A New Algorithm for the Common Solutions of a Generalized Variational Inequality System and a Nonlinear Operator Equation in Banach Spaces

Abstract

We study a new algorithm for the common solutions of a generalized variational inequality system and the fixed points of an asymptotically non-expansive mapping in Banach spaces. Under some specific assumptions imposed on the control parameters, some strong convergence theorems for the sequence generated by our new viscosity iterative scheme to approximate their common solutions are proved. As an application of our main results, we solve the standard constrained convex optimization problem. The results here generalize and improve some other authors’ recently corresponding results.

1. Introduction

In the paper, we assume that X is a real Banach space and $X^{*}$ is the dual space of X. The normalized duality mapping $J:X\to 2^{X^{*}}$ is defined by

$$J\left(v\right)=\{f\in X^{*}:⟨v,f⟩=\parallel v\parallel \parallel f\parallel ,\parallel v\parallel =\parallel f\parallel \},\forall v\in X,$$

where $⟨·,·⟩$ denotes the duality pairing between X and $X^{*}$. As we all know, the mapping J is well defined and J is identity mapping if and only if X is a Hilbert space. In general, J is multiple-valued and nonlinear, $j\in J.$

A Banach space X is called strictly convex if $\frac{\parallel x+y\parallel }{2}<1$ for $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$. The modulus of convexity of X is defined by

$${\delta }_X\left(ϵ\right)=inf\left\{1-\parallel \frac{1}{2}(v+u)\parallel :\parallel v\parallel =\parallel u\parallel \le 1,\parallel v-u\parallel =ϵ\right\},$$

for all $0<ϵ\le 2$. X is said to be uniformly convex if ${\delta }_X\left(ϵ\right)>0$ for all $0<ϵ\le 2$.

Let ${\rho }_X:[0,+\infty )⟶[0,+\infty )$ be the modulus of smoothness of X defined by

$${\rho }_X\left(\zeta \right)=sup\left\{\frac{1}{2}(\parallel v+u\parallel +\parallel v-u\parallel )-1:\parallel v\parallel =1;\parallel u\parallel =\zeta \right\}.$$

A Banach space X is called uniformly smooth if $\frac{{\rho }_X\left(\zeta \right)}{\zeta }\to 0$ as $\zeta \to 0$. Moreover, X is uniformly smooth if the norm of X is uniformly Fr$\acute{e}$chet differentiable.

Suppose $\varnothing \ne C\subset X,T:C\to C$ is an operator and the set of fixed points of T is denoted by $Fix\left(T\right)$, that is $Fix\left(T\right)=\{v\in C:v=Tv\}$. T is said to be asymptotically non-expansive, if there exists a real sequence $\left\{k_n\right\}\subset [1,\infty )$ with $\underset{n\to \infty }{lim}{k}_n=1$ such that

$$\parallel T^{n}v-T^{n}u\parallel \le k_n\parallel v-u\parallel ,\forall v,u\in C.$$

If $k_n\equiv 1$, then we call that T is non-expansive. T is said to be uniformly asymptotically regular iff $\underset{n\to \infty }{lim}\parallel T^{n+1}v-T^{n}v\parallel =0,\forall v\in C$. T is said to be a contraction when there exists a number $\eta \in (0,1)$ such that $\parallel Tv-Tu\parallel \le \eta \parallel v-u\parallel ,\forall v,u\in C$.

A mapping $\widehat{B}:C\to X$ is called inverse strongly accretive if there exists $e\in R^{+}$ such that

$$⟨\widehat{B}v-\widehat{B}u,j(v-u)⟩\ge e{\parallel \widehat{B}v-\widehat{B}u\parallel }^2,\forall v,u\in C.$$

A bounded linear operator $\widehat{B}$ on X is called strongly positive, if there exists a fixed number $\overline{\delta }>0$ such that

$$⟨\widehat{B}v,j\left(v\right)⟩\ge \overline{\delta }{\parallel v\parallel }^2,\parallel e_{1}I-e_2\widehat{B}\parallel =\underset{\parallel v\parallel \le 1}{sup}\left|⟨(e_{1}I-e_2\widehat{B})v,j\left(v\right)⟩\right|,e_1\in [0,1],e_2\in [-1,1],\forall v\in X.$$

Assume $E\ne \varnothing ,E\subset C\subseteq X$ is a convex and closed set. An operator $Q:C\to E$ is called sunny when Q has the following relation:

$$Q(v+\alpha (v-Q\left(v\right)\left)\right)=Q\left(v\right),\forall v\in C,\alpha \ge 0,$$

for $v+\alpha (v-Q(v\left)\right)\in C$. A mapping $Q:C\to E$ is called retraction if $Qv=v,\forall v\in E$. Moreover, Q is called sunny non-expansive retraction from C onto E iff Q is a sunny, retraction, and non-expansive operator from C onto E.

Let C be a nonempty closed convex subset of X and $A,B:C\to X$ be two nonlinear mappings, respectively. Recall the classical variational inequality problem is to find a point $p\in C$ such that

$$\begin{array}{c}\hfill ⟨Ap,j(v-p)⟩\ge 0,\forall v\in C.\end{array}$$

(1)

The set of the solutions of the variational inequality (1) is denoted by $VI(C,A)$. Lots of problems in physics, optimization, differential equation (inclusion), finance and minimax problem reduce to find an element of (1) and relevant numerical analysis methods can be considered to solve the problems, see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16].

In 2011, Katchang et al. [17] introduced the following system of general variational inequalities. They considered the problem of finding $(t,s)\in C\times C$ such that:

$$\left\{\begin{array}{cc}\hfill & ⟨\lambda As+t-s,j(v-t)⟩\ge 0,\forall v\in C,\hfill \\ \hfill & ⟨\mu Bt+s-t,j(v-s)⟩\ge 0,\forall v\in C,\hfill \end{array}\right.$$

(2)

and proved a strong convergence theorem for the following sequence of problem (2):

$$\left\{\begin{array}{cc}\hfill & u_n=Q_C(I-\mu B)v_n,\hfill \\ \hfill & v_{n+1}={\alpha }_{n}f\left(v_n\right)+{\beta }_n{v}_n+{\gamma }_n[\beta S{v}_n+(1-\beta )Q_C(I-\lambda A)u_n],\hfill \end{array}\right.$$

where $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\}\subset (0,1),$ $\lambda$ and $\mu$ are two positive real numbers, $0<\beta <1$, $Q_C$ is the sunny non-expansive retraction from X to C, f is a contractive mapping on C and $S:C\to C$ is a non-expansive mapping.

In 2020, Wang et al. [18] investigated a new general variational inequality system:

$$\left\{\begin{array}{cc}\hfill & ⟨(I-\lambda A)(bt+(1-b\left)s\right)-t,j(v-t)⟩\le 0,\forall v\in C,\hfill \\ \hfill & ⟨(I-\mu B)t-s,j(v-s)⟩\le 0,\forall v\in C,\hfill \end{array}\right.$$

(3)

and proposed the following iterative method:

$$\left\{\begin{array}{cc}\hfill & w_n=Q_C(I-\mu B)v_n,\hfill \\ \hfill & l_n=Q_C(I-\lambda A)(b{v}_n+(1-b)l_n),\hfill \\ \hfill & u_n={\delta }_n{v}_n+(1-{\delta }_n)l_n,\hfill \\ \hfill & v_{n+1}={\alpha }_{n}f\left(v_n\right)+{\beta }_n{v}_n+{\gamma }_n{T}^n{u}_n,\hfill \end{array}\right.$$

where $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\},\left\{{\delta }_n\right\}\subset (0,1)$, $\lambda$ and $\mu$ are two positive real numbers, $0\le b\le 1,$ $Q_C$ is the sunny non-expansive retraction from X to C, f is a contractive mapping on C and $T:C\to C$ is an asymptotically non-expansive mapping. Then they proved that $v_n$ converges to the common element of the fixed points of an asymptotically non-expansive mapping and the solutions of the general variational inequality system (3).

Inspired by the above mentioned results, we study the following new generalized variational inequality system in uniformly convex and uniformly smooth Banach spaces:

$$\left\{\begin{array}{cc}\hfill & ⟨t-(I-{\lambda }_1{B}_1)(bt+(1-b)s),j(v-t)⟩\ge 0,\forall v\in C,\hfill \\ \hfill & ⟨s-(I-{\lambda }_2{B}_2)(bt+(1-b)w),j(v-s)⟩\ge 0,\forall v\in C,\hfill \\ \hfill & ⟨w-(I-{\lambda }_3{B}_3)t,j(v-w)⟩\ge 0,\forall v\in C.\hfill \end{array}\right.$$

(4)

This is the main problem which we seek to solve in this work and construct the suitable abstract iterative method. The new generalized variational inequality system problem (4) contains (1)–(3) as special cases. We construct a new iterative algorithm to approximate the common solutions of the generalized system of variational inequality (4) and an asymptotically non-expansive mapping. Some strong convergence theorems that depend on some suitable conditions are proved. Eventually, we show an application for solving the standard constrained problem of convex optimization to illustrate the efficiency of our main theorem. Some other outcomes proposed by other authors are also improved, see [9,10,17,18,19,20,21,22].

2. Preliminaries

This section contains the useful lemmas that are indispensable for proving our convergence theorem in the following section. Generally speaking, we suppose that space X is uniformly smooth and uniformly convex in this section.

Lemma 1

([18]). Suppose that $X\supset C\ne \varnothing$ and C is a closed convex set. Let J be the normalized duality mapping $J:X\to 2^{X^{*}}$ and $Q:X\to C$ be a retraction. Then the followings are equivalent:

(I) 

Q is sunny and non-expansive;

(II) 

$⟨v-Qv,J(u-Qv)⟩\le 0,\forall v\in X,u\in C$;

(III) 

${\parallel Qv-Qu\parallel }^2\le ⟨v-u,J(Qv-Qu)⟩,\forall v,u\in X$.

Lemma 2

([23]). Assume X is uniformly convex and $V_R\left(0\right)$ is a closed ball of X. Then there exists a continuous strictly increasing convex function $m_1:[0,\infty )\to [0,\infty )$ with $m_1\left(0\right)=0$ such that

$$\parallel a_1{v}_1+a_2{v}_2+\cdots +a_n{v}_n{\parallel }^2=\sum _{i=1}^n{a}_i\parallel v_i{\parallel }^2-a_x{a}_y{m}_1(\parallel v_x-v_y\parallel ),$$

for any $x,y\in \{1,2,...,n\}$ and for $v_i\in V_R\left(0\right):=\{v\in X:\parallel v\parallel \le R\},i=1,2,...,n$ with ${\displaystyle \sum _{i=1}^n}{a}_i=1$.

Lemma 3

([24]). Suppose $\left\{v_n\right\}$ and $\left\{z_n\right\}$ are two bounded sequences of X. Let $\left\{{\mu }_n\right\}\subset [0,1]$ be a number sequence with $0<\underset{n\to \infty }{lim\; inf}{\mu }_n\le \underset{n\to \infty }{lim\; sup}{\mu }_n<1$. Assume that $v_{n+1}=(1-{\mu }_n)v_n+{\mu }_n{z}_n,n\ge 0$ and $\underset{n\to \infty }{lim\; sup}(\parallel z_{n+1}-z_n\parallel -\parallel v_{n+1}-v_n\parallel )\le 0$. Then $\underset{n\to \infty }{lim}\parallel z_n-v_n\parallel =0$.

Lemma 4

([25]). For $\forall v,u\in X,j(v+u)\in J(v+u)$, there holds the relation:

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

Lemma 5

([26]). Let $\left\{c_n\right\}$ be a nonnegative number sequence satisfying:

$$c_{n+1}\le (1-{\sigma }_n)c_n+{\sigma }_n{\theta }_n,\forall n\ge 0,$$

where

$$\left\{{\sigma }_n\right\}\subset [0,1],\sum _{n=0}^{\infty }{\sigma }_n=+\infty ;\underset{n\to \infty }{lim\; sup}{\theta }_n\le 0or\sum _{n=1}^{\infty }\left|{\sigma }_n{\theta }_n\right|<+\infty .$$

Then $\underset{n\to \infty }{lim}{c}_n=0$.

Lemma 6

([27]). Suppose that $\overline{V}$ is a strongly positive linear bounded operator on X with coefficient $\overline{\delta }>0$ and $0<\rho \le \parallel \overline{V}{\parallel }^{-1}.$ Then, $\parallel I-\rho \overline{V}\parallel \le 1-\rho \overline{\delta }.$

Lemma 7

([18]). Assume C is a nonempty convex and closed subset of X. Assume mapping $\widehat{B}:C\to X$ is a $\widehat{b}$-inverse strongly accretive. Then,

$$\parallel (I-\lambda \widehat{B})v-(I-\lambda \widehat{B}){u\parallel }^2\le {\parallel v-u\parallel }^2-\lambda (2\widehat{b}-\lambda ){\parallel \widehat{B}v-\widehat{B}u\parallel }^2,\lambda >0.$$

If $0<\lambda <2\widehat{b}$, then $I-\lambda \widehat{B}$ is non-expansive.

Lemma 8

([28]). Let $C\ne \varnothing$ be a closed convex subset of X. Suppose $Q_C:X\to C$ is a sunny non-expansive mapping and $\widehat{B}$ is an accretive operator from C into X. Then,

$$VI(C,\widehat{B})=Fix\left(Q_C(I-\lambda \widehat{B})\right),\lambda >0,$$

where $VI(C,\widehat{B})=\{t\in C:⟨\widehat{B}t,j(v-t)⟩\ge 0,\forall v\in C\}.$

Lemma 9

([18]). There exists a convex continuous strictly increasing function $m_2:[0,2r]\to R,m_2\left(0\right)=0$ such that

$$m_2(\parallel v-u\parallel )\le {\parallel v\parallel }^2-2⟨v,j\left(u\right)⟩+{\parallel u\parallel }^2,\forall r>0,v,u\in B_r.$$

Lemma 10

([9]). Assume $\varnothing \ne C\subset X$ is a closed convex set. Suppose $T:C\to C$ is an asymptotically non-expansive mapping and $Fix\left(T\right)\ne \varnothing$. If X has a weakly sequentially continuous duality mapping $J,$ then the mapping $I-T$ is demi-closed at zero, where I is the identity mapping, i.e., if $v_n\rightharpoonup v$ and $\parallel v_n-T{v}_n\parallel \to 0$, then $v=Tv$.

Lemma 11.

Assume $X\supset C\ne \varnothing$ is a closed convex set. Let $B_1,B_2,B_3:C\to X$ be three mappings, respectively. When ${\lambda }_1,{\lambda }_2,{\lambda }_3>0$, $b\in [0,1)$, then the following statements are the same:

(i) 

$(t,s,w)\in C\times C\times C$ is a solution of the problem (4);

(ii) 

t is a fixed point of the mapping L, i.e., $t\in Fix\left(L\right)$, where $L:C\to C$ is defined by

$$L\left(v\right)=Q_C(I-{\lambda }_1{B}_1)(bv+(1-b)Q_C(I-{\lambda }_2{B}_2)(bv+(1-b)Q_C(I-{\lambda }_3{B}_3)v)),\forall v\in C,$$

(5)

and $s=Q_C(I-{\lambda }_2{B}_2)(bt+(1-b)w)$, $w=Q_C(I-{\lambda }_3{B}_3)t$.

Proof. 

$\left(i\right)\Rightarrow \left(ii\right)$. Assume that $(t,s,w)\in C\times C\times C$ is a solution of (4). For all $v\in C$, we have

$$\begin{array}{cc}\hfill & ⟨t-(I-{\lambda }_1{B}_1)(bt+(1-b)s),j(v-t)⟩\ge 0,\hfill \\ \hfill & ⟨s-(I-{\lambda }_2{B}_2)(bt+(1-b)w),j(v-s)⟩\ge 0,\hfill \\ \hfill & ⟨w-(I-{\lambda }_3{B}_3)t,j(v-w)⟩\ge 0.\hfill \end{array}$$

From property of $Q_C$, we get

$$\begin{array}{cc}\hfill & t=Q_C(I-{\lambda }_1{B}_1)(bt+(1-b)s),\hfill \\ \hfill & s=Q_C(I-{\lambda }_2{B}_2)(bt+(1-b)w),\hfill \\ \hfill & w=Q_C(I-{\lambda }_3{B}_3)t.\hfill \end{array}$$

It implies that

$$t=Q_C(I-{\lambda }_1{B}_1)(bt+(1-b)Q_C(I-{\lambda }_2{B}_2)(bt+(1-b)Q_C(I-{\lambda }_3{B}_3)t))=L\left(t\right).$$

Then $t\in Fix\left(L\right)$, where $s=Q_C(I-{\lambda }_2{B}_2)(bt+(1-b)w)$ and $w=Q_C(I-{\lambda }_3{B}_3)t$.

$\left(ii\right)\Rightarrow \left(i\right)$. Let $t\in Fix\left(L\right),s=Q_C(I-{\lambda }_2{B}_2)(bt+(1-b)w)$ and $w=Q_C(I-{\lambda }_3{B}_3)t$. Since $t\in Fix\left(L\right)$, we have

$$\begin{array}{cc}\hfill t& =Q_C(I-{\lambda }_1{B}_1)(bt+(1-b)Q_C(I-{\lambda }_2{B}_2)(bt+(1-b)Q_C(I-{\lambda }_3{B}_3)t))\hfill \\ \hfill & =Q_C(I-{\lambda }_1{B}_1)(bt+(1-b)s).\hfill \end{array}$$

In combination with $s=Q_C(I-{\lambda }_2{B}_2)(bt+(1-b)w),w=Q_C(I-{\lambda }_3{B}_3)t$ and the property of $Q_C$, we get

$$\begin{array}{cc}\hfill & ⟨t-(I-{\lambda }_1{B}_1)(bt+(1-b)s),j(v-t)⟩\ge 0,\forall v\in C,\hfill \\ \hfill & ⟨s-(I-{\lambda }_2{B}_2)(bt+(1-b)w),j(v-s)⟩\ge 0,\forall v\in C,\hfill \\ \hfill & ⟨w-(I-{\lambda }_3{B}_3)t,j(v-w)⟩\ge 0,\forall v\in C.\hfill \end{array}$$

Then $(t,s,w)\in C\times C\times C$ is a solution of the problem (4). □

3. Results

Theorem 1.

Suppose X is a uniformly smooth and uniformly convex Banach space and $C\ne \varnothing$ is a closed convex subset of X. Assume $Q_C:X\to C$ is a sunny non-expansive retraction. Let $\widehat{B},B_1,B_2,B_3:C\to X$ be $\widehat{b},b_1,b_2,b_3$-inverse strongly accretive, respectively. L is the mapping defined by (5) in Lemma 11 and $b\in [0,1)$. Assume $\overline{V}:X\to X$ is a bounded linear strongly positive operator on X with coefficient $\overline{\delta }>0$ and $0<\overline{\delta }<{\parallel \overline{V}\parallel }^{-1}.$ Let g be an $\eta \in (0,1)$-contraction mapping on C and $T:C\to C$ be an asymptotically non-expansive mapping with $\Psi =Fix\left(T\right)\bigcap Fix\left(L\right)\bigcap VI(C,\widehat{B})\ne \varnothing$. Assume $\left\{v_n\right\}$ is defined as follows:

$$\left\{\begin{array}{cc}\hfill & v_1\in C,\hfill \\ \hfill & u_n={\beta }_n\delta g\left(v_n\right)+(I-{\beta }_n\overline{V})L{v}_n,\hfill \\ \hfill & v_{n+1}={\mu }_n{v}_n+{\nu }_n{T}^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})+{\omega }_n{Q}_C(I-\lambda \widehat{B})u_n,n\ge 0,\hfill \end{array}\right.$$

(6)

where $\lambda \in (0,2\widehat{b}),\left\{{\beta }_n\right\}\subset (0,1),{\mu }_n+{\nu }_n+{\omega }_n=1$, and the following coefficient conditions are satisfying:

(i) 

$\underset{n\to \infty }{lim}{\beta }_n=0,{\displaystyle \sum _{n=0}^{\infty }}{\beta }_n=\infty ,{\displaystyle \sum _{n=1}^{\infty }}|{\beta }_{n+1}-{\beta }_n|<\infty ;$

(ii) 

$0<h_1\le {\mu }_n,{\nu }_n,{\omega }_n\le h_2<1,{\displaystyle \sum _{n=1}^{\infty }}|{\mu }_{n+1}-{\mu }_n|<\infty ,{\displaystyle \sum _{n=1}^{\infty }}|{\nu }_{n+1}-{\nu }_n|<\infty$, ${\displaystyle \sum _{n=1}^{\infty }}|{\omega }_{n+1}-{\omega }_n|<\infty ;$

(iii) 

$0\le {\xi }_n\le {\xi }_{n+1}\le 1,0<\delta <\frac{h_1\overline{\delta }}{2\eta },k_n^2-1=\epsilon {\beta }_n,0<\epsilon <h_1\overline{\delta }-2h_1\delta \eta ;$

(iv) 

${\nu }_n{k}_n^2(1-{\xi }_n)<1-\tau ,h_2<\tau <1.$

Then the sequence $\left\{v_n\right\}$ converges strongly to $t=Q_{\Psi }(I-\overline{V}+\delta g)t\in \Psi$ and $(t,s,w)$ is a solution of (4), where $s=Q_C(I-{\lambda }_2{B}_2)(bt+(1-b)w)$ and $w=Q_C(I-{\lambda }_3{B}_3)t,{\lambda }_1,{\lambda }_2,{\lambda }_3\in (0,2\overline{b})$ with $\overline{b}=min\{b_1,b_2,b_3\}.$

Proof. 

Our proof process can be divided into 6 steps.

Step 1: Let $\widehat{B},B_1,B_2,B_3$ be $\widehat{b},b_1,b_2,b_3$-inverse strongly accretive mapping, respectively, and $\lambda \in (0,2\widehat{b})$, ${\lambda }_1,{\lambda }_2,{\lambda }_3\in (0,2\overline{b})$ with $\overline{b}=min\{b_1,b_2,b_3\}$. From Lemma 7, we know that $(I-\lambda \widehat{B}),(I-{\lambda }_1{B}_1),(I-{\lambda }_2{B}_2)$ and $(I-{\lambda }_3{B}_3)$ are non-expansive mappings. Moveover, we attain that $Q_C(I-\lambda \widehat{B}),Q_C(I-{\lambda }_1{B}_1),Q_C(I-{\lambda }_2{B}_2)$ and $Q_C(I-{\lambda }_3{B}_3)$ are non-expansive mapping.

So, we have

$$\begin{array}{cc}\hfill \parallel Lv-Lu\parallel & =\parallel Q_C(I-{\lambda }_1{B}_1)(bv+(1-b)Q_C(I-{\lambda }_2{B}_2)(bv+(1-b)Q_C(I-{\lambda }_3{B}_3)v))\hfill \\ \hfill & -Q_C(I-{\lambda }_1{B}_1)(bu+(1-b)Q_C(I-{\lambda }_2{B}_2)(bu+(1-b)Q_C(I-{\lambda }_3{B}_3)u))\parallel \hfill \\ \hfill & \le \parallel (bv+(1-b)Q_C(I-{\lambda }_2{B}_2)(bv+(1-b)Q_C(I-{\lambda }_3{B}_3)v))\hfill \\ \hfill & -(bu+(1-b)Q_C(I-{\lambda }_2{B}_2)(bu+(1-b)Q_C(I-{\lambda }_3{B}_3)u))\parallel \hfill \\ \hfill & \le b\parallel v-u\parallel +(1-b)\parallel Q_C(I-{\lambda }_2{B}_2)(bv+(1-b)Q_C(I-{\lambda }_3{B}_3)v)\hfill \\ \hfill & -Q_C(I-{\lambda }_2{B}_2)(bu+(1-b)Q_C(I-{\lambda }_3{B}_3)u)\parallel \hfill \\ \hfill & \le b\parallel v-u\parallel +(1-b)\parallel (bv+(1-b)Q_C(I-{\lambda }_3{B}_3)v)-(bu+(1-b)Q_C(I-{\lambda }_3{B}_3)u)\parallel \hfill \\ \hfill & \le b\parallel v-u\parallel +(1-b)(b\parallel v-u\parallel +(1-b)\parallel Q_C(I-{\lambda }_3{B}_3)v-Q_C(I-{\lambda }_3{B}_3)u\parallel )\hfill \\ \hfill & \le b\parallel v-u\parallel +(1-b)(b\parallel v-u\parallel +(1-b)\parallel v-u\parallel )\hfill \\ \hfill & =\parallel v-u\parallel .\hfill \end{array}$$

Then, L is non-expansive.

Step 2: For any given $q\in \Psi$, from (6), we have

$$\begin{array}{cc}\hfill \parallel T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})-q\parallel & \le k_n\parallel {\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}-q\parallel \hfill \\ \hfill & \le k_n{\xi }_n\parallel v_n-q\parallel +k_n(1-{\xi }_n)\parallel v_{n+1}-q\parallel ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \parallel u_n-q\parallel & =\parallel {\beta }_n\delta g\left(v_n\right)+(I-{\beta }_n\overline{V})L{v}_n-q\parallel \hfill \\ \hfill & \le {\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}q\parallel +\parallel (I-{\beta }_n\overline{V})(L{v}_n-q)\parallel \hfill \\ \hfill & \le {\beta }_n\parallel \delta g\left(v_n\right)-\delta g\left(q\right)\parallel +{\beta }_n\parallel \delta g\left(q\right)-\overline{V}q\parallel +(1-{\beta }_n\overline{\delta })\parallel v_n-q\parallel \hfill \\ \hfill & \le {\beta }_n\delta \eta \parallel v_n-q\parallel +{\beta }_n\parallel \delta g\left(q\right)-\overline{V}q\parallel +(1-{\beta }_n\overline{\delta })\parallel v_n-q\parallel \hfill \\ \hfill & \le [1-{\beta }_n(\overline{\delta }-\delta \eta )]\parallel v_n-q\parallel +{\beta }_n\parallel \delta g\left(q\right)-\overline{V}q\parallel .\hfill \end{array}$$

Then we compute

$$\begin{array}{cc}\hfill \parallel v_{n+1}-q\parallel & =\parallel {\mu }_n{v}_n+{\nu }_n{T}^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})+{\omega }_n{Q}_C(I-\lambda \widehat{B})u_n-q\parallel \hfill \\ \hfill & \le \parallel {\mu }_n(v_n-q)+{\nu }_n(T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})-q)+{\omega }_n(Q_C(I-\lambda \widehat{B})u_n-q)\parallel \hfill \\ \hfill & \le {\mu }_n\parallel v_n-q\parallel +{\nu }_n\parallel T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})-q\parallel +{\omega }_n\parallel u_n-q\parallel \hfill \\ \hfill & \le {\mu }_n\parallel v_n-q\parallel +{\nu }_n{k}_n{\xi }_n\parallel v_n-q\parallel +{\nu }_n{k}_n(1-{\xi }_n)\parallel v_{n+1}-q\parallel \hfill \\ \hfill & +{\omega }_n[1-{\beta }_n(\overline{\delta }-\delta \eta )]\parallel v_n-q\parallel +{\omega }_n{\beta }_n\parallel \delta g\left(q\right)-\overline{V}q\parallel \hfill \\ \hfill & \le ({\mu }_n+{\nu }_n{k}_n{\xi }_n+{\omega }_n[1-{\beta }_n(\overline{\delta }-\delta \eta )])\parallel v_n-q\parallel +{\nu }_n{k}_n(1-{\xi }_n)\parallel v_{n+1}-q\parallel \hfill \\ \hfill & +{\omega }_n{\beta }_n\parallel \delta g\left(q\right)-\overline{V}q\parallel .\hfill \end{array}$$

It follows that

$$\begin{array}{c}\hfill (1-{\nu }_n{k}_n(1-{\xi }_n))\parallel v_{n+1}-q\parallel \le ({\mu }_n+{\nu }_n{k}_n{\xi }_n+{\omega }_n[1-{\beta }_n(\overline{\delta }-\delta \eta )])\parallel v_n-q\parallel +{\omega }_n{\beta }_n\parallel \delta g\left(q\right)-\overline{V}q\parallel .\end{array}$$

From conditions $\left(ii\right),\left(iii\right)$ and $\left(iv\right)$ in Theorem 1, we get

$$\begin{array}{cc}\hfill \parallel v_{n+1}-q\parallel & \le \frac{{\mu }_n+{\nu }_n{k}_n{\xi }_n+{\omega }_n[1-{\beta }_n(\overline{\delta }-\delta \eta )]}{1-{\nu }_n{k}_n(1-{\xi }_n)}\parallel v_n-q\parallel +\frac{{\omega }_n{\beta }_n}{1-{\nu }_n{k}_n(1-{\xi }_n)}\parallel \delta g\left(q\right)-\overline{V}q\parallel \hfill \\ \hfill & \le (1-\frac{{\omega }_n{\beta }_n(\overline{\delta }-\delta \eta )-{\nu }_n(k_n-1)}{1-{\nu }_n{k}_n(1-{\xi }_n)})\parallel v_n-q\parallel +\frac{{\omega }_n{\beta }_n}{1-{\nu }_n{k}_n(1-{\xi }_n)}\parallel \delta g\left(q\right)-\overline{V}q\parallel \hfill \\ \hfill & \le (1-\frac{h_1{\beta }_n(\overline{\delta }-\delta \eta )-\epsilon {\beta }_n}{1-{\nu }_n{k}_n(1-{\xi }_n)})\parallel v_n-q\parallel +\frac{{\beta }_n}{1-{\nu }_n{k}_n(1-{\xi }_n)}\parallel \delta g\left(q\right)-\overline{V}q\parallel \hfill \\ \hfill & \le (1-\frac{{\beta }_n(h_1\overline{\delta }-h_1\delta \eta -\epsilon )}{1-{\nu }_n{k}_n(1-{\xi }_n)})\parallel v_n-q\parallel +\frac{{\beta }_n(h_1\overline{\delta }-h_1\delta \eta -\epsilon )}{1-{\nu }_n{k}_n(1-{\xi }_n)}\frac{\parallel \delta g\left(q\right)-\overline{V}q\parallel }{h_1\overline{\delta }-h_1\delta \eta -\epsilon }\hfill \\ \hfill & \le max\{\parallel v_n-q\parallel ,\frac{\parallel \delta g\left(q\right)-\overline{V}q\parallel }{h_1\overline{\delta }-h_1\delta \eta -\epsilon }\}.\hfill \end{array}$$

By induction, we readily obtain

$$\parallel v_{n+1}-q\parallel \le max\{\parallel v_1-q\parallel ,\frac{\parallel \delta g\left(q\right)-\overline{V}q\parallel }{h_1\overline{\delta }-h_1\delta \eta -\epsilon }\}.$$

This implies that $\left\{v_n\right\}$ is bounded, and so are $\left\{u_n\right\},\left\{g\left(v_n\right)\right\},\left\{L{v}_n\right\},\left\{T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\right\}$.

Step 3: By (6) and Lemma 6, we gain that

$$\begin{array}{cc}\hfill \parallel u_{n+1}-u_n\parallel & =\parallel {\beta }_{n+1}\delta g\left(v_{n+1}\right)+(I-{\beta }_{n+1}\overline{V})L{v}_{n+1}-({\beta }_n\delta g\left(v_n\right)+(I-{\beta }_n\overline{V})L{v}_n)\parallel \hfill \\ \hfill & =\parallel {\beta }_{n+1}\delta g\left(v_{n+1}\right)-{\beta }_{n+1}\delta g\left(v_n\right)+{\beta }_{n+1}\delta g\left(v_n\right)-{\beta }_n\delta g\left(v_n\right)\hfill \\ \hfill & +(I-{\beta }_{n+1}\overline{V})L{v}_{n+1}-(I-{\beta }_{n+1}\overline{V})L{v}_n+(I-{\beta }_{n+1}\overline{V})L{v}_n-(I-{\beta }_n\overline{V})L{v}_n\parallel \hfill \\ \hfill & =\parallel {\beta }_{n+1}\delta [g\left(v_{n+1}\right)-g\left(v_n\right)]+({\beta }_{n+1}-{\beta }_n)\delta g\left(v_n\right)+(I-{\beta }_{n+1}\overline{V})(L{v}_{n+1}-L{v}_n)\hfill \\ \hfill & -({\beta }_{n+1}-{\beta }_n)\overline{V}L{v}_n\parallel \hfill \\ \hfill & \le {\beta }_{n+1}\delta \parallel g\left(v_{n+1}\right)-g\left(v_n\right)\parallel +|{\beta }_{n+1}-{\beta }_n|\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel +\parallel I-{\beta }_{n+1}\overline{V}\parallel \parallel L{v}_{n+1}\hfill \\ \hfill & -L{v}_n\parallel \hfill \\ \hfill & \le {\beta }_{n+1}\delta \eta \parallel v_{n+1}-v_n\parallel +|{\beta }_{n+1}-{\beta }_n|\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel +(1-{\beta }_{n+1}\overline{\delta })\parallel v_{n+1}-v_n\parallel \hfill \\ \hfill & =(1-{\beta }_{n+1}(\overline{\delta }-\delta \eta ))\parallel v_{n+1}-v_n\parallel +|{\beta }_{n+1}-{\beta }_n|\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \hfill \\ \hfill & \le \parallel v_{n+1}-v_n\parallel +|{\beta }_{n+1}-{\beta }_n|\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel .\hfill \end{array}$$

(7)

By (6) and (7), we can acquire

$$\begin{array}{cc}\hfill \parallel v_{n+2}-v_{n+1}\parallel & =\parallel {\mu }_{n+1}{v}_{n+1}+{\nu }_{n+1}{T}^{n+1}({\xi }_{n+1}{v}_{n+1}+(1-{\xi }_{n+1})v_{n+2})+{\omega }_{n+1}{Q}_C(I-\lambda \widehat{B})u_{n+1}\hfill \\ \hfill & -[{\mu }_n{v}_n+{\nu }_n{T}^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})+{\omega }_n{Q}_C(I-\lambda \widehat{B})u_n]\parallel \hfill \\ \hfill & =\parallel {\mu }_{n+1}(v_{n+1}-v_n)+({\mu }_{n+1}-{\mu }_n)v_n+{\nu }_{n+1}(T^{n+1}({\xi }_{n+1}{v}_{n+1}+(1-{\xi }_{n+1})v_{n+2})\hfill \\ \hfill & -T^{n+1}({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}))+({\nu }_{n+1}-{\nu }_n)T^{n+1}({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\hfill \\ \hfill & +{\nu }_n[T^{n+1}({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})]\hfill \\ \hfill & +{\omega }_{n+1}[Q_C(I-\lambda \widehat{B})u_{n+1}-Q_C(I-\lambda \widehat{B})u_n]+({\omega }_{n+1}-{\omega }_n)Q_C(I-\lambda \widehat{B})u_n\parallel \hfill \\ \hfill & \le {\mu }_{n+1}\parallel v_{n+1}-v_n\parallel +|{\mu }_{n+1}-{\mu }_n|\parallel v_n\parallel +{\nu }_{n+1}{k}_{n+1}\parallel ({\xi }_{n+1}{v}_{n+1}+(1-{\xi }_{n+1})v_{n+2})\hfill \\ \hfill & -({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}))\parallel +|{\nu }_{n+1}-{\nu }_n|\parallel T^{n+1}v\parallel +{\nu }_n\underset{v\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel \hfill \\ \hfill & +{\omega }_{n+1}\parallel Q_C(I-\lambda \widehat{B})u_{n+1}-Q_C(I-\lambda \widehat{B})u_n\parallel +|{\omega }_{n+1}-{\omega }_n|\parallel Q_C(I-\lambda \widehat{B})u_n\parallel \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\mu }_{n+1}\parallel v_{n+1}-v_n\parallel +|{\mu }_{n+1}-{\mu }_n|\parallel v_n\parallel +{\nu }_{n+1}{k}_{n+1}{\xi }_n\parallel v_{n+1}-v_n\parallel \hfill \\ \hfill & +{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})\parallel v_{n+2}-v_{n+1}\parallel +|{\nu }_{n+1}-{\nu }_n|\parallel T^{n+1}v\parallel \hfill \\ \hfill & +{\nu }_n\underset{v\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel +{\omega }_{n+1}\parallel u_{n+1}-u_n\parallel +|{\omega }_{n+1}-{\omega }_n|\parallel Q_C(I-\lambda \widehat{B})u_n\parallel \hfill \\ \hfill & \le {\mu }_{n+1}\parallel v_{n+1}-v_n\parallel +|{\mu }_{n+1}-{\mu }_n|\parallel v_n\parallel +{\nu }_{n+1}{k}_{n+1}{\xi }_n\parallel v_{n+1}-v_n\parallel \hfill \\ \hfill & +{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})\parallel v_{n+2}-v_{n+1}\parallel +|{\nu }_{n+1}-{\nu }_n|\parallel T^{n+1}v\parallel \hfill \\ \hfill & +{\nu }_n\underset{v\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel +{\omega }_{n+1}(\parallel v_{n+1}-v_n\parallel +|{\beta }_{n+1}-{\beta }_n|\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel )\hfill \\ \hfill & +|{\omega }_{n+1}-{\omega }_n|\parallel Q_C(I-\lambda \widehat{B})u_n\parallel \hfill \\ \hfill & \le ({\mu }_{n+1}+{\nu }_{n+1}{k}_{n+1}{\xi }_n+{\omega }_{n+1})\parallel v_{n+1}-v_n\parallel +{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})\parallel v_{n+2}-v_{n+1}\parallel \hfill \\ \hfill & +|{\mu }_{n+1}-{\mu }_n|\parallel v_n\parallel +|{\nu }_{n+1}-{\nu }_n|\parallel T^{n+1}v\parallel +{\nu }_n\underset{v\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel \hfill \\ \hfill & +{\omega }_{n+1}|{\beta }_{n+1}-{\beta }_n|\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel +|{\omega }_{n+1}-{\omega }_n|\parallel Q_C(I-\lambda \widehat{B})u_n\parallel .\hfill \end{array}$$

Which implies that

$$\begin{array}{cc}\hfill & [1-{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})]\parallel v_{n+2}-v_{n+1}\parallel \hfill \\ \hfill & \le ({\mu }_{n+1}+{\nu }_{n+1}{k}_{n+1}{\xi }_n+{\omega }_{n+1})\parallel v_{n+1}-v_n\parallel +|{\mu }_{n+1}-{\mu }_n|\parallel v_n\parallel +|{\nu }_{n+1}-{\nu }_n|\parallel T^{n+1}v\parallel \hfill \\ \hfill & +{\nu }_n\underset{v\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel +{\omega }_{n+1}|{\beta }_{n+1}-{\beta }_n|\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel +|{\omega }_{n+1}-{\omega }_n|\parallel Q_C(I-\lambda \widehat{B})u_n\parallel .\hfill \end{array}$$

By conditions $\left(ii\right)$–$\left(iv\right)$ in Theorem 1, we have

$$\begin{array}{cc}\hfill & \parallel v_{n+2}-v_{n+1}\parallel \hfill \\ \hfill & \le \frac{{\mu }_{n+1}+{\nu }_{n+1}{k}_{n+1}{\xi }_n+{\omega }_{n+1}}{1-{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}\parallel v_{n+1}-v_n\parallel +\frac{|{\mu }_{n+1}-{\mu }_n|}{1-{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}\parallel v_n\parallel \hfill \\ \hfill & +\frac{|{\nu }_{n+1}-{\nu }_n|}{1-{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}\parallel T^{n+1}v\parallel +\frac{{\nu }_n}{1-{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}\underset{v\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel \hfill \\ \hfill & +\frac{{\omega }_{n+1}|{\beta }_{n+1}-{\beta }_n|}{1-{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel +\frac{|{\omega }_{n+1}-{\omega }_n|}{1-{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}\parallel Q_C(I-\lambda \widehat{B})u_n\parallel \hfill \\ \hfill & \le [1+\frac{{\nu }_{n+1}(k_{n+1}-1)-{\nu }_{n+1}{k}_{n+1}({\xi }_{n+1}-{\xi }_n)}{1-{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}]\parallel v_{n+1}-v_n\parallel +\frac{|{\mu }_{n+1}-{\mu }_n|}{1-{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}\parallel v_n\parallel \hfill \\ \hfill & +\frac{|{\nu }_{n+1}-{\nu }_n|}{1-{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}\parallel T^{n+1}v\parallel +\frac{{\nu }_n}{1-{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}\underset{v\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel \hfill \\ \hfill & +\frac{{\omega }_{n+1}|{\beta }_{n+1}-{\beta }_n|}{1-{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel +\frac{|{\omega }_{n+1}-{\omega }_n|}{1-{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}\parallel Q_C(I-\lambda \widehat{B})u_n\parallel \hfill \\ \hfill & \le [1+\frac{{\nu }_{n+1}(k_{n+1}-1)}{\tau }]\parallel v_{n+1}-v_n\parallel +\frac{{\omega }_{n+1}|{\beta }_{n+1}-{\beta }_n|}{\tau }\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \hfill \\ \hfill & +\frac{|{\mu }_{n+1}-{\mu }_n|}{\tau }\parallel v_n\parallel +\frac{|{\nu }_{n+1}-{\nu }_n|}{\tau }\parallel T^{n+1}v\parallel +\frac{{\nu }_n}{\tau }\underset{v\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel \hfill \\ \hfill & +\frac{|{\omega }_{n+1}-{\omega }_n|}{\tau }\parallel Q_C(I-\lambda \widehat{B})u_n\parallel \hfill \\ \hfill & \le (1+\epsilon {\beta }_{n+1})\parallel v_{n+1}-v_n\parallel +M_{n_1},\hfill \end{array}$$

(8)

where

$$\begin{array}{cc}\hfill M_{n_1}& =\frac{|{\mu }_{n+1}-{\mu }_n|}{\tau }\parallel v_n\parallel +\frac{|{\nu }_{n+1}-{\nu }_n|}{\tau }\parallel T^{n+1}v\parallel +\frac{{\nu }_n}{\tau }\underset{v\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel \hfill \\ \hfill & +\frac{{\omega }_{n+1}|{\beta }_{n+1}-{\beta }_n|}{\tau }\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel +\frac{|{\omega }_{n+1}-{\omega }_n|}{\tau }\parallel Q_C(I-\lambda \widehat{B})u_n\parallel .\hfill \end{array}$$

We set

$$\begin{array}{c}\hfill v_{n+1}=(1-{\mu }_n)z_n+{\mu }_n{v}_n,\end{array}$$

where

$$\begin{array}{c}\hfill z_n={\displaystyle \frac{v_{n+1}-{\mu }_n{v}_n}{1-{\mu }_n}},\forall n\ge 0.\end{array}$$

Since

$$\begin{array}{c}\hfill v_{n+1}-{\mu }_n{v}_n={\nu }_n{T}^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})+{\omega }_n{Q}_C(I-\lambda \widehat{B})u_n,\end{array}$$

we attain

$$\begin{array}{cc}\hfill & z_{n+1}-z_n\hfill \\ \hfill & ={\displaystyle \frac{v_{n+2}-{\mu }_n{v}_{n+1}}{1-{\mu }_{n+1}}}-{\displaystyle \frac{v_{n+1}-{\mu }_n{v}_n}{1-{\mu }_n}}\hfill \\ \hfill & ={\displaystyle \frac{{\nu }_{n+1}{T}^{n+1}({\xi }_{n+1}{v}_{n+1}+(1-{\xi }_{n+1})v_{n+2})+{\omega }_{n+1}{Q}_C(I-\lambda \widehat{B})u_{n+1}}{1-{\mu }_{n+1}}}\hfill \\ \hfill & -{\displaystyle \frac{{\nu }_n{T}^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})+{\omega }_n{Q}_C(I-\lambda \widehat{B})u_n}{1-{\mu }_n}}\hfill \\ \hfill & =\frac{{\nu }_{n+1}}{1-{\mu }_{n+1}}{T}^{n+1}({\xi }_{n+1}{v}_{n+1}+(1-{\xi }_{n+1})v_{n+2})-\frac{{\nu }_n}{1-{\mu }_n}{T}^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\hfill \\ \hfill & +\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}{Q}_C(I-\lambda \widehat{B})u_{n+1}-\frac{{\omega }_n}{1-{\mu }_n}{Q}_C(I-\lambda \widehat{B})u_n\hfill \\ \hfill & =\frac{{\nu }_{n+1}}{1-{\mu }_{n+1}}{T}^{n+1}({\xi }_{n+1}{v}_{n+1}+(1-{\xi }_{n+1})v_{n+2})-\frac{{\nu }_{n+1}}{1-{\mu }_{n+1}}{T}^{n+1}({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\hfill \\ \hfill & +\frac{{\nu }_{n+1}}{1-{\mu }_{n+1}}{T}^{n+1}({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})-\frac{{\nu }_n}{1-{\mu }_n}{T}^{n+1}({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\hfill \\ \hfill & +\frac{{\nu }_n}{1-{\mu }_n}{T}^{n+1}({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})-\frac{{\nu }_n}{1-{\mu }_n}{T}^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\hfill \\ \hfill & +\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}{Q}_C(I-\lambda \widehat{B})u_{n+1}-\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}{Q}_C(I-\lambda \widehat{B})u_n+\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}{Q}_C(I-\lambda \widehat{B})u_n\hfill \\ \hfill & -\frac{{\omega }_n}{1-{\mu }_n}{Q}_C(I-\lambda \widehat{B})u_n\hfill \\ \hfill & =\frac{{\nu }_{n+1}}{1-{\mu }_{n+1}}(T^{n+1}({\xi }_{n+1}{v}_{n+1}+(1-{\xi }_{n+1})v_{n+2})-T^{n+1}({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}))\hfill \\ \hfill & +(\frac{{\nu }_{n+1}}{1-{\mu }_{n+1}}-\frac{{\nu }_n}{1-{\mu }_n})T^{n+1}({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})+\frac{{\nu }_n}{1-{\mu }_n}(T^{n+1}({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\hfill \\ \hfill & -T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}))+\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}(Q_C(I-\lambda \widehat{B})u_{n+1}-Q_C(I-\lambda \widehat{B})u_n)\hfill \\ \hfill & +(\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}-\frac{{\omega }_n}{1-{\mu }_n})Q_C(I-\lambda \widehat{B})u_n.\hfill \end{array}$$

Thus, we acquire

$$\begin{array}{cc}\hfill & \parallel z_{n+1}-z_n\parallel \hfill \\ \hfill & \le \frac{{\nu }_{n+1}}{1-{\mu }_{n+1}}\parallel T^{n+1}({\xi }_{n+1}{v}_{n+1}+(1-{\xi }_{n+1})v_{n+2})-T^{n+1}({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel \hfill \\ \hfill & +|\frac{{\nu }_{n+1}}{1-{\mu }_{n+1}}-\frac{{\nu }_n}{1-{\mu }_n}|\parallel T^{n+1}v\parallel +\frac{{\nu }_n}{1-{\mu }_n}\parallel T^{n+1}v-T^{n}v\parallel +\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}\parallel Q_C(I-\lambda \widehat{B})u_{n+1}\hfill \\ \hfill & -Q_C(I-\lambda \widehat{B})u_n\parallel +|\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}-\frac{{\omega }_n}{1-{\mu }_n}|\parallel Q_C(I-\lambda \widehat{B})u_n\parallel \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{{\nu }_{n+1}}{1-{\mu }_{n+1}}{k}_{n+1}\parallel {\xi }_{n+1}{v}_{n+1}+(1-{\xi }_{n+1})v_{n+2}-[{\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}]\parallel +|\frac{{\nu }_{n+1}}{1-{\mu }_{n+1}}\hfill \\ \hfill & -\frac{{\nu }_n}{1-{\mu }_n}|\parallel T^{n+1}v\parallel +\frac{{\nu }_n}{1-{\mu }_n}\parallel T^{n+1}v-T^{n}v\parallel +\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}\parallel u_{n+1}-u_n\parallel \hfill \\ \hfill & +|\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}-\frac{{\omega }_n}{1-{\mu }_n}|\parallel Q_C(I-\lambda \widehat{B})u_n\parallel \hfill \\ \hfill & \le \frac{{\nu }_{n+1}}{1-{\mu }_{n+1}}{k}_{n+1}\parallel {\xi }_n(v_{n+1}-v_n)+(1-{\xi }_{n+1})(v_{n+2}-v_{n+1})\parallel +|\frac{{\nu }_{n+1}}{1-{\mu }_{n+1}}-\frac{{\nu }_n}{1-{\mu }_{n+1}}\hfill \\ \hfill & +\frac{{\nu }_n}{1-{\mu }_{n+1}}-\frac{{\nu }_n}{1-{\mu }_n}|\parallel T^{n+1}v\parallel +\frac{{\nu }_n}{1-{\mu }_n}\underset{v\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel +\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}\parallel u_{n+1}-u_n\parallel \hfill \\ \hfill & +|\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}-\frac{{\omega }_n}{1-{\mu }_{n+1}}+\frac{{\omega }_n}{1-{\mu }_{n+1}}-\frac{{\omega }_n}{1-{\mu }_n}|\parallel Q_C(I-\lambda \widehat{B})u_n\parallel \hfill \\ \hfill & \le \frac{{\nu }_{n+1}{k}_{n+1}{\xi }_n}{1-{\mu }_{n+1}}\parallel v_{n+1}-v_n\parallel +\frac{{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}{1-{\mu }_{n+1}}\parallel v_{n+2}-v_{n+1}\parallel +\frac{|{\nu }_{n+1}-{\nu }_n|}{1-{\mu }_{n+1}}\parallel T^{n+1}v\parallel \hfill \\ \hfill & +{\nu }_n\frac{|{\mu }_{n+1}-{\mu }_n|}{(1-{\mu }_{n+1})(1-{\mu }_n)}\parallel T^{n+1}v\parallel +\frac{{\nu }_n}{1-{\mu }_n}\underset{v\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel +\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}\parallel u_{n+1}-u_n\parallel \hfill \\ \hfill & +\frac{|{\omega }_{n+1}-{\omega }_n|}{1-{\mu }_{n+1}}\parallel Q_C(I-\lambda \widehat{B})u_n\parallel +{\omega }_n\frac{|{\mu }_{n+1}-{\mu }_n|}{(1-{\mu }_{n+1})(1-{\mu }_n)}\parallel Q_C(I-\lambda \widehat{B})u_n\parallel \hfill \\ \hfill & \le \frac{{\nu }_{n+1}{k}_{n+1}{\xi }_n}{1-{\mu }_{n+1}}\parallel v_{n+1}-v_n\parallel +\frac{{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}{1-{\mu }_{n+1}}\parallel v_{n+2}-v_{n+1}\parallel +\frac{|{\nu }_{n+1}-{\nu }_n|}{1-{\mu }_{n+1}}\parallel T^{n+1}v\parallel \hfill \\ \hfill & +\frac{|{\mu }_{n+1}-{\mu }_n|}{(1-{\mu }_{n+1})(1-{\mu }_n)}\parallel T^{n+1}v\parallel +\frac{{\nu }_n}{1-{\mu }_n}\underset{v\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel +\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}\parallel u_{n+1}-u_n\parallel \hfill \\ \hfill & +\frac{|{\omega }_{n+1}-{\omega }_n|}{1-{\mu }_{n+1}}\parallel Q_C(I-\lambda \widehat{B})u_n\parallel +\frac{|{\mu }_{n+1}-{\mu }_n|}{(1-{\mu }_{n+1})(1-{\mu }_n)}\parallel Q_C(I-\lambda \widehat{B})u_n\parallel .\hfill \end{array}$$

From (7), (8) and condition $\left(ii\right),\left(iii\right)$ in Theorem 1, we can compute

$$\begin{array}{cc}\hfill & \parallel z_{n+1}-z_n\parallel \hfill \\ \hfill & \le \frac{{\nu }_{n+1}{k}_{n+1}{\xi }_n}{1-{\mu }_{n+1}}\parallel v_{n+1}-v_n\parallel +\frac{{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}{1-{\mu }_{n+1}}[(1+\epsilon {\beta }_{n+1})\parallel v_{n+1}-v_n\parallel +M_{n_1}]\hfill \\ \hfill & +\frac{|{\nu }_{n+1}-{\nu }_n|}{1-{\mu }_{n+1}}\parallel T^{n+1}v\parallel +\frac{|{\mu }_{n+1}-{\mu }_n|}{(1-{\mu }_{n+1})(1-{\mu }_n)}\parallel T^{n+1}v\parallel +\frac{{\nu }_n}{1-{\mu }_n}\underset{v\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel \hfill \\ \hfill & +\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}(\parallel v_{n+1}-v_n\parallel +|{\beta }_{n+1}-{\beta }_n|\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel )+\frac{|{\omega }_{n+1}-{\omega }_n|}{1-{\mu }_{n+1}}\parallel Q_C(I-\lambda \widehat{B})u_n\parallel \hfill \\ \hfill & +\frac{|{\mu }_{n+1}-{\mu }_n|}{(1-{\mu }_{n+1})(1-{\mu }_n)}\parallel Q_C(I-\lambda \widehat{B})u_n\parallel \hfill \\ \hfill & \le [\frac{{\nu }_{n+1}{k}_{n+1}{\xi }_n}{1-{\mu }_{n+1}}+\frac{{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}{1-{\mu }_{n+1}}(1+\epsilon {\beta }_{n+1})+\frac{{\omega }_{n+1}}{1-{\mu }_{n+1}}]\parallel v_{n+1}-v_n\parallel \hfill \\ \hfill & +\frac{{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}{1-{\mu }_{n+1}}{M}_{n_1}+\frac{|{\nu }_{n+1}-{\nu }_n|}{1-h_2}\parallel T^{n+1}v\parallel +\frac{|{\mu }_{n+1}-{\mu }_n|}{{(1-h_2)}^2}\parallel T^{n+1}v\parallel \hfill \\ \hfill & +\frac{{\nu }_n}{1-h_2}\underset{v\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel +\frac{{\omega }_{n+1}|{\beta }_{n+1}-{\beta }_n|}{1-h_2}\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \hfill \\ \hfill & +\frac{|{\omega }_{n+1}-{\omega }_n|}{1-h_2}\parallel Q_C(I-\lambda \widehat{B})u_n\parallel +\frac{|{\mu }_{n+1}-{\mu }_n|}{{(1-h_2)}^2}\parallel Q_C(I-\lambda \widehat{B})u_n\parallel \hfill \\ \hfill & \le [\frac{{\nu }_{n+1}{k}_{n+1}{\xi }_n+{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})+{\omega }_{n+1}}{1-{\mu }_{n+1}}+\frac{{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})\epsilon {\beta }_{n+1}}{1-{\mu }_{n+1}}]\parallel v_{n+1}-v_n\parallel \hfill \\ \hfill & +\frac{{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}{1-{\mu }_{n+1}}{M}_{n_1}+M_{n_2}\hfill \\ \hfill & \le [1+\frac{{\nu }_{n+1}(k_{n+1}({\xi }_n+1-{\xi }_{n+1})-1)}{1-{\mu }_{n+1}}+\frac{{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})\epsilon {\beta }_{n+1}}{1-{\mu }_{n+1}}]\parallel v_{n+1}-v_n\parallel \hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\frac{{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}{1-{\mu }_{n+1}}{M}_{n_1}+M_{n_2}\hfill \\ \hfill & \le [1+\frac{{\nu }_{n+1}(k_{n+1}-1)}{1-{\mu }_{n+1}}+\frac{{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})\epsilon {\beta }_{n+1}}{1-{\mu }_{n+1}}]\parallel v_{n+1}-v_n\parallel \hfill \\ \hfill & +\frac{{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}{1-{\mu }_{n+1}}{M}_{n_1}+M_{n_2}\hfill \\ \hfill & \le [1+\frac{{\nu }_{n+1}\epsilon {\beta }_{n+1}}{1-{\mu }_{n+1}}+\frac{{\nu }_{n+1}{k}_{n+1}\epsilon {\beta }_{n+1}}{1-{\mu }_{n+1}}]\parallel v_{n+1}-v_n\parallel +\frac{{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}{1-{\mu }_{n+1}}{M}_{n_1}+M_{n_2}\hfill \\ \hfill & \le [1+\frac{(k_{n+1}+1){\nu }_{n+1}\epsilon {\beta }_{n+1}}{1-h_2}]\parallel v_{n+1}-v_n\parallel +\frac{{\nu }_{n+1}{k}_{n+1}(1-{\xi }_{n+1})}{1-h_2}{M}_{n_1}+M_{n_2},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill M_{n_2}& =\frac{|{\nu }_{n+1}-{\nu }_n|}{1-h_2}\parallel T^{n+1}v\parallel +\frac{|{\mu }_{n+1}-{\mu }_n|}{{(1-h_2)}^2}\parallel T^{n+1}v\parallel \hfill \\ \hfill & +\frac{{\nu }_n}{1-h_2}\underset{x\in C}{sup}\parallel T^{n+1}v-T^{n}v\parallel +\frac{{\omega }_{n+1}|{\beta }_{n+1}-{\beta }_n|}{1-h_2}\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \hfill \\ \hfill & +\frac{|{\omega }_{n+1}-{\omega }_n|}{1-h_2}\parallel Q_C(I-\lambda \widehat{B})u_n\parallel +\frac{|{\mu }_{n+1}-{\mu }_n|}{{(1-h_2)}^2}\parallel Q_C(I-\lambda \widehat{B})u_n\parallel .\hfill \end{array}$$

So, by conditions $\left(i\right)$ and $\left(ii\right)$ in Theorem 1, we gain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}(\parallel z_{n+1}-z_n\parallel -\parallel v_{n+1}-v_n\parallel )\le 0.\end{array}$$

Adopting Lemma 3, we gain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel z_n-v_n\parallel =0.\end{array}$$

Taking notice of $v_{n+1}-v_n=(1-{\mu }_n)(z_n-v_n)$. Then

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel v_{n+1}-v_n\parallel =0.\end{array}$$

(9)

From (7), (9) and condition $\left(i\right)$ in Theorem 1, we receive

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel u_{n+1}-u_n\parallel =0.\end{array}$$

Step 4: Again by the definition of $u_n$ and Lemma 4, we get

$$\begin{array}{cc}\hfill \parallel u_n{-q\parallel }^2& =\parallel {\beta }_n\delta g\left(v_n\right)+(I-{\beta }_n\overline{V})L{v}_n{-q\parallel }^2\hfill \\ \hfill & =\parallel (L{v}_n-q)+({\beta }_n\delta g\left(v_n\right)-{\beta }_n\overline{V}L{v}_n){\parallel }^2\hfill \\ \hfill & \le \parallel L{v}_n{-q\parallel }^2+2{\beta }_n⟨\delta g\left(v_n\right)-\overline{V}L{v}_n,j(u_n-q)⟩\hfill \\ \hfill & \le \parallel v_n{-q\parallel }^2+2{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel .\hfill \end{array}$$

(10)

By the non-expansiveness of $Q_C,$ Lemma 7 and (10), we have

$$\begin{array}{cc}\hfill \parallel v_{n+1}{-q\parallel }^2& =\parallel {\mu }_n{v}_n+{\nu }_n{T}^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})+{\omega }_n{Q}_C(I-\lambda \widehat{B})u_n{-q\parallel }^2\hfill \\ \hfill & \le {\mu }_n\parallel v_n{-q\parallel }^2+{\nu }_n\parallel T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}){-q\parallel }^2+{\omega }_n{\parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel }^2\hfill \\ \hfill & \le {\mu }_n\parallel v_n{-q\parallel }^2+{\nu }_n{k}_n^2\parallel ({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}){-q\parallel }^2+{\omega }_n{\parallel (I-\lambda \widehat{B})u_n-(I-\lambda \widehat{B})q\parallel }^2\hfill \\ \hfill & \le {\mu }_n\parallel v_n{-q\parallel }^2+{\nu }_n{k}_n^2{\xi }_n\parallel v_n{-q\parallel }^2+{\nu }_n{k}_n^2(1-{\xi }_n)\parallel v_{n+1}{-q\parallel }^2+{\omega }_n{\parallel u_n-q\parallel }^2\hfill \\ \hfill & -{\omega }_n\lambda (2\widehat{b}-\lambda ){\parallel \widehat{B}{u}_n-\widehat{B}q\parallel }^2\hfill \\ \hfill & \le {\mu }_n\parallel v_n{-q\parallel }^2+{\nu }_n{k}_n^2{\xi }_n\parallel v_n{-q\parallel }^2+{\nu }_n{k}_n^2(1-{\xi }_n)\parallel v_{n+1}{-q\parallel }^2+{\omega }_n{\parallel v_n-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel -{\omega }_n\lambda (2\widehat{b}-\lambda ){\parallel \widehat{B}{u}_n-\widehat{B}q\parallel }^2\hfill \\ \hfill & \le ({\mu }_n+{\nu }_n{k}_n^2{\xi }_n+{\omega }_n)\parallel v_n{-q\parallel }^2+{\nu }_n{k}_n^2(1-{\xi }_n)\parallel v_{n+1}-q{\parallel }^2+2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)\hfill \\ \hfill & -\overline{V}L{v}_n\parallel \parallel u_n-q\parallel -{\omega }_n\lambda (2\widehat{b}-\lambda ){\parallel \widehat{B}{u}_n-\widehat{B}q\parallel }^2.\hfill \end{array}$$

This implies that

$$\begin{array}{cc}\hfill & {\omega }_n\lambda (2\widehat{b}-\lambda )\parallel \widehat{B}{u}_n-\widehat{B}{q\parallel }^2\le ({\mu }_n+{\nu }_n{k}_n^2{\xi }_n+{\omega }_n)\parallel v_n{-q\parallel }^2-[1-{\nu }_n{k}_n^2(1-{\xi }_n)]{\parallel v_{n+1}-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel \hfill \\ \hfill & \le (1-{\nu }_n+{\nu }_n{k}_n^2{\xi }_n)\parallel v_n{-q\parallel }^2-[1-{\nu }_n{k}_n^2(1-{\xi }_n)]\parallel v_{n+1}{-q\parallel }^2+2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel \hfill \\ \hfill & \le (1-{\nu }_n{k}_n^2+{\nu }_n{k}_n^2{\xi }_n+{\nu }_n{k}_n^2-{\nu }_n)\parallel v_n{-q\parallel }^2-[1-{\nu }_n{k}_n^2(1-{\xi }_n)]{\parallel v_{n+1}-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel \hfill \\ \hfill & \le (1-{\nu }_n{k}_n^2(1-{\xi }_n))[\parallel v_n{-q\parallel }^2-\parallel v_{n+1}{-q\parallel }^2]+2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel \hfill \\ \hfill & +{\nu }_n(k_n^2-1){\parallel v_n-q\parallel }^2\hfill \\ \hfill & \le (\parallel v_n-q\parallel +\parallel v_{n+1}-q\parallel )\parallel v_n-v_{n+1}\parallel +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel +{\nu }_n\epsilon {\beta }_n{\parallel v_n-q\parallel }^2.\hfill \end{array}$$

By conditions $\left(i\right),\left(ii\right)$ in Theorem 1 and (9), we can acquire

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel \widehat{B}{u}_n-\widehat{B}q\parallel =0.\end{array}$$

(11)

Applying Lemma 2 and (10), we obtain

$$\begin{array}{cc}\hfill \parallel v_{n+1}{-q\parallel }^2& =\parallel {\mu }_n{v}_n+{\nu }_n{T}^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})+{\omega }_n{Q}_C(I-\lambda \widehat{B})u_n{-q\parallel }^2\hfill \\ \hfill & \le {\mu }_n\parallel v_n{-q\parallel }^2+{\nu }_n\parallel T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}){-q\parallel }^2+{\omega }_n{\parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel }^2\hfill \\ \hfill & -{\mu }_n{\nu }_n{m}_1(\parallel v_n-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel )-{\mu }_n{\omega }_n{m}_1(\parallel v_n-Q_C(I-\lambda \widehat{B})u_n\parallel )\hfill \\ \hfill & \le {\mu }_n\parallel v_n{-q\parallel }^2+{\nu }_n{k}_n^2\parallel ({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}){-q\parallel }^2+{\omega }_n{\parallel u_n-q\parallel }^2\hfill \\ \hfill & -{\mu }_n{\nu }_n{m}_1(\parallel v_n-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel )-{\mu }_n{\omega }_n{m}_1(\parallel v_n-Q_C(I-\lambda \widehat{B})u_n\parallel )\hfill \\ \hfill & \le {\mu }_n\parallel v_n{-q\parallel }^2+{\nu }_n{k}_n^2{\xi }_n\parallel v_n{-q\parallel }^2+{\nu }_n{k}_n^2(1-{\xi }_n)\parallel v_{n+1}{-q\parallel }^2+{\omega }_n{\parallel u_n-q\parallel }^2\hfill \\ \hfill & -{\mu }_n{\nu }_n{m}_1(\parallel v_n-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel )-{\mu }_n{\omega }_n{m}_1(\parallel v_n-Q_C(I-\lambda \widehat{B})u_n\parallel )\hfill \\ \hfill & \le ({\mu }_n+{\nu }_n{k}_n^2{\xi }_n+{\omega }_n)\parallel v_n{-q\parallel }^2+{\nu }_n{k}_n^2(1-{\xi }_n){\parallel v_{n+1}-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel -{\mu }_n{\nu }_n{m}_1(\parallel v_n-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel )\hfill \\ \hfill & -{\mu }_n{\omega }_n{m}_1(\parallel v_n-Q_C(I-\lambda \widehat{B})u_n\parallel ).\hfill \end{array}$$

Which implies that

$$\begin{array}{cc}\hfill & {\mu }_n{\nu }_n{m}_1(\parallel v_n-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}\parallel )+{\mu }_n{\omega }_n{m}_1(\parallel v_n-Q_C(I-\lambda \widehat{B})u_n\parallel )\hfill \\ \hfill & \le ({\mu }_n+{\nu }_n{k}_n^2{\xi }_n+{\omega }_n)\parallel v_n{-q\parallel }^2-[1-{\nu }_n{k}_n^2(1-{\xi }_n)]{\parallel v_{n+1}-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel \hfill \\ \hfill & \le (1-{\nu }_n+{\nu }_n{k}_n^2{\xi }_n)\parallel v_n{-q\parallel }^2-[1-{\nu }_n{k}_n^2(1-{\xi }_n)]{\parallel v_{n+1}-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel \hfill \\ \hfill & \le (1-{\nu }_n{k}_n^2+{\nu }_n{k}_n^2{\xi }_n+{\nu }_n{k}_n^2-{\nu }_n)\parallel v_n{-q\parallel }^2-[1-{\nu }_n{k}_n^2(1-{\xi }_n)]{\parallel v_{n+1}-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel \hfill \\ \hfill & \le [1-{\nu }_n{k}_n^2(1-{\xi }_n)](\parallel v_n{-q\parallel }^2-\parallel v_{n+1}{-q\parallel }^2)+{\nu }_n(k_n^2-1){\parallel v_n-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel \hfill \\ \hfill & \le (\parallel v_n-q\parallel +\parallel v_{n+1}-q\parallel )\parallel v_n-v_{n+1}\parallel +{\nu }_n\epsilon {\beta }_n\parallel v_n{-q\parallel }^2+2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel .\hfill \end{array}$$

From conditions $\left(i\right),\left(ii\right)$ in Theorem 1 and (9), and the properties of the function $m_1$, we acquire

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel v_n-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel =0,\end{array}$$

(12)

and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel v_n-Q_C(I-\lambda \widehat{B})u_n\parallel =0.\end{array}$$

(13)

Applying Lemma 1, Lemma 9 and (10), we gain

$$\begin{array}{cc}\hfill \parallel Q_C(I-\lambda \widehat{B})u_n{-q\parallel }^2& \le ⟨(I-\lambda \widehat{B})u_n-(I-\lambda \widehat{B})q,j(Q_C(I-\lambda \widehat{B})u_n-q)⟩\hfill \\ \hfill & \le ⟨u_n-q,j(Q_C(I-\lambda \widehat{B})u_n-q)⟩-\lambda ⟨\widehat{B}{u}_n-\widehat{B}q,j(Q_C(I-\lambda \widehat{B})u_n-q)⟩\hfill \\ \hfill & \le \frac{1}{2}[\parallel u_n{-q\parallel }^2+\parallel Q_C(I-\lambda \widehat{B})u_n{-q\parallel }^2-m_2(\parallel u_n-q-(Q_C(I-\lambda \widehat{B})u_n-q)\parallel \left)\right]\hfill \\ \hfill & -\lambda \parallel \widehat{B}{u}_n-\widehat{B}q\parallel \parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel \hfill \\ \hfill & \le \frac{1}{2}[\parallel u_n{-q\parallel }^2+\parallel Q_C(I-\lambda \widehat{B})u_n{-q\parallel }^2-m_2(\parallel u_n-Q_C(I-\lambda \widehat{B})u_n\parallel \left)\right]\hfill \\ \hfill & -\lambda \parallel \widehat{B}{u}_n-\widehat{B}q\parallel \parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel \hfill \\ \hfill & \le \frac{1}{2}[\parallel v_n{-q\parallel }^2+2{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel +\parallel Q_C(I-\lambda \widehat{B})u_n{-q\parallel }^2\hfill \\ \hfill & -m_2(\parallel u_n-Q_C(I-\lambda \widehat{B})u_n\parallel \left)\right]-\lambda \parallel \widehat{B}{u}_n-\widehat{B}q\parallel \parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel .\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill \parallel Q_C(I-\lambda \widehat{B})u_n{-q\parallel }^2& \le \parallel v_n{-q\parallel }^2+2{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel \hfill \\ \hfill & -m_2(\parallel u_n-Q_C(I-\lambda \widehat{B})u_n\parallel )-2\lambda \parallel \widehat{B}{u}_n-\widehat{B}q\parallel \parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel .\hfill \end{array}$$

Hence, we know that

$$\begin{array}{cc}\hfill \parallel v_{n+1}{-q\parallel }^2& =\parallel {\mu }_n{v}_n+{\nu }_n{T}^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})+{\omega }_n{Q}_C(I-\lambda \widehat{B})u_n{-q\parallel }^2\hfill \\ \hfill & \le {\mu }_n\parallel v_n{-q\parallel }^2+{\nu }_n\parallel T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}){-q\parallel }^2+{\omega }_n{\parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel }^2\hfill \\ \hfill & \le {\mu }_n\parallel v_n{-q\parallel }^2+{\nu }_n{k}_n^2\parallel ({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}){-q\parallel }^2+{\omega }_n{\parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel }^2\hfill \\ \hfill & \le {\mu }_n\parallel v_n{-q\parallel }^2+{\nu }_n{k}_n^2{\xi }_n\parallel v_n{-q\parallel }^2+{\nu }_n{k}_n^2(1-{\xi }_n){\parallel v_{n+1}-q\parallel }^2\hfill \\ \hfill & +{\omega }_n[\parallel v_n{-q\parallel }^2+2{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel \hfill \\ \hfill & -m_2(\parallel u_n-Q_C(I-\lambda \widehat{B})u_n\parallel )-2\lambda \parallel \widehat{B}{u}_n-\widehat{B}q\parallel \parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel ]\hfill \\ \hfill & \le ({\mu }_n+{\nu }_n{k}_n^2{\xi }_n+{\omega }_n)\parallel v_n{-q\parallel }^2+{\nu }_n{k}_n^2(1-{\xi }_n){\parallel v_{n+1}-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel -{\omega }_n{m}_2(\parallel u_n-Q_C(I-\lambda \widehat{B})u_n\parallel )\hfill \\ \hfill & -2\lambda {\omega }_n\parallel \widehat{B}{u}_n-\widehat{B}q\parallel \parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel .\hfill \end{array}$$

Which implies that

$$\begin{array}{cc}\hfill {\omega }_n{m}_2(\parallel u_n& -Q_C(I-\lambda \widehat{B})u_n\parallel )\le ({\mu }_n+{\nu }_n{k}_n^2{\xi }_n+{\omega }_n)\parallel v_n{-q\parallel }^2-[1-{\nu }_n{k}_n^2(1-{\xi }_n)]{\parallel v_{n+1}-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel -2\lambda {\omega }_n\parallel \widehat{B}{u}_n-\widehat{B}q\parallel \parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel \hfill \\ \hfill & \le (1-{\nu }_n+{\nu }_n{k}_n^2{\xi }_n)\parallel v_n{-q\parallel }^2-[1-{\nu }_n{k}_n^2(1-{\xi }_n)]{\parallel v_{n+1}-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel -2\lambda {\omega }_n\parallel \widehat{B}{u}_n-\widehat{B}q\parallel \parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel \hfill \\ \hfill & \le (1-{\nu }_n{k}_n^2+{\nu }_n{k}_n^2{\xi }_n+{\nu }_n{k}_n^2-{\nu }_n)\parallel v_n{-q\parallel }^2-[1-{\nu }_n{k}_n^2(1-{\xi }_n)]{\parallel v_{n+1}-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel -2\lambda {\omega }_n\parallel \widehat{B}{u}_n-\widehat{B}q\parallel \parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel \hfill \\ \hfill & \le [1-{\nu }_n{k}_n^2(1-{\xi }_n)](\parallel v_n{-q\parallel }^2-\parallel v_{n+1}{-q\parallel }^2)+{\nu }_n(k_n^2-1){\parallel v_n-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel -2\lambda {\omega }_n\parallel \widehat{B}{u}_n-\widehat{B}q\parallel \parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel \hfill \\ \hfill & \le [1-{\nu }_n{k}_n^2(1-{\xi }_n)](\parallel v_n-q\parallel +\parallel v_{n+1}-q\parallel )\parallel v_n-v_{n+1}\parallel +{\nu }_n(k_n^2-1){\parallel v_n-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel -2\lambda {\omega }_n\parallel \widehat{B}{u}_n-\widehat{B}q\parallel \parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel \hfill \\ \hfill & \le (\parallel v_n-q\parallel +\parallel v_{n+1}-q\parallel )\parallel v_n-v_{n+1}\parallel +{\nu }_n\epsilon {\beta }_n{\parallel v_n-q\parallel }^2\hfill \\ \hfill & +2{\omega }_n{\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \parallel u_n-q\parallel -2\lambda {\omega }_n\parallel \widehat{B}{u}_n-\widehat{B}q\parallel \parallel Q_C(I-\lambda \widehat{B})u_n-q\parallel .\hfill \end{array}$$

Similarly, by conditions $\left(i\right),\left(ii\right)$ in Theorem 1 and (9), (11) and the properties of the function $m_2$, we attain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel u_n-Q_C(I-\lambda \widehat{B})u_n\parallel =0.\end{array}$$

(14)

Thinking

$$\begin{array}{c}\hfill \parallel v_n-u_n\parallel \le \parallel v_n-Q_C(I-\lambda \widehat{B})u_n\parallel +\parallel Q_C(I-\lambda \widehat{B})u_n-u_n\parallel ,\end{array}$$

and (13) and (14), we receive

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel v_n-u_n\parallel =0.\end{array}$$

(15)

Adopting condition $\left(i\right)$ in Theorem 1, we can get

$$\begin{array}{cc}\hfill \parallel u_n-L{v}_n\parallel & =\parallel {\beta }_n\delta g\left(v_n\right)+(I-{\beta }_n\overline{V})L{v}_n-L{v}_n\parallel \hfill \\ \hfill & ={\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}L{v}_n\parallel \to 0,asn\to \infty .\hfill \end{array}$$

and also

$$\begin{array}{c}\hfill \parallel v_n-L{v}_n\parallel \le \parallel v_n-u_n\parallel +\parallel u_n-L{v}_n\parallel \to 0,asn\to \infty .\end{array}$$

(16)

Moreover, from (12), (14) and (15), we have

$$\begin{array}{cc}\hfill & \parallel Q_C(I-\lambda \widehat{B})u_n-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel \hfill \\ \hfill & =\parallel Q_C(I-\lambda \widehat{B})u_n-u_n\parallel +\parallel u_n-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel \hfill \\ \hfill & =\parallel Q_C(I-\lambda \widehat{B})u_n-u_n\parallel +\parallel u_n-v_n\parallel +\parallel v_n-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel \to 0asn\to \infty .\hfill \end{array}$$

(17)

Since

$$\begin{array}{cc}\hfill \parallel v_{n+1}& -T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel \hfill \\ \hfill & =\parallel {\mu }_n{v}_n+{\nu }_n{T}^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})+{\omega }_n{Q}_C(I-\lambda \widehat{B})u_n-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel \hfill \\ \hfill & =\parallel {\mu }_n[v_n-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})]+{\omega }_n[Q_C(I-\lambda \widehat{B})u_n-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})]\parallel \hfill \\ \hfill & \le {\mu }_n\parallel v_n-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel +{\omega }_n\parallel Q_C(I-\lambda \widehat{B})u_n-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel .\hfill \end{array}$$

By condition $\left(ii\right)$ in Theorem 1 and (12), (17), we obtain

$$\begin{array}{c}\hfill \parallel v_{n+1}-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel \to 0asn\to \infty .\end{array}$$

(18)

We know that

$$\begin{array}{cc}\hfill \parallel v_n& -T^n{v}_n\parallel =\parallel v_n-v_{n+1}+v_{n+1}-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})+T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})-T^n{v}_n\parallel \hfill \\ \hfill & \le \parallel v_n-v_{n+1}\parallel +\parallel v_{n+1}-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel +\parallel T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})-T^n{v}_n\parallel \hfill \\ \hfill & \le \parallel v_n-v_{n+1}\parallel +\parallel v_{n+1}-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel +k_n\parallel {\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}-v_n\parallel \hfill \\ \hfill & \le \parallel v_n-v_{n+1}\parallel +\parallel v_{n+1}-T^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})\parallel +k_n(1-{\xi }_n)\parallel v_{n+1}-v_n\parallel .\hfill \end{array}$$

By (9) and (18), we acquire

$$\begin{array}{c}\hfill \parallel v_n-T^n{v}_n\parallel \to 0,asn\to \infty .\end{array}$$

(19)

Because T is an asymptotically non-expansive mapping, we attain

$$\begin{array}{cc}\hfill \parallel v_n-T{v}_n\parallel & =\parallel v_n-v_{n+1}+v_{n+1}-T^{n+1}{v}_{n+1}+T^{n+1}{v}_{n+1}-T^{n+1}{v}_n+T^{n+1}{v}_n-T{v}_n\parallel \hfill \\ \hfill & \le \parallel v_n-v_{n+1}\parallel +\parallel v_{n+1}-T^{n+1}{v}_{n+1}\parallel +\parallel T^{n+1}{v}_{n+1}-T^{n+1}{v}_n\parallel +\parallel T^{n+1}{v}_n-T{v}_n\parallel \hfill \\ \hfill & \le \parallel v_n-v_{n+1}\parallel +\parallel v_{n+1}-T^{n+1}{v}_{n+1}\parallel +k_{n+1}\parallel v_{n+1}-v_n\parallel +k_1\parallel T^n{v}_n-v_n\parallel \hfill \\ \hfill & \le (1+k_{n+1})\parallel v_n-v_{n+1}\parallel +\parallel v_{n+1}-T^{n+1}{v}_{n+1}\parallel +k_1\parallel T^n{v}_n-v_n\parallel .\hfill \end{array}$$

By (9) and (19), we acquire

$$\begin{array}{c}\hfill \parallel v_n-T{v}_n\parallel \to 0,asn\to \infty .\end{array}$$

(20)

From (15), (16) and (20), we have

$$\begin{array}{cc}\hfill \parallel u_n-T{u}_n\parallel & \le \parallel u_n-v_n\parallel +\parallel v_n-T{v}_n\parallel +\parallel T{v}_n-T{u}_n\parallel \hfill \\ \hfill & \le \parallel u_n-v_n\parallel +\parallel v_n-T{v}_n\parallel +k_1\parallel v_n-u_n\parallel \to 0,asn\to \infty ,\hfill \end{array}$$

(21)

and

$$\begin{array}{cc}\hfill \parallel u_n-L{u}_n\parallel & \le \parallel u_n-v_n\parallel +\parallel v_n-L{v}_n\parallel +\parallel L{v}_n-L{u}_n\parallel \hfill \\ \hfill & \le \parallel u_n-v_n\parallel +\parallel v_n-L{v}_n\parallel +\parallel v_n-u_n\parallel \to 0,asn\to \infty .\hfill \end{array}$$

(22)

Step 5: We can take a subsequence $u_{n_i}$ of $\left\{u_n\right\}$ and it is satisfied with

$$\begin{array}{c}\hfill \underset{i\to \infty }{lim}⟨\delta g\left(t\right)-\overline{V}t,j(u_{n_i}-t)⟩=\underset{n\to \infty }{lim\; sup}⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩.\end{array}$$

(23)

Since X is a uniformly smooth Banach space and $\left\{v_n\right\}$ is bounded, there exists a subsequence $v_{n_i}\rightharpoonup \overline{v}\in C$ as $i\to \infty$. By (15) and (23), we know

$$\begin{array}{c}\hfill u_{n_i}\rightharpoonup \overline{v},⟨\delta g\left(t\right)-\overline{V}t,j(\overline{v}-t)⟩=\underset{n\to \infty }{lim\; sup}⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩.\end{array}$$

(24)

From (21), (24) and Lemma 10, we can acquire $\overline{v}\in Fix\left(T\right)$. It follows that (22), (24) and Lemma 10, we can attain $\overline{v}\in Fix\left(L\right)$. By (14), (24), Lemma 10 and Lemma 8, we get $\overline{v}\in VI(C,\widehat{B})$. So, we attain $\overline{v}\in \Psi .$ From (24) and property of $Q_C$, we get

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩=⟨\delta g\left(t\right)-\overline{V}t,j(\overline{v}-t)⟩\le 0.\end{array}$$

(25)

Step 6: Finally, from Lemma 4 and Lemma 6, we acquire

$$\begin{array}{cc}\hfill \parallel u_n{-t\parallel }^2& =\parallel {\beta }_n\delta g\left(v_n\right)+(I-{\beta }_n\overline{V})L{v}_n{-t\parallel }^2\hfill \\ \hfill & \le \parallel (I-{\beta }_n\overline{V})(L{v}_n-t){\parallel }^2+2{\beta }_n⟨\delta g\left(v_n\right)-\overline{V}t,j(u_n-t)⟩\hfill \\ \hfill & \le {(1-{\beta }_n\overline{\delta })}^2{\parallel v_n-t\parallel }^2+2{\beta }_n\delta ⟨g\left(v_n\right)-g\left(t\right),j(u_n-t)⟩+2{\beta }_n⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩\hfill \\ \hfill & \le {(1-{\beta }_n\overline{\delta })}^2\parallel v_n{-t\parallel }^2+2{\beta }_n\delta \parallel g\left(v_n\right)-g\left(t\right)\parallel \parallel u_n-t\parallel +2{\beta }_n⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩\hfill \\ \hfill & \le {(1-{\beta }_n\overline{\delta })}^2\parallel v_n{-t\parallel }^2+2{\beta }_n\delta \eta \parallel v_n-t\parallel \parallel u_n-t\parallel +2{\beta }_n⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩\hfill \\ \hfill & \le {(1-{\beta }_n\overline{\delta })}^2\parallel v_n{-t\parallel }^2+2{\beta }_n\delta \eta \parallel v_n-t\parallel ({\beta }_n\parallel \delta g\left(v_n\right)-\overline{V}t\parallel +(1-{\beta }_n\overline{\delta })\parallel L{v}_n-t\parallel )\hfill \\ \hfill & +2{\beta }_n⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩\hfill \\ \hfill & \le {(1-{\beta }_n\overline{\delta })}^2\parallel v_n{-t\parallel }^2+2{\beta }_n\delta \eta \parallel v_n-t\parallel ({\beta }_n\delta \eta \parallel v_n-t\parallel +{\beta }_n\parallel \delta g\left(t\right)-\overline{V}t\parallel \hfill \\ \hfill & +(1-{\beta }_n\overline{\delta })\parallel v_n-t\parallel )+2{\beta }_n⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩\hfill \\ \hfill & \le {(1-{\beta }_n\overline{\delta })}^2\parallel v_n{-t\parallel }^2+2{\beta }_n^2{\delta }^2{\eta }^2\parallel v_n{-t\parallel }^2+2{\beta }_n^2\delta \eta \parallel \delta g\left(t\right)-\overline{V}t\parallel \parallel v_n-t\parallel \hfill \\ \hfill & +2{\beta }_n\delta \eta (1-{\beta }_n\overline{\delta }){\parallel v_n-t\parallel }^2+2{\beta }_n⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩\hfill \\ \hfill & \le (1-{\beta }_n\overline{\delta })\parallel v_n{-t\parallel }^2+2{\beta }_n^2{\overline{\delta }}^2\parallel v_n{-t\parallel }^2+2{\beta }_n^2\overline{\delta }\parallel \delta g\left(t\right)-\overline{V}t\parallel \parallel v_n-t\parallel \hfill \\ \hfill & +2{\beta }_n\delta \eta {\parallel v_n-t\parallel }^2+2{\beta }_n⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \parallel v_{n+1}{-t\parallel }^2& =\parallel {\mu }_n{v}_n+{\nu }_n{T}^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})+{\omega }_n{Q}_C(I-\lambda \widehat{B})u_n{-t\parallel }^2\hfill \\ \hfill & \le {\mu }_n\parallel v_n{-t\parallel }^2+{\nu }_n{k}_n^2\parallel ({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1}){-t\parallel }^2+{\omega }_n{\parallel Q_C(I-\lambda \widehat{B})u_n-t\parallel }^2\hfill \\ \hfill & \le {\mu }_n\parallel v_n{-t\parallel }^2+{\nu }_n{k}_n^2{\xi }_n\parallel v_n{-t\parallel }^2+{\nu }_n{k}_n^2(1-{\xi }_n)\parallel v_{n+1}{-t\parallel }^2+{\omega }_n{\parallel u_n-t\parallel }^2\hfill \\ \hfill & \le {\mu }_n\parallel v_n{-t\parallel }^2+{\nu }_n{k}_n^2{\xi }_n\parallel v_n{-t\parallel }^2+{\nu }_n{k}_n^2(1-{\xi }_n){\parallel v_{n+1}-t\parallel }^2\hfill \\ \hfill & +{\omega }_n[(1-{\beta }_n\overline{\delta })\parallel v_n{-t\parallel }^2+2{\beta }_n^2{\overline{\delta }}^2\parallel v_n{-t\parallel }^2+2{\beta }_n^2\overline{\delta }\parallel \delta g\left(t\right)-\overline{V}t\parallel \parallel v_n-t\parallel \hfill \\ \hfill & +2{\beta }_n\delta \eta \parallel v_n-t{\parallel }^2+2{\beta }_n⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩]\hfill \\ \hfill & \le ({\mu }_n+{\nu }_n{k}_n^2{\xi }_n+{\omega }_n-{\omega }_n{\beta }_n\overline{\delta }+2{\beta }_n\delta \eta {\omega }_n)\parallel v_n{-t\parallel }^2+2{\beta }_n^2{\overline{\delta }}^2{\omega }_n{\parallel v_n-t\parallel }^2\hfill \\ \hfill & +2{\beta }_n^2\overline{\delta }{\omega }_n\parallel \delta g\left(t\right)-\overline{V}t\parallel \parallel v_n-t\parallel +2{\beta }_n{\omega }_n⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩\hfill \\ \hfill & +{\nu }_n{k}_n^2(1-{\xi }_n){\parallel v_{n+1}-t\parallel }^2.\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill (1-{\nu }_n{k}_n^2(1-{\xi }_n)\parallel v_{n+1}-t{\parallel }^2& \le ({\mu }_n+{\nu }_n{k}_n^2{\xi }_n+{\omega }_n-{\omega }_n{\beta }_n(\overline{\delta }-2\delta \eta ))\parallel v_n{-t\parallel }^2+2{\beta }_n^2{\overline{\delta }}^2{\omega }_n{\parallel v_n-t\parallel }^2\hfill \\ \hfill & +2{\beta }_n^2\overline{\delta }{\omega }_n\parallel \delta g\left(t\right)-\overline{V}t\parallel \parallel v_n-t\parallel +2{\beta }_n{\omega }_n⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩.\hfill \end{array}$$

That is

$$\begin{array}{cc}\hfill \parallel v_{n+1}{-t\parallel }^2& \le \frac{{\mu }_n+{\nu }_n{k}_n^2{\xi }_n+{\omega }_n-{\omega }_n{\beta }_n(\overline{\delta }-2\delta \eta )}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}\parallel v_n{-t\parallel }^2+\frac{2{\beta }_n^2{\overline{\delta }}^2{\omega }_n}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}{\parallel v_n-t\parallel }^2\hfill \\ \hfill & +\frac{2{\beta }_n^2\overline{\delta }{\omega }_n}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}\parallel \delta g\left(t\right)-\overline{V}t\parallel \parallel v_n-t\parallel +\frac{2{\beta }_n{\omega }_n}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩\hfill \\ \hfill & \le (1-\frac{{\omega }_n{\beta }_n(\overline{\delta }-2\delta \eta )-{\nu }_n(k_n^2-1)}{1-{\nu }_n{k}_n^2(1-{\xi }_n)})\parallel v_n{-t\parallel }^2+\frac{2{\beta }_n^2{\overline{\delta }}^2{\omega }_n}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}{\parallel v_n-t\parallel }^2\hfill \\ \hfill & +\frac{2{\beta }_n^2\overline{\delta }{\omega }_n}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}\parallel \delta g\left(t\right)-\overline{V}t\parallel \parallel v_n-t\parallel +\frac{2{\beta }_n{\omega }_n}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩\hfill \\ \hfill & \le (1-\frac{h_1{\beta }_n(\overline{\delta }-2\delta \eta )-\epsilon {\beta }_n}{1-{\nu }_n{k}_n^2(1-{\xi }_n)})\parallel v_n{-t\parallel }^2+\frac{2{\beta }_n^2{\overline{\delta }}^2{\omega }_n}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}{\parallel v_n-t\parallel }^2\hfill \\ \hfill & +\frac{2{\beta }_n^2\overline{\delta }{\omega }_n}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}\parallel \delta g\left(t\right)-\overline{V}t\parallel \parallel v_n-t\parallel +\frac{2{\beta }_n{\omega }_n}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩\hfill \\ \hfill & \le (1-\frac{{\beta }_n(h_1\overline{\delta }-2h_1\delta \eta -\epsilon )}{1-{\nu }_n{k}_n^2(1-{\xi }_n)})\parallel v_n{-t\parallel }^2+\frac{2{\beta }_n^2{\overline{\delta }}^2}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}{\parallel v_n-t\parallel }^2\hfill \\ \hfill & +\frac{2{\beta }_n^2\overline{\delta }}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}\parallel \delta g\left(t\right)-\overline{V}t\parallel \parallel v_n-t\parallel +\frac{2{\beta }_n}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩\hfill \\ \hfill & \le (1-\frac{{\beta }_n(h_1\overline{\delta }-2h_1\delta \eta -\epsilon )}{1-{\nu }_n{k}_n^2(1-{\xi }_n)})\parallel v_n-t{\parallel }^2+\frac{{\beta }_n(h_1\overline{\delta }-2h_1\delta \eta -\epsilon )}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}[\frac{2{\beta }_n{\overline{\delta }}^2{\parallel v_n-t\parallel }^2}{h_1\overline{\delta }-2h_1\delta \eta -\epsilon }\hfill \\ \hfill & +\frac{2{\beta }_n\overline{\delta }\parallel \delta g\left(t\right)-\overline{V}t\parallel \parallel v_n-t\parallel }{h_1\overline{\delta }-2h_1\delta \eta -\epsilon }+\frac{2⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩}{h_1\overline{\delta }-2h_1\delta \eta -\epsilon }].\hfill \end{array}$$

Let

$${\sigma }_n=\frac{{\beta }_n(h_1\overline{\delta }-2h_1\delta \eta -\epsilon )}{1-{\nu }_n{k}_n^2(1-{\xi }_n)},$$

and

$${\theta }_n=\frac{2{\beta }_n{\overline{\delta }}^2{\parallel v_n-t\parallel }^2}{h_1\overline{\delta }-2h_1\delta \eta -\epsilon }+\frac{2{\beta }_n\overline{\delta }\parallel \delta g\left(t\right)-\overline{V}t\parallel \parallel v_n-t\parallel }{h_1\overline{\delta }-2h_1\delta \eta -\epsilon }+\frac{2⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩}{h_1\overline{\delta }-2h_1\delta \eta -\epsilon },$$

by conditions $\left(i\right)$, $\left(iii\right)$, $\left(iv\right)$ in Theorem 1 and (25), we can gain

$$\begin{array}{c}\hfill \sum _{n=0}^{\infty }{\sigma }_n=\sum _{n=0}^{\infty }\frac{{\beta }_n(h_1\overline{\delta }-2h_1\delta \eta -\epsilon )}{1-{\nu }_n{k}_n^2(1-{\xi }_n)}\ge \sum _{n=0}^{\infty }{\beta }_n(h_1\overline{\delta }-2h_1\delta \eta -\epsilon )=+\infty ,\end{array}$$

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}{\theta }_n=\underset{n\to \infty }{lim\; sup}[\frac{2{\beta }_n{\overline{\delta }}^2{\parallel v_n-t\parallel }^2}{h_1\overline{\delta }-2h_1\delta \eta -\epsilon }+\frac{2{\beta }_n\overline{\delta }\parallel \delta g\left(t\right)-\overline{V}t\parallel \parallel v_n-t\parallel }{h_1\overline{\delta }-2h_1\delta \eta -\epsilon }+\frac{2⟨\delta g\left(t\right)-\overline{V}t,j(u_n-t)⟩}{h_1\overline{\delta }-2h_1\delta \eta -\epsilon }]\le 0.\end{array}$$

Thus, from Lemma 5, we receive that $\underset{n\to \infty }{lim}\parallel v_n-t\parallel =0.$ So $\left\{v_n\right\}$ converges strongly to $t=Q_{\Psi }(I-\overline{V}+\delta g)t\in \Psi$, and from Lemma 11, we have $(t,s,w)$ is a solution of the problem (4), where $s=Q_C(I-{\lambda }_2{B}_2)(bt+(1-b)w)$ and $w=Q_C(I-{\lambda }_3{B}_3)t.$ The proof is completed.  □

As an application of our main result Theorem 1, we can prove strong convergence theorems for approximating the solution of the standard convex optimization problem.

Assume C is a convex and closed subset of X. The standard constrained convex optimization problem is to find $\tilde{v}\in C$ such that

$$\begin{array}{c}\hfill \tilde{g}\left(\tilde{v}\right)=\underset{v\in C}{min}\tilde{g}\left(v\right),\end{array}$$

(26)

where $\tilde{g}:C\to R$ is a convex, F$\acute{r}$echet differentiable function. ${\Phi }_{\tilde{g}}$ is used to indicate the solution set of (26).

Lemma 12.

([21]) An essential condition of optimality for a point $\tilde{v}\in C$ to be a solution of (26) is that $\tilde{v}$ is a solution of the variational inequality

$$\begin{array}{c}\hfill ⟨\nabla \tilde{g}\left(\tilde{v}\right),v-\tilde{v}⟩\ge 0,\forall v\in C.\end{array}$$

(27)

Equivalently, $v\in C$ is a fixed point of the mapping $Q_C(I-\lambda \nabla \tilde{g}),\forall \lambda >0$. In addition, if $\tilde{g}$ is convex, then the optimality condition (27) is also sufficient.

Theorem 2.

Assume that space X, mappings $Q_C,B_1,B_2,B_3,L,\overline{V},g$ and T are the same as in Theorem 1. Suppose $\tilde{g}:C\to R$ is a convex real-valued function with the gradient $\nabla \tilde{g}=\widehat{B}$ is $1/L_{\tilde{g}}$-inverse strongly accretive. Assume that $\Omega =Fix\left(T\right)\bigcap Fix\left(L\right)\bigcap {\Phi }_{\tilde{g}}\ne \varnothing$. Define the sequence $\left\{v_n\right\}$ in X as follows:

$$\left\{\begin{array}{cc}& v_1\in C,\hfill \\ & u_n={\beta }_n\delta g\left(v_n\right)+(I-{\beta }_n\overline{V})L{v}_n,\hfill \\ & v_{n+1}={\mu }_n{v}_n+{\nu }_n{T}^n({\xi }_n{v}_n+(1-{\xi }_n)v_{n+1})+{\omega }_n{Q}_C(I-\lambda \nabla \tilde{g})u_n,n\ge 1,\hfill \end{array}\right.$$

(28)

where $\lambda \in (0,2/L_{\tilde{g}})$, ${\beta }_n,{\mu }_n,{\nu }_n,{\omega }_n,{\xi }_n$ and δ satisfy the same coefficient conditions as in Theorem 1. Then $\left\{v_n\right\}$ converges strongly to $t=Q_{\Omega }(I-\overline{V}+\delta g)t\in \Omega$ and $(t,s,w)$ is a solution of (4), where $s=Q_C(I-{\lambda }_2{B}_2)(bt+(1-b)w)$ and $w=Q_C(I-{\lambda }_3{B}_3)t,{\lambda }_1,{\lambda }_2,{\lambda }_3\in (0,2\overline{b})$ with $\overline{b}=min\{b_1,b_2,b_3\}.$

Proof. 

From Lemma 12 and Theorem 1, we can gain Theorem 2, where $s=Q_C(I-{\lambda }_2{B}_2)(bt+(1-b)w)$ and $w=Q_C(I-{\lambda }_3{B}_3)t.$ Then, the proof is completed.  □

Remark 1.

$\left(1\right)$ In Theorem 2, we give the iterative approximation method and strong convergence results for the common elements of the solutions ${\Phi }_{\tilde{g}}$ of the standard constrained convex optimization problem (26) and the fixed points $Fix\left(T\right)$ of an asymptotically non-expansive mapping T.

$\left(2\right)$ As we all know, a Hilbert space H is a uniformly smooth and uniformly convex Banach space and the metric projection $P_C:H\to C$ is sunny non-expansive retractive. So, when Hilbert space H takes the place of Banach space X and the sunny non-expansive retraction $Q_C$ becomes to the metric projection $P_C$, the results of Theorem 1 and Theorem 2 still hold. More specifically, if $k_n\equiv 1$ and ${\xi }_n\equiv 1$ in Theorem 2, we can attain the major results of Keerti et al. [19].

4. Numerical Examples

Someone may question whether the coefficients in Theorem 1 are complex enough to be not obtained. Now we give a concrete numerical example to show that the coefficients in Theorem 1 can be obtained.

Let R be the real number set and $\widehat{B},B_1,B_2,B_3$ be a mapping from $[0,20]$ to R defined by $\widehat{B}v=\frac{v}{4},B_{1}v=\frac{v}{2},B_{2}v=\frac{v}{4},B_{3}v=\frac{v}{6}$, respectively. Put $b=\frac{1}{3},\widehat{b}=4,b_1=2,b_2=4,b_3=6,{\lambda }_1=1,{\lambda }_2=2,{\lambda }_3=3,$ then $\overline{b}=min\{b_1,b_2,b_3\}=2,{\lambda }_1,{\lambda }_2,{\lambda }_3\in (0,2\overline{b})=(0,4),$ the mapping $Lv=\frac{5}{18}v$ defined by (5) in Lemma 11. We take $\eta =\frac{1}{3},\delta =\frac{1}{4},{\beta }_n=\frac{1}{5n},$ and let $g,\overline{V}:R\to R$ be defined by $g\left(v\right)=\frac{v}{3},\overline{V}v=\frac{v}{4}$, respectively. So, $\left\{u_n\right\}:u_n=\frac{100n+1}{360n}{v}_n$ is generated by (6). Assume that T is an asymptotically non-expansive mapping defined by $T^{n}v=\sqrt{1+\frac{1}{12n}}v$. Let $v_1\in [0,20]$, and $\left\{v_n\right\}$ is generated by (6) where $h_1=\frac{1}{6},h_2=\frac{1}{2},\lambda =1\in (0,2\widehat{b})=(0,8),{\mu }_n=\frac{3n-1}{12n},{\nu }_n=\frac{1}{2}$ and ${\omega }_n=\frac{3n+1}{12n},{\xi }_n=1-\frac{1}{3n},\epsilon =\frac{5}{12},\tau =\frac{3}{4}.$ From the propose definition, the all conditions of Theorem 1 are satisfied. We know $0\in Fix\left(T\right)\bigcap Fix\left(L\right)\bigcap VI(C,\widehat{B})$. Then $v_n$ is generated by (6) and we simplify the form of (6) to get

$$v_{n+1}=\frac{1440n^2-480n+480n\sqrt{3n(12n+1)}-160\sqrt{3n(12n+1)}+1}{5760n^2-160\sqrt{3n(12n+1)}}{v}_n\approx \frac{4320}{5760}{v}_n.$$

So, clearly, $v_n$ converges strongly to $0\in Fix\left(T\right)\bigcap Fix\left(L\right)\bigcap VI(C,\widehat{B})$.

5. Conclusions

In this paper, we present a new generalized variational inequality system problem (4) which contains many special cases. In the case when we put $B_2\equiv 0$, then the problem (4) reduces to problem (3); If putting $b=0$ in (4), we get

$$\left\{\begin{array}{c}⟨t-(I-{\lambda }_1{B}_1)s,j(v-t)⟩\ge 0,\forall v\in C,\hfill \\ ⟨s-(I-{\lambda }_2{B}_2)w,j(v-s)⟩\ge 0,\forall v\in C,\hfill \\ ⟨w-(I-{\lambda }_3{B}_3)t,j(v-w)⟩\ge 0,\forall v\in C,\hfill \end{array}\right.$$

(29)

which is a variational inequality modified system by Ceng et al. [2]. In the sense that if we take $B_3\equiv 0$ in (29), then the problem (29) reduces to problem (2). We study the common elements of the set of solutions of an asymptotically non-expansive operation equation with the L mapping defined by (5) and the solution set of the generalized proposed system problem (4). The convergence analysis of a new way by using the generalized semi-closure principle supplied by us in Wang et al. [9] for finding the propose common elements in Banach spaces are investigated. Under the suitable conditions imposed on parameters, some strong convergence theorems are attained. We locally generalize some corresponding recent results of Keerti et al. [19] from Hilbert spaces to Banach spaces and from non-expansive mapping to asymptotically non-expansive mapping. As an application, we prove the strong convergence theorem of the standard constrained convex optimization problem by our main result. Eventually, we also give a real numerical example to show that the coefficients in Theorem 1 can be obtained and the theorems are reasonable and valid.