x_n\}$ satisfies Opial’s condition, that is, if $x_n\rightharpoonup x$, then the inequality

$$\begin{aligned} \limsup _{n\rightarrow \infty }\Vert x_n-x\Vert < \limsup _{n\rightarrow \infty }\Vert x_n-y\Vert \end{aligned}$$

holds for every $y\in H$ with $y\ne x$.

Lemma 2.4

([39]) Let H be a Hilbert space. Let $x,y,z\in H$ and $\alpha , \beta , \gamma \in [0,1]$ such that $\alpha + \beta + \gamma = 1$. Then, we have

$$\begin{aligned} \left\| \alpha x + \beta y + \gamma z \right\| ^2 = \alpha \left\| x\right\| ^2 + \beta \left\| y\right\| ^2 + \gamma \left\| z\right\| ^2 - \alpha \beta \left\| x-y\right\| ^2 - \alpha \gamma \left\| x-z\right\| ^2 - \beta \gamma \left\| y-z\right\| ^2. \end{aligned}$$

Lemma 2.5

In a real Hilbert space H, the following inequalities hold:

(i):

$\Vert x-y\Vert ^2 \le \Vert x\Vert ^2-\Vert y\Vert ^2 - 2\langle x-y,y\rangle ,\,\forall x,y\in H$;

(ii):

$\Vert x+y\Vert ^2 \le \Vert x\Vert ^2 + 2\langle y, x + y\rangle ,\,\forall x,y\in H$.

Lemma 2.6

([37]) Let $\left\{ {s_n } \right\}$ be a sequence of nonnegative real numbers satisfying

$$\begin{aligned} s_{n + 1} = \left( {1 - \alpha _n } \right) s_n + \delta _n ,\,\,\forall n \ge 1, \end{aligned}$$

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

(i):

$\sum _{n = 1}^\infty {\alpha _n } = \infty$;

(ii):

$\mathop {\lim \sup }_{n \rightarrow \infty } \frac{\delta _n}{\alpha _n} \le 0$ or $\sum _{n = 1}^\infty {\left| {\delta _n } \right| } < \infty$.

Then $\mathop {\lim }_{n \rightarrow \infty } s_n = 0$.

Lemma 2.7

([35]) Let $\left\{ {x_n } \right\}$ and $\left\{ {w_n } \right\}$ be bounded sequences in a Banach space X and let $\left\{ {\beta _n } \right\}$ be a sequence in [0, 1] with $0< \mathop {\lim \inf }_{n \rightarrow \infty } \beta _n \le \mathop {\lim \sup }_{n \rightarrow \infty } \beta _n < 1$. Suppose $x_{n + 1} = \left( {1 - \beta _n } \right) w_n + \beta _n x_n$ for all integer $n \ge 1$ and

$$\begin{aligned} \mathop {\lim \sup }\limits _{n \rightarrow \infty } \left( {\left\| {w_{n + 1} - w_n } \right\| - \left\| {x_{n + 1} - x_n } \right\| } \right) \le 0. \end{aligned}$$

Then $\mathop {\lim }_{n \rightarrow \infty } \left\| {w_n - x_n } \right\| =0$.

For solving the generalized equilibrium problem, we assume that the bifunctions $F_1:C\times C \rightarrow \mathbb {R}$ and $\varphi _1:C\times C \rightarrow \mathbb {R}$ satisfy the following assumption:

Assumption 2.8

Let C be a nonempty closed convex subset of a Hilbert space $H_1$. Let $F_1:C\times C \rightarrow \mathbb {R}$ and $\varphi _1:C\times C \rightarrow \mathbb {R}$ be two bifunctions satisfy the following conditions:

(A1):

$F_1(x,x)=0$ for all $x\in C;$

(A2):

$F_1$ is monotone, i.e., $F_1(x,y)+F_1(y,x)\le 0,\, \forall x,y \in C;$

(A3):

$F_1$is upper hemicontinuous, i.e., for each $x,y,z \in C,$ $\lim _{t\downarrow 0}F_1(tz+(1-t)x,y)\le F_1(x,y);$

(A4):

For each $x\in C,$ $y\mapsto F_1(x,y)$ is convex and lower semicontinuous;

(A5):

$\varphi _1(x,x)\ge 0$ for all $x\in C$;

(A6):

For each $y\in C,$ $x\mapsto \varphi _1(x,y)$ is upper semicontinuous;

(A7):

For each $x\in C,$ $y\mapsto \varphi _1(x,y)$ is convex and lower semicontinuous,

and assume that for fixed $r>0$ and $z\in C$, there exists a nonempty compact convex subset K of $H_1$ and $x\in C\cap K$ such that

$$\begin{aligned} F_1(y,x)+\varphi _1(y,x)+\frac{1}{r}\langle y-x,x-z\rangle < 0, \ \ \forall y\in C\backslash K. \end{aligned}$$

Lemma 2.9

([24]) Let C be a nonempty closed convex subset of a Hilbert space $H_1$. Let $F_1:C\times C\rightarrow \mathbb {R}$ and $\varphi _1:C\times C \rightarrow \mathbb {R}$ be two bifunctions satisfy Assumption 2.8. Assume $\varphi _1$ is monotone. For $r>0$ and $x \in H_1$. Define a mapping $T^{(F_1,\varphi _1)}_r:H_1\rightarrow C$ as follows:

$$\begin{aligned} T^{(F_1,\varphi _1)}_r(x)=\left\{ z\in C: F_1(z,y)+\varphi _1(z,y)+\frac{1}{r}\langle y-z,z-x\rangle \ge 0, \ \ \forall y\in C\right\} , \end{aligned}$$

for all $x\in H_1$. Then, the following conclusions hold:

(1):

For each $x\in H_1$, $T^{(F_1,\varphi _1)}_r\ne \emptyset$;

(2):

$T^{(F_1,\varphi _1)}_r$ is single-valued;

(3):

$T^{(F_1,\varphi _1)}_r$ is firmly nonexpansive, i.e., for any $x,y\in H_1,$

$$\begin{aligned} \left\| T^{(F_1,\varphi _1)}_rx-T^{(F_1,\varphi _1)}_ry\right\| ^2\le \left\langle T^{(F_1,\varphi _1)}_rx-T^{(F_1,\varphi _1)}_ry,x-y\right\rangle ; \end{aligned}$$

(4):

$F\left( T^{(F_1,\varphi _1)}_r\right) =GEP(F_1,\varphi _1);$

(5):

$GEP(F_1,\varphi _1)$ is compact and convex.

Further, assume that $F_2:Q\times Q\rightarrow \mathbb {R}$ and $\varphi _2:Q\times Q\rightarrow \mathbb {R}$ satisfying Assumption 2.8, where Q is a nonempty closed and convex subset of a Hilbert space $H_2$. For each $s>0$ and $w\in H_2$, define a mapping $T^{(F_2,\varphi _2)}_s:H_2\rightarrow Q$ as follows:

$$\begin{aligned} T^{(F_2,\varphi _2)}_s(v)=\left\{ w\in Q: F_2(w,d)+\varphi _2(w,d)+\frac{1}{r}\langle d-w,w-v\rangle \ge 0, \ \ \forall d\in Q\right\} . \end{aligned}$$

Then we have the following:

(6):

For each $v\in H_2$, $T^{(F_2,\varphi _2)}_s\ne \emptyset$;

(7):

$T^{(F_2,\varphi _2)}_s$ is single-valued;

(8):

$T^{(F_2,\varphi _2)}_s$ is firmly nonexpansive;

(9):

$F\left( T^{(F_2,\varphi _2)}_s\right) =GEP(F_2,\varphi _2);$

(10):

$GEP(F_2,\varphi _2)$ is closed and convex, where $GEP(F_2,\varphi _2)$ is the solution set of the following generalized equilibrium problem:

Find $y^*\in Q$ such that $F_2(y^*,y)+\varphi _2(y^*,y)\ge 0$ for all $y\in Q$.

Further, it is easy to prove that $SGEP(F_1,\varphi _1,F_2,\varphi _2)$ is closed and convex.

Lemma 2.10

([9]) Let C be a nonempty closed convex subset of a Hilbert space $H_1$. Let $F_1:C\times C\rightarrow \mathbb {R}$ and $\varphi _1:C\times C \rightarrow \mathbb {R}$ be two bifunctions satisfy Assumption 2.8 and let $T^{(F_1,\varphi _1)}_r$ be defined as in Lemma 2.9 for $r>0$. Let $x,y\in H_1$ and $r_1,r_2>0$. Then,

$$\begin{aligned} \left\| T^{(F_1,\varphi _1)}_{r_2} y - T^{(F_1,\varphi _1)}_{r_1}x\right\| \le \Vert y-x\Vert + \left| \frac{r_2-r_1}{r_2}\right| \left\| T^{(F_1,\varphi _1)}_{r_2}y -y\right\| . \end{aligned}$$

3 Main results

In this section, we prove the strong convergence theorems for finding a common element of the set of solutions of the split generalized equilibrium problem, the variational inequality problem for a $\tau$-inverse strongly monotone mapping and the fixed point problem for a nonexpansive multivalued mapping in real Hilbert spaces.

Theorem 3.1

Let C be a nonempty closed convex subset of a real Hilbert space $H_1$ and Q be a nonempty closed convex subset of a real Hilbert space $H_2$. Let $A: H_1\rightarrow H_2$ be a bounded linear operator, $D:C\rightarrow H_1$ be a $\tau$-inverse strongly monotone mapping, and $S:C\rightarrow K(C)$ be a nonexpansive multivalued mapping. Let $F_1, \varphi _1:C\times C\rightarrow \mathbb {R}$, $F_2,\varphi _2:Q\times Q\rightarrow \mathbb {R}$ be bifunctions satisfying Assumption 2.8. Let $\varphi _1,\varphi _2$ be monotone, $\varphi _1$ be upper hemicontinuous, and $F_2$ and $\varphi _2$ be upper semicontinuous in the first argument. Assume that $\Gamma =F(S)\cap SGEP(F_1,\varphi _1,F_2,\varphi _2) \cap \text {VI(C,D)}\ne \emptyset$ and $Sp=\{p\}$ for all $p\in F(S)$. Let g be a contraction of C into itself with coefficient $k\in (0,1)$. Let $\{x_n\}$ be a sequence generated by $x_1\in C$ and

$$\begin{aligned} \left\{ \begin{aligned}&u_n=T_{r_n}^{(F_1,\varphi _1)}(I-\xi A^*(I-T_{r_n}^{(F_2,\varphi _2)})A)x_n,\\&y_n=P_C(u_n-\lambda _nDu_n),\\&x_{n+1}=\alpha _ng(x_n)+\beta _n x_n +\gamma _n z_n,\,\,\,n\in \mathbb {N}, \end{aligned} \right. \end{aligned}$$

(3.1)

where $z_n\in Sy_n$ such that $\Vert z_{n+1}-z_n\Vert \le \mathcal {H}(Sy_{n+1},Sy_n)+\varepsilon _n$, $\lim _{n\rightarrow \infty }\varepsilon _n = 0$, and $r_n \in (0,1)$, $\lambda _n\in [a,b]$ for some a, b with $0<a<b<2\tau$, and $\xi \in (0,\frac{1}{L})$ with L is the spectral radius of the operator $A^*A$ and $A^*$ is the adjoint of A and $\{\alpha _n\}$, $\{\beta _n\}$ and $\{\gamma _n\}$ are the sequences in (0, 1) satisfy $\alpha _n+\beta _n+\gamma _n=1$ for all $n\in \mathbb {N}$. Suppose the conditions are satisfied:

(C1):

$\lim _{n\rightarrow \infty }\alpha _n=0$ and $\sum _{n=1}^{\infty }\alpha _n=\infty$;

(C2):

$0<\liminf _{n\rightarrow \infty }\beta _n\le \limsup _{n\rightarrow \infty }\beta _n<1$;

(C3):

$\gamma _n\in [c,1]$ for some $c\in (0,1)$;

(C4):

$\liminf _{n\rightarrow \infty }r_n>0$ and $\sum _{n=1}^{\infty }|r_{n+1}-r_n|<\infty$;

(C5):

$\lim _{n\rightarrow \infty }|\lambda _{n+1}-\lambda _n|=0$.

Then the sequence $\{x_n\}$ converges strongly to $z\in \Gamma$, where $z=P_{\Gamma } g(z)$.

Proof

We shall divide our proof into six steps.

Step 1. We will show that $\{x_n\}$ is bounded. Since D is $\tau$-inverse strongly monotone mapping, we obtain $\langle x-y,Dx-Dy\rangle \ge \tau \Vert Dx-Dy\Vert ^2$. Then for any $x,y \in C$, we have

$$\begin{aligned} \Vert (I-\lambda _nD)x-(I-\lambda _nD)y\Vert ^2&= \Vert (x-y)-\lambda _n(Dx-Dy)\Vert ^2 \nonumber \\&= \Vert x-y\Vert ^2-2\lambda _n\langle x-y,Dx-Dy\rangle +\lambda _n^2\Vert Dx-Dy\Vert ^2 \nonumber \\&\le \Vert x-y\Vert ^2-2\tau \lambda _n \Vert Dx-Dy\Vert ^2 + \lambda _n\Vert Dx-Dy\Vert ^2 \nonumber \\&= \Vert x-y\Vert ^2-\lambda _n(2\tau -\lambda _n)\Vert Dx-Dy\Vert ^2 \nonumber \\&\le \Vert x-y\Vert ^2. \end{aligned}$$

(3.2)

This shows that the mapping $(I-\lambda _nD)$ is a nonexpansive mapping from C to $H_1$.

Let $p\in \Gamma$, that is, $p\in SGEP(F_1,\varphi _1,F_2,\varphi _2)$, we have $p=T_{r_n}^{(F_1,\varphi _1)}p$ and $Ap=T_{r_n}^{(F_2,\varphi _2)}Ap$. Thus, we get that

$$\begin{aligned} \Vert u_n-p\Vert ^2&= \Vert T^{(F_1,\varphi _1)}_{r_n}\left( I-\xi A^*\left( I-T^{(F_2,\varphi _2)}_{r_n}\right) A\right) x_n-T^{(F_1,\varphi _1)}_{r_n}p\Vert ^2\nonumber \\&\le \Vert (I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)x_n-p\Vert ^2\nonumber \\&\le \Vert x_n-p\Vert ^2+\xi ^2\Vert A^*(I-T^{(F_2,\varphi _2)}_{r_n})Ax_n\Vert ^2+2\xi \langle p-x_n, A^*(I-T^{(F_2,\varphi _2)}_{r_n})Ax_n\rangle \nonumber \\&\le \Vert x_n-p\Vert ^2+\xi ^2\langle Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n, AA^*(I-T^{(F_2,\varphi _2)}_{r_n})Ax_n \rangle \nonumber \\&\quad +\,2\xi \langle A(p-x_n), Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n\rangle \nonumber \\&\le \Vert x_n-p\Vert ^2+L\xi ^2\langle Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n, Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n \rangle \nonumber \\&\quad +\,2\xi \langle A(p-x_n)+(Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n) \nonumber \\&\quad -(Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n), Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n\rangle \nonumber \\&\le \Vert x_n-p\Vert ^2+L\xi ^2\Vert Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n\Vert ^2\nonumber \\&\quad +\,2\xi \left( \langle Ap-T^{(F_2,\varphi _2)}_{r_n}Ax_n, Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n\rangle -\Vert Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n\Vert ^2\right) \nonumber \\&\le \Vert x_n-p\Vert ^2+L\xi ^2\Vert Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n\Vert ^2\nonumber \\&\quad +\,2\xi \left( \frac{1}{2}\Vert Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n\Vert ^2-\Vert Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n\Vert ^2\right) \nonumber \\&= \Vert x_n-p\Vert ^2+\xi (L\xi -1)\Vert Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n\Vert ^2. \end{aligned}$$

(3.3)

Since $\xi \in (0,\frac{1}{L})$, we obtain

$$\begin{aligned} \Vert u_n-p\Vert ^2\le \Vert x_n-p\Vert ^2. \end{aligned}$$

(3.4)

Now, we estimate

$$\begin{aligned} \Vert y_n-p\Vert ^2&= \Vert P_C(u_n-\lambda _nDu_n)-P_C(p-\lambda _nDp)\Vert ^2 \nonumber \\&\le \Vert (u_n-\lambda _nDu_n)-(p-\lambda _nDp)\Vert ^2 \nonumber \\&\le \Vert u_n-p\Vert ^2-\lambda _n(2\tau -\lambda _n)\Vert Du_n-Dp\Vert ^2 \nonumber \\&\le \Vert u_n-p\Vert ^2 \nonumber \\&\le \Vert x_n-p\Vert ^2. \end{aligned}$$

(3.5)

Further, we estimate

$$\begin{aligned} \Vert x_{n+1}-p\Vert ^2&=\Vert \alpha _ng(x_n)+\beta _n x_n +\gamma _n z_n-p\Vert \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert +\beta _n\Vert x_n-p\Vert +\gamma _n\Vert z_n-p\Vert \nonumber \\&\le \alpha _n(\Vert g(x_n)-g(p)\Vert +\Vert g(p)-p\Vert )+\beta _n\Vert x_n-p\Vert +\gamma _n\text {dist}(z_n,Sp) \nonumber \\&\le \alpha _n(k\Vert x_n-p\Vert +\Vert g(p)-p\Vert )+\beta _n\Vert x_n-p\Vert +\gamma _n\text {dist}(z_n,Sp) \nonumber \\&=(k\alpha _n+\beta _n)\Vert x_n-p\Vert +\alpha _n\Vert g(p)-p\Vert +\gamma _n\text {dist}(z_n,Sp) \nonumber \\&\le (k\alpha _n+\beta _n)\Vert x_n-p\Vert +\alpha _n\Vert g(p)-p\Vert +\gamma _n\mathcal {H}(Sy_n,Sp) \nonumber \\&\le (k\alpha _n+\beta _n)\Vert x_n-p\Vert +\alpha _n\Vert g(p)-p\Vert +\gamma _n\Vert y_n-p\Vert \nonumber \\&\le (k\alpha _n+\beta _n)\Vert x_n-p\Vert +\alpha _n\Vert g(p)-p\Vert +\gamma _n\Vert x_n-p\Vert \nonumber \\&= (k\alpha _n+\beta _n+\gamma _n)\Vert x_n-p\Vert +\alpha _n\Vert g(p)-p\Vert \nonumber \\&= (1-(\alpha _{n}(1-k)))\Vert x_n-p\Vert +\alpha _n(1-k)\frac{\Vert g(p)-p\Vert }{1-k} \nonumber \\&\le \max \left\{ \Vert x_n-p\Vert ,\frac{\Vert g(p)-p\Vert }{1-k}\right\} .\nonumber \end{aligned}$$

For every $n\ge 1$, we can conclude that

$$\begin{aligned} \Vert x_n-p\Vert \le \max \left\{ \Vert x_1-p\Vert ,\frac{\Vert g(p)-p\Vert }{1-k}\right\} \end{aligned}$$

for a fixed element $x_1\in C$ by using the mathematical induction. Hence $\{x_n\}$ is bounded; so are $\{u_n\}$, $\{y_n\}$ and $\{z_n\}$.

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

From the nonexpansivity of the mapping $(I-\lambda _nD)$, we have

$$\begin{aligned} \Vert y_{n+1}-y_n\Vert&=\Vert P_C(u_{n+1}-\lambda _{n+1}Du_{n+1})-P_C(u_n-\lambda _nDu_n)\Vert \nonumber \\&\le \Vert (u_{n+1}-\lambda _{n+1}Du_{n+1})-(u_n-\lambda _nDu_n)\Vert \nonumber \\&=\Vert (u_{n+1}-u_n)-\lambda _{n+1}(Du_{n+1}-Du_n)+(\lambda _{n+1}-\lambda _{n})Du_n\Vert \nonumber \\&\le \Vert (u_{n+1}-u_n)-\lambda _{n+1}(Du_{n+1}-Du_n)\Vert +|(\lambda _{n+1}-\lambda _{n})|\Vert Du_n\Vert \nonumber \\&\le \Vert u_{n+1}-u_n\Vert +|\lambda _{n+1}-\lambda _{n}|\Vert Du_n\Vert . \end{aligned}$$

(3.6)

Since $T^{(F_1,\varphi _1)}_{r_n}(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)$ is nonexpansive, $u_n=T^{(F_1,\varphi _1)}_{r_n}(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)x_n$ and $u_{n+1}=T^{(F_1,\varphi _1)}_{r_{n+1}}(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_{n+1}})A)x_{n+1}$, it follows from Lemmma 2.10 that

$$\begin{aligned} \Vert u_{n+1}-u_n\Vert&= \Vert T^{(F_1,\varphi _1)}_{r_{n+1}}(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_{n+1}})A)x_{n+1}-T^{(F_1,\varphi _1)}_{r_n}(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)x_n\Vert \nonumber \\&\le \Vert T^{(F_1,\varphi _1)}_{r_{n+1}}(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_{n+1}})A)x_{n+1}-T^{(F_1,\varphi _1)}_{r_{n+1}}(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_{n+1}})A)x_{n} \Vert \nonumber \\&\quad +\Vert T^{(F_1,\varphi _1)}_{r_{n+1}}(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_{n+1}})A)x_{n}- T^{(F_1,\varphi _1)}_{r_n}(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)x_n\Vert \nonumber \\&\le \Vert x_{n+1}-x_n\Vert +\Vert (I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_{n+1}})A)x_{n}-(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)x_n\Vert \nonumber \\&\quad +\left| 1-\frac{r_n}{r_{n+1}}\right| \Vert T^{(F_1,\varphi _1)}_{r_{n+1}}(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)x_n-(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_{n+1}})A)x_n\Vert \nonumber \\&\le \Vert x_{n+1}-x_n\Vert +\xi \Vert A\Vert \Vert T^{(F_2,\varphi _2)}_{r_{n+1}}Ax_n-T^{(F_2,\varphi _2)}_{r_{n}}Ax_n\Vert +\eta _n \nonumber \\&\le \Vert x_{n+1}-x_n\Vert +\xi \Vert A\Vert \left| 1-\frac{r_n}{r_{n+1}}\right| \Vert T^{(F_2,\varphi _2)}_{r_{n+1}}Ax_n-Ax_n\Vert +\eta _n \nonumber \\&=\Vert x_{n+1}-x_n\Vert +\xi \Vert A\Vert \kappa _n+\eta _n, \end{aligned}$$

(3.7)

where

$$\begin{aligned} \kappa _n:=\left| 1-\frac{r_n}{r_{n+1}}\right| \Vert T^{(F_2,\varphi _2)}_{r_{n+1}}Ax_n-Ax_n\Vert \end{aligned}$$

and

$$\begin{aligned} \eta _n:=\left| 1-\frac{r_n}{r_{n+1}}\right| \Vert T^{(F_1,\varphi _1)}_{r_{n+1}}(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)x_n-(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_{n+1}})A)x_n\Vert . \end{aligned}$$

By using (3.6) and (3.7), we get

$$\begin{aligned} \Vert y_{n+1}-y_n\Vert \le \Vert x_{n+1}-x_n\Vert +\xi \Vert A\Vert \kappa _n +\eta _n + |\lambda _{n+1}-\lambda _{n}|\Vert Du_n\Vert . \end{aligned}$$

(3.8)

Setting $x_{n+1}=\beta _nx_n+(1-\beta _n)w_n$, which implies from (3.1) that

$$\begin{aligned} w_n=\frac{x_{n+1}-\beta _nx_n}{1-\beta _n}=\frac{\alpha _{n}g(x_n)+\gamma _nz_n}{1-\beta _n}. \end{aligned}$$

Therefore, by using (3.8), we obtain that

$$\begin{aligned} \Vert w_{n+1}-w_n\Vert&=\left\| \frac{\alpha _{n+1}g(x_{n+1})+\gamma _{n+1}z_{n+1}}{1-\beta _{n+1}}-\frac{\alpha _{n}g(x_n)+\gamma _nz_n}{1-\beta _n}\right\| \\&=\left\| \frac{\alpha _{n+1}}{1-\beta _{n+1}}(g(x_{n+1})-g(x_n))+ \left( \frac{\alpha _{n+1}}{1-\beta _{n+1}}-\frac{\alpha _{n}}{1-\beta _{n}}\right) g(x_n)\right. \\&\qquad \left. +\frac{\gamma _{n+1}}{1-\beta _{n+1}}(z_{n+1}-z_n)+ \left( \frac{\gamma _{n+1}}{1-\beta _{n+1}}-\frac{\gamma _{n}}{1-\beta _{n}}\right) z_n\right\| \\&\le \frac{\alpha _{n+1}}{1-\beta _{n+1}}\Vert g(x_{n+1})-g(x_n)\Vert + \left| \frac{\alpha _{n+1}}{1-\beta _{n+1}}-\frac{\alpha _{n}}{1-\beta _{n}}\right| \Vert g(x_n)\Vert \\&\qquad +\frac{\gamma _{n+1}}{1-\beta _{n+1}}\Vert z_{n+1}-z_n\Vert +\left| \frac{\gamma _{n+1}}{1-\beta _{n+1}}-\frac{\gamma _{n}}{1-\beta _{n}}\right| \Vert z_n\Vert \\&\le \frac{k\alpha _{n+1}}{1-\beta _{n+1}}\Vert x_{n+1}-x_n\Vert + \left| \frac{\alpha _{n+1}}{1-\beta _{n+1}}-\frac{\alpha _{n}}{1-\beta _{n}}\right| (\Vert g(x_n)\Vert +\Vert z_n\Vert )\\&\qquad +\frac{\gamma _{n+1}}{1-\beta _{n+1}}\Vert z_{n+1}-z_n\Vert \\&\le \frac{k\alpha _{n+1}}{1-\beta _{n+1}}\Vert x_{n+1}-x_n\Vert + \left| \frac{\alpha _{n+1}}{1-\beta _{n+1}}-\frac{\alpha _{n}}{1-\beta _{n}}\right| (\Vert g(x_n)\Vert +\Vert z_n\Vert )\\&\qquad +\frac{\gamma _{n+1}}{1-\beta _{n+1}}(\mathcal {H}(Sy_{n+1},Sy_n)+\varepsilon _n)\\&\le \frac{k\alpha _{n+1}}{1-\beta _{n+1}}\Vert x_{n+1}-x_n\Vert + \left| \frac{\alpha _{n+1}}{1-\beta _{n+1}}-\frac{\alpha _{n}}{1-\beta _{n}}\right| (\Vert g(x_n)\Vert +\Vert z_n\Vert )\\&\qquad +\frac{\gamma _{n+1}}{1-\beta _{n+1}}(\Vert y_{n+1}-y_n\Vert +\varepsilon _n)\\&\le \frac{k\alpha _{n+1}}{1-\beta _{n+1}}\Vert x_{n+1}-x_n\Vert + \left| \frac{\alpha _{n+1}}{1-\beta _{n+1}}-\frac{\alpha _{n}}{1-\beta _{n}}\right| (\Vert g(x_n)\Vert +\Vert z_n\Vert )\\&\qquad +\frac{\gamma _{n+1}}{1-\beta _{n+1}}(\Vert x_{n+1}-x_n\Vert +\xi \Vert A\Vert \kappa _n +\eta _n +|\lambda _{n+1}-\lambda _{n}|\Vert Du_n\Vert +\varepsilon _n)\\&=\left( 1-\frac{(1-k)\alpha _{n+1}}{1-\beta _{n+1}}\right) \Vert x_{n+1}-x_n\Vert +\left| \frac{\alpha _{n+1}}{1-\beta _{n+1}}-\frac{\alpha _{n}}{1-\beta _{n}}\right| (\Vert g(x_n)\Vert +\Vert z_n\Vert )\\&\qquad +\frac{\gamma _{n+1}}{1-\beta _{n+1}}(\xi \Vert A\Vert \kappa _n +\eta _n +|\lambda _{n+1}-\lambda _{n}|\Vert Du_n\Vert +\varepsilon _n)\\&\le \Vert x_{n+1}-x_n\Vert +\left| \frac{\alpha _{n+1}}{1-\beta _{n+1}}-\frac{\alpha _{n}}{1-\beta _{n}}\right| (\Vert g(x_n)\Vert +\Vert z_n\Vert )\\&\qquad +\frac{\gamma _{n+1}}{1-\beta _{n+1}}(\xi \Vert A\Vert \kappa _n +\eta _n +|\lambda _{n+1}-\lambda _{n}|\Vert Du_n\Vert +\varepsilon _n). \end{aligned}$$

It follows that

$$\begin{aligned} \Vert w_{n+1}-w_n\Vert -\Vert x_{n+1}-x_n\Vert&\le \left| \frac{\alpha _{n+1}}{1-\beta _{n+1}}-\frac{\alpha _{n}}{1-\beta _{n}}\right| (\Vert g(x_n)\Vert +\Vert z_n\Vert )\\&\qquad +\frac{\gamma _{n+1}}{1-\beta _{n+1}}(\xi \Vert A\Vert \kappa _n +\eta _n +|\lambda _{n+1}-\lambda _{n}|\Vert Du_n\Vert +\varepsilon _n). \end{aligned}$$

By the conditions (C1), (C2), (C4) and (C5), we have

$$\begin{aligned} \limsup _{n\rightarrow \infty }(\Vert w_{n+1}-w_n\Vert -\Vert x_{n+1}-x_n\Vert )\le 0. \end{aligned}$$

This implies by Lemma 2.7 that $\lim _{n\rightarrow \infty }\Vert w_n-x_n\Vert =0$ and

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert x_{n+1}-x_n\Vert =\lim _{n\rightarrow \infty }(1-\beta _{n})\Vert w_{n}-x_n\Vert = 0. \end{aligned}$$

(3.9)

Step 3. We will show that $\lim _{n\rightarrow \infty }\Vert x_n-u_n\Vert =\lim _{n\rightarrow \infty }\Vert z_n-x_n\Vert =0$.

It follows from (3.3), (3.4), (3.5), and Lemma 2.4 that

$$\begin{aligned} \Vert x_{n+1}-p\Vert ^2&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n \Vert z_n-p\Vert ^2 \nonumber \\&= \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n \text {dist}(z_n,Sp)^2 \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n \mathcal {H}(Sy_n,Sp)^2 \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n \Vert y_n-p\Vert ^2 \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n \Vert u_n-p\Vert ^2 \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 \nonumber \\&\qquad +\gamma _n (\Vert x_n-p\Vert ^2+\xi (L\xi -1)\Vert Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n\Vert ^2) \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+(1-\alpha _{n}) \Vert x_n-p\Vert ^2 \nonumber \\&\qquad -\xi (1-L\xi )\gamma _n\Vert Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n\Vert ^2. \end{aligned}$$

(3.10)

Then we have

$$\begin{aligned} \xi (1-L\xi )\gamma _n\Vert Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n\Vert ^2)&\le \alpha _n\Vert g(x_n)-p\Vert ^2+(\Vert x_n-p\Vert ^2 -\Vert x_{n+1}-p\Vert ^2) \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+(\Vert x_n-p\Vert +\Vert x_{n+1}-p\Vert )\Vert x_{n+1}-x_{n}\Vert . \end{aligned}$$

By the conditions (C1), (C3), $\xi (1-L\xi )>0$, $\Vert x_{n+1}-x_{n}\Vert \rightarrow 0$ as $n\rightarrow \infty$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert Ax_n-T^{(F_2,\varphi _2)}_{r_n}Ax_n\Vert =0. \end{aligned}$$

(3.11)

For $p\in \Gamma$, $p=T^{(F_1,\varphi _1)}_{r_n}p$, $T^{(F_1,\varphi _1)}_{r_n}$ is firmly nonexpansive, and $I-\gamma A^*(I-T^{(F_2,\varphi _2)}_{r_n})A$ is nonexpansive, we obtain that

$$\begin{aligned} \Vert u_n-p\Vert ^2&= \Vert T^{(F_1,\varphi _1)}_{r_n}(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)x_n-T^{(F_1,\varphi _1)}_{r_n}p\Vert ^2\nonumber \\&\le \langle T^{(F_1,\varphi _1)}_{r_n}(I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)x_n-T^{(F_1,\varphi _1)}_{r_n}p, (I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)x_n-p\rangle \nonumber \\&\le \langle u_n-p, (I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)x_n-p\rangle \nonumber \\&= \frac{1}{2}\left( \Vert u_n-p\Vert ^2+\Vert (I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)x_n-p\Vert ^2\right. \nonumber \\&\quad \left. -\Vert u_n-x_n -\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})Ax_n\Vert ^2\right) \nonumber \\&= \frac{1}{2}\left( \Vert u_n-p\Vert ^2+\Vert (I-\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)x_n-p\Vert ^2\right. \nonumber \\&\quad \left. -\Vert (u_n-p) -(x_n +\xi A^*(I-T^{(F_2,\varphi _2)}_{r_n})Ax_n-p)\Vert ^2\right) \nonumber \\&\le \frac{1}{2}\left( \Vert u_n-p\Vert ^2+\Vert x_n-p\Vert ^2-(\Vert u_n-x_n\Vert ^2+\xi ^2\Vert A^*(I-T^{(F_2,\varphi _2)}_{r_n})Ax_n\Vert ^2\right. \nonumber \\&\quad \left. -2\xi \langle u_n-x_n,A^*(I-T^{(F_2,\varphi _2)}_{r_n})Ax_n \rangle )\right) ,\nonumber \end{aligned}$$

which implies that

$$\begin{aligned} \Vert u_n-p\Vert ^2&\le \Vert x_n-p\Vert ^2-\Vert u_n-x_n\Vert ^2+2\xi \langle u_n-x_n,A^*(I-T^{(F_2,\varphi _2)}_{r_n})Ax_n \rangle \nonumber \\&\le \Vert x_n-p\Vert ^2-\Vert u_n-x_n\Vert ^2+2\xi \Vert A(u_n-x_n)\Vert \Vert (I-T^{(F_2,\varphi _2)}_{r_n})Ax_n\Vert . \end{aligned}$$

(3.12)

It follows from (3.10) and (3.12) that

$$\begin{aligned} \Vert x_{n+1}-p\Vert ^2&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n \Vert u_n-p\Vert ^2 \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n (\Vert x_n-p\Vert ^2-\Vert u_n-x_n\Vert ^2 \nonumber \\&\quad +2\xi \Vert A(u_n-x_n)\Vert \Vert (I-T^{(F_2,\varphi _2)}_{r_n})Ax_n\Vert ) \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+(1-\alpha _{n}) \Vert x_n-p\Vert ^2- \gamma _{n}\Vert u_n-x_n\Vert ^2 \nonumber \\&\quad +2\xi \gamma _{n}\Vert A(u_n-x_n)\Vert \Vert (I-T^{(F_2,\varphi _2)}_{r_n})Ax_n\Vert \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\Vert x_n-p\Vert ^2- \gamma _{n}\Vert u_n-x_n\Vert ^2 \nonumber \\&\quad +2\xi \gamma _{n}\Vert A(u_n-x_n)\Vert \Vert (I-T^{(F_2,\varphi _2)}_{r_n})Ax_n\Vert . \end{aligned}$$

Therefore, we get that

$$\begin{aligned} \gamma _{n}\Vert u_n-x_n\Vert ^2&\le \alpha _n\Vert g(x_n)-p\Vert ^2+(\Vert x_n-p\Vert ^2-\Vert x_{n+1}-p\Vert ^2) \\&\quad +2\xi \gamma _{n}\Vert A(u_n-x_n)\Vert \Vert (I-T^{(F_2,\varphi _2)}_{r_n})Ax_n\Vert \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+(\Vert x_n-p\Vert +\Vert x_{n+1}-p\Vert )\Vert x_{n}-x_{n+1}\Vert \\&\quad +2\xi \gamma _{n}\Vert A(u_n-x_n)\Vert \Vert (I-T^{(F_2,\varphi _2)}_{r_n})Ax_n\Vert . \end{aligned}$$

By the conditions (C1), (C3), $\Vert x_{n}-x_{n+1}\Vert \rightarrow 0$, and $\Vert (I-T^{(F_2,\varphi _2)}_{r_n})Ax_n\Vert \rightarrow 0$ as $n\rightarrow \infty$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert u_n-x_n\Vert =0. \end{aligned}$$

(3.13)

Consider

$$\begin{aligned} \Vert x_n-z_n\Vert&\le \Vert x_n-x_{n+1}\Vert +\Vert x_{n+1}-z_n\Vert \\&= \Vert x_n-x_{n+1}\Vert +\Vert \alpha _ng(x_n)+\beta _n x_n +\gamma _n z_n-z_n\Vert \\&\le \Vert x_n-x_{n+1}\Vert +\alpha _{n}\Vert g(x_n)-z_n\Vert +\beta _{n}\Vert z_n-x_n\Vert \end{aligned}$$

and then,

$$\begin{aligned} \Vert z_n-x_n\Vert&\le \frac{1}{1-\beta _{n}}\Vert x_n-x_{n+1}\Vert +\frac{\alpha _{n}}{1-\beta _{n}}\Vert g(x_n)-z_n\Vert . \end{aligned}$$

By the conditions (C1), (C2) and $\Vert x_{n}-x_{n+1}\Vert \rightarrow 0$ as $n\rightarrow \infty$, we obtain

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert z_n-x_n\Vert =0. \end{aligned}$$

(3.14)

Step 4. We will show that $\lim _{n\rightarrow \infty }\Vert z_n-y_n\Vert =0$.

For each $p\in \Gamma$, we have

$$\begin{aligned} \Vert x_{n+1}-p\Vert ^2&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n \Vert z_n-p\Vert ^2 \nonumber \\&= \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n \text {dist}(z_n,Sp)^2 \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n \mathcal {H}(Sy_n,Sp)^2 \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n \Vert y_n-p\Vert ^2 \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n (\Vert P_C(u_n-\lambda _{n}Du_n)-P_C(p-\lambda _{n}Dp)\Vert ) \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n (\Vert u_n-p\Vert ^2+\lambda _{n}(\lambda _{n}-2\tau )\Vert Du_n-Dp\Vert ^2) \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n (\Vert x_n-p\Vert ^2+\lambda _{n}(\lambda _{n}-2\tau )\Vert Du_n-Dp\Vert ^2) \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+(1-\alpha _{n}) \Vert x_n-p\Vert ^2 +\gamma _n \lambda _{n}(\lambda _{n}-2\tau )\Vert Du_n-Dp\Vert ^2 \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\Vert x_n-p\Vert ^2 +\gamma _n \lambda _{n}(\lambda _{n}-2\tau )\Vert Du_n-Dp\Vert ^2\\&= \alpha _n\Vert g(x_n)-p\Vert ^2+\Vert x_n-p\Vert ^2 -\gamma _n \lambda _{n}(2\tau -\lambda _{n})\Vert Du_n-Dp\Vert ^2\\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\Vert x_n-p\Vert ^2 -\gamma _n a(2\tau -b)\Vert Du_n-Dp\Vert ^2, \nonumber \end{aligned}$$

which yields

$$\begin{aligned} -\gamma _n a(2\tau -b)\Vert Du_n-Dp\Vert ^2&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\Vert x_n-p\Vert ^2 -\Vert x_{n+1}-p\Vert ^2 \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+(\Vert x_n-p\Vert +\Vert x_{n+1}-p\Vert )\Vert x_n-x_{n+1}\Vert . \end{aligned}$$

By the conditions (C1), (C3), $\Vert x_n-x_{n+1}\Vert \rightarrow 0$ as $n\rightarrow \infty$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert Du_n-Dp\Vert =0. \end{aligned}$$

(3.15)

Furthermore, we observe that

$$\begin{aligned} \Vert y_{n}-p\Vert ^2&=\Vert P_C(u_n-\lambda _nDu_n)-P_C(p-\lambda _nDp)\Vert ^2\\&\le \langle y_n-p,(u_n-\lambda _nDu_n)-(p-\lambda _nDp) \rangle \\&\le \frac{1}{2} (\Vert y_n-p\Vert ^2+\Vert (u_n-\lambda _nDu_n)-(p-\lambda _nDp)\Vert ^2-\Vert (y_n-u_n)+\lambda _{n}(Du_n-Dp)\Vert ^2) \nonumber \\&\le \frac{1}{2} (\Vert y_n-p\Vert ^2+\Vert (u_n-p\Vert ^2-\Vert (y_n-u_n)+\lambda _{n}(Du_n-Dp)\Vert ^2). \nonumber \end{aligned}$$

Thus, we have

$$\begin{aligned} \Vert y_{n}-p\Vert ^2&\le \Vert u_n-p\Vert ^2-\Vert y_n-u_n\Vert ^2-\lambda _{n}^2\Vert Du_n-Dp\Vert ^2+2\lambda _{n}\langle y_n-u_n,Du_n-Dp\rangle \\&\le \Vert u_n-p\Vert ^2-\Vert y_n-u_n\Vert ^2+2\lambda _{n}\Vert y_n-u_n\Vert \Vert Du_n-Dp\Vert \\&\le \Vert x_n-p\Vert ^2-\Vert y_n-u_n\Vert ^2+2\lambda _{n}\Vert y_n-u_n\Vert \Vert Du_n-Dp\Vert . \end{aligned}$$

It follows that

$$\begin{aligned} \Vert x_{n+1}-p\Vert ^2&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n \Vert y_n-p\Vert ^2 \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\beta _n \Vert x_n-p\Vert ^2 +\gamma _n(\Vert x_n-p\Vert ^2-\Vert y_n-u_n\Vert ^2\\&\quad +2\lambda _{n}\Vert y_n-u_n\Vert \Vert Du_n-Dp\Vert ) \nonumber \\&= \alpha _n\Vert g(x_n)-p\Vert ^2+(1-\alpha _{n})\Vert x_n-p\Vert ^2 -\gamma _n\Vert y_n-u_n\Vert ^2\\&\quad +2\gamma _n\lambda _{n}\Vert y_n-u_n\Vert \Vert Du_n-Dp\Vert \nonumber \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\Vert x_n-p\Vert ^2 -\gamma _n\Vert y_n-u_n\Vert ^2 +2\gamma _n\lambda _{n}\Vert y_n-u_n\Vert \Vert Du_n-Dp\Vert . \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} \gamma _n\Vert y_n-u_n\Vert ^2&\le \alpha _n\Vert g(x_n)-p\Vert ^2+\Vert x_n-p\Vert ^2 -\Vert x_{n+1}-p\Vert ^2 +2\gamma _n\lambda _{n}\Vert y_n-u_n\Vert \Vert Du_n-Dp\Vert \\&\le \alpha _n\Vert g(x_n)-p\Vert ^2+(\Vert x_n-p\Vert +\Vert x_{n+1}-p\Vert )\Vert x_n-x_{n+1}\Vert \\&\quad +2\gamma _n\lambda _{n}\Vert y_n-u_n\Vert \Vert Du_n-Dp\Vert . \end{aligned}$$

By the conditions (C1), (C3), $\Vert x_n-x_{n+1}\Vert \rightarrow 0$, $\Vert Du_n-Dp\Vert \rightarrow 0$ as $n\rightarrow \infty$, we obtain

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert y_n-u_n\Vert =0. \end{aligned}$$

(3.16)

Observe that

$$\begin{aligned} \Vert z_n-y_n\Vert \le \Vert z_n-x_n\Vert +\Vert x_n-u_n\Vert +\Vert u_n-y_n\Vert , \end{aligned}$$

from (3.13), (3.14) and (3.16), we get

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert z_n-y_n\Vert =0. \end{aligned}$$

(3.17)

Step 5. We will show that $\limsup _{n\rightarrow \infty }\langle g(z)-z,x_n-z\rangle \le 0$ where $z=P_{\Gamma }g(z)$.

To show this, we choose a subsequence $\{z_{n_i}\}$ of $\{z_n\}$ such that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle g(z)-z,z_n-z\rangle =\lim _{i\rightarrow \infty }\langle g(z)-z,z_{n_i}-z\rangle . \end{aligned}$$

(3.18)

Since $\{z_{n_i}\}$ is bounded, there exists a subsequence $\{z_{n_{i_j}}\}$ of $\{z_{n_{i}}\}$ which converges weakly to some $w\in C$. Without loss of generality, we can assume that $z_{n_i} \rightharpoonup w$. Since $\Vert z_n-y_n\Vert \rightarrow 0$, we obtain $y_{n_i}\rightharpoonup w$ as $i\rightarrow \infty$.

Next, we show that $w\in \Gamma$, that is, $w\in F(S)\cap SGEP(F_1,\varphi _1,F_2,\varphi _2) \cap VI(C,D)$.

Step 5.1. We will show that $w\in F(S)$. Since Sw is compact, we can choose $q'_{n} \in Sw$ such that $\Vert z_{n} - q'_{n} \Vert = \text {dist}(z_{n}, Sw)$ and the sequence $\{q'_{n}\}$ has a convergent subsequence $\{q'_{n_i}\}$ with $\lim _{i \rightarrow \infty } q'_{n_i} = q' \in Sw$. By nonexpansiveness of S, we obtain that

$$\begin{aligned} \Vert y_{n_i} - q' \Vert&\le \Vert y_{n_i} - z_{n_i} \Vert + \Vert z_{n_i} - q'_{n_i} \Vert + \Vert q'_{n_i} - q' \Vert \\&\le \Vert y_{n_i} - z_{n_i} \Vert + \text {dist}(z_{n_i},Sw) + \Vert q'_{n_i}- q' \Vert \\&\le \Vert y_{n_i} - z_{n_i} \Vert + \mathcal {H}(Sy_{n_i},Sw) + \Vert q'_{n_i}- q' \Vert \\&\le \Vert y_{n_i} - z_{n_i} \Vert + \Vert y_{n_i} - w \Vert + \Vert q'_{n_i} - q'\Vert . \end{aligned}$$

This implies by (3.17) and $\lim _{i \rightarrow \infty } q'_{n_i} = q'$ that

$$\begin{aligned} \limsup _{i \rightarrow \infty } \Vert y_{n_i} - q' \Vert \le \limsup _{i \rightarrow \infty } \Vert y_{n_i} - w \Vert . \end{aligned}$$

By Opial’s condition, we get $w=q' \in Tw$. Hence, $w\in F(S)$.

Step 5.2. We will show that $w\in SGEP(F_1,\varphi _1,F_2,\varphi _2)$. First, we will show that $w\in GEP(F_1,\varphi _1)$. Since $u_n=T^{(F_1,\varphi _1)}_{r_n}(I-\gamma A^*(I-T^{(F_2,\varphi _2)}_{r_n})A)x_n$, we have

$$\begin{aligned} F_1(u_n,y)+\varphi _1(u_n,y)+\frac{1}{r_n}\left\langle y-u_n,u_n-x_n -\gamma A^*\left( I-T^{(F_2,\varphi _2)}_{r_n}\right) Ax_n\right\rangle \ge 0, \end{aligned}$$

for all $y\in C$, which implies that

$$\begin{aligned} F_1(u_n,y)+\varphi _1(u_n,y)+\frac{1}{r_n}\langle y-u_n,u_n-x_n \rangle -\frac{1}{r_n}\left\langle y-u_n, \gamma A^*\left( I-T^{(F_2,\varphi _2)}_{r_n}\right) Ax_n\right\rangle \ge 0, \end{aligned}$$

for all $y\in C$. It follows from the monotonicity of $F_1$ and $\varphi _1$ that

$$\begin{aligned} \frac{1}{r_n}\langle y-u_n,u_n-x_n \rangle -\frac{1}{r_n}\left\langle y-u_n, \gamma A^*\left( I-T^{(F_2,\varphi _2)}_{r_n}\right) Ax_n\right\rangle \ge F_1(y,u_n)+\varphi _1(y,u_n), \end{aligned}$$

for all $y\in C$. Since $\Vert u_n-x_n\Vert \rightarrow 0$, $\Vert z_n-x_n\Vert \rightarrow 0$, $\Vert z_n-y_n\Vert \rightarrow 0$, and $y_{n_i}\rightharpoonup w$, we have $u_{n_{i}}\rightharpoonup w$ and $u_{n_{i}}-x_{n_{i}}\rightarrow 0$ as $i\rightarrow \infty$. It follows by the condition (C4), (3.11), (3.13), Assumption 2.8 (A4) and (A7) that $0\ge F_1(y,w)+\varphi _1(y,w)$ for all $y\in C$. Put $y_t=ty+(1-t)w$ for all $t\in (0,1]$ and $y\in C$. Consequently, we get $y_t\in C$ and hence $F_1(y_t,w)+\varphi _1(y_t,w)\le 0$. So, by Assumption 2.8 (A1)-(A7), we have

$$\begin{aligned} 0&\le F_1(y_t,y_t)+\varphi _1(y_t,y_t) \\&\le t(F_1(y_t,y)+\varphi _1(y_t,y)) + (1-t)(F_1(y_t,w) +\varphi _1(y_t,w))\\&\le t(F_1(y_t,y)+\varphi _1(y_t,y)) + (1-t)(F_1(w,y_t) +\varphi _1(w,y_t))\\&\le F_1(y_t,y)+\varphi _1(y_t,y). \end{aligned}$$

Hence, we have $F_1(y_t,y)+\varphi _1(y_t,y)\ge 0$ for all $y\in C$. Letting $t\rightarrow 0$, by Assumption 2.8 (A3) and upper hemicontinuity of $\varphi _1$, we have $F_1(q,y)+\varphi _1(q,y)\ge 0$ for all $y\in C$. This implies that $w\in GEP(F_1,\varphi _1)$.

Next, we show that $Aw\in GEP(F_2,\varphi _2)$. Since $\Vert z_n-x_n\Vert \rightarrow 0$, $\Vert z_n-y_n\Vert \rightarrow 0$, and $y_{n_i}\rightharpoonup w$, we have $x_{n_{i}}\rightharpoonup w$. Since A is a bounded linear operator, we get $Ax_{n_{i}}\rightarrow Aw$.

Now, setting $v_{n_{i}}=Ax_{n_{i}}-T_{r_{n_i}}^{(F_2,\varphi _2)}Ax_{n_i}$. It follows from (3.11) that

$$\begin{aligned} \lim _{i\rightarrow \infty }v_{n_{i}}=0 \text { and } Ax_{n_{i}}-v_{n_{i}}=T_{r_{n_i}}^{(F_2,\varphi _2)}Ax_{n_i}. \end{aligned}$$

Therefore, from Lemma 2.9, we have

$$\begin{aligned} 0\le F_2(Ax_{n_{i}}-v_{n_{i}},z)+\varphi _2(Ax_{n_{i}}-v_{n_{i}},z)+\dfrac{1}{r_{n_i}}\langle z-(Ax_{n_{i}}-v_{n_{i}}), (Ax_{n_{i}}-v_{n_{i}})-Ax_{n_{i}}\rangle , \end{aligned}$$

for all $z\in Q$. Since $F_2$ and $\varphi _2$ are upper semicontinuous in the first argument, it follows that

$$\begin{aligned} F_2(Aw,z)+\varphi _2(Aw,z)\ge 0 \text { for all } z\in Q. \end{aligned}$$

This means that $Aw\in GEP(F_2,\varphi _2)$ and hence $w\in SGEP(F_1,\varphi _1,F_2,\varphi _2)$.

Step 5.3. We will show that $w\in VI(C,D)$. Let $U:H_1 \rightarrow 2^{H_1}$ be a multivalued mapping defined by

$$\begin{aligned} Uw = \left\{ {\begin{array}{*{20}l} {Dw+N_Cw,} &{} {w\in C} \\ {\emptyset ,} &{} {w\notin C} \end{array} } \right. \end{aligned}$$

where $N_Cw$ is the normal cone to C at $w\in C$. Then U is maximal monotone, and $0\in Uw$ if and only if $w\in VI(C,D)$. Let G(U) be the graph of U and let $(w,y)\in G(U)$. Then we have $y\in Uw=Dw+N_Cw$ and hence $y-Dw \in N_Cw.$ Since $y_n \in C$ for all $n\in \mathbb {N}$, we have

$$\begin{aligned} \langle w-y_n,y-Dw\rangle \ge 0. \end{aligned}$$

(3.19)

On the other hand, from $y_n=P_C(u_n-\lambda _nDu_n)$, we have

$$\begin{aligned} \langle w-y_n,y_n-(u_n-\lambda _nDu_n)\rangle \ge 0, \end{aligned}$$

that is,

$$\begin{aligned} \left\langle w-y_n,\dfrac{y_n-u_n}{\lambda _n}+Du_n\right\rangle \ge 0. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \langle w-y_{n_i},y\rangle&\ge \langle w-y_{n_i},Dw\rangle \nonumber \\&\ge \langle w-y_{n_i},Dw\rangle -\left\langle w-y_{n_i},\dfrac{y_{n_i}-u_{n_i}}{\lambda _{n_i}}+Du_{n_i}\right\rangle \nonumber \\&= \left\langle w-y_{n_i},Dw-\dfrac{y_{n_i}-u_{n_i}}{\lambda _{n_i}}-Du_{n_i}\right\rangle \nonumber \\&= \langle w-y_{n_i},Dw-Dy_{n_i}\rangle +\langle w-y_{n_i},Dy_{n_i}-Du_{n_i}\rangle \nonumber \\&\quad - \left\langle w-y_{n_i},\dfrac{y_{n_i}-u_{n_i}}{\lambda _{n_i}}\right\rangle \nonumber \\&\ge \langle w-y_{n_i},Dy_{n_i}\rangle - \left\langle w-y_{n_i},\dfrac{y_{n_i}-u_{n_i}}{\lambda _{n_i}}+Du_{n_i}\right\rangle \nonumber \\&\ge \Vert w-y_{n_i}\Vert \Vert Dy_{n_i}-Du_{n_i}\Vert - \Vert w-y_{n_i}\Vert \left\| \dfrac{y_{n_i}-u_{n_i}}{\lambda _{n_i}}\right\| . \end{aligned}$$

(3.20)

Noting that $\Vert y_{n_i}-u_{n_i}\Vert \rightarrow 0$ as $i\rightarrow \infty$ and D is $\tau$-inverse strongly monotone, hence from the inequality (3.20), we have $\langle w-z,y\rangle \ge 0$ as $i\rightarrow \infty$. Since U is maximal monotone, we have $w\in U^{-1}0$, and hence $w\in VI(C,D)$. By Steps 5.1, 5.2 and 5.3, we can conclude that $w\in F(S)\cap SGEP(F_1,\varphi _1,F_2,\varphi _2) \cap VI(C,D)$, that is, $w\in \Gamma$.

Since $z=P_{\Gamma }g(z)$ and $z_{n_i} \rightharpoonup w$ as $i\rightarrow \infty$, it implies by (2.1) that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle g(z)-z,x_n-z\rangle&=\limsup _{n\rightarrow \infty }\langle g(z)-z,z_n-z\rangle \nonumber \\&=\limsup _{i\rightarrow \infty }\langle g(z)-z,z_{n_i}-z\rangle \nonumber \\&=\langle g(z)-z,w-z\rangle \nonumber \\&\le 0. \end{aligned}$$

(3.21)

Step 6. Finally, we will show that $\{x_n\}$ converges strongly to z.

Consider

$$\begin{aligned} \Vert x_{n+1}-z\Vert ^2&=\langle x_{n+1}-z,x_{n+1}-z \rangle \nonumber \\&=\langle \alpha _ng(x_n)+\beta _n x_n +\gamma _n z_n-(\alpha _{n}+\beta _{n}+\gamma _{n})z,x_{n+1}-z\rangle \nonumber \\&=\alpha _n\langle g(x_n)-z,x_{n+1}-z\rangle +\beta _n \langle x_n-z,x_{n+1}-z\rangle +\gamma _n \langle z_n-z,x_{n+1}-z\rangle \nonumber \\&\le \alpha _n\Vert g(x_n)-g(z)\Vert \Vert x_{n+1}-z\Vert + \alpha _n\langle g(z)-z,x_{n+1}-z\rangle \nonumber \\&\quad +\dfrac{\beta _n}{2}(\Vert x_n-z\Vert ^2+\Vert x_{n+1}-z\Vert ^2)+\dfrac{\gamma _n}{2}(\Vert z_n-z\Vert ^2+\Vert x_{n+1}-z\Vert ^2)\nonumber \\&\le k\alpha _n\Vert x_n-z\Vert \Vert x_{n+1}-z\Vert + \alpha _n\langle g(z)-z,x_{n+1}-z\rangle \nonumber \\&\quad +\dfrac{\beta _n}{2}(\Vert x_n-z\Vert ^2+\Vert x_{n+1}-z\Vert ^2)+\dfrac{\gamma _n}{2}(\text {dist}(z_n,Sz)^2+\Vert x_{n+1}-z\Vert ^2)\nonumber \\&\le k\alpha _n\Vert x_n-z\Vert \Vert x_{n+1}-z\Vert + \alpha _n\langle g(z)-z,x_{n+1}-z\rangle \nonumber \\&\quad +\dfrac{\beta _n}{2}(\Vert x_n-z\Vert ^2+\Vert x_{n+1}-z\Vert ^2)+\dfrac{\gamma _n}{2}(\mathcal {H}(Sy_n,Sz)^2+\Vert x_{n+1}-z\Vert ^2)\nonumber \\&\le k\alpha _n\Vert x_n-z\Vert \Vert x_{n+1}-z\Vert +\alpha _n\langle g(z)-z,x_{n+1}-z\rangle \nonumber \\&\quad +\dfrac{\beta _n}{2}(\Vert x_n-z\Vert ^2+\Vert x_{n+1}-z\Vert ^2)+\dfrac{\gamma _n}{2}(\Vert y_n-z\Vert ^2+\Vert x_{n+1}-z\Vert ^2) \nonumber \\&\le k\alpha _n\Vert x_n-z\Vert \Vert x_{n+1}-z\Vert +\alpha _n\langle g(z)-z,x_{n+1}-z\rangle \nonumber \\&\quad +\dfrac{\beta _n+\gamma _n}{2}(\Vert x_n-z\Vert ^2+\Vert x_{n+1}-z\Vert ^2)\nonumber \\&= k\alpha _n\Vert x_n-z\Vert \Vert x_{n+1}-z\Vert +\alpha _n\langle g(z)-z,x_{n+1}-z\rangle \nonumber \\&\quad +\dfrac{(1-\alpha _{n})}{2}(\Vert x_n-z\Vert ^2+\Vert x_{n+1}-z\Vert ^2) \nonumber \\&\le k\alpha _n\Vert x_n-z\Vert \Vert x_{n+1}-z\Vert +\alpha _n\langle g(z)-z,x_{n+1}-z\rangle \nonumber \\&\quad +\dfrac{(1-\alpha _{n})}{2}\Vert x_n-z\Vert ^2+\dfrac{1}{2}\Vert x_{n+1}-z\Vert ^2. \nonumber \end{aligned}$$

This implies that

$$\begin{aligned} \Vert x_{n+1}-z\Vert ^2&\le (1-\alpha _{n})\Vert x_n-z\Vert ^2+2\alpha _n\langle g(z)-z,x_{n+1}-z\rangle \nonumber \\&\quad +2k\alpha _n\Vert x_n-z\Vert \Vert x_{n+1}-z\Vert \\&\le (1-\alpha _{n})\Vert x_n-z\Vert ^2+\delta _n, \end{aligned}$$

where

$$\begin{aligned} \delta _n=2\alpha _n\langle g(z)-z,x_{n+1}-z\rangle +2k\alpha _n\Vert x_n-z\Vert \Vert x_{n+1}-z\Vert . \end{aligned}$$

By the inequality (3.21) and condition (C1), we get $\delta _n\rightarrow 0$ as $n\rightarrow \infty$. By using Lemma 2.6, it implies that $x_n\rightarrow z$ as $n\rightarrow \infty$. This completes the proof.$\square$

Remark 3.2

Theorem 3.1 extends the corresponding one of Kazmi and Rizvi [16, 17] and Moudafi [26] to a nonexpansive multivalued mapping and to a split generalized equilibrium problem. In fact, we present a new viscosity approximation method for finding a common solution of three problems including the split generalized equilibrium problem, the variational inequality problem for a $\tau$-inverse strongly monotone mapping and the fixed point problem for a nonexpansive multivalued mapping.

If $\varphi _1= \varphi _2=0$, then the split generalized equilibrium problem reduces to split equilibrium problem. So, the following result can be obtained from Theorem 3.1 immediately.

Corollary 3.3

Let C be a nonempty closed convex subset of a real Hilbert space $H_1$ and Q be a nonempty closed convex subset of a real Hilbert space $H_2$. Let $A: H_1\rightarrow H_2$ be a bounded linear operator, $D:C\rightarrow H_1$ be $\tau$-inverse strongly monotone mapping, and $S:C\rightarrow K(C)$ be a nonexpansive multivalued mapping. Let $F_1:C\times C\rightarrow \mathbb {R}$, $F_2:Q\times Q\rightarrow \mathbb {R}$ be bifunctions satisfying Assumption 2.8. Let $F_2$ be upper semicontinuous in the first argument. Assume that $\Gamma =F(S)\cap SEP(F_1,F_2) \cap \text {VI(C,D)}\ne \emptyset$ and $Sp=\{p\}$ for all $p\in F(S)$. Let g be a contraction of C into itself with coefficient $k\in (0,1)$. Let $\{x_n\}$ be a sequence generated by $x_1\in C$ and

$$\begin{aligned} \left\{ \begin{aligned}&u_n=T_{r_n}^{F_1}(I-\xi A^*(I-T_{r_n}^{F_2})A)x_n,\\&y_n=P_C(u_n-\lambda _nDu_n),\\&x_{n+1}=\alpha _ng(x_n)+\beta _n x_n +\gamma _n z_n,\,\,\,n\in \mathbb {N}, \end{aligned} \right. \end{aligned}$$

where $z_n\in Sy_n$ such that $\Vert z_{n+1}-z_n\Vert \le \mathcal {H}(Sy_{n+1},Sy_n)+\varepsilon _n$, $\lim _{n\rightarrow \infty }\varepsilon _n = 0$, and $r_n \in (0,1)$, $\lambda _n\in [a,b]$ for some a, b with $0<a<b<2\tau$, and $\xi \in (0,\frac{1}{L})$ with L is the spectral radius of the operator $A^*A$ and $A^*$ is the adjoint of A and $\{\alpha _n\}$, $\{\beta _n\}$ and $\{\gamma _n\}$ are the sequences in (0, 1) satisfy $\alpha _n+\beta _n+\gamma _n=1$ for all $n\in \mathbb {N}$. Suppose the conditions are satisfied:

(C1):

$\lim _{n\rightarrow \infty }\alpha _n=0$ and $\sum _{n=1}^{\infty }\alpha _n=\infty$;

(C2):

$0<\liminf _{n\rightarrow \infty }\beta _n\le \limsup _{n\rightarrow \infty }\beta _n<1$;

(C3):

$\gamma _n\in [c,1]$ for some $c\in (0,1)$;

(C4):

$\liminf _{n\rightarrow \infty }r_n>0$ and $\sum _{n=1}^{\infty }|r_{n+1}-r_n|<\infty$;

(C5):

$\lim _{n\rightarrow \infty }|\lambda _{n+1}-\lambda _n|=0$.

Then the sequence $\{x_n\}$ converges strongly to $z\in \Gamma$, where $z=P_{\Gamma } g(z)$.

Recall that a multivalued mapping $S:C\subseteq H_1\rightarrow CB(C)$ is said to satisfy Condition (*) if $\Vert x-p\Vert =\text {dist}(x,Sp)$ for all $x\in H_1$ and $p\in F(S)$; see [31]. We see that S satisfies Condition (*) if and only if $Sp=\{p\}$ for all $p\in F(S)$. Then the following results can be obtained from Theorem 3.1 and Corollary 3.3 immediately.

Corollary 3.4

Let C be a nonempty closed convex subset of a real Hilbert space $H_1$ and Q be a nonempty closed convex subset of a real Hilbert space $H_2$. Let $A: H_1\rightarrow H_2$ be a bounded linear operator, $D:C\rightarrow H_1$ be a $\tau$-inverse strongly monotone mapping, and $S:C\rightarrow K(C)$ be a nonexpansive multivalued mapping. Let $F_1, \varphi _1:C\times C\rightarrow \mathbb {R}$, $F_2,\varphi _2:Q\times Q\rightarrow \mathbb {R}$ be bifunctions satisfying Assumption 2.8. Let $\varphi _1,\varphi _2$ be monotone, $\varphi _1$ be upper hemicontinuous, and $F_2$ and $\varphi _2$ be upper semicontinuous in the first argument. Assume that $\Gamma =F(S)\cap SGEP(F_1,\varphi _1,F_2,\varphi _2) \cap \text {VI(C,D)}\ne \emptyset$ and S satisfies Condition (*). Let g be a contraction of C into itself with coefficient $k\in (0,1)$. Let $\{x_n\}$ be a sequence generated by $x_1\in C$ and

$$\begin{aligned} \left\{ \begin{aligned}&u_n=T_{r_n}^{(F_1,\varphi _1)}(I-\xi A^*(I-T_{r_n}^{(F_2,\varphi _2)})A)x_n,\\&y_n=P_C(u_n-\lambda _nDu_n),\\&x_{n+1}=\alpha _ng(x_n)+\beta _n x_n +\gamma _n z_n,\,\,\,n\in \mathbb {N}, \end{aligned} \right. \end{aligned}$$

where $z_n\in Sy_n$ such that $\Vert z_{n+1}-z_n\Vert \le \mathcal {H}(Sy_{n+1},Sy_n)+\varepsilon _n$, $\lim _{n\rightarrow \infty }\varepsilon _n = 0$, and $r_n \in (0,1)$, $\lambda _n\in [a,b]$ for some a, b with $0<a<b<2\tau$, and $\xi \in (0,\frac{1}{L})$ with L is the spectral radius of the operator $A^*A$ and $A^*$ is the adjoint of A and $\{\alpha _n\}$, $\{\beta _n\}$ and $\{\gamma _n\}$ are the sequences in (0, 1) satisfy $\alpha _n+\beta _n+\gamma _n=1$ for all $n\in \mathbb {N}$. Suppose the conditions are satisfied:

(C1):

$\lim _{n\rightarrow \infty }\alpha _n=0$ and $\sum _{n=1}^{\infty }\alpha _n=\infty$;

(C2):

$0<\liminf _{n\rightarrow \infty }\beta _n\le \limsup _{n\rightarrow \infty }\beta _n<1$;

(C3):

$\gamma _n\in [c,1]$ for some $c\in (0,1)$;

(C4):

$\liminf _{n\rightarrow \infty }r_n>0$ and $\sum _{n=1}^{\infty }|r_{n+1}-r_n|<\infty$;

(C5):

$\lim _{n\rightarrow \infty }|\lambda _{n+1}-\lambda _n|=0$.

Then the sequence $\{x_n\}$ converges strongly to $z\in \Gamma$, where $z=P_{\Gamma } g(z)$.

Corollary 3.5

Let C be a nonempty closed convex subset of a real Hilbert space $H_1$ and Q be a nonempty closed convex subset of a real Hilbert space $H_2$. Let $A: H_1\rightarrow H_2$ be a bounded linear operator, $D:C\rightarrow H_1$ be a $\tau$-inverse strongly monotone mapping, and $S:C\rightarrow K(C)$ be a nonexpansive multivalued mapping. Let $F_1:C\times C\rightarrow \mathbb {R}$, $F_2:Q\times Q\rightarrow \mathbb {R}$ be bifunctions satisfying Assumption 2.8. Let $F_2$ be upper semicontinuous in the first argument. Assume that $\Gamma =F(S)\cap SEP(F_1,F_2) \cap \text {VI(C,D)}\ne \emptyset$ and S satisfies Condition (*). Let g be a contraction of C into itself with coefficient $k\in (0,1)$. Let $\{x_n\}$ be a sequence generated by $x_1\in C$ and

$$\begin{aligned} \left\{ \begin{aligned}&u_n=T_{r_n}^{F_1}(I-\xi A^*(I-T_{r_n}^{F_2})A)x_n,\\&y_n=P_C(u_n-\lambda _nDu_n),\\&x_{n+1}=\alpha _ng(x_n)+\beta _n x_n +\gamma _n z_n,\,\,\,n\in \mathbb {N}, \end{aligned} \right. \end{aligned}$$

where $z_n\in Sy_n$ such that $\Vert z_{n+1}-z_n\Vert \le \mathcal {H}(Sy_{n+1},Sy_n)+\varepsilon _n$, $\lim _{n\rightarrow \infty }\varepsilon _n = 0$, and $r_n \in (0,1)$, $\lambda _n\in [a,b]$ for some a, b with $0<a<b<2\tau$, and $\xi \in (0,\frac{1}{L})$ with L is the spectral radius of the operator $A^*A$ and $A^*$ is the adjoint of A and $\{\alpha _n\}$, $\{\beta _n\}$ and $\{\gamma _n\}$ are the sequences in (0, 1) satisfy $\alpha _n+\beta _n+\gamma _n=1$ for all $n\in \mathbb {N}$. Suppose the conditions are satisfied:

(C1):

$\lim _{n\rightarrow \infty }\alpha _n=0$ and $\sum _{n=1}^{\infty }\alpha _n=\infty$;

(C2):

$0<\liminf _{n\rightarrow \infty }\beta _n\le \limsup _{n\rightarrow \infty }\beta _n<1$;

(C3):

$\gamma _n\in [c,1]$ for some $c\in (0,1)$;

(C4):

$\liminf _{n\rightarrow \infty }r_n>0$ and $\sum _{n=1}^{\infty }|r_{n+1}-r_n|<\infty$;

(C5):

$\lim _{n\rightarrow \infty }|\lambda _{n+1}-\lambda _n|=0$.

Then the sequence $\{x_n\}$ converges strongly to $z\in \Gamma$, where $z=P_{\Gamma } g(z)$.

We now present a numerical example to demonstrate the performance and convergence of our theoretical results. All codes were written in Scilab.

Example 3.6

Let $H_1=H_2=\mathbb {R}$, $C=Q=[0,15]$. Let $A:H_1\rightarrow H_2$ be defined by $Ax=x$ for each $x\in H_1$. Then $A^*y=y$ for each $y\in H_2$. Let $D:C\rightarrow H_2$ defined by $Dx=\frac{x}{5}$ for each $x\in C$. For each $x\in C$, we define a multivalued mapping S on C as follows:

$$\begin{aligned} Sx = \left[ 0, \frac{7x}{10}\right] . \end{aligned}$$

For each $x,y\in C$, define bifunctions $F_1,\varphi _1:C\times C\rightarrow \mathbb {R}$ by

$$\begin{aligned} F_1(x,y)=3y^2+6xy-9x^2\,\,\text {and }\,\varphi _1(x,y)=y^2-x^2. \end{aligned}$$

For each $w,v\in Q$, define $F_2,\varphi _2:Q\times Q\rightarrow \mathbb {R}$ by

$$\begin{aligned} F_2(w,v)=4v^2+2wv-6w^2\,\,\text {and }\,\varphi _2(w,v)=w-v. \end{aligned}$$

Choose $r_n=\frac{n}{n+1}$, $\gamma =\frac{1}{4}$. It is easy to check that S, A, D, $F_1$, $F_2$, $\varphi _1$, $\varphi _2$, and $\{r_n\}$ satisfy all conditions in Theorem 3.1 with $\Gamma =\{0\}$.

For each $x\in C$ and each $n\in \mathbb {N}$, we compute $T^{(F_2,\varphi _2)}_{r}Ax$. Find w such that

$$\begin{aligned} 0&\le F_2(w,v)+\varphi _2(w,v)+\frac{1}{r}\langle v-w,w-Ax\rangle \\&= 4v^2+2wv-6w^2+w-v + \frac{1}{r}(v-w)(w-x)\\&\Leftrightarrow \\ 0&\le 4rv^2+2rwv-6rw^2+rw-rv + (v-w)(w-x)\\&=4rv^2+2rwv-6rw^2+rw-rv+wv-vx-w^2+wx\\&=4rv^2+(2rw-r+w-x)v + (-6rw^2+rw-w^2+wx) \end{aligned}$$

for all $v\in Q$. Let $J_2(v)=4rv^2+(2rw-r+w-x)v + (-6rw^2+rw-w^2+wx)$. $J_2(v)$ is s a quadratic function of v with coefficient $a=4r$, $b=2rw-r-x-w$, and $c=-6rw^2+rw-w^2+wx$. Determine the discriminant $\Delta$ of $J_2$ as follows:

$$\begin{aligned} \Delta&=b^2-4ac\\&= (2rw-r+w-x)^2 -4(4r)(-6rw^2+rw-w^2+wx)\\&= 100r^2w^2-20r^2w+20rw^2-20rwx+r^2-2rw+2rx+w^2-2wx+x^2\\&=(100r^2+20r+1)w^2+(-20r^2-20rx-2r-2x)w+(2rx+x^2+r^2)\\&=(10r+1)^2w^2-2w(10r+1)(x+r)+(x+r)^2\\&=((10r+1)w-(x+r))^2. \end{aligned}$$

We know that $J_2(v)\ge 0$ for all $v\in \mathbb {R}$. If it has at most one solution in $\mathbb {R}$, then $\Delta \le 0$, so we have $w=\frac{x+r}{10r+1}.$ This implies that

$$\begin{aligned} T^{(F_2,\varphi _2)}_{r}Ax=\frac{x+r}{10r+1}. \end{aligned}$$

Furthermore, we can get

$$\begin{aligned} \left( I-\gamma A^*\left( I-T^{(F_2,\varphi _2)}_{r}\right) A\right) x&=x-\gamma A^*(Ax-T^{(F_2,\varphi _2)}_{r}Ax)\\&=x-\frac{1}{4}A^*\left( x-\frac{x+r}{10r+1}\right) \\&=x-\frac{1}{4}\left( \frac{10rx-r}{10r+1}\right) \\&=\frac{30xr+4x+r}{40r+4}. \end{aligned}$$

Next, we find $u\in C$ such that $F_1(u,z)+\varphi _1(u,z)+\frac{1}{r}\langle z-u,u-s\rangle \ge 0$ for all $z\in C$, where $s=\left( I-\gamma A^*\left( I-T^{(F_2,\varphi _2)}_{r}\right) A\right) x$. Note that

$$\begin{aligned} 0&\le F_1(u,z)+\varphi _1(u,z)+\frac{1}{r}\langle z-u,u-s\rangle \\&= 4z^2+6uz-10u^2 + \frac{1}{r}\left\langle v-u,u-s\right\rangle \\&\Leftrightarrow \\ 0&\le 4rz^2+6ruz-10ru^2+(z-u)(u-s)\\&= 4rz^2+6ruz-10ru^2+uz-sz-u^2+us\\&= 4rz^2+(6ru+u-s)z+(-10ru^2-u^2+us) \end{aligned}$$

for all $z\in C$. Let $J_1(z)=4rz^2+(6ru+u-s)z+(-10ru^2-u^2+us)$. $J_1(z)$ is s a quadratic function of z with coefficient $a=4r$, $b=6ru+u-s$, and $c=-10ru^2-u^2+us$. Determine the discriminant $\Delta$ of $J_1$ as follows:

$$\begin{aligned} \Delta&= (6ru+u-s)^2 -4(4r)(-10ru^2-u^2+us)\\&= 196r^2u^2+28ru^2-28rus+u^2-2us+s^2\\&=((14r+1)u-s)^2. \end{aligned}$$

We know that $J_1(z)\ge 0$ for all $z\in \mathbb {R}$. If it has at most one solution in $\mathbb {R}$, then $\Delta \le 0$, so we have $u=\frac{s}{14r+1}.$ This implies that

$$\begin{aligned} u_n&=T^{(F_1,\varphi _1)}_{r_n}\left( I-\gamma A^*\left( I-T^{(F_2,\varphi _2)}_{r_n}\right) A\right) x_n,\\&=\frac{30x_nr_n+4x_n+r_n}{(40r_n+4)(14r_n+1)}\\&=\frac{30x_nr_n+4x_n+r_n}{560r_n^2+96r_n+4}. \end{aligned}$$

We put $z_n=\frac{7y_n}{10}$ for all $n\in \mathbb {N}$. Then the algorithm (3.1) becomes:

$$\begin{aligned} \left\{ \begin{aligned}&r_n=\frac{n}{n+1},\\&u_n=\frac{30x_nr_n+4x_n+r_n}{560r_n^2+96r_n+4},\\&x_{n+1}=\alpha _ng(x_n)+\beta _n x_n + \frac{7\gamma _nP_C\left( u_n-\frac{\lambda _nu_n}{5}\right) }{10},\,\,\,n\in \mathbb {N}. \end{aligned} \right. \end{aligned}$$

(3.22)

In this example, we set the parameter on algorithm (3.22) by $\lambda _n=\frac{1}{20}$, $\alpha _n=\frac{1}{n+1}$, $\beta _n=\frac{4n}{10n+10}$ and $\gamma _n=\frac{5n}{10n+10}$ for all $n\in \mathbb {N}$.

Figure 1 indicates the behavior of $x_n$ for algorithm (3.22) with $g(x)=0.5x$ that converges to the same solution, that is, $0\in \Gamma$ as a solution of this example.

Moreover, we test the effect of the different contraction mappings g on the convergence of the algorithm (3.22). In this test, Figure 2 presents the behaviour of $x_n$ by choosing three different contraction mappings $g(x)=0.1x$, $g(x)=0.5x$ and $g(x)=0.9x$. We see that the sequence $\{x_n\}$ by choosing the contraction $g(x)=0.1x$ converges to the solution $0\in \Gamma$ faster than the others.

Fig. 1

Behaviours of $x_n$ with three random initial points $x_1$

Fig. 2

Behaviours of $x_n$ with three different contraction mappings g

4 Conclusion

The results presented in this paper modify, extend, and improve the corresponding results of Kazmi and Rizvi [16, 17] and Moudafi [26], and others. The main aim of this paper is to propose an iterative algorithm based on the modified viscosity approximation method to find an element for solving a class of split generalized equilibrium problems, the variational inequality problems, and fixed point problems for nonexpansive multivalued mappings in real Hilbert spaces.