On a General Extragradient Implicit Method and Its Applications to Optimization

Abstract

Let X be a Banach space with both q-uniformly smooth and uniformly convex structures. This article introduces and considers a general extragradient implicit method for solving a general system of variational inequalities (GSVI) with the constraints of a common fixed point problem (CFPP) of a countable family of nonlinear mappings ${\left\{S_n\right\}}_{n=0}^{\infty }$ and a monotone variational inclusion, zero-point, problem. Here, the constraints are symmetrical and the general extragradient implicit method is based on Korpelevich’s extragradient method, implicit viscosity approximation method, Mann’s iteration method, and the W-mappings constructed by ${\left\{S_n\right\}}_{n=0}^{\infty }$.

1. Introduction

Let X be Banach space and J be duality set-valued mapping on X. Let $A_1:C\to X$ and $A_2:C\to X$ be two nonlinear nonself mappings of accretive type. In this work, we investigate the following symmetrical system problem:

$$\left\{\begin{array}{c}⟨{\mu }_1{A}_1{y}^{*}+x^{*}-y^{*},J(x-x^{*})⟩\ge 0,\forall x\in C,\hfill \\ ⟨{\mu }_2{A}_2{x}^{*}+y^{*}-x^{*},J(x-y^{*})⟩\ge 0,\forall x\in C,\hfill \end{array}\right.$$

(1)

with two real constants ${\mu }_1$ and ${\mu }_2>0$. This is called a symmetrical variational system. This system was first introduced and studied in [1]. The symmetry system is quite applicable in lots of convex optimizations and finds a lot of applications in applied sciences, such as intensity modulated radiation therapy, signal processing, image reconstruction, and so on. Indeed, the model of these problems can be rewritten as a variational inequality, which is a special case of the system that is, the unconstrained minimization problem

$$\underset{x\in H}{min}\overline{f}\left(x\right):=f\left(x\right)+I_C\left(x\right),$$

where $f:H\to R$ is a real-valued convex function that is assumed to be continuously differentiable and $I_C\left(x\right)$ is the indicator of C:

$$I_C\left(x\right)=\left\{\begin{array}{cc}0,\hfill & x\in C,\hfill \\ \infty ,\hfill & x\notin C.\hfill \end{array}\right.$$

There are lot of numerical techniques for dealing with it; see, e.g., [2,3,4,5,6,7,8,9]. In addition, $x^{*}=y^{*}$, $A_1=A_2=A$ yield that Equation (1) becomes the generalized variational inequality, which consists of numerically getting $x^{*}\in C$ with $⟨J(x-x^{*}),A{x}^{*}⟩\ge 0$, where x is any vector in its subset C. The (generalized) variational inequality models lots of real applications, such as image reconstruction in emission tomography. In addition, one knows that projection methods are efficient for such a problem [10]. In 2006, Aoyama, Iiduka, and Takahashi [11] proposed and focused on a process and proved the norm convergence of the sequences defined by their process with the aid of the weak topology.

In 2013, in order to solve the above symmetrical variational system with common fixed points of a family of non-expansive self-mappings ${\left\{S_n\right\}}_{n=0}^{\infty }$ on C, Ceng et al. [12] investigated an implicit two-step iterative process via a relaxed gradient technique in a class of Banach spaces with restricted geometry structures. Let ${\Pi }_C$ be a sunny non-expansive retraction operator onto set C, $A_1$ be ${\alpha }_1$-inverse-strongly accretive nonself operator, $A_2$ be ${\alpha }_2$-inverse-strongly accretive nonself operator from C to X, and f be a contraction self operator on C. Under the restriction $\Omega ={\cap }_{n=0}^{\infty }\mathrm{Fix}\left(S_n\right)\cap \mathrm{Fix}\left({\Pi }_C(I-{\mu }_1{A}_1){\Pi }_C(I-{\mu }_2{A}_2)\right)\ne \varnothing$, let $\left\{x_n\right\}$ be the vector sequence devised by

$$\left\{\begin{array}{c}{y}_n=(1-{\alpha }_n){\Pi }_C(I-{\mu }_1{A}_1){\Pi }_C(I-{\mu }_2{A}_2)x_n+{\alpha }_{n}f\left(y_n\right),\hfill \\ x_{n+1}=(1-{\beta }_n)S_n{y}_n+{\beta }_n{x}_n\forall n\ge 0,\hfill \end{array}\right.$$

(2)

with $0<{\kappa }_2{\mu }_i<2{\alpha }_i$ for $i=1,2$, where $\left\{{\alpha }_n\right\}$ and $\left\{{\beta }_n\right\}$ are number sequences in $(0,1)$ satisfying the conditions: ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$, ${lim}_{n\to \infty }{\alpha }_n=0$, ${lim\; sup}_{n\to \infty }{\beta }_n<1$ and ${lim\; inf}_{n\to \infty }{\beta }_n>0$. They proved norm convergence of $\left\{x_n\right\}$ to $x^{*}\in \Omega$. Recently, this problem has attracted much attention from the authors working on convex believel problems; see, e.g., [13,14,15,16,17,18,19]

Meantime, in order to solve the Equation (1) with the common fixed point problem constraint of a countable family of non-expansive self-mappings ${\left\{S_n\right\}}_{n=0}^{\infty }$ on C, Song and Ceng [20] found a general iterative scheme in a Banach space with both uniformly convex and q-uniformly smooth structures (whose smoothness constant is ${\kappa }_q$, where $1<q\le 2$). Let ${\Pi }_C,A_1,A_2,G$ be the same operators as above. One lets $\Omega ={\cap }_{n=0}^{\infty }\mathrm{Fix}\left(S_n\right)\cap \mathrm{GSVI}(C,A_1,A_2)\ne \varnothing$ and suppose that f is L-Lipschitzian nonself mapping with constant $L\ge 0$ and F is a k-Lipschitz $\eta$-strongly accretive singel-valued noself operator. Let $\tau =\rho (\eta -\frac{{\kappa }_q{\rho }^{q-1}{k}^q}{q})$ and assume $0<{\rho }^{q-1}<\frac{q\eta }{{\kappa }_q{k}^q},0<{\mu }_i^{q-1}<\frac{q{\alpha }_i}{{\kappa }_q}$, $\tau >L\gamma >0$. For arbitrarily given $x_0\in C$, let $\left\{x_n\right\}$ be the sequence generated by

$$\left\{\begin{array}{c}{y}_n=(1-{\beta }_n)x_n+{\beta }_n{\Pi }_C(I-{\mu }_1{A}_1){\Pi }_C(I-{\mu }_2{A}_2)x_n,\hfill \\ x_{n+1}={\Pi }_C[{\alpha }_n\gamma f\left(x_n\right)+{\gamma }_n{x}_n+((1-{\gamma }_n)I-{\alpha }_n\rho F)S_n{y}_n]\forall n\ge 0,\hfill \end{array}\right.$$

where $\left\{{\gamma }_n\right\},\left\{{\beta }_n\right\},\left\{{\alpha }_n\right\}$ are real control sequences processing parameter conditions. They claimed convergence of $\left\{x_n\right\}$ to $x^{*}\in \Omega$ in the sense of norm.

Suppose that A is a q-order $\alpha$-inverse-strongly accretive self operator on X and $B:X\to 2^X$ is an accretive operator with the range of ${(I+B)}^{-1}$ filling the full space. In 2017, in order to solve the variational inclusion (VI) of obtaining $x^{*}\in X$ such that $0\in (A+B)x^{*}$, Chang et al. [21] suggested and devised a viscosity implicit generalized rule in the setting of smooth Banach spaces that also processes uniform convex structures. They claimed that $\left\{x_n\right\}$ converges to $x^{*}\in \Omega$ in norm. The method employed by Chang et al. [21] has been applied to popular equilibrium problems; see, e.g., [22,23,24,25,26,27]

Motivated by the above research results, the purpose of this research is to obtain, on the Banach space with uniform convexness and q-uniform smoothness, for example, $L_p$ with $p>1$, a feasibility point in the solution set of the Equation (1) involving a CFPP of nonlinear operator ${\left\{S_n\right\}}_{n=0}^{\infty }$ and a variational inclusion (VI). We suggest and investigate a general method of gradient implicit typ, which is based on Korpelevich’s extragradient method, the implicit viscosity approximation method, and the W-mappings constructed by ${\left\{S_n\right\}}_{n=0}^{\infty }$. We then prove the vector sequences devised and generated by the proposed implicit method to a solution of the symmetrical variational Equation (1) with the VI and CFPP constraints in the norm. Finally, our results are applied for solving the CFPP of non-expansive and strict pseudocontractive operators, and convex minimization problems in Hilbert spaces. Our results improve and extend some related recent results in [12,20,21,28,29].

2. Preliminaries

Let $q>1$ be a real number. The set-valued duality mapping $J_q:X\to 2^{X^{*}}$ is defined as

$$J_q\left(x\right):=\{\varphi \in X^{*}{:\parallel x\parallel }^q={⟨x,\varphi ⟩\mathrm{and}\parallel x\parallel }^{q-1}=\parallel \varphi \parallel \}\forall x\in X.$$

It is known that the duality mapping $J_q$ defined above from X into the family of nonempty (by Hahn–Banach’s theorem) weak${}^{*}$ compact subsets of $X^{*}$ satisfies, for all $x\in X$, $J_q(-x)=-J_q\left(x\right)$. Under the structures of smoothness and uniform convexness, one knows that there exists a continuous convex and strictly increasing function $g:[0,2r]\to \mathbf{R}$ such that $0=g\left(0\right)$ and $g(\parallel x-y\parallel )\le {\parallel x\parallel }^2+{\parallel y\parallel }^2-2⟨x,J\left(y\right)⟩$ for all $x,y\in B_r=\{y\in X:\parallel y\parallel \le r\}$. We suppose that $\Pi$ maps C into some subset D. One recalls that $\Pi$ is called a sunny provided that $t(x-\Pi (x\left)\right)+\Pi \left(x\right)\in C$ for $x\in C$ and $t\ge 0$, $\Pi [\Pi (x)+t(x-\Pi \left(x\right)\left)\right]=\Pi \left(x\right)$. $\Pi$ is a retraction provided $\Pi ={\Pi }^2$.

Lemma 1.

[30] We suppose that $q>1$ and X is a q-uniformly smooth Banach space with the generalized duality mapping $J_q$. Then, for any given $x,y\in X$, the inequality holds: ${\parallel x+y\parallel }^q\le {\parallel x\parallel }^q+q⟨y,j_q(x+y)⟩\forall j_q(x+y)\in J_q(x+y)$ and $\mathrm{Fix}\left({(I+\lambda B)}^{-1}(I-\lambda A)\right)={(A+B)}^{-1}0\forall \lambda >0$. Let α, β, and γ be three position real constants with $\alpha +\beta +\gamma =1$. In addition, if X is uniformly convex, then there exists a continuous convex and strictly increasing function $g:[0,\infty )\to [0,\infty )$ with the restraint that ${\parallel \alpha x+\beta y+\gamma y\parallel }^2+\alpha \beta g(\parallel x-y\parallel )\le {\alpha \parallel x\parallel }^2+{\beta \parallel y\parallel }^2+\gamma {\parallel y\parallel }^2$ for all $\alpha ,\beta ,\gamma \in [0,1]$.

Proposition 1.

[31] We suppose that X is q-uniformly smooth space with $q\in (1,2]$. Then, ${\kappa }_q{\parallel y\parallel }^q+{\parallel x\parallel }^q\ge {\parallel x+y\parallel }^q-q⟨y,J_q\left(x\right)⟩$ for any vectors $x\in X$, $y\in X$. If $q=2$, the special case, then ${\kappa }_2{\parallel y\parallel }^2+{\parallel x\parallel }^2\ge {\parallel x+y\parallel }^2-2⟨y,J_2\left(x\right)⟩$ for any vectors $x\in X$, $y\in X$.

From now on, one assumes that A is a set-valued operator from C to $2^X$. A is called an accretive operator if $⟨j_q(x-y),u-v⟩\ge 0,$ where $j_q(x-y)\in J_q(x-y)$, $\forall u\in Ax,v\in Ay$. A is called an $\alpha$-inverse-strongly accretive operator $⟨j_q(x-y),u-v⟩\ge \alpha {\parallel Ax-Ay\parallel }^q$, where $j_q(x-y)\in J_q(x-y)$, $\alpha >0$, $\forall u\in Ax,v\in Ay$. For all $\lambda >0$, $X=(I+\lambda A)C$. Then, A is called m-accretive. On the class of m-accretive operators, one can get a back-ward operator $J_{\lambda }^A={(I+\lambda A)}^{-1}$, which is commonly called the resolvent operator of A.

Lemma 2.

[32] In a Banach space X, one has $J_{\lambda }x=J_{\mu }(\frac{\mu }{\lambda }x+(1-\frac{\mu }{\lambda })J_{\lambda }x)$, $\forall x\in X,\mu ,\lambda >0$. Let $J_{\lambda }^A$ be the associated resolvent operator of A. Thus, $J_{\lambda }^A$ is a single-valued Lipschitz continuous operator $\mathrm{Fix}\left(J_{\lambda }^A\right)=A^{-1}0$, where $A^{-1}0=\{x\in C:0\in Ax\}$; if the setting is reduced to Hilbert spaces, m-accretiveness is equivalent to the maximal monotonicity.

Proposition 2.

[33] Let X be a uniformly convex and q-uniformly smooth Banach space. Assume that $r>0$ is some positive real number and A is a single-valued accretive with the inverse-strongly accretiveness and B is an accretive operator with $X=(I+\lambda B)C$. α-inverse-strongly accretive mapping of order q and $B:C\to 2^X$ is an m-accretive operator. Thus, there exists a continuous convex and strictly increasing function $\varphi :{\mathbf{R}}^{+}\to {\mathbf{R}}^{+}$ with $\varphi \left(0\right)=0$ such that

$$\parallel T_{\lambda }x-T_{\lambda }{y\parallel }^q+\lambda (\alpha q-{\lambda }^{q-1}{\kappa }_q){\parallel Ax-Ay\parallel }^q\le {\parallel x-y\parallel }^q-\varphi (\parallel (I-J_{\lambda }^B)(I-\lambda A)y-(I-J_{\lambda }^B)(I-\lambda A)x\parallel ),$$

for all $x,y\in {\tilde{B}}_r$, a ball in C, where ${\kappa }_q$ is the q-uniformly smooth constant of X. In particular, if $0<{\lambda }^{q-1}\le \frac{\alpha q}{{\kappa }_q}$, then $T_{\lambda }$ is non-expansive.

Lemma 3.

[20] Let X be q-uniformly smooth and $A:C\to X$ be q-order α-inverse-strongly accretive. Then, the following inequality holds:

$$\lambda (q\alpha -{\kappa }_q{\lambda }^{q-1}){\parallel Ax-Ay\parallel }^q+\parallel (I-\lambda A)x-(I-\lambda A){y\parallel }^q\le {\parallel x-y\parallel }^q.$$

In particular, if $0<{\lambda }^{q-1}\le \frac{q\alpha }{{\kappa }_q}$, then the complimentary operator of A processes the nonexpansivity. Suppose that ${\Pi }_C$ is a non-expansive sunny retraction from X onto C. Let both the mapping $A_1:C\to X$ and $A_2:C\to X$ be inverse-strongly accretive and let G be a self-mapping on C defined by $Gx:={\Pi }_C(I-{\mu }_1{A}_1){\Pi }_C(I-{\mu }_2{A}_2)\forall x\in C$. If $0<{\mu }_i^{q-1}\le \frac{q{\alpha }_i}{{\kappa }_q}$ for $i=1,2$, G is a non-expansive self-mapping C. $(x^{*},y^{*}),$ where both $x^{*}$ and $y^{*}$ are in C, solves the variational system (1) if and only if $x^{*}={\Pi }_C(y^{*}-{\mu }_1{A}_1{y}^{*})$, where $y^{*}={\Pi }_C(x^{*}-{\mu }_2{A}_2{x}^{*})$.

Lemma 4.

[34] Let $x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)y_n\forall n\ge 0$ and ${lim\; sup}_{n\to \infty }(\parallel y_{n+1}-y_n\parallel -\parallel x_{n+1}-x_n\parallel )\le 0$, where $\left\{{\alpha }_n\right\}$ is a sequence satisfying the condition ${lim\; sup}_{n\to \infty }{\alpha }_n<1$ and ${lim\; inf}_{n\to \infty }{\alpha }_n>0$ and $\left\{x_n\right\}$ and $\left\{y_n\right\}$ are bounded sequences in a Banach space. Then, ${lim}_{n\to \infty }\parallel y_n-x_n\parallel =0$.

Let $\left\{{\zeta }_n\right\}$ be a real sequence in (0,1) and $S_i$ a non-expansive mapping defined on C for each $i\in \{1,2,·,\}$. Next, one defines a mapping associated with n by

$$\left\{\begin{array}{c}{U}_{n,n}={\zeta }_n{S}_n{U}_{n,n+1}+(1-{\zeta }_n)I,\hfill \\ U_{n,n-1}={\zeta }_{n-1}{S}_{n-1}{U}_{n,n}+(1-{\zeta }_{n-1})I,\hfill \\ \cdots \hfill \\ U_{n,k}={\zeta }_k{S}_k{U}_{n,k+1}+(1-{\zeta }_k)I,\hfill \\ \cdots \hfill \\ W_n=U_{n,0}={\zeta }_0{S}_0{U}_{n,1}+(1-{\zeta }_0)I,\hfill \end{array}\right.$$

(3)

where $U_{n,n+1}=I$. The $W_n$, called W-mapping, is a non-expansive mapping.

Lemma 5.

[35] Let ${\left\{S_n\right\}}_{n=0}^{\infty }$ be a countable family of non-expansive self-mappings on C, which is a subset of strictly convex space with ${\cap }_{n=0}^{\infty }\mathrm{Fix}\left(S_n\right)\ne \varnothing$, and ${\left\{{\zeta }_n\right\}}_{n=0}^{\infty }$ be a real sequence such that $0<{\zeta }_n\le b<1\forall n\ge 0$. Then, the following statements hold:

(i) 

the limit ${lim}_{n\to \infty }{U}_{n,k}x$ exists for all $x\in C$ and $k\ge 0$;

(ii) 

$W_n$ is non-expansive and $\mathrm{Fix}\left(W_n\right)={\cap }_{i=0}^n\mathrm{Fix}\left(S_i\right)\forall n\ge 0$;

(iii) 

the mapping $W:C\to C$ defined by $Wx:={lim}_{n\to \infty }{W}_{n}x={lim}_{n\to \infty }{U}_{n,0}x\forall x\in C$, is a non-expansive mapping satisfying $\mathrm{Fix}\left(W\right)={\cap }_{n=0}^{\infty }\mathrm{Fix}\left(S_n\right)$ and it is called the W-mapping generated by $S_0,S_1,\dots$ and ${\zeta }_0,{\zeta }_1,\dots$.

Using the same arguments as in the proof of [[36], Lemma 4], we obtain the following.

Proposition 3.

Let ${\left\{S_n\right\}}_{n=0}^{\infty }$ and ${\left\{{\zeta }_n\right\}}_{n=0}^{\infty }$ be as in Lemma 5. Let D be any bounded set in C. One has ${lim}_{n\to \infty }{sup}_{x\in D}\parallel W_{n}x-Wx\parallel =0$.

Lemma 6.

[37] Let $a_{n+1}\le a_n+{\lambda }_n{\gamma }_n-a_n{\lambda }_n\forall n\ge 0$, where $\left\{{\lambda }_n\right\}$ and $\left\{{\gamma }_n\right\}$ are sequences of real numbers such that ${lim\; sup}_{n\to \infty }{\gamma }_n\le 0$ or ${\sum }_{n=0}^{\infty }\left|{\lambda }_n{\gamma }_n\right|<\infty$; $\left\{{\lambda }_n\right\}\subset [0,1]$ and ${\sum }_{n=0}^{\infty }{\lambda }_n=\infty$. Hence, ${lim}_{n\to \infty }{a}_n=0$.

3. Convergence Results

Theorem 1.

Suppose that X is uniformly convex and q-uniformly, where $1<q\le 2$, smooth space. Suppose that B is a set-valued m-accretive operator and A is a single-valued α-inverse-strongly accretive operator. Suppose that $A_1$ is a single-valued ${\alpha }_1$-inverse-strongly accretive operator and $A_2$ is a single-valued ${\alpha }_2$-inverse-strongly accretive operator. Suppose that f is a contraction defined on set C with contractive efficient $\delta \in (0,1)$ and $\left\{W_n\right\}$ is the sequence defined by Equation (3). Suppose that ${\Pi }_C$ is a non-expansive sunny retraction from X onto set C and $\Omega ={\cap }_{n=0}^{\infty }\mathrm{Fix}\left(S_n\right)\cap \mathrm{GSVI}(C,A_1,A_2)\cap {(B+A)}^{-1}0\ne \varnothing$, where $\mathrm{GSVI}(C,A_1,A_2)$ is the fixed point set of $G:={\Pi }_C(I-{\mu }_1{A}_1){\Pi }_C(I-{\mu }_2{A}_2)$ with $0<{\mu }_1^{q-1}{\kappa }_q<q{\alpha }_1$ and $0<{\mu }_2^{q-1}{\kappa }_q<q{\alpha }_2$. Define a sequence $\left\{x_n\right\}$ as follows:

$$\left\{\begin{array}{c}{u}_n={\Pi }_C(y_n-{\mu }_2{A}_2{y}_n),\hfill \\ v_n={\Pi }_C(u_n-{\mu }_1{A}_1{u}_n),\hfill \\ y_n={\beta }_n{x}_n+{\gamma }_n{W}_n(t_n{x}_n+(1-t_n)J_{{\lambda }_n}^B(I-{\lambda }_{n}A)v_n)+{\alpha }_{n}f\left(x_n\right),\hfill \\ x_{n+1}=(1-{\delta }_n)W_n{y}_n+{\delta }_n{x}_{n}n\ge 0,\hfill \end{array}\right.$$

(4)

where $\left\{{\lambda }_n\right\}\subset (0,{(\frac{q\alpha }{{\kappa }_q})}^{\frac{1}{q-1}})$ and $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\},\left\{{\delta }_n\right\},\left\{t_n\right\}\subset (0,1)$ satisfy the following conditions:

(i) 

${\alpha }_n+{\beta }_n+{\gamma }_n=1,$ and ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$;

(ii) 

${lim}_{n\to \infty }|{\beta }_n-{\beta }_{n-1}|={lim}_{n\to \infty }|{\gamma }_n-{\gamma }_{n-1}|={lim}_{n\to \infty }{\alpha }_n={lim}_{n\to \infty }|t_n-t_{n-1}|=0$;

(iii) 

${lim\; inf}_{n\to \infty }{\gamma }_n{t}_n(1-t_n)>0$ and ${lim\; sup}_{n\to \infty }{\gamma }_n(1-t_n)<1$;

(iv) 

${lim\; inf}_{n\to \infty }{\beta }_n{\gamma }_n>0$, ${lim\; inf}_{n\to \infty }{\delta }_n>0$ and ${lim\; sup}_{n\to \infty }{\delta }_n<1$;

(v) 

$0<\overline{\lambda }\le {\lambda }_n\forall n\ge 0$ and ${lim}_{n\to \infty }{\lambda }_n=\lambda <{\left(\frac{q\alpha }{{\kappa }_q}\right)}^{\frac{1}{q-1}}$.

Then, $x_n\to x^{*}\in \Omega$ strongly.

Proof. 

Re-write process Equation (4) as

$$\left\{\begin{array}{c}{y}_n={\beta }_n{x}_n+{\gamma }_n{W}_n(t_n{x}_n+(1-t_n)T_{n}G{y}_n)+{\alpha }_{n}f\left(x_n\right),\hfill \\ x_{n+1}=(1-{\delta }_n)W_n{y}_n+{\delta }_n{x}_{n}n\ge 0,\hfill \end{array}\right.$$

(5)

where $T_n:=J_{{\lambda }_n}^B(I-{\lambda }_{n}A)\forall n\ge 0$. From $\left\{{\lambda }_n\right\}\subset (0,{(\frac{q\alpha }{{\kappa }_q})}^{\frac{1}{q-1}})$ and Proposition 2, we observe, for each n, that $T_n$ is a non-expansive self-mapping on C. Since ${\alpha }_n+{\beta }_n+{\gamma }_n=1$, we know that

$${\alpha }_n\delta +{\beta }_n+{\gamma }_n{t}_n+{\gamma }_n(1-t_n)={\alpha }_n\delta +{\gamma }_n+{\beta }_n=1-{\alpha }_n(1-\delta ).$$

For each n, one defines a self-mapping $F_n$ on C by $F_n\left(x\right)={\alpha }_{n}f\left(x_n\right)+{\beta }_n{x}_n+{\gamma }_n{W}_n(t_n{x}_n+(1-t_n)T_{n}Gx)\forall x\in C$. Thus,

$$\begin{array}{cc}\parallel F_n\left(x\right)-F_n\left(y\right)\parallel \hfill & ={\gamma }_n\parallel W_n(t_n{x}_n+(1-t_n)T_{n}Gx)-W_n(t_n{x}_n+(1-t_n)T_{n}Gy)\parallel \hfill \\ \hfill & \le {\gamma }_n(1-t_n)\parallel T_{n}Gx-T_{n}Gy\parallel \le {\gamma }_n(1-t_n)\parallel x-y\parallel .\hfill \end{array}$$

Since $0<{\gamma }_n(1-t_n)<1$, one has a unique vector $y_n\in C$ satisfying

$$y_n={\alpha }_{n}f\left(x_n\right)+{\beta }_n{x}_n+{\gamma }_n{W}_n(t_n{x}_n+(1-t_n)T_{n}G{y}_n).$$

The following proof is split to complete this conclusion. □

Step 1. We show that iterative sequence $\left\{x_n\right\}$ is bounded. Take a fixed $p\in \Omega ={\cap }_{n=0}^{\infty }\mathrm{Fix}\left(S_n\right)\cap \mathrm{GSVI}(C,A_1,A_2)\cap {(A+B)}^{-1}0$ arbitrarily. Lemma 5 guarantees $W_{n}p=p,Gp=p$ and $T_{n}p=p$. Moreover, the nonexpansivity of $T_n$ and G (due to Proposition 2) sends us to

$$\begin{array}{c}\parallel p-y_n\parallel =\parallel {\beta }_n(p-x_n)+{\gamma }_n[p-W_n(t_n{x}_n+(1-t_n)T_{n}G{y}_n)]+{\alpha }_n(p-f\left(x_n\right))\parallel \hfill \\ \le {\beta }_n\parallel p-x_n\parallel +{\gamma }_n\parallel p-W_n(t_n{x}_n+(1-t_n)T_{n}G{y}_n)\parallel +{\alpha }_n(\parallel f\left(x_n\right)-f\left(p\right)\parallel +\parallel p-f\left(p\right)\parallel )\hfill \\ \le {\beta }_n\parallel p-x_n\parallel +{\gamma }_n[t_n\parallel p-x_n\parallel +(1-t_n)\parallel p-T_{n}G{y}_n\parallel ]+{\alpha }_n(\delta \parallel p-x_n\parallel +\parallel f\left(p\right)-p\parallel )\hfill \\ \le {\gamma }_n(1-t_n)\parallel p-y_n\parallel +({\alpha }_n\delta +{\beta }_n+{\gamma }_n{t}_n)\parallel p-x_n\parallel +{\alpha }_n\parallel f\left(p\right)-p\parallel ,\hfill \end{array}$$

which therefore implies that

$$\begin{array}{cc}\parallel p-y_n\parallel \hfill & \le \frac{{\alpha }_n\delta +{\beta }_n+{\gamma }_n{t}_n}{1-{\gamma }_n(1-t_n)}\parallel p-x_n\parallel +\frac{{\alpha }_n}{1-{\gamma }_n(1-t_n)}\parallel f\left(p\right)-p\parallel \hfill \\ \hfill & =\frac{1-{\alpha }_n(1-\delta )-{\gamma }_n(1-t_n)}{1-{\gamma }_n(1-t_n)}\parallel p-x_n\parallel +\frac{{\alpha }_n}{1-{\gamma }_n(1-t_n)}\parallel f\left(p\right)-p\parallel \hfill \\ \hfill & =(1-\frac{{\alpha }_n(1-\delta )}{1-{\gamma }_n(1-t_n)})\parallel p-x_n\parallel +\frac{{\alpha }_n}{1-{\gamma }_n(1-t_n)}\parallel f\left(p\right)-p\parallel .\hfill \end{array}$$

(6)

Thus, from Equation (5), Equation (6), and Lemma 5 (i), we have

$$\begin{array}{cc}\parallel p-x_{n+1}\parallel \hfill & \le (1-{\delta }_n)\parallel p-W_n{y}_n\parallel +{\delta }_n\parallel p-x_n\parallel \le (1-{\delta }_n)\parallel p-y_n\parallel +{\delta }_n\parallel p-x_n\parallel \hfill \\ \hfill & \le {\delta }_n\parallel p-x_n\parallel +(1-{\delta }_n)\{(1-\frac{{\alpha }_n(1-\delta )}{1-{\gamma }_n(1-t_n)})\parallel p-x_n\parallel +\frac{{\alpha }_n}{1-{\gamma }_n(1-t_n)}\parallel f\left(p\right)-p\parallel \}\hfill \\ \hfill & =[1-\frac{(1-{\delta }_n)(1-\delta )}{1-{\gamma }_n(1-t_n)}{\alpha }_n]\parallel p-x_n\parallel +\frac{(1-{\delta }_n)(1-\delta )}{1-{\gamma }_n(1-t_n)}{\alpha }_n\frac{\parallel f\left(p\right)-p\parallel }{1-\delta }\le max\{\frac{\parallel f\left(p\right)-p\parallel }{1-\delta },\parallel p-x_n\parallel \}.\hfill \end{array}$$

One claims that all the iterative sequences are bounded.

Step 2. One proves that $\parallel x_{n+1}-x_n\parallel$ goes to 0 as n goes to ∞. By borrowing Equation (5), we have

$$z_n-z_{n-1}=t_n(x_n-x_{n-1})+(t_n-t_{n-1})(x_{n-1}-T_{n-1}G{y}_{n-1})+(1-t_n)(T_{n}G{y}_n-T_{n-1}G{y}_{n-1}),$$

and

$$\begin{array}{cc}{y}_n-y_{n-1}\hfill & =({\alpha }_n-{\alpha }_{n-1})f\left(x_{n-1}\right)+{\alpha }_n(f\left(x_n\right)-f\left(x_{n-1}\right))+{\beta }_n(x_n-x_{n-1})\hfill \\ \hfill & +({\beta }_n-{\beta }_{n-1})x_{n-1}+{\gamma }_n(W_n{z}_n-W_{n-1}{z}_{n-1})+({\gamma }_n-{\gamma }_{n-1})W_{n-1}{z}_{n-1}.\hfill \end{array}$$

(7)

By using Lemma 2 and Proposition 2, one deduces that

$$\begin{array}{c}\parallel T_{n}G{y}_n-T_{n-1}G{y}_{n-1}\parallel \le \parallel T_{n}G{y}_n-T_{n}G{y}_{n-1}\parallel +\parallel T_{n}G{y}_{n-1}-T_{n-1}G{y}_{n-1}\parallel \hfill \\ \le \parallel y_n-y_{n-1}\parallel +\parallel J_{{\lambda }_n}^B(I-{\lambda }_{n}A)G{y}_{n-1}-J_{{\lambda }_{n-1}}^B(I-{\lambda }_{n-1}A)G{y}_{n-1}\parallel \hfill \\ \le \parallel y_n-y_{n-1}\parallel +\parallel J_{{\lambda }_n}^B(I-{\lambda }_{n}A)G{y}_{n-1}-J_{{\lambda }_{n-1}}^B(I-{\lambda }_{n}A)G{y}_{n-1}\parallel \hfill \\ +\parallel J_{{\lambda }_{n-1}}^B(I-{\lambda }_{n}A)G{y}_{n-1}-J_{{\lambda }_{n-1}}^B(I-{\lambda }_{n-1}A)G{y}_{n-1}\parallel \hfill \\ =\parallel y_n-y_{n-1}\parallel +\parallel J_{{\lambda }_{n-1}}^B(\frac{{\lambda }_{n-1}}{{\lambda }_n}I+(1-\frac{{\lambda }_{n-1}}{{\lambda }_n})J_{{\lambda }_n}^B)(I-{\lambda }_{n}A)G{y}_{n-1}\hfill \\ -J_{{\lambda }_{n-1}}^B(I-{\lambda }_{n}A)G{y}_{n-1}\parallel +\parallel J_{{\lambda }_{n-1}}^B(I-{\lambda }_{n}A)G{y}_{n-1}-J_{{\lambda }_{n-1}}^B(I-{\lambda }_{n-1}A)G{y}_{n-1}\parallel \hfill \\ \le \parallel y_n-y_{n-1}\parallel +|1-\frac{{\lambda }_{n-1}}{{\lambda }_n}|\parallel J_{{\lambda }_n}^B(I-{\lambda }_{n}A)G{y}_{n-1}-(I-{\lambda }_{n}A)G{y}_{n-1}\parallel \hfill \\ +|{\lambda }_n-{\lambda }_{n-1}|\parallel AG{y}_{n-1}\parallel \hfill \\ \le |{\lambda }_n-{\lambda }_{n-1}|M_1+\parallel y_n-y_{n-1}\parallel ,\hfill \end{array}$$

(8)

where ${sup}_{n\ge 1}\{\frac{1}{\overline{\lambda }}\parallel J_{{\lambda }_n}^B(I-{\lambda }_{n}A)G{y}_{n-1}-(I-{\lambda }_{n}A)G{y}_{n-1}\parallel +\parallel AG{y}_{n-1}\parallel \}\le M_1$ for some $M_1>0$. Thus, it follows from Equation (8) that

$$\begin{array}{cc}\parallel W_n{z}_n-W_{n-1}{z}_{n-1}\parallel \hfill & \le \parallel W_n{z}_{n-1}-W_{n-1}{z}_{n-1}\parallel +\parallel W_n{z}_n-W_n{z}_{n-1}\parallel \hfill \\ \hfill & \le |t_n-t_{n-1}|\parallel x_{n-1}-T_{n-1}G{y}_{n-1}\parallel +t_n\parallel x_n-x_{n-1}\parallel \hfill \\ \hfill & +(1-t_n)\parallel T_{n}G{y}_n-T_{n-1}G{y}_{n-1}\parallel +\parallel W_n{z}_{n-1}-W_{n-1}{z}_{n-1}\parallel \hfill \\ \hfill & \le |t_n-t_{n-1}|\parallel x_{n-1}-T_{n-1}G{y}_{n-1}\parallel +t_n\parallel x_n-x_{n-1}\parallel \hfill \\ \hfill & +(1-t_n)[\parallel y_n-y_{n-1}\parallel +|{\lambda }_n-{\lambda }_{n-1}|M_1]+\parallel W_n{z}_{n-1}-W_{n-1}{z}_{n-1}\parallel .\hfill \end{array}$$

This inequality together with Equation (7), implies that

$$\begin{array}{c}\parallel y_n-y_{n-1}\parallel \le |{\alpha }_n-{\alpha }_{n-1}|\parallel f\left(x_{n-1}\right)\parallel +{\beta }_n\parallel x_n-x_{n-1}\parallel +{\alpha }_n\parallel f\left(x_n\right)-f\left(x_{n-1}\right)\parallel \hfill \\ +|{\beta }_n-{\beta }_{n-1}|\parallel x_{n-1}\parallel +{\gamma }_n\parallel W_n{z}_n-W_{n-1}{z}_{n-1}\parallel +|{\gamma }_n-{\gamma }_{n-1}|\parallel W_{n-1}{z}_{n-1}\parallel \hfill \\ \le |{\alpha }_n-{\alpha }_{n-1}|\parallel f\left(x_{n-1}\right)\parallel +{\alpha }_n\delta \parallel x_n-x_{n-1}\parallel +{\beta }_n\parallel x_n-x_{n-1}\parallel \hfill \\ +|{\beta }_n-{\beta }_{n-1}|\parallel x_{n-1}\parallel +{\gamma }_n\{t_n\parallel x_n-x_{n-1}\parallel +|t_n-t_{n-1}|\parallel x_{n-1}-T_{n-1}G{y}_{n-1}\parallel \hfill \\ +(1-t_n)[\parallel y_n-y_{n-1}\parallel +|{\lambda }_n-{\lambda }_{n-1}|M_1]+\parallel W_n{z}_{n-1}-W_{n-1}{z}_{n-1}\parallel \}\hfill \\ +|{\gamma }_n-{\gamma }_{n-1}|\parallel W_{n-1}{z}_{n-1}\parallel \hfill \\ \le {\gamma }_n(1-t_n)\parallel y_n-y_{n-1}\parallel +({\alpha }_n\delta +{\beta }_n+{\gamma }_n{t}_n)\parallel x_{n-1}-x_n\parallel +\left(\right|{\alpha }_{n-1}-{\alpha }_n|\hfill \\ +|{\beta }_{n-1}-{\beta }_n|+|t_{n-1}-t_n|+|{\gamma }_{n-1}-{\gamma }_n|+|{\lambda }_{n-1}-{\lambda }_n\left|\right)M_2+\parallel W_n{z}_{n-1}-W_{n-1}{z}_{n-1}\parallel ,\hfill \end{array}$$

where ${sup}_{n\ge 0}\{\parallel f\left(x_n\right)\parallel +\parallel x_n\parallel +\parallel T_{n}G{y}_n\parallel +M_1+\parallel W_n{z}_n\parallel \}\le M_2$ for some $M_2>0$. Then,

$$\begin{array}{cc}\parallel y_n-y_{n-1}\parallel \hfill & \le \frac{{\alpha }_n\delta +{\beta }_n+{\gamma }_n{t}_n}{1-{\gamma }_n(1-t_n)}\parallel x_{n-1}-x_n\parallel +\frac{1}{1-{\gamma }_n(1-t_n)}\left[\right(|{\alpha }_{n-1}-{\alpha }_n|+|{\beta }_{n-1}-{\beta }_n|\hfill \\ \hfill & +|t_{n-1}-t_n|+|{\gamma }_{n-1}-{\gamma }_n|+|{\lambda }_{n-1}|-{\lambda }_n)M_2+\parallel W_{n-1}{z}_{n-1}-W_n{z}_{n-1}\parallel ]\hfill \\ \hfill & =(1-\frac{{\alpha }_n(1-\delta )}{1-{\gamma }_n(1-t_n)})\parallel x_n-x_{n-1}\parallel +\frac{1}{1-{\gamma }_n(1-t_n)}\left[\right(|{\alpha }_n-{\alpha }_{n-1}|+|{\beta }_n-{\beta }_{n-1}|\hfill \\ \hfill & +|{\gamma }_n-{\gamma }_{n-1}|+|t_n-t_{n-1}|+|{\lambda }_n-{\lambda }_{n-1}\left|\right)M_2+\parallel W_n{z}_{n-1}-W_{n-1}{z}_{n-1}\parallel ]\hfill \\ \hfill & \le \parallel x_n-x_{n-1}\parallel +\frac{1}{1-{\gamma }_n(1-t_n)}\left[\right(|{\gamma }_{n-1}-{\gamma }_n|+|{\beta }_{n-1}-{\beta }_n|+|{\alpha }_{n-1}-{\alpha }_n|\hfill \\ \hfill & +|t_{n-1}-t_n|+|{\lambda }_{n-1}-{\lambda }_n\left|\right)M_2+\parallel W_n{z}_{n-1}-W_{n-1}{z}_{n-1}\parallel ],\hfill \end{array}$$

and hence

$$\begin{array}{c}\parallel W_n{y}_n-W_{n-1}{y}_{n-1}\parallel \le \parallel W_n{y}_n-W_n{y}_{n-1}\parallel +\parallel W_n{y}_{n-1}-W_{n-1}{y}_{n-1}\parallel \hfill \\ \le \parallel x_n-x_{n-1}\parallel +\frac{1}{1-{\gamma }_n(1-t_n)}\left[\right(|{\gamma }_{n-1}-{\gamma }_n|+|{\beta }_{n-1}-{\beta }_n|+|{\alpha }_{n-1}-{\alpha }_n|\hfill \\ +|t_{n-1}-t_n|+|{\lambda }_{n-1}-{\lambda }_n\left|\right)M_2+\parallel W_n{z}_{n-1}-W_{n-1}{z}_{n-1}\parallel ]+\parallel W_n{y}_{n-1}-W_{n-1}{y}_{n-1}\parallel .\hfill \end{array}$$

Consequently,

$$\begin{array}{c}\parallel W_n{y}_n-W_{n-1}{y}_{n-1}\parallel -\parallel x_n-x_{n-1}\parallel \le \frac{1}{1-{\gamma }_n(1-t_n)}\left[\right(|{\gamma }_{n-1}-{\gamma }_n|+|{\beta }_{n-1}-{\beta }_n|+|{\alpha }_{n-1}-{\alpha }_n|\hfill \\ +|{\lambda }_{n-1}-{\lambda }_n|+|t_{n-1}-t_n\left|\right)M_2+\parallel W_{n-1}{z}_{n-1}-W_n{z}_{n-1}\parallel ]+\parallel W_n{y}_{n-1}-W_{n-1}{y}_{n-1}\parallel .\hfill \end{array}$$

Since ${lim}_{n\to \infty }{sup}_{x\in D}\parallel W_{n}x-Wx\parallel =0$, where $D=\{y_n:n\ge 0\}\cup \{z_n:n\ge 0\}$ of C (due to Proposition 3), we know that

$$\underset{n\to \infty }{lim}\parallel W_n{y}_{n-1}-W_{n-1}{y}_{n-1}\parallel =\underset{n\to \infty }{lim}\parallel W_n{z}_{n-1}-W_{n-1}{z}_{n-1}\parallel =0.$$

Note that ${lim}_{n\to \infty }{\alpha }_n=0,{lim}_{n\to \infty }{\lambda }_n=\lambda$ and ${lim\; inf}_{n\to \infty }(1-{\gamma }_n(1-t_n))>0$. Since $|{\beta }_n-{\beta }_{n-1}|$, $|{\gamma }_n-{\gamma }_{n-1}|$ and $|t_n-t_{n-1}|$ all go to 0 as n goes to the infinity (due to conditions (ii), (iii)), one says

$$\underset{n\to \infty }{lim\; sup}(\parallel W_n{y}_n-W_{n-1}{y}_{n-1}\parallel -\parallel x_n-x_{n-1}\parallel )\le 0.$$

Lemma 4 guarantees ${lim}_{n\to \infty }\parallel W_n{y}_n-x_n\parallel =0$. Hence, we obtain

$$\underset{n\to \infty }{lim}\parallel x_{n+1}-x_n\parallel =\underset{n\to \infty }{lim}(1-{\delta }_n)\parallel W_n{y}_n-x_n\parallel =0.$$

(9)

Step 3. We show that $\parallel x_n-y_n\parallel \to 0$ and $\parallel x_n-G{x}_n\parallel \to 0$ as $n\to \infty$. Indeed, for simplicity, we denote $\overline{p}:={\Pi }_C(I-{\mu }_2{A}_2)p$. Note that $u_n={\Pi }_C(I-{\mu }_2{A}_2)y_n$ and $v_n={\Pi }_C(I-{\mu }_1{A}_1)u_n$. Then, $v_n=G{y}_n$. From Lemma 3, we have

$$\begin{array}{cc}\parallel u_n-\overline{p}{\parallel }^q\hfill & \le \parallel (I-{\mu }_2{A}_2)y_n-(I-{\mu }_2{A}_2){p\parallel }^q\hfill \\ \hfill & \le \parallel y_n{-p\parallel }^q-{\mu }_2(q{\alpha }_2-{\kappa }_q{\mu }_2^{q-1}){\parallel A_2{y}_n-A_{2}p\parallel }^q.\hfill \end{array}$$

(10)

In the same way, we get

$$\parallel v_n{-p\parallel }^q+{\mu }_1(q{\alpha }_1-{\kappa }_q{\mu }_1^{q-1})\parallel A_1{u}_n-A_1\overline{p}{\parallel }^q\le {\parallel u_n-\overline{p}\parallel }^q.$$

(11)

Substituting Equation (10) for Equation (11), we obtain

$$\begin{array}{cc}\parallel v_n{-p\parallel }^q+{\mu }_1(q{\alpha }_1-{\kappa }_q{\mu }_1^{q-1}){\parallel A_1{u}_n-A_1\overline{p}\parallel }^q\hfill & \le \parallel y_n{-p\parallel }^q-{\mu }_2(q{\alpha }_2-{\kappa }_q{\mu }_2^{q-1}){\parallel A_2{y}_n-A_{2}p\parallel }^q.\hfill \end{array}$$

(12)

By Lemma 1, we infer from Equation (5) and Equation (12) that $\parallel z_n{-p\parallel }^q\le t_n\parallel x_n{-p\parallel }^q+(1-t_n){\parallel v_n-p\parallel }^q$, and hence

$$\begin{array}{c}\parallel y_n{-p\parallel }^q={\parallel {\beta }_n(x_n-p)+{\gamma }_n(W_n{z}_n-p)+{\alpha }_n(f\left(p\right)-p)+{\alpha }_n(f\left(x_n\right)-f\left(p\right))\parallel }^q\hfill \\ \le \parallel {\beta }_n(x_n-p)+{\gamma }_n(W_n{z}_n-p)+{\alpha }_n(f\left(x_n\right)-f\left(p\right)){\parallel }^q+q{\alpha }_n⟨f\left(p\right)-p,J_q(y_n-p)⟩\hfill \\ \le {\alpha }_n\parallel f\left(x_n\right)-f\left(p\right){\parallel }^q+{\beta }_n\parallel x_n{-p\parallel }^q+{\gamma }_n{\parallel W_n{z}_n-p\parallel }^q+q{\alpha }_n⟨f\left(p\right)-p,J_q(y_n-p)⟩\hfill \\ \le {\alpha }_n\delta \parallel x_n{-p\parallel }^q+{\beta }_n\parallel x_n{-p\parallel }^q+{\gamma }_n[t_n\parallel x_n{-p\parallel }^q+(1-t_n)\parallel v_n-p{\parallel }^q]\hfill \\ +q{\alpha }_n\parallel f\left(p\right)-p\parallel \parallel y_n{-p\parallel }^{q-1}\hfill \\ \le ({\alpha }_n\delta +{\beta }_n+{\gamma }_n{t}_n)\parallel x_n{-p\parallel }^q+{\gamma }_n(1-t_n)[\parallel p-y_n{\parallel }^q-{\mu }_2(q{\alpha }_2-{\kappa }_q{\mu }_2^{q-1}){\parallel A_2{y}_n-A_{2}p\parallel }^q\hfill \\ -{\mu }_1(q{\alpha }_1-{\kappa }_q{\mu }_1^{q-1})\parallel A_1{u}_n-A_1\overline{p}{\parallel }^q]+q{\alpha }_n\parallel p-y_n{\parallel }^{q-1}\parallel p-f\left(p\right)\parallel .\hfill \end{array}$$

It yields that

$$\begin{array}{cc}\parallel y_n{-p\parallel }^q\hfill & \le (1-\frac{{\alpha }_n(1-\delta )}{1-{\gamma }_n(1-t_n)})\parallel x_n{-p\parallel }^q-\frac{{\gamma }_n(1-t_n)}{1-{\gamma }_n(1-t_n)}[{\mu }_2(q{\alpha }_2-{\kappa }_q{\mu }_2^{q-1}){\parallel A_2{y}_n-A_{2}p\parallel }^q\hfill \\ \hfill & +{\mu }_1(q{\alpha }_1-{\kappa }_q{\mu }_1^{q-1})\parallel A_1{u}_n-A_1\overline{p}{\parallel }^q]+\frac{q{\alpha }_n}{1-{\gamma }_n(1-t_n)}\parallel p-y_n{\parallel }^{q-1}\parallel p-f\left(p\right)\parallel .\hfill \end{array}$$

Combing this with Equation (5), one says

$$\begin{array}{c}\parallel x_{n+1}{-p\parallel }^q\le {\delta }_n\parallel x_n{-p\parallel }^q+(1-{\delta }_n){\parallel y_n-p\parallel }^q\hfill \\ \le {\delta }_n\parallel x_n{-p\parallel }^q+(1-{\delta }_n)\{(1-\frac{{\alpha }_n(1-\delta )}{1-{\gamma }_n(1-t_n)})\parallel x_n-p{\parallel }^q-\frac{{\gamma }_n(1-t_n)}{1-{\gamma }_n(1-t_n)}[{\mu }_2(q{\alpha }_2-{\kappa }_q{\mu }_2^{q-1})\times \hfill \\ \times \parallel A_2{y}_n-A_2{p\parallel }^q+{\mu }_1(q{\alpha }_1-{\kappa }_q{\mu }_1^{q-1})\parallel A_1{u}_n-A_1\overline{p}{\parallel }^q]+\frac{q{\alpha }_n}{1-{\gamma }_n(1-t_n)}\parallel f\left(p\right)-p\parallel \parallel y_n-p{\parallel }^{q-1}\}\hfill \\ =(1-\frac{{\alpha }_n(1-{\delta }_n)(1-\delta )}{1-{\gamma }_n(1-t_n)})\parallel x_n-p{\parallel }^q-\frac{{\gamma }_n(1-{\delta }_n)(1-t_n)}{1-{\gamma }_n(1-t_n)}[{\mu }_2(q{\alpha }_2-{\kappa }_q{\mu }_2^{q-1})\times \hfill \\ \times \parallel A_2{y}_n-A_2{p\parallel }^q+{\mu }_1(q{\alpha }_1-{\kappa }_q{\mu }_1^{q-1})\parallel A_1{u}_n-A_1\overline{p}{\parallel }^q]+\frac{q(1-{\delta }_n){\alpha }_n}{1-{\gamma }_n(1-t_n)}\parallel f\left(p\right)-p\parallel \parallel y_n{-p\parallel }^{q-1}\hfill \\ \le \parallel x_n{-p\parallel }^q-\frac{(1-{\delta }_n){\gamma }_n(1-t_n)}{1-{\gamma }_n(1-t_n)}[{\mu }_2(q{\alpha }_2-{\kappa }_q{\mu }_2^{q-1}){\parallel A_2{y}_n-A_{2}p\parallel }^q\hfill \\ +{\mu }_1(q{\alpha }_1-{\kappa }_q{\mu }_1^{q-1})\parallel A_1{u}_n-A_1\overline{p}{\parallel }^q]+{\alpha }_n{M}_3,\hfill \end{array}$$

(13)

where ${sup}_{n\ge 0}\{\frac{q(1-{\delta }_n)}{1-{\gamma }_n(1-t_n)}\parallel p-y_n{\parallel }^{q-1}\parallel p-f\left(p\right)\parallel \}\le M_3$, where $M_3>0$ is a real. Thus, it follows from Equation (13) and Proposition 1 that

$$\begin{array}{c}\frac{(1-{\delta }_n){\gamma }_n(1-t_n)}{1-{\gamma }_n(1-t_n)}[{\mu }_2(q{\alpha }_2-{\kappa }_q{\mu }_2^{q-1})\parallel A_2{y}_n-A_2{p\parallel }^q+{\mu }_1(q{\alpha }_1-{\kappa }_q{\mu }_1^{q-1})\parallel A_1{u}_n-A_1\overline{p}{\parallel }^q]\hfill \\ \le \parallel x_n{-p\parallel }^q-{\parallel x_{n+1}-p\parallel }^q+{\alpha }_n{M}_3\hfill \\ \le q\parallel x_n-x_{n+1}\parallel \parallel x_{n+1}{-p\parallel }^{q-1}+{\kappa }_q{\parallel x_n-x_{n+1}\parallel }^q+{\alpha }_n{M}_3.\hfill \end{array}$$

Since $0<{\mu }_i^{q-1}<\frac{q{\alpha }_i}{{\kappa }_q}$ for $i=1,2$, from Equation (9), ${lim\; inf}_{n\to \infty }{\gamma }_n(1-t_n)>0,{lim\; inf}_{n\to \infty }(1-{\delta }_n)>0$ and ${lim}_{n\to \infty }{\alpha }_n=0$, we get

$$\underset{n\to \infty }{lim}\parallel A_2{y}_n-A_{2}p\parallel =0\mathrm{and}\underset{n\to \infty }{lim}\parallel A_1{u}_n-A_1\overline{p}\parallel =0.$$

(14)

Utilizing Propositions 1 and 3, we have

$$\begin{array}{c}\parallel u_n-\overline{p}{\parallel }^2={\parallel {\Pi }_C(I-{\mu }_2{A}_2)y_n-{\Pi }_C(I-{\mu }_2{A}_2)p\parallel }^2\hfill \\ \le ⟨(I-{\mu }_2{A}_2)y_n-(I-{\mu }_2{A}_2)p,J(u_n-\overline{p})⟩\hfill \\ =⟨y_n-p,J(u_n-\overline{p})⟩+{\mu }_2⟨A_{2}p-A_2{y}_n,J(u_n-\overline{p})⟩\hfill \\ \le \frac{1}{2}[\parallel y_n{-p\parallel }^2+\parallel u_n-\overline{p}{\parallel }^2-g_1(\parallel y_n-u_n-(p-\overline{p})\parallel \left)\right]+{\mu }_2\parallel A_{2}p-A_2{y}_n\parallel \parallel u_n-\overline{p}\parallel ,\hfill \end{array}$$

which implies that

$$\parallel u_n-\overline{p}{\parallel }^2+g_1(\parallel y_n-u_n-(p-\overline{p})\parallel )\le \parallel y_n{-p\parallel }^2+2{\mu }_2\parallel A_{2}p-A_2{y}_n\parallel \parallel u_n-\overline{p}\parallel .$$

(15)

Following the above line, one can derive

$$\parallel v_n{-p\parallel }^2+g_2(\parallel u_n-v_n+(p-\overline{p})\parallel )\le \parallel u_n-\overline{p}{\parallel }^2+2{\mu }_1\parallel A_1\overline{p}-A_1{u}_n\parallel \parallel v_n-p\parallel .$$

(16)

Combining Equation (15) and Equation (16), one further derives

$$\begin{array}{c}\parallel v_n{-p\parallel }^2+g_1(\parallel y_n-u_n-(p-\overline{p})\parallel )+g_2(\parallel u_n-v_n+(p-\overline{p})\parallel )\hfill \\ \le \parallel y_n{-p\parallel }^2+2{\mu }_2\parallel A_{2}p-A_2{y}_n\parallel \parallel u_n-\overline{p}\parallel +2{\mu }_1\parallel A_1\overline{p}-A_1{u}_n\parallel \parallel v_n-p\parallel .\hfill \end{array}$$

(17)

Utilizing Lemma 1, we obtain from Equation (5) and Equation (17) that

$$\begin{array}{cc}\parallel z_n{-p\parallel }^2\hfill & \le (1-t_n)\parallel T_{n}G{y}_n{-p\parallel }^2-t_n(1-t_n)g_3(\parallel x_n-T_{n}G{y}_n\parallel )+t_n{\parallel x_n-p\parallel }^2\hfill \\ \hfill & \le (1-t_n)\parallel v_n{-p\parallel }^2-t_n(1-t_n)g_3(\parallel x_n-T_{n}G{y}_n\parallel )+t_n{\parallel x_n-p\parallel }^2,\hfill \end{array}$$

and hence

$$\begin{array}{c}\parallel y_n{-p\parallel }^2\le +2{\alpha }_n⟨J(y_n-p),f\left(p\right)-p⟩+{\parallel {\beta }_n(x_n-p)+{\gamma }_n(W_n{z}_n-p)+{\alpha }_n(f\left(x_n\right)-f\left(p\right))\parallel }^2\hfill \\ \le {\beta }_n\parallel x_n{-p\parallel }^2+{\gamma }_n\parallel p-W_n{z}_n{\parallel }^2+{\alpha }_n\parallel f\left(p\right)-f\left(x_n\right){\parallel }^2-{\beta }_n{\gamma }_n{g}_4(\parallel x_n-W_n{z}_n\parallel )\hfill \\ +2{\alpha }_n⟨f\left(p\right)-p,J(y_n-p)⟩\hfill \\ \le {\alpha }_n\delta \parallel x_n{-p\parallel }^2+{\beta }_n\parallel x_n{-p\parallel }^2+{\gamma }_n[t_n\parallel x_n{-p\parallel }^2+(1-t_n){\parallel v_n-p\parallel }^2\hfill \\ -t_n(1-t_n)g_3(\parallel x_n-T_{n}G{y}_n\parallel \left)\right]+2{\alpha }_n\parallel f\left(p\right)-p\parallel \parallel y_n-p\parallel -{\beta }_n{\gamma }_n{g}_4(\parallel x_n-W_n{z}_n\parallel )\hfill \\ \le {\alpha }_n\delta \parallel x_n{-p\parallel }^2+{\beta }_n\parallel x_n{-p\parallel }^2+{\gamma }_n\{t_n\parallel x_n{-p\parallel }^2+(1-t_n)[\parallel y_n{-p\parallel }^2\hfill \\ -g_1(\parallel y_n-u_n-(p-\overline{p})\parallel )-g_2(\parallel u_n-v_n+(p-\overline{p})\parallel )+2{\mu }_2\parallel A_{2}p-A_2{y}_n\parallel \parallel u_n-\overline{p}\parallel \hfill \\ +2{\mu }_1\parallel A_1\overline{p}-A_1{u}_n\parallel \parallel v_n-p\parallel ]-t_n(1-t_n)g_3(\parallel x_n-T_{n}G{y}_n\parallel \left)\right\}+2{\alpha }_n\parallel f\left(p\right)-p\parallel \parallel y_n-p\parallel \hfill \\ -{\beta }_n{\gamma }_n{g}_4(\parallel x_n-W_n{z}_n\parallel )\hfill \\ \le ({\alpha }_n\delta +{\beta }_n+{\gamma }_n{t}_n)\parallel x_n{-p\parallel }^2+{\gamma }_n(1-t_n)\parallel y_n{-p\parallel }^2-{\gamma }_n(1-t_n)[g_1(\parallel y_n-u_n-(p-\overline{p})\parallel )\hfill \\ +g_2(\parallel u_n-v_n+(p-\overline{p})\parallel \left)\right]+2{\mu }_2\parallel A_{2}p-A_2{y}_n\parallel \parallel u_n-\overline{p}\parallel +2{\mu }_1\parallel A_1\overline{p}-A_1{u}_n\parallel \parallel v_n-p\parallel \hfill \\ +2{\alpha }_n\parallel f\left(p\right)-p\parallel \parallel y_n-p\parallel -{\gamma }_n{t}_n(1-t_n)g_3(\parallel x_n-T_{n}G{y}_n\parallel )-{\beta }_n{\gamma }_n{g}_4(\parallel x_n-W_n{z}_n\parallel ),\hfill \end{array}$$

which immediately yields

$$\begin{array}{c}\parallel y_n{-p\parallel }^2\le (1-\frac{{\alpha }_n(1-\delta )}{1-{\gamma }_n(1-t_n)})\parallel x_n{-p\parallel }^2-\frac{{\gamma }_n(1-t_n)}{1-{\gamma }_n(1-t_n)}[g_1(\parallel y_n-u_n-(p-\overline{p})\parallel )\hfill \\ +g_2(\parallel u_n-v_n+(p-\overline{p})\parallel \left)\right]+\frac{2}{1-{\gamma }_n(1-t_n)}[{\mu }_2\parallel A_{2}p-A_2{y}_n\parallel \parallel u_n-\overline{p}\parallel \hfill \\ +{\mu }_1\parallel A_1\overline{p}-A_1{u}_n\parallel \parallel v_n-p\parallel +{\alpha }_n\parallel p-y_n\parallel \parallel f\left(p\right)-p\parallel ]\hfill \\ -\frac{1}{1-{\gamma }_n(1-t_n)}[{\gamma }_n{t}_n(1-t_n)g_3(\parallel x_n-T_{n}G{y}_n\parallel )+{\beta }_n{\gamma }_n{g}_4(\parallel x_n-W_n{z}_n\parallel \left)\right].\hfill \end{array}$$

This together with Equation (5) leads to

$$\begin{array}{c}\parallel x_{n+1}{-p\parallel }^2\le {\delta }_n\parallel x_n{-p\parallel }^2+(1-{\delta }_n){\parallel y_n-p\parallel }^2\hfill \\ \le {\delta }_n\parallel x_n{-p\parallel }^2+(1-{\delta }_n)\{(1-\frac{{\alpha }_n(1-\delta )}{1-{\gamma }_n(1-t_n)})\parallel x_n{-p\parallel }^2-\frac{{\gamma }_n(1-t_n)}{1-{\gamma }_n(1-t_n)}[g_1(\parallel y_n-u_n-(p-\overline{p})\parallel )\hfill \\ +g_2(\parallel u_n-v_n+(p-\overline{p})\parallel \left)\right]+\frac{2}{1-{\gamma }_n(1-t_n)}[{\mu }_2\parallel A_{2}p-A_2{y}_n\parallel \parallel u_n-\overline{p}\parallel \hfill \\ +{\mu }_1\parallel A_1\overline{p}-A_1{u}_n\parallel \parallel v_n-p\parallel +{\alpha }_n\parallel f\left(p\right)-p\parallel \parallel y_n-p\parallel ]\hfill \\ -\frac{1}{1-{\gamma }_n(1-t_n)}[{\gamma }_n{t}_n(1-t_n)g_3(\parallel x_n-T_{n}G{y}_n\parallel )+{\beta }_n{\gamma }_n{g}_4(\parallel x_n-W_n{z}_n\parallel \left)\right]\}\hfill \\ \le (1-\frac{{\alpha }_n(1-{\delta }_n)(1-\delta )}{1-{\gamma }_n(1-t_n)})\parallel x_n{-p\parallel }^2-\frac{1-{\delta }_n}{1-{\gamma }_n(1-t_n)}[{\gamma }_n(1-t_n)(g_1(\parallel y_n-u_n-(p-\overline{p})\parallel )\hfill \\ +g_2(\parallel u_n-v_n+(p-\overline{p})\parallel \left)\right)+{\gamma }_n{t}_n(1-t_n)g_3(\parallel x_n-T_{n}G{y}_n\parallel )+{\beta }_n{\gamma }_n{g}_4(\parallel x_n-W_n{z}_n\parallel \left)\right]\hfill \\ +\frac{2}{1-{\gamma }_n(1-t_n)}[{\mu }_2\parallel A_{2}p-A_2{y}_n\parallel \parallel u_n-\overline{p}\parallel +{\mu }_1\parallel A_1\overline{p}-A_1{u}_n\parallel \parallel v_n-p\parallel \hfill \\ +{\alpha }_n\parallel f\left(p\right)-p\parallel \parallel y_n-p\parallel ]\hfill \\ \le \parallel x_n{-p\parallel }^2-\frac{1-{\delta }_n}{1-{\gamma }_n(1-t_n)}[{\gamma }_n(1-t_n)(g_1(\parallel y_n-u_n-(p-\overline{p})\parallel )+g_2(\parallel u_n-v_n+(p-\overline{p})\parallel \left)\right)\hfill \\ +{\gamma }_n{t}_n(1-t_n)g_3(\parallel x_n-T_{n}G{y}_n\parallel )+{\beta }_n{\gamma }_n{g}_4(\parallel x_n-W_n{z}_n\parallel \left)\right]+\frac{2}{1-{\gamma }_n(1-t_n)}[{\mu }_2\parallel A_{2}p-A_2{y}_n\parallel \times \hfill \\ \times \parallel u_n-\overline{p}\parallel +{\mu }_1\parallel A_1\overline{p}-A_1{u}_n\parallel \parallel v_n-p\parallel +{\alpha }_n\parallel p-y_n\parallel \parallel p-f\left(p\right)\parallel ].\hfill \end{array}$$

It yields that

$$\begin{array}{c}\frac{1-{\delta }_n}{1-{\gamma }_n(1-t_n)}[{\gamma }_n(1-t_n)(g_1(\parallel y_n-u_n-(p-\overline{p})\parallel )+g_2(\parallel u_n-v_n+(p-\overline{p})\parallel \left)\right)\hfill \\ +{\gamma }_n{t}_n(1-t_n)g_3(\parallel x_n-T_{n}G{y}_n\parallel )+{\beta }_n{\gamma }_n{g}_4(\parallel x_n-W_n{z}_n\parallel \left)\right]\hfill \\ \le \parallel x_n{-p\parallel }^2-\parallel x_{n+1}{-p\parallel }^2+\frac{2}{1-{\gamma }_n(1-t_n)}[{\mu }_2\parallel A_{2}p-A_2{y}_n\parallel \parallel u_n-\overline{p}\parallel \hfill \\ +{\mu }_1\parallel A_1\overline{p}-A_1{u}_n\parallel \parallel v_n-p\parallel +{\alpha }_n\parallel y_n-p\parallel \parallel f\left(p\right)-p\parallel ]\hfill \\ \le (\parallel x_{n+1}-p\parallel +\parallel x_n-p\parallel )\parallel x_n-x_{n+1}\parallel +\frac{2}{1-{\gamma }_n(1-t_n)}[{\mu }_2\parallel A_{2}p-A_2{y}_n\parallel \parallel u_n-\overline{p}\parallel \hfill \\ +{\mu }_1\parallel A_1\overline{p}-A_1{u}_n\parallel \parallel v_n-p\parallel +{\alpha }_n\parallel f\left(p\right)-p\parallel \parallel y_n-p\parallel ].\hfill \end{array}$$

Utilizing Equation (9) and Equation (14), from ${lim\; inf}_{n\to \infty }(1-{\delta }_n)>0,{lim\; inf}_{n\to \infty }{\gamma }_n{t}_n(1-t_n)>0$ and ${lim\; inf}_{n\to \infty }{\beta }_n{\gamma }_n>0$, we conclude that ${lim}_{n\to \infty }{g}_1(\parallel y_n-u_n-(p-\overline{p})\parallel )=0$, ${lim}_{n\to \infty }{g}_2(\parallel u_n-v_n+(p-\overline{p})\parallel )=0$, ${lim}_{n\to \infty }{g}_3(\parallel x_n-T_{n}G{y}_n\parallel )=0$ and ${lim}_{n\to \infty }{g}_4(\parallel x_n-W_n{z}_n\parallel )=0$. Utilizing the properties of $g_1,g_2,g_3$ and $g_4$, we deduce that

$$\begin{array}{c}\underset{n\to \infty }{lim}\parallel y_n-u_n-(p-\overline{p})\parallel =\underset{n\to \infty }{lim}\parallel u_n-v_n+(p-\overline{p})\parallel =\underset{n\to \infty }{lim}\parallel x_n-T_{n}G{y}_n\parallel =\underset{n\to \infty }{lim}\parallel x_n-W_n{z}_n\parallel =0.\end{array}$$

(18)

From Equation (18), we get

$$\parallel y_n-G{y}_n\parallel =\parallel y_n-v_n\parallel \le \parallel y_n-u_n-(p-\overline{p})\parallel +\parallel u_n-v_n+(p-\overline{p})\parallel \to 0(n\to \infty ).$$

(19)

In the meantime, again from Equation (5), we have $y_n-x_n={\alpha }_n(f\left(x_n\right)-x_n)+{\gamma }_n(W_n{z}_n-x_n)$. Hence, from Equation (18), we get $\parallel y_n-x_n\parallel \le {\alpha }_n\parallel f\left(x_n\right)-x_n\parallel +\parallel W_n{z}_n-x_n\parallel \to 0(n\to \infty )$. This together with Equation (19) implies that

$$\begin{array}{cc}\parallel x_n-G{x}_n\parallel \hfill & \le \parallel x_n-y_n\parallel +\parallel y_n-G{y}_n\parallel +\parallel G{y}_n-G{x}_n\parallel \hfill \\ \hfill & \le 2\parallel x_n-y_n\parallel +\parallel y_n-G{y}_n\parallel \to 0(n\to \infty ).\hfill \end{array}$$

(20)

Step 4. We show that $\parallel x_n-W{x}_n\parallel \to 0$, $\parallel x_n-T_{\lambda }{x}_n\parallel \to 0$ and $\parallel x_n-\Gamma x_n\parallel \to 0$ as $n\to \infty$, where $Wx={lim}_{n\to \infty }{W}_{n}x\forall x\in C$, $T_{\lambda }=J_{\lambda }^B(I-\lambda A)$ and $\Gamma x={\theta }_{1}Wx+{\theta }_{2}Gx+{\theta }_3{T}_{\lambda }x\forall x\in C$ for constants ${\theta }_1,{\theta }_2,{\theta }_3\in (0,1)$ satisfying ${\theta }_1+{\theta }_2+{\theta }_3=1$. Indeed, since $x_{n+1}-x_n+x_n-y_n={\delta }_n(x_n-y_n)+(1-{\delta }_n)(W_n{y}_n-y_n)$, from $x_n-x_{n+1}\to 0$ and $x_n-y_n\to 0$, we obtain

$$\parallel W_n{y}_n-y_n\parallel =\frac{1}{1-{\delta }_n}\parallel x_{n+1}-x_n+(1-{\delta }_n)(x_n-y_n)\parallel \le \frac{\parallel x_{n+1}-x_n\parallel +\parallel x_n-y_n\parallel }{1-{\delta }_n}\to 0(n\to \infty ),$$

which together with Proposition 3 and $x_n-y_n\to 0$ implies that

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

(21)

Furthermore, utilizing the same method used for Equation (8), one arrives at

$$\begin{array}{cc}\parallel T_n{y}_n-T_{\lambda }{y}_n\parallel \hfill & \le |1-\frac{\lambda }{{\lambda }_n}|\parallel J_{{\lambda }_n}^B(I-{\lambda }_{n}A)y_n-(I-{\lambda }_{n}A)y_n\parallel +|{\lambda }_n-\lambda |\parallel A{y}_n\parallel \hfill \\ \hfill & =|1-\frac{\lambda }{{\lambda }_n}|\parallel T_n{y}_n-(I-{\lambda }_{n}A)y_n\parallel +|{\lambda }_n-\lambda |\parallel A{y}_n\parallel .\hfill \end{array}$$

Since ${lim}_{n\to \infty }{\lambda }_n=\lambda$ and the sequences $\left\{y_n\right\},\left\{T_n{y}_n\right\},\left\{A{y}_n\right\}$ are bounded, we get

$$\underset{n\to \infty }{lim}\parallel T_n{y}_n-T_{\lambda }{y}_n\parallel =0.$$

(22)

By utilizing Lemma 1, we deduce from Equation (18), Equation (19), Equation (22), and $x_n-y_n\to 0$ that

$$\underset{n\to \infty }{lim}\parallel T_{\lambda }{x}_n-x_n\parallel =0.$$

(23)

We now define the mapping $\Gamma x={\theta }_{1}Wx+{\theta }_{2}Gx+{\theta }_3{T}_{\lambda }x\forall x\in C$ for constants ${\theta }_1,{\theta }_2,{\theta }_3\in (0,1)$ satisfying ${\theta }_1+{\theta }_2+{\theta }_3=1$. Lemma 4 further sends us to

$$\begin{array}{cc}\parallel x_n-\Gamma x_n\parallel \hfill & =\parallel {\theta }_1(x_n-W{x}_n)+{\theta }_2(x_n-G{x}_n)+{\theta }_3(x_n-T_{\lambda }{x}_n)\parallel \hfill \\ \hfill & \le {\theta }_1\parallel x_n-W{x}_n\parallel +{\theta }_2\parallel x_n-G{x}_n\parallel +{\theta }_3\parallel x_n-T_{\lambda }{x}_n\parallel .\hfill \end{array}$$

(24)

From Equation (20), Equation (21), Equation (23), and Equation (24), we get

$$\underset{n\to \infty }{lim}\parallel x_n-\Gamma x_n\parallel =0.$$

(25)

Step 5. We show that

$$\underset{n\to \infty }{lim\; sup}⟨f\left(x^{*}\right)-x^{*},J(x_n-x^{*})⟩\le 0,$$

(26)

where $x^{*}=$ s-${lim}_{n\to \infty }{x}_t$ with $x_t$ being a fixed point of the contraction $(1-t)\Gamma +tf$ for each $t\in (0,1)$. By Lemma 1, we conclude that

$$\begin{array}{c}\parallel x_n-x_t{\parallel }^2\le f_n\left(t\right)+2t\parallel x_t-x_n{\parallel }^2+(1+t^2-2t){\parallel x_t-x_n\parallel }^2+2t⟨f\left(x_t\right)-x_t,J(x_t-x_n)⟩,\hfill \end{array}$$

(27)

where

$$f_n\left(t\right)=(\parallel \Gamma x_n-x_n\parallel +2\parallel x_t-x_n\parallel )\parallel x_n-\Gamma x_n{\parallel (1-t)}^2\to 0(n\to \infty ).$$

(28)

Equation (27) yields that

$$2t⟨J(x_t-x_n),x_t-f\left(x_t\right)⟩\le f_n\left(t\right)+t^2{\parallel x_n-x_t\parallel }^2.$$

(29)

Letting $n\to \infty$ in Equation (29), one arrives at

$$\underset{n\to \infty }{lim\; sup}2⟨x_t-f\left(x_t\right),J(x_t-x_n)⟩\le t{M}_4,$$

(30)

where $sup\{\parallel x_t-x_n{\parallel }^2\}\le M_4$, where $M_4>0$. Further letting t go to 0 in Equation (30), we have

$$\underset{t\to 0}{lim\; sup}\underset{n\to \infty }{lim\; sup}⟨x_t-f\left(x_t\right),J(x_t-x_n)⟩\le 0.$$

Thus,

$$\begin{array}{c}{\displaystyle \underset{n\to \infty }{lim\; sup}}⟨f\left(x^{*}\right)-x^{*},J(x_n-x^{*})⟩\le {\displaystyle \underset{n\to \infty }{lim\; sup}}⟨f\left(x^{*}\right)-x^{*},J(x_n-x^{*})-J(x_n-x_t)⟩\hfill \\ +(1+\delta )\parallel x_t-x^{*}\parallel {\displaystyle \underset{n\to \infty }{lim\; sup}}\parallel x_n-x_t\parallel +{\displaystyle \underset{n\to \infty }{lim\; sup}}⟨f\left(x_t\right)-x_t,J(x_n-x_t)⟩.\hfill \end{array}$$

Taking into account that ${lim}_{t\to 0}\parallel x_t-x^{*}\parallel =0$, we have

$$\begin{array}{c}{\displaystyle \underset{n\to \infty }{lim\; sup}}⟨f\left(x^{*}\right)-x^{*},J(x_n-x^{*})⟩={\displaystyle \underset{t\to 0}{lim\; sup}\underset{n\to \infty }{lim\; sup}}⟨f\left(x^{*}\right)-x^{*},J(x_n-x^{*})⟩\hfill \\ \le {\displaystyle \underset{t\to 0}{lim\; sup}\underset{n\to \infty }{lim\; sup}}⟨f\left(x^{*}\right)-x^{*},J(x_n-x^{*})-J(x_n-x_t)⟩.\hfill \end{array}$$

(31)

Since the space is smooth, we conclude from Equation (26) that

$$\begin{array}{c}{\displaystyle \underset{n\to \infty }{lim\; sup}}⟨f\left(x^{*}\right)-x^{*},J(y_n-x^{*})⟩={\displaystyle \underset{n\to \infty }{lim\; sup}}\{⟨J(x_n-x^{*}),f\left(x^{*}\right)-x^{*}⟩\hfill \\ +⟨J(y_n-x^{*})-J(x_n-x^{*}),f\left(x^{*}\right)-x^{*}⟩\}={\displaystyle \underset{n\to \infty }{lim\; sup}}⟨J(x_n-x^{*}),f\left(x^{*}\right)-x^{*}⟩\le 0.\hfill \end{array}$$

(32)

Step 6. We show that $\parallel x_n-x^{*}\parallel \to 0$ as $n\to \infty$. Indeed, we observe that

$$\begin{array}{c}\parallel y_n-x^{*}{\parallel }^2={\parallel {\alpha }_n(f\left(x_n\right)-f\left(x^{*}\right))+{\beta }_n(x_n-x^{*})+{\gamma }_n(W_n{z}_n-x^{*})+{\alpha }_n(f\left(x^{*}\right)-x^{*})\parallel }^2\hfill \\ \le {\alpha }_n\parallel f\left(x_n\right)-f\left(x^{*}\right){\parallel }^2+{\beta }_n\parallel x_n-x^{*}{\parallel }^2+{\gamma }_n{\parallel z_n-x^{*}\parallel }^2+2{\alpha }_n⟨f\left(x^{*}\right)-x^{*},J(y_n-x^{*})⟩\hfill \\ \le {\alpha }_n\delta \parallel x_n-x^{*}{\parallel }^2+{\beta }_n\parallel x_n-x^{*}{\parallel }^2+{\gamma }_n(t_n\parallel x_n-x^{*}{\parallel }^2+(1-t_n)\parallel y_n-x^{*}{\parallel }^2)\hfill \\ +2{\alpha }_n⟨f\left(x^{*}\right)-x^{*},J(y_n-x^{*})⟩,\hfill \end{array}$$

which hence yields

$$\begin{array}{cc}\parallel y_n-x^{*}{\parallel }^2\hfill & \le (1-\frac{{\alpha }_n(1-\delta )}{1-{\gamma }_n(1-t_n)}){\parallel x_n-x^{*}\parallel }^2+\frac{2{\alpha }_n}{1-{\gamma }_n(1-t_n)}⟨f\left(x^{*}\right)-x^{*},J(y_n-x^{*})⟩.\hfill \end{array}$$

(33)

Thus,

$$\begin{array}{c}\parallel x_{n+1}-x^{*}{\parallel }^2\le {\delta }_n\parallel x_n-x^{*}{\parallel }^2+(1-{\delta }_n){\parallel W_n{y}_n-x^{*}\parallel }^2\hfill \\ \le {\delta }_n\parallel x_n-x^{*}{\parallel }^2+(1-{\delta }_n)\{(1-\frac{{\alpha }_n(1-\delta )}{1-{\gamma }_n(1-t_n)}){\parallel x_n-x^{*}\parallel }^2\hfill \\ +\frac{2{\alpha }_n}{1-{\gamma }_n(1-t_n)}⟨f\left(x^{*}\right)-x^{*},J(y_n-x^{*})⟩\}\hfill \\ =[1-\frac{{\alpha }_n(1-{\delta }_n)(1-\delta )}{1-{\gamma }_n(1-t_n)}]{\parallel x_n-x^{*}\parallel }^2+\frac{{\alpha }_n(1-{\delta }_n)(1-\delta )}{1-{\gamma }_n(1-t_n)}·\frac{2⟨f\left(x^{*}\right)-x^{*},J(y_n-x^{*})⟩}{1-\delta }.\hfill \end{array}$$

(34)

Since ${lim\; inf}_{n\to \infty }\frac{(1-{\delta }_n)(1-\delta )}{1-{\gamma }_n(1-t_n)}>0,\left\{\frac{{\alpha }_n(1-\delta )}{1-{\gamma }_n(1-t_n)}\right\}\subset (0,1)$ and ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$, we know that $\left\{\frac{{\alpha }_n(1-{\delta }_n)(1-\delta )}{1-{\gamma }_n(1-t_n)}\right\}\subset (0,1)$ and ${\sum }_{n=0}^{\infty }\frac{{\alpha }_n(1-{\delta }_n)(1-\delta )}{1-{\gamma }_n(1-t_n)}=\infty$. Utilizing Lemma 6 and Equation (32), one from Equation (34) gets that $\parallel x_n-x^{*}\parallel \to 0$ as n tends to the infinity. This completes the proof.

Let $q>1$. A mapping $T:C\to C$ is said to be $\eta$-strictly pseudocontractive of order q if for each $x,y\in C$, there exists $j_q(x-y)\in J_q(x-y)$ such that $⟨Tx-Ty,j_q(x-y)⟩\le {\parallel x-y\parallel }^q-\eta {\parallel x-y-(Tx-Ty)\parallel }^q$ for some $\eta \in (0,1)$. It is clear that $T:C\to C$ is $\eta$-strictly pseudocontractive of order q iff $I-T$ is q-order $\eta$-inverse-strongly accretive.

Corollary 1.

Let X be uniformly convex and q-uniformly, where $1<q\le 2$, smooth space. Let $B:C\to 2^X$ be an m-accretive operator and $A:C\to X$ be a q-order α-inverse-strongly accretive operator Let ${\Pi }_C$ be a non-expansive sunny retraction onto C and let T be a q-order η-strictly pseudocontractive self-mapping defined on C such that $\Omega ={\cap }_{n=0}^{\infty }\mathrm{Fix}\left(S_n\right)\cap \mathrm{Fix}\left(T\right)\cap {(A+B)}^{-1}0\ne \varnothing$. Let f be a δ-contractive self-mapping defined on C with constant $\delta \in (0,1)$ and $\left\{W_n\right\}$ be the vector sequence defined by Equation (3). Define a sequence $\left\{x_n\right\}$ as follows:

$$\left\{\begin{array}{c}{y}_n={\beta }_n{x}_n+{\gamma }_n{W}_n(t_n{x}_n+(1-t_n)J_{{\lambda }_n}^B(I-{\lambda }_{n}A)((1-l)I+lT)y_n)+{\alpha }_{n}f\left(x_n\right),\hfill \\ x_{n+1}=(1-{\delta }_n)W_n{y}_n+{\delta }_n{x}_{n}n\ge 0,\hfill \end{array}\right.$$

(35)

where $0<l<min\{1,{(\frac{q\eta }{{\kappa }_q})}^{\frac{1}{q-1}}\}$, $\left\{{\lambda }_n\right\}\subset (0,{(\frac{q\alpha }{{\kappa }_q})}^{\frac{1}{q-1}})$ and $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\},\left\{{\delta }_n\right\},\left\{t_n\right\}\subset (0,1)$ satisfy the following conditions:

(i) 

${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$ and ${\alpha }_n+{\beta }_n+{\gamma }_n=1;$

(ii) 

${lim}_{n\to \infty }|{\beta }_n-{\beta }_{n-1}|={lim}_{n\to \infty }|{\gamma }_n-{\gamma }_{n-1}|={lim}_{n\to \infty }{\alpha }_n={lim}_{n\to \infty }|t_n-t_{n-1}|=0$;

(iii) 

${lim\; sup}_{n\to \infty }{\gamma }_n{t}_n(1-t_n)>0$ and ${lim\; inf}_{n\to \infty }{\gamma }_n(1-t_n)<1$;

(iv) 

${lim\; inf}_{n\to \infty }{\beta }_n{\gamma }_n>0$, ${lim\; inf}_{n\to \infty }{\delta }_n>0$ and ${lim\; sup}_{n\to \infty }{\delta }_n<1$;

(v) 

$0<\overline{\lambda }\le {\lambda }_n\forall n\ge 0$ and ${lim}_{n\to \infty }{\lambda }_n=\lambda <{\left(\frac{q\alpha }{{\kappa }_q}\right)}^{\frac{1}{q-1}}$.

Then, $x_n\to x^{*}\in \Omega$ strongly.

Proof. 

In Theorem 1, we put $A_1=I-T,A_2=0$ and ${\mu }_1=l$, where $0<l<min\{1,{(\frac{q\eta }{{\kappa }_q})}^{\frac{1}{q-1}}\}$. Then, GSVI (1) is equivalent to the variational inequality: $⟨A_1{x}^{*},J(x-x^{*})⟩\ge 0\forall x\in C$. In this case, $A_1:C\to X$ is q-order $\eta$-inverse-strongly accretive. It is not hard to see that $\mathrm{Fix}\left(T\right)=\mathrm{VI}(C,A_1)$. Indeed, for $l\in (0,1)$, we observe that

$$\begin{array}{cc}p\in \mathrm{VI}(C,A_1)\hfill & \iff ⟨A_{1}p,J(x-p)⟩\ge 0\forall x\in C\iff p={\Pi }_C(p-l{A}_{1}p)\hfill \\ \hfill & \iff p={\Pi }_C(p-l(I-T)p)=(1-l)p+lTp\iff p\in \mathrm{Fix}\left(T\right).\hfill \end{array}$$

Thus, we obtain that $\Omega ={\cap }_{n=0}^{\infty }\mathrm{Fix}\left(S_n\right)\cap \mathrm{GSVI}(C,A_1,A_2)\cap {(A+B)}^{-1}0={\cap }_{n=0}^{\infty }\mathrm{Fix}\left(S_n\right)\cap \mathrm{Fix}\left(T\right)\cap {(A+B)}^{-1}0$, and ${\Pi }_C(I-{\mu }_1{A}_1){\Pi }_C(I-{\mu }_2{A}_2)y_n={\Pi }_C(I-{\mu }_1{A}_1)y_n=((1-l)I+lT)y_n$. Thus, Equation (4) reduces to Equation (35). Therefore, the desired result follows from Theorem 3.1. □

4. Subresults

4.1. Variational Inequality Problem

The framework of potential spaces will be restricted into a Hilbert space H in this section. Let $A:C\to H$, where C is a nonempty subset, be a single-valued operator. Let us recall the classical variational inequality problem (VIP): $⟨A{x}^{*},x-x^{*}⟩\ge 0$ for any $x\in C$. The set of solutions of the VIP is denoted by the notation $\mathrm{VI}(C,A)$. Let $I_C$ be an indicator operator of C given by

$$I_C\left(x\right)=\left\{\begin{array}{c}0,\mathrm{if}x\in C,\hfill \\ \infty ,\mathrm{if}x\notin C.\hfill \end{array}\right.$$

One finds that $I_C$ is a proper convex and lower semicontinuous function and $\partial I_C$, the subdifferential, is a maximally monotone operator. For $\lambda >0$, the resolvent of $\partial I_C$ is denoted by $J_{\lambda }^{\partial I_C}$, i.e., $J_{\lambda }^{\partial I_C}={(I+\lambda \partial I_C)}^{-1}$. We denote the normal cone of C at u by $N_C\left(u\right)$, i.e., $N_C\left(u\right)=\{w\in H:⟨w,v-u⟩\le 0\forall v\in C\}$. Note that

$$\begin{array}{cc}\partial I_C\left(u\right)\hfill & =\{w\in H:I_C\left(v\right)+⟨w,v-u⟩\le I_C\left(u\right)\}\hfill \\ \hfill & =\{w\in H:⟨w,v-u⟩\le 0\}=N_C\left(u\right).\hfill \end{array}$$

Thus, we know that $x-u\in \lambda N_C\left(u\right)\iff u=J_{\lambda }^{\partial I_C}\left(x\right)\iff u=P_C\left(x\right)\forall v\in C\iff ⟨x-u,v-u⟩\le 0$. Hence, we get $\mathrm{VI}(C,A)={(A+\partial I_C)}^{-1}0$.

Next, putting $B=\partial I_C$ in Corollary 1, we can obtain the following convergence theorem.

Theorem 2.

Let the mapping $A:C\to H$ be α-inverse-strongly monotone, and $T:C\to C$ be a 2-order η-strictly pseudocontractive mapping such that $\Omega ={\cap }_{n=0}^{\infty }\mathrm{Fix}\left(S_n\right)\cap \mathrm{VI}(C,A)\cap \mathrm{Fix}\left(T\right)\ne \varnothing$. Let $\left\{W_n\right\}$ be the mapping sequence defined by (2.1) and f be a δ-contractive self-mapping with contractive constant $\delta \in (0,1)$. Define a sequence $\left\{x_n\right\}$ by

$$\left\{\begin{array}{c}{y}_n={\beta }_n{x}_n+{\gamma }_n{W}_n(t_n{x}_n+(1-t_n)P_C(I-{\lambda }_{n}A)((1-l)I+lT)y_n)+{\alpha }_{n}f\left(x_n\right),\hfill \\ x_{n+1}=(1-{\delta }_n)W_n{y}_n+{\delta }_n{x}_{n}n\ge 0,\hfill \end{array}\right.$$

where $0<l<min\{1,2\eta \}$, $\left\{{\lambda }_n\right\}\subset (0,2\alpha )$ and $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\},\left\{{\delta }_n\right\},\left\{t_n\right\}\subset (0,1)$ satisfy the following conditions:

(i) 

${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$ and ${\alpha }_n+{\beta }_n+{\gamma }_n=1;$

(ii) 

${lim}_{n\to \infty }|{\beta }_n-{\beta }_{n-1}|={lim}_{n\to \infty }|{\gamma }_n-{\gamma }_{n-1}|={lim}_{n\to \infty }{\alpha }_n={lim}_{n\to \infty }|t_n-t_{n-1}|=0$;

(iii) 

${lim\; sup}_{n\to \infty }{\gamma }_n{t}_n(1-t_n)\ge 0$ and ${lim\; inf}_{n\to \infty }{\gamma }_n(1-t_n)<1$;

(iv) 

${lim\; inf}_{n\to \infty }{\beta }_n{\gamma }_n>0$, ${lim\; inf}_{n\to \infty }{\delta }_n>0$ and ${lim\; sup}_{n\to \infty }{\delta }_n<1$;

(v) 

$0<\overline{\lambda }\le {\lambda }_n\forall n\ge 0$ and ${lim}_{n\to \infty }{\lambda }_n=\lambda <2\alpha$.

Then, $x_n\to x^{*}\in \Omega$ strongly.

4.2. Convex Minimization Problem

Let $g:C\to \mathbf{R}$ be a convex smooth function and $h:C\to \mathbf{R}$ be a proper convex and lower semicontinuous function. The convex minimization problem is

$$g\left(x^{*}\right)+h\left(x^{*}\right)=\underset{x\in C}{min}\{g\left(x\right)+h\left(x\right)\}.$$

(36)

This is equivalent to the problem $0\in \partial h\left(x^{*}\right)+\nabla g\left(x^{*}\right)$, where $\partial h$ is the subdifferential of h and $\nabla g$ is the gradient of g. Next, setting $A=\nabla g$ and $B=\partial h$ in Corollary 1, we can obtain the following.

Theorem 3.

Let $g:C\to \mathbf{R}$ be a convex and differentiable function with $\frac{1}{\alpha }$-Lipschitz continuous gradient $\nabla g$ and $h:C\to \mathbf{R}$ be a convex and lower semicontinuous function. Let f be a δ-contractive self-mapping defined on C and $\left\{W_n\right\}$ be the sequence defined by Equation (3). Let T be an η-strictly pseudocontractive self-mapping defined on C with order 2 such that $\Omega ={\cap }_{n=0}^{\infty }\mathrm{Fix}\left(S_n\right)\cap \mathrm{Fix}\left(T\right)\cap {(\nabla g+\partial h)}^{-1}0\ne \varnothing$, where ${(\nabla g+\partial h)}^{-1}0$ is the set of minimizers attained by $g+h$. Define a sequence $\left\{x_n\right\}$ by

$$\left\{\begin{array}{c}{y}_n={\beta }_n{x}_n+{\gamma }_n{W}_n(t_n{x}_n+(1-t_n)J_{{\lambda }_n}^{\partial h}(I-{\lambda }_n\nabla g)((1-l)I+lT)y_n)+{\alpha }_{n}f\left(x_n\right),\hfill \\ x_{n+1}=(1-{\delta }_n)W_n{y}_n+{\delta }_n{x}_{n}n\ge 0,\hfill \end{array}\right.$$

where $0<l<min\{1,2\eta \}$, $\left\{{\lambda }_n\right\}\subset (0,2\alpha )$ and $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\},\left\{{\delta }_n\right\},\left\{t_n\right\}\subset (0,1)$ satisfy the following conditions:

(i) 

${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$, ${\alpha }_n+{\beta }_n+{\gamma }_n=1;$

(ii) 

${lim}_{n\to \infty }|{\beta }_n-{\beta }_{n-1}|={lim}_{n\to \infty }|{\gamma }_n-{\gamma }_{n-1}|={lim}_{n\to \infty }{\alpha }_n={lim}_{n\to \infty }|t_n-t_{n-1}|=0$;

(iii) 

${lim\; sup}_{n\to \infty }{\gamma }_n{t}_n(1-t_n)<1$ and ${lim\; inf}_{n\to \infty }{\gamma }_n(1-t_n)>0$;

(iv) 

${lim\; inf}_{n\to \infty }{\beta }_n{\gamma }_n>0$, ${lim\; inf}_{n\to \infty }{\delta }_n>0$ and ${lim\; sup}_{n\to \infty }{\delta }_n<1$;

(v) 

$0<\overline{\lambda }\le {\lambda }_n\forall n\ge 0$ and ${lim}_{n\to \infty }{\lambda }_n=\lambda <2\alpha$.

Then, $x_n\to x^{*}\in \Omega$ strongly. Indeed, ${}^{*}$ also solves the inequality: $⟨(I-f)x^{*},x^{*}-p⟩\le 0\forall p\in \Omega$ uniquely.

4.3. Split Feasibility Problem

Let $\Gamma :H_1\to H_2$ be a linear bounded operator with its adjoint ${\Gamma }^{*}$. Let C and Q be convex closed sets in Hilbert spaces $H_1$ and $H_2$, respectively. One considers the split feasibility problem (SFP): $x^{*}\in C\mathrm{and}\Gamma x^{*}\in Q$. The solution set of the SFP is $C\cap {\Gamma }^{-1}Q$. To solve the SFP, one can set it as the following convexly minimization problem:

$$\underset{x\in C}{min}g\left(x\right):=\frac{1}{2}{\parallel \Gamma x-P_Q\Gamma x\parallel }^2.$$

Here, g has a Lipschitz gradient given by $\nabla g={\Gamma }^{*}(I-P_Q)\Gamma$. In addition, $\nabla g$ is $\frac{1}{{\parallel \Gamma \parallel }^2}$-inverse-strongly monotone, where ${\parallel \Gamma \parallel }^2$ is the spectral radius of ${\Gamma }^{*}\Gamma$. Thus, $x^{*}$ solves the SFP iff $x^{*}$ satisfies the inclusion problem:

$$\begin{array}{cc}0\in \partial I_C\left(x^{*}\right)+\nabla g\left(x^{*}\right)\hfill & \iff x^{*}-\lambda \nabla g\left(x^{*}\right)\in (I+\lambda \partial I_C)x^{*}\hfill \\ \hfill & \iff x^{*}=J_{\lambda }^{\partial I_C}(x^{*}-\lambda \nabla g\left(x^{*}\right))\hfill \\ \hfill & \iff x^{*}=P_C(x^{*}-\lambda \nabla g\left(x^{*}\right)).\hfill \end{array}$$

Theorem 4.

Let $\Gamma :H_1\to H_2$ be a linear bounded operator with its adjoint ${\Gamma }^{*}$, and T be an η-strictly pseudocontractive self-mapping defined on C with order 2 such that $\Omega ={\cap }_{n=0}^{\infty }\mathrm{Fix}\left(S_n\right)\cap \mathrm{Fix}\left(T\right)\cap (C\cap {\Gamma }^{-1}Q)\ne \varnothing$. Let f be a δ-contractive self-mapping defined on C and $\left\{W_n\right\}$ be the sequence defined by (2.1). Define a sequence $\left\{x_n\right\}$ by

$$\left\{\begin{array}{c}{y}_n={\beta }_n{x}_n+{\gamma }_n{W}_n(t_n{x}_n+(1-t_n)P_C(I-{\lambda }_n{\Gamma }^{*}(I-P_Q)\Gamma )((1-l)I+lT)y_n)+{\alpha }_{n}f\left(x_n\right),\hfill \\ x_{n+1}=(1-{\delta }_n)W_n{y}_n+{\delta }_n{x}_{n}n\ge 0,\hfill \end{array}\right.$$

where $0<l<min\{1,2\eta \}$, $\left\{{\lambda }_n\right\}\subset (0,\frac{2}{{\parallel \Gamma \parallel }^2})$ and $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\},\left\{{\delta }_n\right\},\left\{t_n\right\}\subset (0,1)$ satisfy the following conditions:

(i) 

${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$, ${\alpha }_n+{\beta }_n+{\gamma }_n=1;$

(ii) 

${lim}_{n\to \infty }|{\beta }_{n-1}-{\beta }_n|={lim}_{n\to \infty }|{\gamma }_{n-1}-{\gamma }_n|={lim}_{n\to \infty }{\alpha }_n={lim}_{n\to \infty }|t_{n-1}-t_n|=0$;

(iii) 

${lim\; sup}_{n\to \infty }{\gamma }_n{t}_n(1-t_n)<1$ and ${lim\; inf}_{n\to \infty }{\gamma }_n(1-t_n)>0$;

(iv) 

${lim\; inf}_{n\to \infty }{\beta }_n{\gamma }_n>0$, ${lim\; inf}_{n\to \infty }{\delta }_n>0$ and ${lim\; sup}_{n\to \infty }{\delta }_n<1$;

(v) 

$0<\overline{\lambda }\le {\lambda }_n\forall n\ge 0$ and ${lim}_{n\to \infty }{\lambda }_n=\lambda <\frac{2}{{\parallel \Gamma \parallel }^2}$.

Then, $x_n\to x^{*}\in \Omega$ strongly.

5. Conclusions

In this paper, we established norm convergence theorems of solutions for a general symmetrical variational system, which can be acted as a framework for many real world problems arising in engineering and medical imaging, which involves some convex optimization subproblems. There is no compact assumption on the operators of accretive type and the sets in the whole space. The restrictions, which are also mild, imposed on the control parameters. Our results provide an outlet for viscosity type algorithms without compact assumptions in infinite-dimensional spaces. From the space frameworks’ point of view, the space in our convergence theorems is still not general; however, it is Banach now. It is of interest to further relax the convex restrictions in the future research.