Open Access

Implicit and explicit iterative methods for mixed equilibria with constraints of system of generalized equilibria and hierarchical fixed point problem

Journal of Inequalities and Applications20152015:280

https://doi.org/10.1186/s13660-015-0805-2

Received: 8 May 2015

Accepted: 28 August 2015

Published: 17 September 2015

Abstract

In this paper, we introduce one composite implicit relaxed extragradient-like scheme and another composite explicit relaxed extragradient-like scheme for finding a common solution of a finite family of generalized mixed equilibrium problems (GMEPs) with the constraints of a system of generalized equilibrium problems (SGEP) and the hierarchical fixed point problem (HFPP) for a strictly pseudocontractive mapping in a real Hilbert space. We establish the strong convergence of these two composite relaxed extragradient-like schemes to the same common solution of finitely many GMEPs and the SGEP, which is the unique solution of the HFPP for a strictly pseudocontractive mapping. In particular, we make use of weaker control conditions than previous ones for the sake of proving strong convergence. Utilizing these results, we first propose the composite implicit and explicit relaxed extragradient-like schemes for finding a common fixed point of a finite family of strictly pseudocontractive mappings, and then we derive their strong convergence to the unique common solution of the SGEP and some HFPP. Our results complement, develop, improve, and extend the corresponding ones given by some authors recently in this area.

Keywords

composite relaxed extragradient-like method generalized mixed equilibrium problem system of generalized equilibrium problems inverse strongly monotone mapping strictly pseudocontractive mapping fixed point

MSC

49J30 47H09 47J20 49M05

1 Introduction

Let H be a real Hilbert space with inner product \(\langle\cdot,\cdot \rangle\) and induced norm \(\|\cdot\|\), C be a nonempty, closed, and convex subset of H, and \(P_{C}\) be the metric projection of H onto C. Let \(T:C\to C\) be a self-mapping on C. We denote by \(\operatorname{Fix}(T)\) the set of fixed points of T and by R the set of all real numbers. A mapping \(A:H\to H\) is called γ̄-strongly positive on H if there exists a constant \(\bar{\gamma}>0\) such that
$$\langle Ax,x\rangle\geq\bar{\gamma}\|x\|^{2},\quad \forall x\in H. $$
A mapping \(F:C\to H\) is called L-Lipschitz-continuous if there exists a constant \(L\geq0\) such that
$$\|Fx-Fy\|\leq L\|x-y\|,\quad \forall x,y\in C. $$
In particular, if \(L=1\) then F is called a nonexpansive mapping; if \(L\in[0,1)\) then F is called a contraction. A mapping \(T:C\to C\) is called k-strictly pseudocontractive (or a k-strict pseudocontraction) if there exists a constant \(k\in[0,1)\) such that
$$\|Tx-Ty\|^{2}\leq\|x-y\|^{2}+k\bigl\| (I-T)x-(I-T)y \bigr\| ^{2},\quad \forall x,y\in C. $$
In particular, if \(k=0\), then T is a nonexpansive mapping. The mapping T is pseudocontractive if and only if
$$\langle Tx-Ty,x-y\rangle\leq\|x-y\|^{2},\quad \forall x,y\in C. $$
T is strongly pseudocontractive if and only if there exists a constant \(\lambda\in(0,1)\) such that
$$\langle Tx-Ty,x-y\rangle\leq\lambda\|x-y\|^{2},\quad \forall x,y\in C. $$
Note that the class of strictly pseudocontractive mappings includes the class of nonexpansive mappings as a subclass. That is, T is nonexpansive if and only if T is 0-strictly pseudocontractive. The mapping T is also said to be pseudocontractive if \(k=1\) and T is said to be strongly pseudocontractive if there exists a positive constant \(\lambda\in(0,1)\) such that \(T+(1-\lambda)I\) is pseudocontractive. Clearly, the class of strictly pseudocontractive mappings falls into the one between the classes of nonexpansive mappings and of pseudocontractive mappings. Also it is clear that the class of strongly pseudocontractive mappings is independent of the class of strictly pseudocontractive mappings (see [1]). The class of pseudocontractive mappings is one of the most important classes of mappings among nonlinear mappings. Recently, many authors have been devoting to the study of the problem of finding fixed points of pseudocontractive mappings; see e.g., [29] and the references therein.
Let \({\mathcal{A}}:C\to H\) be a nonlinear mapping on C. The variational inequality problem (VIP) associated with the set C and the mapping \({\mathcal{A}}\) is stated as follows: find \(x^{*}\in C\) such that
$$ \bigl\langle {\mathcal{A}}x^{*},x-x^{*}\bigr\rangle \geq0,\quad \forall x\in C. $$
(1.1)
The solution set of VIP (1.1) is denoted by \(\operatorname{VI}(C,{\mathcal{A}})\).
The VIP (1.1) was first discussed by Lions [10]. There are many applications of VIP (1.1) in various fields; see, e.g., [4, 5, 7, 11]. It is well known that, if \({\mathcal{A}}\) is a strongly monotone and Lipschitz-continuous mapping on C, then VIP (1.1) has a unique solution. In 1976, Korpelevich [12] proposed an iterative algorithm for solving VIP (1.1) in Euclidean space \(\mathbf{R}^{n}\):
$$ \left \{ \textstyle\begin{array}{@{}l} y_{n}=P_{C}(x_{n}-\tau{\mathcal{A}}x_{n}),\\ x_{n+1}=P_{C}(x_{n}-\tau{\mathcal{A}}y_{n}), \quad\forall n\geq0, \end{array}\displaystyle \right . $$
(1.2)
with \(\tau>0\) a given number, which is known as the extragradient method. The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see, e.g., [5, 11, 1329] and references therein, to name but a few.
In 2011, Ceng et al. [30] also introduced the following iterative method:
$$ x_{n+1}=P_{C}\bigl[\alpha_{n}\gamma Vx_{n}+(I-\alpha_{n}\mu F)Tx_{n}\bigr],\quad \forall n\geq0, $$
(1.3)
where \(T:C\to C\) is a nonexpansive mapping such that \(\operatorname{Fix}(T)\neq \emptyset\), \(F:C\to H\) is a κ-Lipschitzian and η-strongly monotone operator with positive constants \(\kappa,\eta>0\), \(V:C\to H\) is an l-Lipschitzian mapping with constant \(l \geq0\) and \(0<\mu<\frac{2\eta}{\kappa^{2}}\). They proved that, under mild conditions, the sequence \(\{x_{n}\}\) generated by (1.3) converges strongly to a point \(\tilde{x}\in\operatorname{Fix}(T)\) which is the unique solution to the VIP
$$ \bigl\langle (\mu F-\gamma V)\tilde{x},p-\tilde{x}\bigr\rangle \geq0,\quad \forall p \in \operatorname{Fix}(T). $$
(1.4)
Their results also improve Tian’s results [31] from the contractive mapping f to the Lipschitzian mapping V.
In 2011, Ceng et al. [32] introduced one general composite implicit scheme that generates a net \(\{x_{t}\}_{t\in(0,\min\{1, \frac{2-\bar{\gamma}}{\tau-\gamma\alpha}\})}\) in an implicit way
$$ x_{t}=(I-\theta_{t}A)Tx_{t}+ \theta_{t}\bigl[Tx_{t}-t\bigl(\mu FTx_{t}-\gamma f(x_{t})\bigr)\bigr], $$
(1.5)
and also proposed another general composite explicit scheme that generates a sequence \(\{x_{n}\}\) in an explicit way
$$ \left \{ \textstyle\begin{array}{@{}l} y_{n}=(I-\alpha_{n}\mu F)Tx_{n}+\alpha_{n}\gamma f(x_{n}),\\ x_{n+1}=(I-\beta_{n}A)Tx_{n}+\beta_{n}y_{n}, \quad\forall n\geq0, \end{array}\displaystyle \right . $$
(1.6)
where \(x_{0}\in H\) is an arbitrary initial guess, \(F:H\to H\) is a κ-Lipschitzian and η-strongly monotone operator with positive constants \(\kappa,\eta>0\), \(T:H\to H\) is a nonexpansive mapping, \(A:H\to H\) is a γ̄-strongly positive bounded linear operator, and \(f:H\to H\) is an α-contractive mapping with \(\alpha\in(0,1)\). They proved that, under appropriate conditions, the net \(\{x_{t}\}\) and the sequence \(\{x_{n}\}\) generated by (1.5) and (1.6), respectively, converge strongly to the same point \(\tilde{x}\in\operatorname{Fix}(T)\), which is the unique solution to the VIP
$$ \bigl\langle (A-I)\tilde{x},p-\tilde{x}\bigr\rangle \geq0, \quad\forall p\in \operatorname{Fix}(T). $$
(1.7)
Their results supplement and develop the corresponding ones of Marino and Xu [33], Yamada [34] and Tian [31].
Very recently, inspired by Ceng et al. [32], Jung [1] introduced one general composite implicit scheme that generates a net \(\{x_{t}\}_{t\in(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})}\) in an implicit way
$$ x_{t}=(I-\theta_{t}A)T_{t}x_{t}+ \theta_{t}\bigl[t\gamma Vx_{t}+(I-t\mu F)T_{t}x_{t} \bigr], $$
(1.8)
and also proposed another general composite explicit scheme that generates a sequence \(\{x_{n}\}\) in an explicit way,
$$ \left \{ \textstyle\begin{array}{@{}l} y_{n}=\alpha_{n}\gamma Vx_{n}+(I-\alpha_{n}\mu F)T_{n}x_{n},\\ x_{n+1}=(I-\beta_{n}A)T_{n}x_{n}+\beta_{n}y_{n},\quad \forall n\geq0, \end{array}\displaystyle \right . $$
(1.9)
where \(x_{0}\in H\) is an arbitrary initial guess and the following conditions are satisfied:
  • \(T:H\to H\) is a k-strictly pseudocontractive mapping with \(\operatorname{Fix}(T)\neq\emptyset\);

  • A is a γ̄-strongly positive bounded linear operator on H with \(\bar{\gamma}\in(1,2)\);

  • \(F:H\to H\) is a κ-Lipschitzian and η-strongly monotone operator with \(0<\mu<\frac{2\eta}{\kappa^{2}}\);

  • \(V:H\to H\) is an l-Lipschitzian mapping with \(0\leq\gamma l<\tau\) and \(\tau=1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\);

  • \(T_{t}:H\to H\) is a mapping defined by \(T_{t}x=\lambda_{t}x+(1-\lambda _{t})Tx\), \(t\in(0,1)\), for \(0\leq k\leq\lambda_{t}\leq\lambda<1\) and \(\lim_{t\to0}\lambda_{t}=\lambda\);

  • \(T_{n}:H\to H\) is a mapping defined by \(T_{n}x=\lambda_{n}x+(1-\lambda_{n})Tx\) for \(0\leq k\leq\lambda_{n}\leq\lambda<1\) and \(\lim_{n\to\infty}\lambda_{n}=\lambda\);

  • \(\{\alpha_{n}\}\subset[0,1]\), \(\{\beta_{n}\}\subset(0,1]\) and \(\{\theta_{t}\} _{t\in(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})} \subset(0,1)\).

The author of [1] proved that, under weaker control conditions than the previous ones, the net \(\{x_{t}\}\) and the sequence \(\{x_{n}\}\) generated by (1.8) and (1.9), respectively, converge strongly to the same point \(\tilde{x}\in\operatorname{Fix}(T)\), which is the unique solution to the VIP
$$ \bigl\langle (A-I)\tilde{x},p-\tilde{x}\bigr\rangle \geq0, \quad\forall p\in \operatorname{Fix}(T). $$
(1.10)
His results extend and improve Ceng et al.’s corresponding ones [32] from the nonexpansive mapping T to the strictly pseudocontractive mapping T and from the contractive mapping f to the Lipschitzian mapping V.
On the other hand, let \(\varphi:C\to\mathbf{R}\) be a real-valued function, \({\mathcal{A}}:C\to H\) be a nonlinear mapping and \({ \varTheta }:C\times C\to\mathbf{R}\) be a bifunction. In 2008, Peng and Yao [13] introduced the generalized mixed equilibrium problem (GMEP) of finding \(x\in C\) such that
$$ { \varTheta }(x,y)+\varphi(y)-\varphi(x)+\langle{\mathcal{A}}x,y-x\rangle \geq0, \quad\forall y\in C. $$
(1.11)
We denote the set of solutions of GMEP (1.11) by \(\operatorname{GMEP}({ \varTheta },\varphi,{\mathcal{A}})\). The GMEP (1.11) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others. Recently, many authors have been devoting to the study of the GMEP (1.11) and its special cases, e.g., generalized equilibrium problem (GEP), mixed equilibrium problem (MEP),equilibrium problem (EP), etc.; see, e.g., [15, 18, 2329, 31, 3538] and the references therein.
It was assumed in [13] that \({ \varTheta }:C\times C\to\mathbf{R}\) is a bifunction satisfying conditions (A1)-(A4) and \(\varphi:C \to\mathbf{R}\) is a lower semicontinuous and convex function with restriction (B1) or (B2), where
  1. (A1)

    \({ \varTheta }(x,x)=0\) for all \(x\in C\);

     
  2. (A2)

    Θ is monotone, i.e., \({ \varTheta }(x,y)+{ \varTheta }(y,x)\leq0\) for any \(x,y\in C\);

     
  3. (A3)
    Θ is upper-hemicontinuous, i.e., for each \(x,y,z\in C\),
    $$\limsup_{t\to0^{+}}{ \varTheta }\bigl(tz+(1-t)x,y\bigr)\leq{ \varTheta }(x,y); $$
     
  4. (A4)

    \({ \varTheta }(x,\cdot)\) is convex and lower semicontinuous for each \(x\in C\);

     
  5. (B1)
    for each \(x\in H\) and \(r>0\), there exists a bounded subset \(D_{x}\subset C\) and \(y_{x}\in C\) such that for any \(z\in C\setminus D_{x}\),
    $${ \varTheta }(z,y_{x})+\varphi(y_{x})-\varphi(z)+ \frac{1}{r}\langle y_{x}-z,z-x\rangle< 0; $$
     
  6. (B2)

    C is a bounded set.

     
Given a positive number \(r>0\). Let \(T^{({ \varTheta },\varphi)}_{r}:H\to C\) be the solution set of the auxiliary mixed equilibrium problem, that is, for each \(x\in H\),
$$T^{({ \varTheta },\varphi)}_{r}(x):=\biggl\{ y\in C:{ \varTheta }(y,z)+\varphi (z)- \varphi(y)+\frac{1}{r}\langle y-x, z-y\rangle\geq0,\forall z\in C\biggr\} . $$
In particular, if \(\varphi\equiv0\) then \(T^{({ \varTheta },\varphi)}_{r}\) is rewritten as \(T^{ \varTheta }_{r}:H\to C\), i.e.,
$$T^{ \varTheta }_{r}(x):=\biggl\{ y\in C:{ \varTheta }(y,z)+ \frac{1}{r}\langle y-x,z-y\rangle\geq0,\forall z\in C\biggr\} . $$
Let \({ \varPhi }_{1},{ \varPhi }_{2}:C\times C\to\mathbf{R}\) be two bifunctions and \(F_{1},F_{2}:C\to H\) be two mappings. Consider the problem of finding \((x^{*},y^{*})\in C\times C\) such that
$$ \left \{ \textstyle\begin{array}{@{}l} { \varPhi }_{1}(x^{*},x)+\langle F_{1}y^{*},x-x^{*}\rangle+\frac {1}{\nu_{1}}\langle x^{*}-y^{*},x-x^{*}\rangle \geq0, \quad\forall x\in C,\\ { \varPhi }_{2}(y^{*},y)+\langle F_{2}x^{*},y-y^{*}\rangle+\frac{1}{\nu_{2}}\langle y^{*}-x^{*},y-y^{*}\rangle \geq0, \quad\forall y\in C, \end{array}\displaystyle \right . $$
(1.12)
which is called a system of generalized equilibrium problems (SGEP) where \(\nu_{1}>0\) and \(\nu_{2}>0\) are two constants. In 2010, Ceng and Yao [23] transformed the SGEP (1.12) into the fixed point problem of the mapping \(G=T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu_{1} F_{1})T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})\), that is, \(Gx^{*}=x^{*}\), where \(y^{*}=T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})x^{*}\). Throughout this paper, the fixed point set of the mapping G is denoted by Ξ.
In particular, if \({ \varPhi }_{1}\equiv{ \varPhi }_{2}\equiv0\), then problem (1.12) reduces to the system of variational inequalities (SVI) of finding \((x^{*},y^{*})\in C\times C\) such that
$$ \left \{ \textstyle\begin{array}{@{}l} \langle\nu_{1}F_{1}y^{*}+x^{*}-y^{*},x-x^{*}\rangle\geq0,\quad \forall x\in C,\\ \langle\nu_{2}F_{2}x^{*}+y^{*}-x^{*},y-y^{*}\rangle\geq0,\quad \forall y\in C, \end{array}\displaystyle \right . $$
(1.13)
where \(\nu_{1}>0\) and \(\nu_{2}>0\) are two constants. Recently, many authors have addressed the study of the SVI (1.13); see, e.g., [11, 14, 15, 1720, 3941] and the references therein.
Let \(T:C\to C\) be a k-strictly pseudocontractive mapping. In 2010, Ceng and Yao [23] proposed and analyzed the following relaxed extragradient-like iterative scheme for finding a common solution \(x^{*}\in{ \varOmega }:=\operatorname{Fix}(T)\cap\operatorname{GMEP}({ \varTheta },\varphi,{\mathcal{A}})\cap{ \varXi }\) of the GMEP (1.11), the SGEP (1.12), and the fixed point problem of T:
$$ \left \{ \textstyle\begin{array}{@{}l} z_{n}=T^{({ \varTheta },\varphi)}_{\lambda_{n}}(I-\lambda _{n}{\mathcal{A}})x_{n},\\ y_{n}=T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu_{1}F_{1})T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu _{2}F_{2})z_{n},\\ x_{n+1}=\alpha_{n}u+\beta_{n}x_{n}+\gamma_{n}y_{n}+\delta_{n}Ty_{n}, \quad\forall n\geq0, \end{array}\displaystyle \right . $$
(1.14)
where \(0<\nu_{j}<2\zeta_{j}\) for \(j=1,2\), and \(\{\lambda_{n}\}\subset[0,2\eta ]\), \(\{\alpha_{n}\},\{\beta_{n}\},\{\gamma_{n}\},\{\delta_{n}\}\subset [0,1]\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}+\delta_{n}=1\) and \((\gamma _{n}+\delta_{n})k\leq\gamma_{n}\), \(\forall n\geq0\). Under some mild assumptions, the authors [23] proved that \(\{x_{n}\} \) converges strongly to \(x^{*}=P_{ \varOmega }u\) and \((x^{*},y^{*})\) is a solution of the SGEP (1.12), where \(y^{*}=T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})x^{*}\).

In this paper, we introduce one composite implicit relaxed extragradient-like scheme and another composite explicit relaxed extragradient-like scheme for finding a common solution of a finite family of generalized mixed equilibrium problems (GMEP) with the constraints of the SGEP (1.12) and the hierarchical fixed point problem (HFPP) for a strictly pseudocontractive mapping in a real Hilbert space. We establish the strong convergence of these two composite relaxed extragradient-like schemes to the same common solution of finitely many GMEPs and the SGEP (1.12), which is the unique solution of the HFPP for a strictly pseudocontractive mapping. In particular, we make use of weaker control conditions than the previous ones for the sake of proving strong convergence. Utilizing these results, we first propose the composite implicit and explicit relaxed extragradient-like schemes for finding a common fixed point of a finite family of strictly pseudocontractive mappings, and then derive their strong convergence to the unique common solution of the SGEP (1.12) and some HFPP. Our results complement, develop, improve, and extend the corresponding ones given by some authors recently in this area. See, e.g., Ceng et al. [32], Jung [1], and Ceng and Yao [23].

2 Preliminaries

Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by \(\langle\cdot, \cdot\rangle\) and \(\|\cdot\|\), respectively. Let C be a nonempty, closed, and convex subset of H. We write \(x_{n}\rightharpoonup x\) to indicate that the sequence \(\{x_{n}\}\) converges weakly to x and \(x_{n}\to x\) to indicate that the sequence \(\{x_{n}\}\) converges strongly to x. Moreover, we use \(\omega_{w}(x_{n})\) to denote the weak ω-limit set of the sequence \(\{x_{n}\}\), i.e.,
$$\omega_{w}(x_{n}):=\bigl\{ x\in H:x_{n_{i}} \rightharpoonup x \mbox{ for some subsequence }\{x_{n_{i}}\} \mbox{ of } \{x_{n}\}\bigr\} . $$
The metric (or nearest point) projection from H onto C is the mapping \(P_{C}:H\to C\) which assigns to each point \(x\in H\) the unique point \(P_{C}x\in C\) satisfying the property
$$\|x-P_{C}x\|=\inf_{y\in C}\|x-y\|=:d(x,C). $$

The following properties of projections are useful and pertinent to our purpose.

Proposition 2.1

Given any \(x\in H\) and \(z\in C\). One has
  1. (i)

    \(z=P_{C}x \Leftrightarrow \langle x-z,y-z\rangle\leq0\), \(\forall y\in C\);

     
  2. (ii)

    \(z=P_{C}x \Leftrightarrow \|x-z\|^{2}\leq\|x-y\|^{2}-\|y-z\|^{2}\), \(\forall y\in C\);

     
  3. (iii)

    \(\langle P_{C}x-P_{C}y,x-y\rangle\geq\|P_{C}x-P_{C}y\|^{2}\), \(\forall y\in H\), which hence implies that \(P_{C}\) is nonexpansive and monotone.

     

Definition 2.1

A mapping \(T:H\to H\) is said to be firmly nonexpansive if \(2T-I\) is nonexpansive, or equivalently, if T is 1-inverse strongly monotone (1-ism),
$$\langle x-y,Tx-Ty\rangle\geq\|Tx-Ty\|^{2}, \quad\forall x,y\in H; $$
alternatively, T is firmly nonexpansive if and only if T can be expressed as
$$T=\frac{1}{2}(I+S), $$
where \(S:H\to H\) is nonexpansive; projections are firmly nonexpansive.

Definition 2.2

A mapping \(F:C\to H\) is said to be
  1. (i)
    monotone if
    $$\langle Fx-Fy,x-y\rangle\geq0,\quad \forall x,y\in C; $$
     
  2. (ii)
    η-strongly monotone if there exists a constant \(\eta>0\) such that
    $$\langle Fx-Fy,x-y\rangle\geq\eta\|x-y\|^{2}, \quad\forall x,y\in C; $$
     
  3. (iii)
    α-inverse strongly monotone if there exists a constant \(\alpha>0\) such that
    $$\langle Fx-Fy,x-y\rangle\geq\alpha\|Fx-Fy\|^{2},\quad \forall x,y\in C. $$
     

It can easily be seen that if T is nonexpansive, then \(I-T\) is monotone. It is also easy to see that the projection \(P_{C}\) is 1-ism. Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields.

On the other hand, it is obvious that if \(F:C\to H\) is α-inverse strongly monotone, then F is monotone and \(\frac{1}{\alpha}\)-Lipschitz-continuous. Moreover, we also have, for all \(u,v\in C\) and \(\lambda>0\),
$$ \bigl\| (I-\lambda F)u-(I-\lambda F)v\bigr\| ^{2}\leq\|u-v\|^{2}+ \lambda(\lambda-2\alpha )\|Fu-Fv\|^{2}. $$
(2.1)
Consequently, if \(\lambda\leq2\alpha\), then \(I-\lambda F\) is a nonexpansive mapping from C to H.

Next we list some elementary conclusions for the MEP.

Proposition 2.2

(see [35])

Assume that \({ \varTheta }:C\times C\to\mathbf{R}\) satisfies (A1)-(A4) and let \(\varphi: C\to\mathbf{R}\) be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For \(r>0\) and \(x\in H\), define a mapping \(T^{({ \varTheta },\varphi)}_{r}:H\to C\) as follows:
$$T^{({ \varTheta },\varphi)}_{r}(x)=\biggl\{ z\in C:{ \varTheta }(z,y)+\varphi (y)- \varphi(z)+\frac{1}{r}\langle y-z,z-x\rangle \geq0,\forall y\in C\biggr\} $$
for all \(x\in H\). Then the following hold:
  1. (i)

    for each \(x\in H, T^{({ \varTheta },\varphi)}_{r}(x)\) is nonempty and single-valued;

     
  2. (ii)
    \(T^{({ \varTheta },\varphi)}_{r}\) is firmly nonexpansive, that is, for any \(x,y\in H\),
    $$\bigl\| T^{({ \varTheta },\varphi)}_{r}x-T^{({ \varTheta },\varphi)}_{r}y \bigr\| ^{2}\leq \bigl\langle T^{({ \varTheta }, \varphi)}_{r}x-T^{({ \varTheta },\varphi)}_{r}y,x-y \bigr\rangle ; $$
     
  3. (iii)

    \(\operatorname{Fix}(T^{({ \varTheta },\varphi)}_{r}) =\operatorname{MEP}({ \varTheta },\varphi)\);

     
  4. (iv)

    \(\operatorname{MEP}({ \varTheta },\varphi)\) is closed and convex;

     
  5. (v)

    \(\|T^{({ \varTheta },\varphi)}_{s}x-T^{({ \varTheta },\varphi)}_{t}x\| ^{2}\leq\frac{s-t}{s}\langle T^{({ \varTheta },\varphi)}_{s}x -T^{({ \varTheta },\varphi)}_{t}x,T^{({ \varTheta },\varphi)}_{s}x-x\rangle\) for all \(s,t>0\) and \(x\in H\).

     

In 2010, Ceng and Yao [23] transformed the SGEP (1.12) into a fixed point problem in the following way:

Proposition 2.3

(see [23])

Let \({ \varPhi }_{1},{ \varPhi }_{2}:C\times C\to\mathbf{R}\) be two bifunctions satisfying conditions (A1)-(A4). Then \((x^{*},y^{*})\in C\times C\) is a solution of the SGEP (1.12) if and only if \(x^{*}\) is a fixed point of the mapping \(G:C\to C\) defined by
$$Gx=T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu_{1}F_{1})T^{{ \varPhi }_{2}}_{\nu_{2}}(I- \nu _{2}F_{2})x, \quad\forall x\in C, $$
where \(y^{*}=T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})x^{*}\).

In particular, if the mapping \(F_{j}:C\to H\) is \(\zeta_{j}\)-inverse strongly monotone for \(j=1,2\), then the mapping G is nonexpansive provided \(\nu_{j}\in(0,2\zeta_{j}]\) for \(j=1,2\). We denote by Ξ the fixed point set of the mapping G.

In Proposition 2.3, putting \({ \varPhi }_{1}\equiv{ \varPhi }_{2}\equiv0\), we get the following.

Corollary 2.1

(see [15], Lemma 2.1)

For given \(x^{*},y^{*}\in C\), \((x^{*},y^{*})\) is a solution of the SVI (1.13) if and only if \(x^{*}\) is a fixed point of the mapping \(G:C\to C\) defined by \(Gx=P_{C}(I-\nu_{1}F_{1})P_{C}(I-\nu_{2}F_{2})x\) for all \(x\in C\), where \(y^{*}=P_{C}(I-\nu_{2}F_{2})x^{*}\).

In particular, if the mapping \(F_{j}:C\to H\) is \(\zeta_{j}\)-inverse strongly monotone for \(j=1,2\), then the mapping G is nonexpansive provided \(\nu_{j}\in(0,2\zeta_{j}]\) for \(j=1,2\). We denote by Ξ the fixed point set of the mapping G.

We need some facts and tools in a real Hilbert space H; these are listed as lemmas below.

Lemma 2.1

Let X be a real inner product space. Then we have the following inequality:
$$\|x+y\|^{2}\leq\|x\|^{2}+2\langle y,x+y\rangle,\quad \forall x,y\in X. $$

Lemma 2.2

Let H be a real Hilbert space. Then the following hold:
  1. (a)

    \(\|x-y\|^{2}=\|x\|^{2}-\|y\|^{2}-2\langle x-y,y\rangle\) for all \(x,y\in H\);

     
  2. (b)

    \(\|\lambda x+\mu y\|^{2}=\lambda\|x\|^{2}+\mu\|y\|^{2}-\lambda\mu\|x-y\| ^{2}\) for all \(x,y\in H\) and \(\lambda,\mu\in[0,1]\) with \(\lambda+\mu=1\);

     
  3. (c)
    if \(\{x_{n}\}\) is a sequence in H such that \(x_{n}\rightharpoonup x\), it follows that
    $$\limsup_{n\to\infty}\|x_{n}-y\|^{2}=\limsup _{n\to\infty}\|x_{n}-x\|^{2}+\|x-y\| ^{2}, \quad\forall y\in H. $$
     
It is clear that, in a real Hilbert space H, \(T:C\to C\) is k-strictly pseudocontractive if and only if the following inequality holds:
$$\langle Tx-Ty,x-y\rangle\leq\|x-y\|^{2}-\frac{1-k}{2} \bigl\| (I-T)x-(I-T)y\bigr\| ^{2}, \quad\forall x,y\in C. $$
This immediately implies that if T is a k-strictly pseudocontractive mapping, then \(I-T\) is \(\frac{1-k}{2}\)-inverse strongly monotone; for further detail, we refer to [42] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings and that the class of pseudocontractions strictly includes the class of strict pseudocontractions.

Lemma 2.3

(see [42], Proposition 2.1)

Let C be a nonempty, closed, and convex subset of a real Hilbert space H and \(T: C\to C\) be a mapping.
  1. (i)
    If T is a k-strictly pseudocontractive mapping, then T satisfies the Lipschitzian condition
    $$\|Tx-Ty\|\leq\frac{1+k}{1-k}\|x-y\|,\quad \forall x,y\in C. $$
     
  2. (ii)

    If T is a k-strictly pseudocontractive mapping, then the mapping \(I-T\) is semiclosed at 0, that is, if \(\{x_{n}\}\) is a sequence in C such that \(x_{n}\rightharpoonup\tilde{x}\) and \((I-T)x_{n}\to0\), then \((I-T)\tilde{x}=0\).

     
  3. (iii)

    If T is k-(quasi-)strict pseudocontraction, then the fixed point set \(\operatorname{Fix}(T)\) of T is closed and convex so that the projection \(P_{\operatorname{Fix}(T)}\) is well defined.

     

Lemma 2.4

(see [15])

Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let \(T:C\to C\) be a k-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers such that \((\gamma+\delta)k \leq\gamma\). Then
$$\bigl\| \gamma(x-y)+\delta(Tx-Ty)\bigr\| \leq(\gamma+\delta)\|x-y\|, \quad\forall x,y\in C. $$

Lemma 2.5

(see [43], Demiclosedness principle)

Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let S be a nonexpansive self-mapping on C. Then \(I-S\) is demiclosed. That is, whenever \(\{x_{n}\}\) is a sequence in C weakly converging to some \(x\in C\) and the sequence \(\{(I-S)x_{n}\}\) strongly converges to some y, it follows that \((I-S)x=y\). Here I is the identity operator of H.

Lemma 2.6

Let \(F:C\to H\) be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 2.1(i)) implies
$$u\in\operatorname{VI}(C,F)\quad \Leftrightarrow \quad u=P_{C}(u- \lambda Fu),\quad \lambda>0. $$
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. We introduce some notations. Let λ be a number in \((0,1]\) and let \(\mu>0\). Associating with a nonexpansive mapping \(T:C\to C\), we define the mapping \(T^{\lambda}:C\to H\) by
$$T^{\lambda}x:=Tx-\lambda\mu F(Tx), \quad\forall x\in C, $$
where \(F:C\to H\) is an operator such that, for some positive constants \(\kappa,\eta>0\), F is κ-Lipschitzian and η-strongly monotone on C; that is, F satisfies the conditions:
$$\|Fx-Fy\|\leq\kappa\|x-y\| \quad\textit{and}\quad \langle Fx-Fy,x-y\rangle \geq \eta\|x-y\|^{2} $$
for all \(x,y\in C\).

Lemma 2.7

(see [37], Lemma 3.1)

\(T^{\lambda}\) is a contraction provided \(0<\mu<\frac{2\eta}{\kappa^{2}}\); that is,
$$\bigl\| T^{\lambda}x-T^{\lambda}y\bigr\| \leq(1-\lambda\tau)\|x-y\|,\quad \forall x,y\in C, $$
where \(\tau=1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\in(0,1]\).

Lemma 2.8

(see [44], Lemma 2.1)

Let \(\{a_{n}\}\) be a sequence of nonnegative real numbers satisfying
$$a_{n+1}\leq(1-\omega_{n})a_{n}+\omega_{n} \delta_{n}+r_{n},\quad \forall n\geq0, $$
where \(\{\omega_{n}\}\), \(\{\delta_{n}\}\), and \(\{r_{n}\}\) satisfy the following conditions:
  1. (i)

    \(\{\omega_{n}\}\subset[0,1]\) and \(\sum^{\infty}_{n=0}\omega_{n}=\infty\);

     
  2. (ii)

    either \(\limsup_{n\to\infty}\delta_{n}\leq0\) or \(\sum^{\infty}_{n=0}\omega_{n}|\delta_{n}|<\infty\);

     
  3. (iii)

    \(r_{n}\geq0\) for all \(n\geq0\), and \(\sum^{\infty}_{n=1}r_{n}<\infty\).

     
Then \(\lim_{n\to\infty}a_{n}=0\).

Lemma 2.9

(see [33])

Assume that A is a γ̄-strongly positive bounded linear operator on H with \(0< \rho\leq\|A\|^{-1}\). Then \(\|I-\rho A\|\leq1-\rho\bar{\gamma}\).

Let LIM be a Banach limit. According to time and circumstances, we use \(\operatorname{LIM}_{n}a_{n}\) instead of LIMa for every \(a=\{a_{n}\}\in l^{\infty}\). The following properties are well known:
  1. (i)

    for all \(n\geq1\), \(a_{n}\leq c_{n}\) implies \(\operatorname{LIM}_{n}a_{n} \leq\operatorname{LIM}_{n}c_{n}\);

     
  2. (ii)

    \(\operatorname{LIM}_{n}a_{n+N}=\operatorname{LIM}_{n}a_{n}\) for any fixed positive integer N;

     
  3. (iii)

    \(\liminf_{n\to\infty}a_{n}\leq\operatorname{LIM}_{n}a_{n} \leq\limsup_{n\to\infty }a_{n}\) for all \(\{a_{n}\}\in l^{\infty}\).

     
The following lemma was given in [39], Proposition 2.

Lemma 2.10

Let \(a\in\mathbf{R}\) be a real number and let a sequence \(\{a_{n}\}\in l^{\infty}\) satisfy the condition \(\operatorname{LIM}_{n}a_{n}\leq a\) for all Banach limit LIM. If \(\limsup_{n\to \infty}(a_{n+1}-a_{n})\leq0\), then \(\limsup_{n\to\infty} a_{n}\leq a\).

Recall that a set-valued mapping \(\widetilde{T}:D(\widetilde{T})\subset H\to2^{H}\) is called monotone if for all \(x,y\in D (\widetilde{T})\), \(f\in{\widetilde{T}}x\), and \(g\in{\widetilde{T}}y\) imply
$$\langle f-g,x-y\rangle\geq0. $$
A set-valued mapping is called maximal monotone if is monotone and \((I+\lambda\widetilde{T})D (\widetilde{T})=H\) for each \(\lambda>0\), where I is the identity mapping of H. We denote by \(G(\widetilde{T})\) the graph of . It is well known that a monotone mapping is maximal if and only if, for \((x,f)\in H\times H\), \(\langle f-g,x-y\rangle\geq0\) for every \((y,g)\in G(\widetilde{T})\) implies \(f\in \widetilde{T}x\). Next we provide an example to illustrate the concept of a maximal monotone mapping.
Let \({ \varGamma }:C\to H\) be a monotone and Lipschitz-continuous mapping and let \(N_{C}v\) be the normal cone to C at \(v\in C\), i.e.,
$$N_{C}v=\bigl\{ u\in H:\langle v-p,u\rangle\geq0, \forall p\in C\bigr\} . $$
Define
$$\widetilde{T}v= \left \{ \textstyle\begin{array}{@{}l@{\quad}ll} { \varGamma }v+N_{C}v,& \mbox{if }v\in C,\\ \emptyset, &\mbox{if }v\notin C. \end{array}\displaystyle \right . $$
Then it is well known [27] that is maximal monotone and \(0\in\widetilde{T}v\) if and only if \(v\in\operatorname{VI}(C,{ \varGamma })\).

3 Main results

Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Throughout this section, we always assume the following:
  • \(F:C\to H\) is a κ-Lipschitzian and η-strongly monotone operator with positive constants \(\kappa,\eta>0\), and \(F_{j}:C\to H\) is \(\zeta_{j}\)-inverse strongly monotone for \(j=1,2\);

  • \(T:C\to C\) is a k-strictly pseudocontractive mapping and \({\mathcal {A}}_{i}:C\to H\) is \(\eta_{i}\)-inverse strongly monotone for each \(i=1,\ldots,N\);

  • A is a γ̄-strongly positive bounded linear operator on H with \(\bar{\gamma}\in(1,2)\) and \(V:C\to H\) is an l-Lipschitzian mapping with \(l\geq0\);

  • \({ \varTheta }_{i},{ \varPhi }_{j}:C\times C\to\mathbf{R}\) are the bifunctions satisfying conditions (A1)-(A4) and \(\varphi_{i}:C\to \mathbf{R}\cup\{+\infty\}\) be a proper lower semicontinuous and convex function with restrictions (B1) or (B2) for each \(i =1,\ldots,N\) and \(j=1,2\);

  • \(0<\mu<\frac{2\eta}{\kappa^{2}}\) and \(0\leq\gamma l<\tau\) with \(\tau =1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\);

  • \(S:C\to C\) is a mapping defined by \(Sx=\lambda x+(1-\lambda)Tx\) for \(0\leq k\leq\lambda<1\);

  • \(G:C\to C\) is a mapping defined by \(Gx=T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu _{1}F_{1})T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})x\) with \(0< \nu_{j}<2\zeta_{j}\) for \(j=1,2\);

  • \({ \varDelta }^{N}_{t}:C\to C\) is a mapping defined by \({ \varDelta }^{N}_{t}x=T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t}}(I-r_{N,t}{\mathcal {A}}_{N})\cdots T^{({ \varTheta }_{1},\varphi _{1})}_{r_{1,t}}(I-r_{1,t}{\mathcal{A}}_{1})x\), \(t\in(0,1)\), for \(\{r_{i,t}\}\subset[a_{i},b_{i}]\subset(0,2\eta_{i})\), \(i=1,\ldots,N\);

  • \({ \varDelta }^{N}_{n}:C\to C\) is a mapping defined by \({ \varDelta }^{N}_{n}x=T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,n}}(I-r_{N,n}{\mathcal {A}}_{N})\cdots T^{({ \varTheta }_{1},\varphi _{1})}_{r_{1,n}}(I-r_{1,n}{\mathcal{A}}_{1})x\) with \(\{r_{i,n}\}\subset[a_{i},b_{i}]\subset(0,2\eta_{i})\) and \(\lim_{n\to \infty}r_{i,n}=r_{i}\), for each \(i=1,\ldots,N\);

  • \({ \varOmega }:=(\bigcap^{N}_{i=1}\operatorname{GMEP}({ \varTheta }_{i},\varphi _{i},{\mathcal{A}}_{i}))\cap\operatorname{Fix}(T)\cap{ \varXi }\neq\emptyset\) and \(P_{{ \varOmega }}\) is the metric projection of H onto Ω;

  • \(\{\alpha_{n}\}\subset[0,1]\), \(\{\beta_{n}\}\subset(0,1]\) and \(\{\theta_{t}\} _{t\in(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})} \subset(0,1)\).

Next, put
$${ \varDelta }^{i}_{t}=T^{({ \varTheta }_{i},\varphi _{i})}_{r_{i,t}}(I-r_{i,t}{ \mathcal{A}}_{i})T^{({ \varTheta }_{i-1},\varphi_{i-1})}_{r_{i-1,t}}(I-r_{i-1,t}{ \mathcal{A}}_{i-1})\cdots T^{({ \varTheta }_{1},\varphi_{1})}_{r_{1,t}}(I-r_{1,t}{ \mathcal{A}}_{1}),\quad \forall t\in(0,1), $$
and
$${ \varDelta }^{i}_{n}=T^{({ \varTheta }_{i},\varphi _{i})}_{r_{i,n}}(I-r_{i,n}{ \mathcal{A}}_{i})T^{({ \varTheta }_{i-1},\varphi _{i-1})}_{r_{i-1,n}} (I-r_{i-1,n}{ \mathcal{A}}_{i-1})\cdots T^{({ \varTheta }_{1},\varphi _{1})}_{r_{1,n}}(I-r_{1,n}{ \mathcal{A}}_{1}),\quad \forall n\geq0, $$
for all \(i\in\{1,\ldots,N\}\), and \({ \varDelta }^{0}_{t}={ \varDelta }^{0}_{n}=I\), where I is the identity mapping on H.
By Lemma 2.4, we know that S is nonexpansive. It is clear that \(\operatorname{Fix}(S)=\operatorname{Fix}(T)\). Since \(\{\lambda_{i,t}\}\subset [a_{i},b_{i}]\subset(0,2\eta_{i})\), utilizing (2.1) and Proposition 2.2(ii) we have for all \(x,y\in C\)
$$\begin{aligned}[b] \bigl\| { \varDelta }^{N}_{t}x-{ \varDelta }^{N}_{t}y \bigr\| &=\bigl\| T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t}}(I-r_{N,t}{\mathcal {A}}_{N}){ \varDelta }^{N-1}_{t}x -T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t}}(I-r_{N,t}{ \mathcal{A}}_{N}){ \varDelta }^{N-1}_{t}y\bigr\| \\ &\leq\bigl\| (I-r_{N,t}{\mathcal{A}}_{N}){ \varDelta }^{N-1}_{t}x-(I-r_{N,t}{\mathcal{A}}_{N}){ \varDelta }^{N-1}_{t}y\bigr\| \\ &\leq\bigl\| { \varDelta }^{N-1}_{t}x-{ \varDelta }^{N-1}_{t}y \bigr\| \\ &\leq\cdots \\ &\leq\bigl\| { \varDelta }^{i}_{t}x-{ \varDelta }^{i}_{t}y \bigr\| \\ &\leq\cdots \\ &\leq\bigl\| { \varDelta }^{0}_{t}x-{ \varDelta }^{0}_{t}y \bigr\| \\ &=\|x-y\|, \end{aligned} $$
which implies that \({ \varDelta }^{i}_{t}:C\to C\) is a nonexpansive mapping for all \(t\in(0,1)\). Also, since \(\{r_{i,n}\}\subset [a_{i},b_{i}]\subset(0,2\eta_{i})\), utilizing (2.1) and Proposition 2.2(ii) we have for all \(x,y\in C\)
$$\begin{aligned} \bigl\| { \varDelta }^{N}_{n}x-{ \varDelta }^{N}_{n}y \bigr\| &=\bigl\| T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,n}}(I-r_{N,n}{\mathcal {A}}_{N}){ \varDelta }^{N-1}_{n}x -T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,n}}(I-r_{N,n}{ \mathcal{A}}_{N}){ \varDelta }^{N-1}_{n}y\bigr\| \\ &\leq\bigl\| (I-r_{N,n}{\mathcal{A}}_{N}){ \varDelta }^{N-1}_{n}x-(I-r_{N,n}{\mathcal{A}}_{N}){ \varDelta }^{N-1}_{n}y\bigr\| \\ &\leq\bigl\| { \varDelta }^{N-1}_{n}x-{ \varDelta }^{N-1}_{n}y \bigr\| \\ &\leq\cdots \\ &\leq\bigl\| { \varDelta }^{i}_{n}x-{ \varDelta }^{i}_{n}y \bigr\| \\ &\leq\cdots \\ &\leq\bigl\| { \varDelta }^{0}_{n}x-{ \varDelta }^{0}_{n}y \bigr\| \\ &=\|x-y\|, \end{aligned}$$
which implies that \({ \varDelta }^{i}_{n}:C\to C\) is a nonexpansive mapping for all \(n\geq0\).
In this section, we introduce the first composite relaxed extragradient-like scheme that generates a net \(\{x_{t}\}_{t\in(0,\min\{1, \frac{2-\bar{\gamma}}{\tau-\gamma l}\})}\) in an implicit manner:
$$ x_{t}=P_{C}\bigl[(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}+\theta_{t}\bigl(t \gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t} \bigr)\bigr]. $$
(3.1)
We prove the strong convergence of \(\{x_{t}\}\) as \(t\to0\) to a point \(\tilde{x}\in{ \varOmega }\) which is a unique solution to the VIP
$$ \bigl\langle (A-I)\tilde{x},p-\tilde{x}\bigr\rangle \geq0,\quad \forall p\in{ \varOmega }. $$
(3.2)
For arbitrarily given \(x_{0}\in C\), we also propose the second composite relaxed extragradient-like scheme, which generates a sequence \(\{x_{n}\}\) in an explicit way:
$$ \left \{ \textstyle\begin{array}{@{}l} y_{n}=\alpha_{n}\gamma Vx_{n}+(I-\alpha_{n}\mu F)S{ \varDelta }^{N}_{n}Gx_{n},\\ x_{n+1}=P_{C}[(I-\beta_{n}A)S{ \varDelta }^{N}_{n}Gx_{n}+\beta_{n}y_{n}],\quad \forall n\geq0, \end{array}\displaystyle \right . $$
(3.3)
and establish the strong convergence of \(\{x_{n}\}\) as \(n\to\infty\) to the same point \(\tilde{x}\in{ \varOmega }\), which is also the unique solution to VIP (3.2).
Now, for \(t\in(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})\), and \(\theta_{t}\in(0,\|A\|^{-1}]\), consider a mapping \(Q_{t} :C\to C\) defined by
$$Q_{t}x=P_{C}\bigl[(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx+\theta_{t}\bigl(t\gamma Vx+(I-t \mu F)S{ \varDelta }^{N}_{t}Gx\bigr)\bigr],\quad \forall x\in C. $$
It is easy to see that \(Q_{t}\) is a contractive mapping with constant \(1-\theta_{t}(\bar{\gamma}-1+t(\tau-\gamma l))\). Indeed, by Proposition 2.3 and Lemmas 2.7 and 2.9, we have
$$\begin{aligned} \bigl\| Q_{t}x-Q_{t}y\bigr\| \leq&\bigl\| (I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx+\theta_{t}\bigl(t\gamma Vx+(I-t \mu F)S{ \varDelta }^{N}_{t}Gx\bigr) \\ &{} -(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gy- \theta_{t}\bigl(t\gamma Vx+(I-t\mu F)S{ \varDelta }^{N}_{t}Gy \bigr)\bigr\| \\ \leq&\bigl\| (I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx-(I- \theta_{t}A)S{ \varDelta }^{N}_{t}Gy\bigr\| \\ &{} +\theta_{t}\bigl\| \bigl(t\gamma Vx+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx \bigr)-\bigl(t\gamma Vy+(I-t\mu F)S{ \varDelta }^{N}_{t}Gy\bigr) \bigr\| \\ \leq&(1-\theta_{t}\bar{\gamma})\bigl\| S{ \varDelta }^{N}_{t}Gx-S{ \varDelta }^{N}_{t}Gy\bigr\| +\theta_{t}\bigl[t\gamma \|Vx-Vy\| \\ &{} +\bigl\| (I-t\mu F)S{ \varDelta }^{N}_{t}Gx-(I-t\mu F)S{ \varDelta }^{N}_{t}Gy\bigr\| \bigr] \\ \leq&(1-\theta_{t}\bar{\gamma})\|x-y\|+\theta_{t}\bigl[t \gamma l\|x-y\|+(1-t\tau )\|x-y\|\bigr] \\ =&\bigl[1-\theta_{t}\bigl(\bar{\gamma}-1+t(\tau-\gamma l)\bigr)\bigr] \|x-y\|. \end{aligned}$$
Since \(\bar{\gamma}\in(1,2)\), \(\tau-\gamma l>0\), and
$$0< t< \min\biggl\{ 1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\biggr\} \leq\frac{2-\bar{\gamma}}{\tau-\gamma l}, $$
it follows that
$$0< \bar{\gamma}-1+t(\tau-\gamma l)< 1, $$
which together with \(0<\theta_{t}\leq\|A\|^{-1}<1\) yields
$$0< 1-\theta_{t}\bigl(\bar{\gamma}-1+t(\tau-\gamma l)\bigr)< 1. $$
Hence \(Q_{t}:C\to C\) is a contractive mapping. By the Banach contraction principle, \(Q_{t}\) has a unique fixed point, denoted by \(x_{t}\), which uniquely solves the fixed point equation (3.1).

We summarize the basic properties of \(\{x_{t}\}\). The argument techniques in [1, 22, 32] extend to developing the new argument ones for these basic properties. We include the argument process for the sake of completeness.

Proposition 3.1

Let \(\{x_{t}\}\) be defined via (3.1). Then
  1. (i)

    \(\{x_{t}\}\) is bounded for \(t\in(0,\min\{1,\frac{2-\bar{\gamma}}{\tau -\gamma l}\})\);

     
  2. (ii)

    \(\lim_{t\to0}\|x_{t}-Sx_{t}\|=0\), \(\lim_{t\to0}\|x_{t}-Gx_{t}\|=0\) and \(\lim_{t\to0}\|x_{t}-{ \varDelta }^{N}_{t}x_{t}\|=0\) provided \(\lim_{t\to0}\theta_{t}=0\);

     
  3. (iii)

    \(x_{t}:(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})\to H\) is locally Lipschitzian provided \(\theta_{t}:(0,\min\{1,\frac {2-\bar{\gamma}}{\tau-\gamma l}\})\to(0,\|A\|^{-1}]\) is locally Lipschitzian, and \(\lambda_{i,t}:(0,\min\{1,\frac{2-\bar{\gamma}}{ \tau-\gamma l}\})\to[a_{i},b_{i}]\) is locally Lipschitzian for each \(i=1,\ldots,N\);

     
  4. (iv)

    \(x_{t}\) defines a continuous path from \((0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})\) into H provided \(\theta_{t}:(0, \min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\}) \to(0,\|A\|^{-1}]\) is continuous, and \(\lambda_{i,t}:(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})\to[a_{i},b_{i}]\) is continuous for each \(i=1,\ldots,N\).

     

Proof

(i) Let \(p\in{ \varOmega }\). Noting that \(\operatorname{Fix}(S)=\operatorname{Fix}(T)\), \(Sp=p\), \(Gp=p\), and \({ \varDelta }^{i}_{t}p=p\) for each \(i=1,\ldots,N\), by the nonexpansivity of S, G, and \({ \varDelta }^{i}_{t}\), and Lemmas 2.7 and 2.9 we get
$$\begin{aligned} &\|x_{t}-p\| \\ &\quad\leq\bigl\| (I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}+ \theta_{t}\bigl(t\gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}\bigr)-p\bigr\| \\ &\quad=\bigl\| (I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}-(I- \theta_{t}A)S{ \varDelta }^{N}_{t}Gp \\ &\qquad{} +\theta_{t}\bigl(t\gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}-p\bigr)+\theta _{t}(I-A)p\bigr\| \\ &\quad\leq\bigl\| (I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}-(I- \theta_{t}A)S{ \varDelta }^{N}_{t}Gp\bigr\| \\ &\qquad{} +\theta_{t}\bigl\| t\gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}-p\bigr\| +\theta _{t} \bigl\| (I-A)p\bigr\| \\ &\quad=\bigl\| (I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}-(I- \theta_{t}A)S{ \varDelta }^{N}_{t}Gp\bigr\| \\ &\qquad{}+\theta_{t}\bigl\| (I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}-(I-t \mu F)S{ \varDelta }^{N}_{t}Gp \\ &\qquad{} +t(\gamma Vx_{t}-\mu Fp)\bigr\| +\theta_{t}\bigl\| (I-A)p\bigr\| \\ &\quad\leq(1-\theta_{t}\bar{\gamma})\bigl\| S{ \varDelta }^{N}_{t}Gx_{t}-S{ \varDelta }^{N}_{t}Gp\bigr\| \\ &\qquad{} +\theta_{t}\bigl[\bigl\| (I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}-(I-t \mu F)S{ \varDelta }^{N}_{t}Gp\bigr\| \\ &\qquad{} +t\bigl(\gamma\|Vx_{t}-Vp\|+\|\gamma Vp-\mu Fp\|\bigr)\bigr]+ \theta_{t}\bigl\| (I-A)p\bigr\| \\ &\quad\leq(1-\theta_{t}\bar{\gamma})\|x_{t}-p\|+ \theta_{t}\bigl[(1-t\tau)\|x_{t}-p\| \\ &\qquad{} +t\bigl(\gamma l\|x_{t}-p\|+\bigl\| (\gamma V-\mu F)p\bigr\| \bigr) \bigr]+\theta_{t}\|I-A\|\|p\| \\ &\quad=\bigl[1-\theta_{t}\bigl(\bar{\gamma}-1+t(\tau-\gamma l)\bigr) \bigr]\|x_{t}-p\|+\theta_{t}\bigl[\|I-A\| \|p\|+t\bigl\| (\gamma V-\mu F)p\bigr\| \bigr]. \end{aligned}$$
So, it follows that
$$\begin{aligned} \|x_{t}-p\|&\leq\frac{\|I-A\|\|p\|+t\|(\gamma V-\mu F)p\| }{\bar{\gamma}-1+t(\tau-\gamma l)} \\ &\leq\frac{\|I-A\|\|p\|+t\|(\gamma V-\mu F)p\|}{\bar{\gamma}-1} \\ &\leq\frac{\|I-A\|\|p\|+\|(\gamma V-\mu F)p\|}{\bar{\gamma}-1}. \end{aligned}$$
Hence \(\{x_{t}\}\) is bounded and so are \(\{Vx_{t}\}\), \(\{{ \varDelta }^{N}_{t}x_{t}\}\), \(\{S{ \varDelta }^{N}_{t}Gx_{t}\}\), and \(\{FS{ \varDelta }^{N}_{t}Gx_{t}\}\).
(ii) By the definition of \(\{x_{t}\}\), we have
$$\begin{aligned} &\bigl\| x_{t}-S{ \varDelta }^{N}_{t}Gx_{t}\bigr\| \\ &\quad=\bigl\| P_{C}\bigl[(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}+\theta_{t}\bigl(t \gamma Vx_{t} +(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t} \bigr)\bigr]-P_{C}S{ \varDelta }^{N}_{t}Gx_{t} \bigr\| \\ &\quad\leq\bigl\| (I-\theta_{t}A)S{ \varLambda }^{N}_{t}Gx_{t}+ \theta_{t}\bigl(t\gamma Vx_{t}+(I-t\mu F)S{ \varLambda }^{N}_{t}Gx_{t}\bigr)-S{ \varLambda }^{N}_{t}Gx_{t}\bigr\| \\ &\quad=\bigl\| \theta_{t}\bigl[(I-A)S{ \varDelta }^{N}_{t}Gx_{t}+t \bigl(\gamma Vx_{t}-\mu FS{ \varDelta }^{N}_{t}Gx_{t} \bigr)\bigr]\bigr\| \\ &\quad=\theta_{t}\bigl\| (I-A)S{ \varDelta }^{N}_{t}Gx_{t}+t \bigl(\gamma Vx_{t}-\mu FS{ \varDelta }^{N}_{t}Gx_{t} \bigr)\bigr\| \\ &\quad\leq\theta_{t}\|I-A\|\bigl\| S{ \varDelta }^{N}_{t}Gx_{t} \bigr\| +t\bigl\| \gamma Vx_{t}-\mu FS{ \varDelta }^{N}_{t}Gx_{t} \bigr\| \to0 \quad\mbox{as } t\to0, \end{aligned}$$
by the boundedness of \(\{Vx_{t}\}\), \(\{S{ \varDelta }^{N}_{t}Gx_{t}\}\), and \(\{FS{ \varDelta }^{N}_{t}Gx_{t}\}\) in the assertion (i). That is,
$$ \lim_{t\to0}\bigl\| x_{t}-S{ \varDelta }^{N}_{t}Gx_{t} \bigr\| =0. $$
(3.4)
Since \(p=Gp=T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu_{1}F_{1})T^{{ \varPhi }_{2}}_{\nu _{2}}(I-\nu_{2}F_{2})p\) and \(F_{j}\) is \(\zeta_{j}\)-inverse strongly monotone with \(0<\nu_{j}<2\zeta_{j}\) for \(j=1,2\), by Proposition 2.2(ii) we deduce that
$$\begin{aligned} &\|Gx_{t}-p\|^{2} \\ &\quad=\bigl\| T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu_{1}F_{1})T^{{ \varPhi }_{2}}_{\nu_{2}}(I- \nu_{2}F_{2})x_{t} -T^{{ \varPhi }_{1}}_{\nu_{1}}(I- \nu_{1}F_{1})T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu _{2}F_{2})p\bigr\| ^{2} \\ &\quad\leq\bigl\| (I-\nu_{1}F_{1})T^{{ \varPhi }_{2}}_{\nu_{2}}(I- \nu_{2}F_{2})x_{t}-(I-\nu _{1}F_{1})T^{{ \varPhi }_{2}}_{\nu_{2}}(I- \nu_{2}F_{2})p\bigr\| ^{2} \\ &\quad=\bigl\| \bigl[T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})x_{t}-T^{{ \varPhi }_{2}}_{\nu _{2}}(I- \nu_{2}F_{2})p\bigr] \\ &\qquad{}-\nu_{1}\bigl[F_{1}T^{{ \varPhi }_{2}}_{\nu_{2}}(I- \nu_{2}F_{2})x_{t}-F_{1}T^{{ \varPhi }_{2}}_{\nu_{2}}(I- \nu_{2}F_{2})p\bigr]\bigr\| ^{2} \\ &\quad\leq\bigl\| T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})x_{t}-T^{{ \varPhi }_{2}}_{\nu _{2}}(I- \nu_{2}F_{2})p\bigr\| ^{2} \\ &\qquad{} +\nu_{1}(\nu_{1}-2\zeta_{1}) \bigl\| F_{1}T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu _{2}F_{2})x_{t}-F_{1}T^{{ \varPhi }_{2}}_{\nu_{2}}(I- \nu_{2}F_{2})p\bigr\| ^{2} \\ &\quad\leq\bigl\| T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})x_{t}-T^{{ \varPhi }_{2}}_{\nu _{2}}(I- \nu_{2}F_{2})p\bigr\| ^{2} \\ &\quad\leq\bigl\| (I-\nu_{2}F_{2})x_{t}-(I- \nu_{2}F_{2})p\bigr\| ^{2} \\ &\quad=\bigl\| (x_{t}-p)-\nu_{2}(F_{2}x_{t}-F_{2}p) \bigr\| ^{2} \\ &\quad\leq\|x_{t}-p\|^{2}+\nu_{2}( \nu_{2}-2\zeta_{2})\|F_{2}x_{t}-F_{2}p \|^{2} \\ &\quad\leq\|x_{t}-p\|^{2}. \end{aligned}$$
(3.5)
In the meantime, utilizing the \(\eta_{i}\)-inverse strong monotonicity of \({\mathcal{A}}_{i}\), we obtain
$$\begin{aligned} \bigl\| { \varDelta }^{i}_{t}Gx_{t}-p \bigr\| ^{2}&=\bigl\| T^{({ \varTheta }_{i},\varphi_{i})}_{r_{i,t}}(I-r_{i,t}{ \mathcal{A}}_{i}){ \varDelta }^{i-1}_{t}Gx_{t} -T^{({ \varTheta }_{i},\varphi_{i})}_{r_{i,t}}(I-r_{i,t}{\mathcal{A}}_{i})p \bigr\| ^{2} \\ &\leq\bigl\| (I-r_{i,t}{\mathcal{A}}_{i}){ \varDelta }^{i-1}_{t}Gx_{t}-(I-r_{i,t}{ \mathcal{A}}_{i})p\bigr\| ^{2} \\ &=\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-p-r_{i,t} \bigl({\mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr)\bigr\| ^{2} \\ &\leq\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-p \bigr\| ^{2}+r_{i,t}(r_{i,t}-2\eta_{i})\bigl\| { \mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| ^{2} \\ &\leq\|Gx_{t}-p\|^{2}+r_{i,t}(r_{i,t}-2 \eta_{i})\bigl\| {\mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| ^{2}, \end{aligned}$$
(3.6)
for each \(i\in\{1,2,\ldots,N\}\). Simple calculations show that
$$\begin{aligned} &x_{t}-p \\ &\quad=x_{t}-w_{t}+w_{t}-p \\ &\quad=x_{t}-w_{t}+(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}+\theta_{t}\bigl(t \gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t} \bigr)-p \\ &\quad=x_{t}-w_{t}+(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}-(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gp+\theta_{t}\bigl[t\gamma Vx_{t} \\ &\qquad{}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}-p \bigr]+\theta_{t}(I-A)p \\ &\quad=x_{t}-w_{t}+(I-\theta_{t}A) \bigl(S{ \varDelta }^{N}_{t}Gx_{t}-S{ \varDelta }^{N}_{t}Gp \bigr)+\theta _{t}\bigl[t(\gamma Vx_{t}-\mu Fp) \\ &\qquad{}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}-(I-t \mu F)p\bigr]+\theta_{t}(I-A)p, \end{aligned}$$
(3.7)
where \(w_{t}=(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}+\theta_{t}(t\gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t})\).
For simplicity, we write \(\tilde{x}_{t}=T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu _{2}F_{2})x_{t}\), \(\tilde{p}=T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})p\) and \(y_{t}=T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu_{1}F_{1})\tilde{x}_{t}\). Then we have \(y_{t}=T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu_{1}F_{1})T^{{ \varPhi }_{2}}_{\nu _{2}}(I-\nu_{2}F_{2})x_{t}\) and \(p=Gp=T^{{ \varPhi }_{1}}_{\nu_{1}} (I-\nu_{1}F_{1})\tilde{p}\). Then, by Propositions 2.1(i) and 2.3, and Lemmas 2.7 and 2.9, from (3.5)-(3.7) we obtain
$$\begin{aligned} &\|x_{t}-p\|^{2} \\ &\quad=\langle x_{t}-w_{t},x_{t}-p\rangle+\bigl\langle (I-\theta_{t}A) \bigl(S{ \varDelta }^{N}_{t}Gx_{t}-S{ \varDelta }^{N}_{t}Gp\bigr),x_{t}-p\bigr\rangle \\ &\qquad{} +\theta_{t}\bigl[t\langle\gamma Vx_{t}-\mu Fp,x_{t}-p\rangle +\bigl\langle (I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}-(I-t \mu F)p,x_{t}-p\bigr\rangle \bigr] \\ &\qquad{} +\theta_{t}\bigl\langle (I-A)p,x_{t}-p\bigr\rangle \\ &\quad\leq\bigl\langle (I-\theta_{t}A) \bigl(S{ \varDelta }^{N}_{t}Gx_{t}-S{ \varDelta }^{N}_{t}Gp\bigr),x_{t}-p\bigr\rangle + \theta_{t}\bigl[t\langle\gamma Vx_{t}-\mu Fp,x_{t}-p\rangle \\ &\qquad{} +\bigl\langle (I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}-(I-t \mu F)p,x_{t}-p\bigr\rangle \bigr]+\theta_{t}\bigl\langle (I-A)p,x_{t}-p\bigr\rangle \\ &\quad=\bigl\langle (I-\theta_{t}A) \bigl(S{ \varDelta }^{N}_{t}Gx_{t}-S{ \varDelta }^{N}_{t}Gp\bigr),x_{t}-p\bigr\rangle \\ &\qquad{} +\theta_{t}\bigl[\bigl\langle (I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}-(I-t\mu F)p,x_{t}-p \bigr\rangle \\ &\qquad{} +t\bigl(\gamma\langle Vx_{t}-Vp,x_{t}-p\rangle+ \langle\gamma Vp-\mu Fp,x_{t}-p\rangle\bigr)\bigr] \\ &\qquad{} +\theta_{t}\bigl\langle (I-A)p,x_{t}-p\bigr\rangle \\ &\quad\leq\bigl\| (I-\theta_{t}A) \bigl(S{ \varDelta }^{N}_{t}Gx_{t}-S{ \varDelta }^{N}_{t}Gp\bigr)\bigr\| \| x_{t}-p\| \\ &\qquad{} +\theta_{t}\bigl[\bigl\| (I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}-(I-t \mu F)p\bigr\| \|x_{t}-p\| \\ &\qquad{} +t\bigl(\gamma\|Vx_{t}-Vp\|\|x_{t}-p\|+\|\gamma Vp- \mu Fp\|\|x_{t}-p\| \bigr)\bigr]+\theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq(1-\theta_{t}\bar{\gamma})\bigl\| S{ \varDelta }^{N}_{t}Gx_{t}-S{ \varDelta }^{N}_{t}Gp\bigr\| \|x_{t}-p\|+ \theta_{t}\bigl[(1-t\tau)\bigl\| { \varDelta }^{N}_{t}Gx_{t}-p \bigr\| \|x_{t}-p\| \\ &\qquad{} +t\bigl(\gamma l\|x_{t}-p\|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\|\bigr)\bigr]+\theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq(1-\theta_{t}\bar{\gamma})\bigl\| { \varDelta }^{N}_{t}Gx_{t}-p \bigr\| \|x_{t}-p\|+\theta _{t}\bigl[(1-t\tau)\bigl\| { \varDelta }^{N}_{t}Gx_{t}-p\bigr\| \|x_{t}-p\| \\ &\qquad{} +t\bigl(\gamma l\|x_{t}-p\|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\|\bigr)\bigr]+\theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad=\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr)\bigl\| { \varDelta }^{N}_{t}Gx_{t}-p\bigr\| \|x_{t}-p\| \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl(\bigl\| { \varDelta }^{N}_{t}Gx_{t}-p \bigr\| ^{2}+\|x_{t}-p\|^{2}\bigr) \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl(\bigl\| { \varDelta }^{i}_{t}Gx_{t}-p \bigr\| ^{2}+\|x_{t}-p\|^{2}\bigr) \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl(\|Gx_{t}-p\|^{2} \\ &\qquad{}+r_{i,t}(r_{i,t} -2\eta_{i})\bigl\| {\mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| ^{2}+\|x_{t}-p\|^{2} \bigr) \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl(\|x_{t}-p\|^{2}+\nu_{2}(\nu _{2}-2\zeta_{2})\|F_{2}x_{t}-F_{2}p \|^{2} \\ &\qquad{} +\nu_{1}(\nu_{1}-2\zeta_{1}) \|F_{1}\tilde{x}_{t}-F_{1}\tilde{p}\|^{2} \\ &\qquad{} +r_{i,t}(r_{i,t}-2\eta_{i})\bigl\| { \mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| ^{2}+\|x_{t}-p\|^{2} \bigr) \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad=\bigl[1-\theta_{t}(\bar{\gamma}-1+t(\tau-\gamma l)\bigr] \|x_{t}-p\|^{2}-\frac{1-\theta _{t}(\bar{\gamma}-1+t\tau)}{2}\bigl[\nu_{2}(2 \zeta_{2}-\nu_{2})\|F_{2}x_{t}-F_{2}p \|^{2} \\ &\qquad{} +\nu_{1}(2\zeta_{1}-\nu_{1}) \|F_{1}\tilde{x}_{t}-F_{1}\tilde{p} \|^{2}+r_{i,t}(2\eta _{i}-r_{i,t})\bigl\| { \mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| ^{2}\bigr] \\ &\qquad{} +\theta_{t}\bigl(t\|\gamma Vp-\mu Fp\|\|x_{t}-p \|+\bigl\| (I-A)p\bigr\| \|x_{t}-p\|\bigr) \\ &\quad\leq\|x_{t}-p\|^{2}-\frac{1-\theta_{t}(\bar{\gamma}-1+t\tau)}{2}\bigl[ \nu_{2}(2\zeta _{2}-\nu_{2})\|F_{2}x_{t}-F_{2}p \|^{2} \\ &\qquad{} +\nu_{1}(2\zeta_{1}-\nu_{1}) \|F_{1}\tilde{x}_{t}-F_{1}\tilde{p} \|^{2}+r_{i,t}(2\eta _{i}-r_{i,t})\bigl\| { \mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| ^{2}\bigr] \\ &\qquad{} +\theta_{t}\bigl(t\|\gamma Vp-\mu Fp\|\|x_{t}-p \|+\bigl\| (I-A)p\bigr\| \|x_{t}-p\|\bigr), \end{aligned}$$
(3.8)
which together with \(\nu_{j}\in(0,2\zeta_{j})\), \(j=1,2\), and \(\{r_{i,t}\} \subset[a_{i},b_{i}]\subset(0,2\eta_{i})\), \(i=1,\ldots,N\), implies that
$$\begin{aligned} &\frac{1-\theta_{t}(\bar{\gamma}-1+t\tau)}{2}\bigl[\nu_{2}(2\zeta _{2}- \nu_{2})\|F_{2}x_{t}-F_{2}p \|^{2} \\ &\qquad{} +\nu_{1}(2\zeta_{1}-\nu_{1}) \|F_{1}\tilde{x}_{t}-F_{1}\tilde{p} \|^{2}+a_{i}(2\eta _{i}-b_{i})\bigl\| { \mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| ^{2}\bigr] \\ &\quad\leq\frac{1-\theta_{t}(\bar{\gamma}-1+t\tau)}{2}\bigl[\nu_{2}(2\zeta_{2}- \nu_{2})\| F_{2}x_{t}-F_{2}p \|^{2} \\ &\qquad{} +\nu_{1}(2\zeta_{1}-\nu_{1}) \|F_{1}\tilde{x}_{t}-F_{1}\tilde{p} \|^{2}+r_{i,t}(2\eta _{i}-r_{i,t})\bigl\| { \mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{{ \mathcal{A}}}_{i}p\bigr\| ^{2}\bigr] \\ &\quad\leq\theta_{t}\bigl(t\|\gamma Vp-\mu Fp\|\|x_{t}-p \|+\bigl\| (I-A)p\bigr\| \|x_{t}-p\|\bigr). \end{aligned}$$
Since \(\lim_{t\to0}\theta_{t}=0\) and \(\{x_{t}\}\) is bounded, we have
$$ \begin{aligned} &\lim_{t\to0}\|F_{2}x_{t}-F_{2}p \|=0,\qquad \lim_{t\to0}\|F_{1}\tilde{x}_{t}-F_{1}\tilde{p}\|=0 \quad\mbox{and}\\ & \lim _{t\to0}\bigl\| {\mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| =0 \end{aligned} $$
(3.9)
for each \(i=1,\ldots,N\).
On the other hand, in terms of the firm nonexpansivity of \(T^{{ \varPhi }_{j}}_{\nu_{j}}\) and the \(\zeta_{j}\)-inverse strong monotonicity of \(F_{j}\) for \(j=1,2\), we obtain from \(\nu_{j}\in(0,2\zeta_{j})\), \(j=1,2\), and (3.5)
$$\begin{aligned} \|\tilde{x}_{t}-\tilde{p}\|^{2} =&\bigl\| T^{{ \varPhi }_{2}}_{\nu_{2}}(I- \nu_{2}F_{2})x_{t}-T^{{ \varPhi }_{2}}_{\nu _{2}}(I- \nu_{2}F_{2})p\bigr\| ^{2} \\ \leq&\bigl\langle (I-\nu_{2}F_{2})x_{t}-(I- \nu_{2}F_{2})p,\tilde{x}_{t}-\tilde{p}\bigr\rangle \\ =&\frac{1}{2}\bigl[\bigl\| (I-\nu_{2}F_{2})x_{t}-(I- \nu_{2}F_{2})p\bigr\| ^{2}+\|\tilde{x}_{t}- \tilde{p}\|^{2} \\ &{} -\bigl\| (I-\nu_{2}F_{2})x_{t}-(I- \nu_{2}F_{2})p-(\tilde{x}_{t}-\tilde{p}) \bigr\| ^{2}\bigr] \\ \leq&\frac{1}{2}\bigl[\|x_{t}-p\|^{2}+\|\tilde{x}_{t}-\tilde{p}\|^{2}-\bigl\| (x_{t}-\tilde{x}_{t})-\nu_{2}(F_{2}x_{t}-F_{2}p)-(p- \tilde{p})\bigr\| ^{2}\bigr] \\ =&\frac{1}{2}\bigl[\|x_{t}-p\|^{2}+\|\tilde{x}_{t}-\tilde{p}\|^{2}-\bigl\| (x_{t}-\tilde{x}_{t})-(p-\tilde{p})\bigr\| ^{2} \\ &{} +2\nu_{2}\bigl\langle (x_{t}-\tilde{x}_{t})-(p-\tilde{p}),F_{2}x_{t}-F_{2}p \bigr\rangle -\nu ^{2}_{2}\|F_{2}x_{t}-F_{2}p \|^{2}\bigr] \end{aligned}$$
and
$$\begin{aligned} \|y_{t}-p\|^{2} =&\bigl\| T^{{ \varPhi }_{1}}_{\nu_{1}}(I- \nu_{1}F_{1})\tilde{x}_{t}-T^{{ \varPhi }_{1}}_{\nu _{1}}(I- \nu_{1}F_{1}) (I-\nu_{1}F_{1})\tilde{p} \bigr\| ^{2} \\ \leq&\bigl\langle (I-\nu_{1}F_{1})\tilde{x}_{t}-(I-\nu_{1}F_{1})\tilde{p},y_{t}-p \bigr\rangle \\ =&\frac{1}{2}\bigl[\bigl\| (I-\nu_{1}F_{1})\tilde{x}_{t}-(I-\nu_{1}F_{1})\tilde{p}\bigr\| ^{2}+\| y_{t}-p\|^{2} \\ &{} -\bigl\| (I-\nu_{1}F_{1})\tilde{x}_{t}-(I- \nu_{1}F_{1})\tilde{p}-(y_{t}-p)\bigr\| ^{2} \bigr] \\ \leq&\frac{1}{2}\bigl[\|\tilde{x}_{t}-\tilde{p}\|^{2}+ \|y_{t}-p\|^{2}-\bigl\| (\tilde{x}_{t}-y_{t})+(p- \tilde{p})\bigr\| ^{2} \\ &{} +2\nu_{1}\bigl\langle F_{1}\tilde{x}_{t}-F_{1} \tilde{p},(\tilde{x}_{t}-y_{t})+(p-\tilde{p})\bigr\rangle - \nu^{2}_{1}\|F_{1}\tilde{x}_{t}-F_{1} \tilde{p}\|^{2}\bigr] \\ \leq&\frac{1}{2}\bigl[\|x_{t}-p\|^{2}+ \|y_{t}-p\|^{2}-\bigl\| (\tilde{x}_{t}-y_{t})+(p- \tilde{p})\bigr\| ^{2} \\ &{} +2\nu_{1}\bigl\langle F_{1}\tilde{x}_{t}-F_{1} \tilde{p},(\tilde{x}_{t}-y_{t})+(p-\tilde{p})\bigr\rangle \bigr]. \end{aligned}$$
Thus, we have
$$\begin{aligned} &\|\tilde{x}_{t}-\tilde{p}\|^{2} \\ &\quad\leq\|x_{t}-p\|^{2}-\bigl\| (x_{t}-\tilde{x}_{t})-(p-\tilde{p})\bigr\| ^{2}+2\nu_{2}\bigl\langle (x_{t}-\tilde{x}_{t})-(p-\tilde{p}),F_{2}x_{t}-F_{2}p \bigr\rangle \\ &\qquad{} -\nu^{2}_{2}\|F_{2}x_{t}-F_{2}p \|^{2} \end{aligned}$$
(3.10)
and
$$ \|y_{t}-p\|^{2}\leq\|x_{t}-p\|^{2}-\bigl\| ( \tilde{x}_{t}-y_{t})+(p-\tilde{p})\bigr\| ^{2}+2 \nu_{1}\| F_{1}\tilde{x}_{t}-F_{1}\tilde{p} \|\bigl\| (\tilde{x}_{t}-y_{t})+(p-\tilde{p})\bigr\| . $$
(3.11)
Consequently, from (3.5), (3.8), and (3.10) it follows that
$$\begin{aligned} &\|x_{t}-p\|^{2} \\ &\quad\leq\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl(\|Gx_{t}-p\|^{2}\\ &\qquad{}+r_{i,t}(r_{i,t} -2\eta_{i})\bigl\| {\mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| ^{2}+\|x_{t}-p\|^{2} \bigr) \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl(\|Gx_{t}-p\|^{2}+\|x_{t}-p \| ^{2}\bigr) \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl[\|\tilde{x}_{t}-\tilde{p}\|^{2}+ \nu_{1}(\nu_{1}-2\zeta_{1}) \|F_{1}\tilde{x}_{t}-F_{1}\tilde{p}\|^{2}+\|x_{t}-p \|^{2}\bigr] \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl[\|\tilde{x}_{t}-\tilde{p}\|^{2}+ \|x_{t}-p\|^{2}\bigr] \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl[\|x_{t}-p\|^{2}-\bigl\| (x_{t}- \tilde{x}_{t})-(p-\tilde{p})\bigr\| ^{2} \\ &\qquad{} +2\nu_{2}\bigl\langle (x_{t}-\tilde{x}_{t})-(p-\tilde{p}),F_{2}x_{t}-F_{2}p \bigr\rangle -\nu ^{2}_{2}\|F_{2}x_{t}-F_{2}p \|^{2}+\|x_{t}-p\|^{2}\bigr] \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq\bigl[1-\theta_{t}\bigl(\bar{\gamma}-1+t(\tau-\gamma l)\bigr) \bigr]\|x_{t}-p\|^{2} \\ &\qquad{} -\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl\| (x_{t}-\tilde{x}_{t})-(p-\tilde{p}) \bigr\| ^{2} \\ &\qquad{} +\nu_{2}\bigl\| (x_{t}-\tilde{x}_{t})-(p- \tilde{p})\bigr\| \|F_{2}x_{t}-F_{2}p\| \\ &\qquad{} +\theta_{t}\bigl(t\|\gamma Vp-\mu Fp\|\|x_{t}-p \|+\bigl\| (I-A)p\bigr\| \|x_{t}-p\|\bigr) \\ &\quad\leq\|x_{t}-p\|^{2}-\bigl(1-\theta_{t}( \bar{\gamma}-1+t\tau)\bigr)\frac{1}{2}\bigl\| (x_{t}-\tilde{x}_{t})-(p-\tilde{p})\bigr\| ^{2} \\ &\qquad{} +\nu_{2}\bigl\| (x_{t}-\tilde{x}_{t})-(p- \tilde{p})\bigr\| \|F_{2}x_{t}-F_{2}p\| \\ &\qquad{} +\theta_{t}\bigl(t\|\gamma Vp-\mu Fp\|\|x_{t}-p \|+\bigl\| (I-A)p\bigr\| \|x_{t}-p\|\bigr), \end{aligned}$$
which hence leads to
$$\begin{aligned} &\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr)\frac{1}{2}\bigl\| (x_{t}-\tilde{x}_{t})-(p-\tilde{p})\bigr\| ^{2} \\ &\quad\leq\nu_{2}\bigl\| (x_{t}-\tilde{x}_{t})-(p- \tilde{p})\bigr\| \|F_{2}x_{t}-F_{2}p\| \\ &\qquad{} +\theta_{t}\bigl(t\|\gamma Vp-\mu Fp\|\|x_{t}-p \|+\bigl\| (I-A)p\bigr\| \|x_{t}-p\|\bigr). \end{aligned}$$
Since \(\lim_{t\to0}\theta_{t}=0\) and \(\lim_{t\to0}\|F_{2}x_{t}-F_{2}p\|=0\) (due to (3.9)), we deduce from the boundedness of \(\{x_{t}\}\) and \(\{\tilde{x}_{t}\}\) that
$$ \lim_{t\to0}\bigl\| (x_{t}-\tilde{x}_{t})-(p- \tilde{p})\bigr\| =0. $$
(3.12)
Furthermore, from (3.5), (3.8), and (3.11) it follows that
$$\begin{aligned} &\|x_{t}-p\|^{2} \\ &\quad\leq\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl(\|Gx_{t}-p\|^{2}\\ &\qquad{}+r_{i,t}(r_{i,t} -2\eta_{i})\bigl\| {\mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| ^{2}+\|x_{t}-p\|^{2} \bigr) \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl(\|Gx_{t}-p\|^{2}+\|x_{t}-p \| ^{2}\bigr) \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad=\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr)\frac{1}{2} \bigl(\|y_{t}-p\|^{2}+\|x_{t}-p\| ^{2} \bigr) \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl[\|x_{t}-p\|^{2}-\bigl\| (\tilde{x}_{t}-y_{t})+(p-\tilde{p})\bigr\| ^{2} \\ &\qquad{} +2\nu_{1}\|F_{1}\tilde{x}_{t}-F_{1} \tilde{p}\|\bigl\| (\tilde{x}_{t}-y_{t})+(p-\tilde{p})\bigr\| + \|x_{t}-p\|^{2}\bigr] \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq\bigl[1-\theta_{t}\bigl(\bar{\gamma}-1+t(\tau-\gamma l)\bigr) \bigr]\|x_{t}-p\|^{2} \\ &\qquad{} -\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl\| (\tilde{x}_{t}-y_{t})+(p-\tilde{p}) \bigr\| ^{2} +\nu_{1}\|F_{1}\tilde{x}_{t}-F_{1} \tilde{p}\|\bigl\| (\tilde{x}_{t}-y_{t})+(p-\tilde{p})\bigr\| \\ &\qquad{} +\theta_{t}\bigl(t\|\gamma Vp-\mu Fp\|\|x_{t}-p \|+\bigl\| (I-A)p\bigr\| \|x_{t}-p\|\bigr) \\ &\quad\leq\|x_{t}-p\|^{2}-\bigl(1-\theta_{t}( \bar{\gamma}-1+t\tau)\bigr)\frac{1}{2}\bigl\| (\tilde{x}_{t}-y_{t})+(p- \tilde{p})\bigr\| ^{2} \\ &\qquad{} +\nu_{1}\|F_{1}\tilde{x}_{t}-F_{1} \tilde{p}\|\bigl\| (\tilde{x}_{t}-y_{t})+(p-\tilde{p})\bigr\| \\ &\qquad{} +\theta_{t}\bigl(t\|\gamma Vp-\mu Fp\|\|x_{t}-p \|+\bigl\| (I-A)p\bigr\| \|x_{t}-p\|\bigr), \end{aligned}$$
which hence yields
$$\begin{aligned} &\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr)\frac{1}{2}\bigl\| ( \tilde{x}_{t}-y_{t})+(p-\tilde{p})\bigr\| ^{2} \\ &\quad\leq\nu_{1}\|F_{1}\tilde{x}_{t}-F_{1} \tilde{p}\|\bigl\| (\tilde{x}_{t}-y_{t})+(p-\tilde{p})\bigr\| \\ &\qquad{} +\theta_{t}\bigl(t\|\gamma Vp-\mu Fp\|\|x_{t}-p \|+\bigl\| (I-A)p\bigr\| \|x_{t}-p\|\bigr). \end{aligned}$$
Since \(\lim_{t\to0}\theta_{t}=0\) and \(\lim_{t\to0}\|F_{1}\tilde{x}_{t}-F_{1}\tilde{p}\|=0\) (due to (3.9)), we deduce from the boundedness of \(\{x_{t}\}\), \(\{y_{t}\}\), and \(\{\tilde{x}_{t}\}\) that
$$ \lim_{t\to0}\bigl\| (\tilde{x}_{t}-y_{t})+(p- \tilde{p})\bigr\| =0. $$
(3.13)
Note that
$$\|x_{t}-y_{t}\|\leq\bigl\| (x_{t}-\tilde{x}_{t})-(p-\tilde{p})\bigr\| +\bigl\| (\tilde{x}_{t}-y_{t})+(p- \tilde{p})\bigr\| . $$
Hence from (3.12) and (3.13) we get
$$ \lim_{t\to0}\|x_{t}-Gx_{t}\|=\lim _{t\to0}\|x_{t}-y_{t}\|=0. $$
(3.14)
Utilizing Proposition 2.2(ii) and Lemma 2.2(a), we obtain for each \(i\in\{1,\ldots,N\}\)
$$\begin{aligned} &\bigl\| { \varDelta }^{i}_{t}Gx_{t}-p\bigr\| ^{2} \\ &\quad=\bigl\| T^{({ \varTheta }_{i},\varphi_{i})}_{r_{i,t}}(I-r_{i,t}{\mathcal {A}}_{i}){ \varDelta }^{i-1}_{t}Gx_{t} -T^{({ \varTheta }_{i},\varphi_{i})}_{r_{i,t}}(I-r_{i,t}{\mathcal{A}}_{i})p \bigr\| ^{2} \\ &\quad\leq\bigl\langle (I-r_{i,t}{\mathcal{A}}_{i}){ \varDelta }^{i-1}_{t}Gx_{t}-(I-r_{i,t}{ \mathcal{A}}_{i})p,{ \varDelta }^{i}_{t}Gx_{t}-p \bigr\rangle \\ &\quad=\frac{1}{2}\bigl(\bigl\| (I-r_{i,t}{\mathcal{A}}_{i}){ \varDelta }^{i-1}_{t}Gx_{t}-(I-r_{i,t}{ \mathcal{A}}_{i})p\bigr\| ^{2}+\bigl\| { \varDelta }^{i}_{t}Gx_{t}-p \bigr\| ^{2} \\ &\qquad{} -\bigl\| (I-r_{i,t}{\mathcal{A}}_{i}){ \varDelta }^{i-1}_{t}Gx_{t}-(I-r_{i,t}{ \mathcal{A}}_{i})p-\bigl({ \varDelta }^{i}_{t}Gx_{t}-p \bigr)\bigr\| ^{2}\bigr) \\ &\quad\leq\frac{1}{2}\bigl(\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-p \bigr\| ^{2}+\bigl\| { \varDelta }^{i}_{t}Gx_{t}-p \bigr\| ^{2} -\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t} \\ &\qquad{} -r_{i,t}\bigl({\mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{\mathcal{A}}_{i}p \bigr)\bigr\| ^{2}\bigr) \\ &\quad\leq\frac{1}{2}\bigl(\|x_{t}-p\|^{2}+\bigl\| { \varDelta }^{i}_{t}Gx_{t}-p\bigr\| ^{2} -\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t}-r_{i,t}\bigl({ \mathcal {A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr)\bigr\| ^{2}\bigr), \end{aligned}$$
which immediately leads to
$$\begin{aligned} &\bigl\| { \varDelta }^{i}_{t}Gx_{t}-p \bigr\| ^{2} \\ &\quad\leq\|x_{t}-p\|^{2}-\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t}-r_{i,t}\bigl({ \mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal {A}}_{i}p\bigr)\bigr\| ^{2} \\ &\quad=\|x_{t}-p\|^{2}-\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t}\bigr\| ^{2}-r^{2}_{i,t} \bigl\| {\mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| ^{2} \\ &\qquad{} +2r_{i,t}\bigl\langle { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t},{\mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{\mathcal{A}}_{i}p \bigr\rangle \\ &\quad\leq\|x_{t}-p\|^{2}-\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t}\bigr\| ^{2} \\ &\qquad{} +2r_{i,t}\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t}\bigr\| \bigl\| {\mathcal {A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| . \end{aligned}$$
(3.15)
Combining (3.8) and (3.15) we conclude that
$$\begin{aligned} &\|x_{t}-p\|^{2} \\ &\quad\leq\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl(\bigl\| { \varDelta }^{i}_{t}Gx_{t}-p \bigr\| ^{2}+\|x_{t}-p\|^{2}\bigr) \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2}\bigl[\|x_{t}-p\|^{2}-\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t} \bigr\| ^{2} \\ &\qquad{} +2r_{i,t}\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t}\bigr\| \bigl\| {\mathcal {A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| +\|x_{t}-p\|^{2}\bigr] \\ &\qquad{} +\theta_{t}t\bigl(\gamma l\|x_{t}-p \|^{2}+\|\gamma Vp-\mu Fp\|\|x_{t}-p\| \bigr)+ \theta_{t}\bigl\| (I-A)p\bigr\| \|x_{t}-p\| \\ &\quad\leq\bigl[1-\theta_{t}\bigl(\bar{\gamma}-1+t(\tau-\gamma l)\bigr) \bigr]\|x_{t}-p\|^{2} \\ &\qquad{} -\bigl(1-\theta_{t}(\bar{\gamma}-1+t\tau)\bigr) \frac{1}{2} \bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t}\bigr\| ^{2} \\ &\qquad{} +r_{i,t}\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t}\bigr\| \bigl\| {\mathcal {A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| \\ &\qquad{} +\theta_{t}\bigl(t\|\gamma Vp-\mu Fp\|\|x_{t}-p \|+\bigl\| (I-A)p\bigr\| \|x_{t}-p\|\bigr) \\ &\quad\leq\|x_{t}-p\|^{2}-\frac{1-\theta_{t}(\bar{\gamma}-1+t\tau)}{2}\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t} \bigr\| ^{2} \\ &\qquad{} +r_{i,t}\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t}\bigr\| \bigl\| {\mathcal {A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| \\ &\qquad{} +\theta_{t}\bigl(t\|\gamma Vp-\mu Fp\|\|x_{t}-p \|+\bigl\| (I-A)p\bigr\| \|x_{t}-p\|\bigr), \end{aligned}$$
which hence yields
$$\begin{aligned} &\frac{1-\theta_{t}(\bar{\gamma}-1+t\tau)}{2}\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t}\bigr\| ^{2} \\ &\quad\leq r_{i,t}\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t}\bigr\| \bigl\| {\mathcal {A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t}-{ \mathcal{A}}_{i}p\bigr\| \\ &\qquad{} +\theta_{t}\bigl(t\|\gamma Vp-\mu Fp\|\|x_{t}-p \|+\bigl\| (I-A)p\bigr\| \|x_{t}-p\|\bigr). \end{aligned}$$
Since \(\{r_{i,t}\}\subset[a_{i},b_{i}]\subset(0,2\eta_{i})\), \(\lim_{t\to0}\theta _{t}=0\) and \(\lim_{t\to0}\|{\mathcal{A}}_{i}{ \varDelta }^{i-1} _{t}Gx_{t}-{\mathcal{A}}_{i}p\|=0\) (due to (3.9)), we deduce from the boundedness of \(\{x_{t}\}\) and \(\{{ \varDelta }^{i}_{t}Gx_{t}\}\) that
$$ \lim_{t\to0}\bigl\| { \varDelta }^{i-1}_{t}Gx_{t}-{ \varDelta }^{i}_{t}Gx_{t}\bigr\| =0,\quad \forall i\in\{1, \ldots,N\}. $$
(3.16)
Note that
$$\begin{aligned} \bigl\| Gx_{t}-{ \varDelta }^{N}_{t}Gx_{t} \bigr\| ={}&\bigl\| { \varDelta }^{0}_{t}Gx_{t}-{ \varDelta }^{N}_{t}Gx_{t}\bigr\| \\ \leq{}&\bigl\| { \varDelta }^{0}_{t}Gx_{t}-{ \varDelta }^{1}_{t}Gx_{t}\bigr\| +\bigl\| { \varDelta }^{1}_{t}Gx_{t}-{ \varDelta }^{2}_{t}Gx_{t} \bigr\| +\cdots \\ &{} +\bigl\| { \varDelta }^{N-1}_{t}Gx_{t}-{ \varDelta }^{N}_{t}Gx_{t}\bigr\| . \end{aligned}$$
Hence, from (3.16) we get
$$ \lim_{t\to0}\bigl\| Gx_{t}-{ \varDelta }^{N}_{t}Gx_{t} \bigr\| =0. $$
(3.17)
Also, observe that
$$\begin{aligned} \bigl\| x_{t}-{ \varDelta }^{N}_{t}x_{t}\bigr\| &\leq \|x_{t}-Gx_{t}\|+\bigl\| Gx_{t}-{ \varDelta }^{N}_{t}Gx_{t} \bigr\| +\bigl\| { \varDelta }^{N}_{t}Gx_{t}-{ \varDelta }^{N}_{t}x_{t}\bigr\| \\ &\leq2\|x_{t}-Gx_{t}\|+\bigl\| Gx_{t}-{ \varDelta }^{N}_{t}Gx_{t}\bigr\| . \end{aligned}$$
So, it follows from (3.14) and (3.17) that
$$ \lim_{t\to0}\bigl\| x_{t}-{ \varDelta }^{N}_{t}x_{t} \bigr\| =0. $$
(3.18)
In addition, it is not hard to find that
$$\begin{aligned} \|x_{t}-Sx_{t}\|&\leq\bigl\| x_{t}-S{ \varDelta }^{N}_{t}Gx_{t}\bigr\| +\bigl\| S{ \varDelta }^{N}_{t}Gx_{t}-S{ \varDelta }^{N}_{t}x_{t}\bigr\| +\bigl\| S{ \varDelta }^{N}_{t}x_{t}-Sx_{t} \bigr\| \\ &\leq\bigl\| x_{t}-S{ \varDelta }^{N}_{t}Gx_{t} \bigr\| +\|Gx_{t}-x_{t}\|+\bigl\| { \varDelta }^{N}_{t}x_{t}-x_{t} \bigr\| . \end{aligned}$$
Consequently, from (3.4), (3.14), and (3.18) we deduce that
$$ \lim_{t\to0}\|x_{t}-Sx_{t} \|=0. $$
(3.19)
(iii) Let \(t,t_{0}\in(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})\). Utilizing Proposition 2.2(ii), (v), we deduce that
$$\begin{aligned} &\bigl\| { \varDelta }^{N}_{t}Gx_{t}-{ \varDelta }^{N}_{t_{0}}Gx_{t_{0}}\bigr\| \\ &\quad=\bigl\| T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t}}(I-r_{N,t}{\mathcal {A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t} -T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t_{0}}}(I-r_{N,t_{0}}{\mathcal {A}}_{N}){ \varDelta }^{N-1}_{t_{0}}Gx_{t_{0}}\bigr\| \\ &\quad\leq\bigl\| T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t}}(I-r_{N,t}{\mathcal {A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t} -T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t_{0}}}(I-r_{N,t_{0}}{\mathcal {A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t}\bigr\| \\ &\qquad{} +\bigl\| T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t_{0}}}(I-r_{N,t_{0}}{\mathcal {A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t} -T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t_{0}}}(I-r_{N,t_{0}}{\mathcal {A}}_{N}){ \varDelta }^{N-1}_{t_{0}}Gx_{t_{0}}\bigr\| \\ &\quad\leq\bigl\| T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t}}(I-r_{N,t}{\mathcal {A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t} -T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t_{0}}}(I-r_{N,t}{\mathcal {A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t}\bigr\| \\ &\qquad{} +\bigl\| T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t_{0}}}(I-r_{N,t}{\mathcal {A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t} -T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t_{0}}}(I-r_{N,t_{0}}{\mathcal {A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t}\bigr\| \\ &\qquad{} +\bigl\| (I-r_{N,t_{0}}{\mathcal{A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t}-(I-r_{N,t_{0}}{ \mathcal{A}}_{N}){ \varDelta }^{N-1}_{t_{0}}Gx_{t_{0}} \bigr\| \\ &\quad\leq\frac{|r_{N,t}-r_{N,t_{0}}|}{r_{N,t}}\bigl\| T^{({ \varTheta }_{N},\varphi _{N})}_{r_{N,t}}(I-r_{N,t}{ \mathcal{A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t} -(I-r_{N,t}{\mathcal{A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t} \bigr\| \\ &\qquad{} +|r_{N,t}-r_{N,t_{0}}|\bigl\| {\mathcal{A}}_{N}{ \varDelta }^{N-1}_{t}Gx_{t}\bigr\| +\bigl\| { \varDelta }^{N-1}_{t}Gx_{t}-{ \varDelta }^{N-1}_{t_{0}}Gx_{t_{0}} \bigr\| \\ &\quad=|r_{N,t}-r_{N,t_{0}}|\biggl[\bigl\| {\mathcal{A}}_{N}{ \varDelta }^{N-1}_{t}Gx_{t}\bigr\| + \frac{1}{r_{N,t}} \bigl\| T^{({ \varTheta }_{N},\varphi _{N})}_{r_{N,t}}(I-r_{N,t}{\mathcal{A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t} \\ &\qquad{} -(I-r_{N,t}{\mathcal{A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t}\bigr\| \biggr]+\bigl\| { \varDelta }^{N-1}_{t}Gx_{t}-{ \varDelta }^{N-1}_{t_{0}}Gx_{t_{0}} \bigr\| \\ &\quad\leq\cdots \\ &\quad\leq|r_{N,t}-r_{N,t_{0}}|\biggl[\bigl\| {\mathcal{A}}_{N}{ \varDelta }^{N-1}_{t}Gx_{t}\bigr\| +\frac{1}{r_{N,t}} \bigl\| T^{({ \varTheta }_{N},\varphi _{N})}_{r_{N,t}}(I-r_{N,t}{\mathcal{A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t} \\ &\qquad{} -(I-r_{N,t}{\mathcal{A}}_{N}){ \varDelta }^{N-1}_{t}Gx_{t}\bigr\| \biggr]+\cdots +|r_{1,t}-r_{1,t_{0}}|\biggl[\bigl\| {\mathcal{A}}_{1}{ \varDelta }^{0}_{t}Gx_{t}\bigr\| \\ &\qquad{} +\frac{1}{r_{1,t}}\bigl\| T^{({ \varTheta }_{1},\varphi _{1})}_{r_{1,t}}(I-r_{1,t}{ \mathcal{A}}_{1}){ \varDelta }^{0}_{t}Gx_{t} -(I-r_{1,t}{\mathcal{A}}_{1}){ \varDelta }^{0}_{t}Gx_{t} \bigr\| \biggr] \\ &\qquad{} +\bigl\| { \varDelta }^{0}_{t}Gx_{t}-{ \varDelta }^{0}_{t_{0}}Gx_{t_{0}}\bigr\| \\ &\quad\leq\widetilde{M}_{0}{ \sum^{N}_{i=1}}|r_{i,t}-r_{i,t_{0}}|+ \| x_{t}-x_{t_{0}}\|, \end{aligned}$$
(3.20)
where \(\widetilde{M}_{0}>0\) is a constant such that for each \(t\in(0,\min\{ 1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})\)
$$\sum^{N}_{i=1}\biggl[\bigl\| { \mathcal{A}}_{i}{ \varDelta }^{i-1}_{t}Gx_{t} \bigr\| +\frac{1}{r_{i,t}}\bigl\| T^{({ \varTheta }_{i},\varphi _{i})}_{r_{i,t}}(I-r_{i,t}{ \mathcal{A}}_{i}){ \varDelta }^{i-1}_{t}Gx_{t} -(I-r_{i,t}{\mathcal{A}}_{i}){ \varDelta }^{i-1}_{t}Gx_{t} \bigr\| \biggr]\leq\widetilde{M}_{0}. $$
In terms of (3.20) we calculate
$$\begin{aligned} &\|x_{t}-x_{t_{0}}\| \\ &\quad\leq\bigl\| (I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}+ \theta_{t}\bigl(t\gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}\bigr) \\ &\qquad{} -(I-\theta_{t_{0}}A)S{ \varDelta }^{N}_{t_{0}}Gx_{t_{0}}- \theta _{t_{0}}\bigl({t_{0}}\gamma Vx_{t_{0}}+(I-t_{0} \mu F)S{ \varDelta }^{N}_{t_{0}}Gx_{t_{0}}\bigr)\bigr\| \\ &\quad\leq\bigl\| (I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}-(I- \theta_{t_{0}}A)S{ \varDelta }^{N}_{t}Gx_{t}\bigr\| \\ &\qquad{} +\bigl\| (I-\theta_{t_{0}}A)S{ \varDelta }^{N}_{t}Gx_{t}-(I- \theta_{t_{0}}A)S{ \varDelta }^{N}_{t_{0}}Gx_{t_{0}}\bigr\| \\ &\qquad{} +|\theta_{t}-\theta_{t_{0}}|\bigl\| t\gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t} \bigr\| \\ &\qquad{} +\theta_{t_{0}}\bigl\| \bigl[t\gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}\bigr]-\bigl[{t_{0}} \gamma Vx_{t_{0}}+(I-t_{0}\mu F)S{ \varDelta }^{N}_{t_{0}}Gx_{t_{0}} \bigr]\bigr\| \\ &\quad\leq|\theta_{t}-\theta_{t_{0}}|\|A\|\bigl\| S{ \varDelta }^{N}_{t}Gx_{t}\bigr\| +(1-\theta _{t_{0}} \bar{\gamma})\bigl\| S{ \varDelta }^{N}_{t}Gx_{t}-S{ \varDelta }^{N}_{t_{0}}Gx_{t_{0}}\bigr\| \\ & \qquad{} +|\theta_{t}-\theta_{t_{0}}|\bigl\| t\gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t} \bigr\| +\theta_{t_{0}}\bigl\| (t-{t_{0}})\gamma Vx_{t} \\ &\qquad{} +t_{0}\gamma(Vx_{t}-Vx_{t_{0}})-(t-t_{0}) \mu FS{ \varDelta }^{N}_{t}Gx_{t}+(I-t_{0} \mu F)S{ \varDelta }^{N}_{t}Gx_{t} \\ &\qquad{} -(I-t_{0}\mu F)S{ \varDelta }^{N}_{t_{0}}Gx_{t_{0}} \bigr\| \\ &\quad\leq|\theta_{t}-\theta_{t_{0}}|\|A\|\bigl\| S{ \varDelta }^{N}_{t}Gx_{t}\bigr\| +(1-\theta _{t_{0}} \bar{\gamma})\bigl\| { \varDelta }^{N}_{t}Gx_{t}-{ \varDelta }^{N}_{t_{0}}Gx_{t_{0}}\bigr\| \\ &\qquad{} +|\theta_{t}-\theta_{t_{0}}|\bigl\| t\gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t} \bigr\| +\theta_{t_{0}}\bigl\| (t-{t_{0}})\gamma Vx_{t} \\ &\qquad{} +t_{0}\gamma(Vx_{t}-Vx_{t_{0}})-(t-t_{0}) \mu FS{ \varDelta }^{N}_{t}Gx_{t}+(I-t_{0} \mu F)S{ \varDelta }^{N}_{t}Gx_{t} \\ & \qquad{} -(I-t_{0}\mu F)S{ \varDelta }^{N}_{t_{0}}Gx_{t_{0}} \bigr\| \\ &\quad\leq|\theta_{t}-\theta_{t_{0}}|\|A\|\bigl\| S{ \varDelta }^{N}_{t}Gx_{t}\bigr\| +(1-\theta _{t_{0}} \bar{\gamma}) \Biggl[\|x_{t}-x_{t_{0}}\|+\widetilde{M}_{0}{ \sum^{N}_{i=1}}|r_{i,t}-r_{i,t_{0}}| \Biggr] \\ & \qquad{} +|\theta_{t}-\theta_{t_{0}}|\bigl[\bigl\| S{ \varDelta }^{N}_{t}Gx_{t}\bigr\| +t\bigl(\gamma \|Vx_{t}\| +\mu\bigl\| FS{ \varDelta }^{N}_{t}Gx_{t} \bigr\| \bigr)\bigr] \\ & \qquad{} +\theta_{t_{0}}\bigl[\bigl(\gamma\|Vx_{t}\|+\mu \bigl\| FS{ \varDelta }^{N}_{t}Gx_{t}\bigr\| \bigr)|t-{t_{0}}|+t_{0}\gamma l\|x_{t}-x_{t_{0}} \| \\ &\qquad{} +(1-t_{0}\tau)\bigl\| { \varDelta }^{N}_{t}Gx_{t}-{ \varDelta }^{N}_{t_{0}}Gx_{t_{0}}\bigr\| \bigr] \\ &\quad\leq|\theta_{t}-\theta_{t_{0}}|\|A\|\bigl\| S{ \varDelta }^{N}_{t}Gx_{t}\bigr\| +(1-\theta _{t_{0}} \bar{\gamma}) \Biggl(\|x_{t}-x_{t_{0}}\|+\widetilde{M}_{0}{ \sum^{N}_{i=1}}|r_{i,t}-r_{i,t_{0}}| \Biggr) \\ &\qquad{} +|\theta_{t}-\theta_{t_{0}}|\bigl(\bigl\| S{ \varDelta }^{N}_{t}Gx_{t}\bigr\| +\gamma\|Vx_{t} \| +\mu\bigl\| FS{ \varDelta }^{N}_{t}Gx_{t}\bigr\| \bigr) \\ &\qquad{} +\theta_{t_{0}}\bigl(\gamma\|Vx_{t}\|+\mu\bigl\| FS{ \varDelta }^{N}_{t}Gx_{t}\bigr\| \bigr)|t-t_{0}|+ \theta_{t_{0}}t_{0}\gamma l\|x_{t}-x_{t_{0}}\| \\ & \qquad{} +\theta_{t_{0}}(1-t_{0}\tau) \Biggl( \|x_{t}-x_{t_{0}}\|+\widetilde{M}_{0}{ \sum ^{N}_{i=1}}|r_{i,t}-r_{i,t_{0}}|\Biggr). \end{aligned}$$
This immediately implies that
$$\begin{aligned} \|x_{t}-x_{t_{0}}\| \leq&\frac{\|A\|\|S{ \varDelta }^{N}_{t}Gx_{t}\| +\|S{ \varDelta }^{N}_{t}Gx_{t}\|+\gamma\|Vx_{t}\|+\mu\|FS{ \varDelta }^{N}_{t}Gx_{t}\|}{ \theta_{t_{0}}(\bar{\gamma}-1+t_{0}(\tau-\gamma l))}| \theta_{t}-\theta _{t_{0}}| \\ &{} +\frac{\gamma\|Vx_{t}\|+\mu\|FS{ \varDelta }^{N}_{t}Gx_{t}\|}{\bar{\gamma}-1+t_{0}(\tau-\gamma l)}|t-t_{0}| \\ &{}+\frac{[1-\theta_{t_{0}}(\bar{\gamma}-1+t_{0}\tau)] \widetilde{M}_{0}}{\theta _{t_{0}}(\bar{\gamma}-1+t_{0}(\tau-\gamma l))} { \sum ^{N}_{i=1}}|r_{i,t}-r_{i,t_{0}}|. \end{aligned}$$
Since \(\theta_{t}:(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})\to(0,\| A\|^{-1}]\) is locally Lipschitzian, and \(r_{i,t}: (0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})\to[a_{i},b_{i}]\) is locally Lipschitzian for each \(i=1,\ldots,N\), we conclude that \(x_{t}:(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})\to C\) is locally Lipschitzian.

(iv) From the last inequality in (iii), the result follows immediately. □

We prove the following theorem for strong convergence of the net \(\{ x_{t}\}\) as \(t\to0\), which guarantees the existence of solutions of the variational inequality (3.2).

Theorem 3.1

Let the net \(\{x_{t}\}\) be defined via (3.1). If \(\lim_{t\to0}\theta_{t}=0\), then \(x_{t}\) converges strongly to a point \(\tilde{x}\in{ \varOmega }\) as \(t\to0\), which solves the VIP (3.2). Equivalently, we have \(P_{{ \varOmega }}(2I-A) \tilde{x}=\tilde{x}\).

Proof

We first show the uniqueness of solutions of the VIP (3.2), which is indeed a consequence of the strong monotonicity of \(A-I\). In fact, since A is a γ̄-strongly positive bounded linear operator with \(\bar{\gamma}\in (1,2)\), we know that \(A-I\) is \((\bar{\gamma}-1)\)-strongly monotone with constant \(\bar{\gamma}-1\in(0,1)\). Suppose that \(\tilde{x}\in{ \varOmega }\) and \(\hat{x}\in{ \varOmega }\) both are solutions to the VIP (3.2). Then we have
$$ \bigl\langle (A-I)\tilde{x},\tilde{x}-\hat{x}\bigr\rangle \leq0 $$
(3.21)
and
$$ \bigl\langle (A-I)\hat{x},\hat{x}-\tilde{x}\bigr\rangle \leq0. $$
(3.22)
Adding (3.21) and (3.22) yields
$$\bigl\langle (A-I)\tilde{x}-(A-I)\hat{x},\tilde{x}-\hat{x}\bigr\rangle \leq0. $$
The strong monotonicity of \(A-I\) implies that \(\tilde{x}=\hat{x}\) and the uniqueness is proved.
Next, we prove that \(x_{t}\to\tilde{x}\) as \(t\to0\). Observing \(\operatorname{Fix}(T)=\operatorname{Fix}(S)\), from (3.1), we write, for given \(p\in{ \varOmega }\),
$$\begin{aligned} &x_{t}-p \\ &\quad=x_{t}-w_{t}+w_{t}-p \\ &\quad=x_{t}-w_{t}+(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}+\theta_{t}\bigl(t \gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t} \bigr)-p \\ &\quad=x_{t}-w_{t}+(I-\theta_{t}A) \bigl(S{ \varDelta }^{N}_{t}Gx_{t}-S{ \varDelta }^{N}_{t}Gp \bigr) \\ &\qquad{} +\theta_{t}\bigl[t\gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}-p\bigr]+\theta _{t}(I-A)p \\ &\quad=x_{t}-w_{t}+(I-\theta_{t}A) \bigl(S{ \varDelta }^{N}_{t}Gx_{t}-S{ \varDelta }^{N}_{t}Gp \bigr) \\ &\qquad{} +\theta_{t}\bigl[t(\gamma Vx_{t}-\mu Fp)+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}-(I-t\mu F)p\bigr]+ \theta_{t}(I-A)p, \end{aligned}$$
where \(w_{t}=(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}+\theta_{t}(t\gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t})\). Then, by Proposition 2.1(i), we have
$$\begin{aligned} &\|x_{t}-p\|^{2} \\ &\quad=\langle x_{t}-w_{t},x_{t}-p\rangle+\bigl\langle (I-\theta_{t}A) (S{ \varDelta }^{N}_{t}Gx_{t}-S{ \varDelta }^{N}_{t}Gp,x_{t}-p\bigr\rangle \\ &\qquad{} +\theta_{t}\bigl[t\langle\gamma Vx_{t}-\mu Fp,x_{t}-p\rangle +\bigl\langle (I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t}-(I-t \mu F)p,x_{t}-p\bigr\rangle \bigr] \\ &\qquad{} +\theta_{t}\bigl\langle (I-A)p,x_{t}-p\bigr\rangle \\ &\quad\leq(1-\theta_{t}\bar{\gamma})\|x_{t}-p \|^{2}+\theta_{t}\bigl[(1-t\tau)\|x_{t}-p\| ^{2}+t\gamma l\|x_{t}-p\|^{2} \\ &\qquad{} +t\bigl\langle (\gamma V-\mu F)p,x_{t}-p\bigr\rangle \bigr]+ \theta_{t}\bigl\langle (I-A)p,x_{t}-p\bigr\rangle \\ &\quad=\bigl[1-\theta_{t}\bigl(\bar{\gamma}-1+t(\tau-\gamma l)\bigr) \bigr]\|x_{t}-p\|^{2}+\theta _{t}\bigl(t\bigl\langle (\gamma V-\mu F)p,x_{t}-p\bigr\rangle \\ &\qquad{} +\bigl\langle (I-A)p,x_{t}-p\bigr\rangle \bigr). \end{aligned}$$
Therefore,
$$ \|x_{t}-p\|^{2}\leq\frac{1}{\bar{\gamma}-1+t(\tau-\gamma l)}\bigl(t\bigl\langle ( \gamma V-\mu F)p,x_{t}-p\bigr\rangle +\bigl\langle (I-A)p,x_{t}-p \bigr\rangle \bigr). $$
(3.23)
Since the net \(\{x_{t}\}_{t\in(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})}\) is bounded (due to Proposition 3.1(i)), we know that if \(\{t_{n}\}\) is a subsequence in \((0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})\) such that \(t_{n}\to0\) and \(x_{t_{n}}\rightharpoonup x^{*}\), then from (3.23), we obtain \(x_{t_{n}}\to x^{*}\). Let us show that \(x^{*}\in{ \varOmega }\). Indeed, by Proposition 3.1(ii), we know that \(\lim_{n\to\infty}\| x_{t_{n}}-Sx_{t_{n}}\|=0\) and \(\lim_{n\to\infty}\|x_{t_{n}}- Gx_{t_{n}}\|=0\). Hence, according to Lemma 2.5, we get \(x^{*}\in\operatorname{Fix}(S)\cap{ \varXi }\). It is clear from the definition of S that \(x^{*}\in\operatorname{Fix}(T)\cap{ \varXi }\). Next we prove that \(x^{*}\in \bigcap^{N}_{m=1}\operatorname{GMEP}({ \varTheta }_{m},\varphi_{m}, {\mathcal{A}}_{m})\). As a matter of fact, utilizing (3.14) and (3.16), we obtain from \(x_{t_{n}}\to x^{*}\), \(y_{t_{n}}=Gx_{t_{n}}\to x^{*}\) and \({ \varDelta }^{m}_{t_{n}}y_{t_{n}}\to x^{*}\) for each \(m=1,\ldots,N\). Since \({ \varDelta }^{m}_{t_{n}}y_{t_{n}}=T^{({ \varTheta }_{m},\varphi_{m})}_{r_{m,t_{n}}}(I-r_{m,t_{n}}{\mathcal{A}}_{m}){ \varDelta }^{m-1}_{t_{n}}y_{t_{n}}\), \(n\geq0\), \(m\in\{1,\ldots,N\}\), we have
$$\begin{aligned} 0 \leq&{ \varTheta }_{m}\bigl({ \varDelta }^{m}_{t_{n}}y_{t_{n}},y \bigr)+\varphi_{m}(y)-\varphi_{m}\bigl({ \varDelta }^{m}_{t_{n}}y_{t_{n}} \bigr) \\ &{} +\bigl\langle {\mathcal{A}}_{m}{ \varDelta }^{m-1}_{t_{n}}y_{t_{n}},y-{ \varDelta }^{m}_{t_{n}}y_{t_{n}}\bigr\rangle + \frac{1}{r_{m,t_{n}}} \bigl\langle y-{ \varDelta }^{m}_{t_{n}}y_{t_{n}},{ \varDelta }^{m}_{t_{n}}y_{t_{n}}-{ \varDelta }^{m-1}_{t_{n}}y_{t_{n}} \bigr\rangle . \end{aligned}$$
By (A2), we have
$$\begin{aligned} { \varTheta }_{m}\bigl(y,{ \varDelta }^{m}_{t_{n}}y_{t_{n}} \bigr) \leq&\varphi_{m}(y)-\varphi_{m}\bigl({ \varDelta }^{m}_{t_{n}}y_{t_{n}}\bigr)+\bigl\langle { \mathcal{A}}_{m}{ \varDelta }^{m-1}_{t_{n}}y_{t_{n}},y-{ \varDelta }^{m}_{t_{n}}y_{t_{n}}\bigr\rangle \\ &{} +\frac{1}{r_{m,t_{n}}}\bigl\langle y-{ \varDelta }^{m}_{t_{n}}y_{t_{n}},{ \varDelta }^{m}_{t_{n}}y_{t_{n}}-{ \varDelta }^{m-1}_{t_{n}}y_{t_{n}} \bigr\rangle . \end{aligned}$$
Let \(z_{t}=ty+(1-t)x^{*}\) for all \(t\in(0,1]\) and \(y\in C\). This implies that \(z_{t}\in C\). Then we have
$$\begin{aligned} &\bigl\langle z_{t}-{ \varDelta }^{m}_{t_{n}}y_{t_{n}},{ \mathcal {A}}_{m}z_{t}\bigr\rangle \\ &\quad\geq\varphi_{m}\bigl({ \varDelta }^{m}_{t_{n}}y_{t_{n}} \bigr)-\varphi_{m}(z_{t})+\bigl\langle z_{t}-{ \varDelta }^{m}_{t_{n}}y_{t_{n}},{\mathcal{A}}_{m}z_{t} \bigr\rangle -\bigl\langle z_{t}-{ \varDelta }^{m}_{t_{n}}y_{t_{n}}, {\mathcal{A}}_{m}{ \varDelta }^{m-1}_{t_{n}}y_{t_{n}} \bigr\rangle \\ &\qquad{} -\biggl\langle z_{t}-{ \varDelta }^{m}_{t_{n}}y_{t_{n}}, \frac{{ \varDelta }^{m}_{t_{n}}y_{t_{n}}-{ \varDelta }^{m-1}_{t_{n}}y_{t_{n}}}{r_{m,t_{n}}}\biggr\rangle +{ \varTheta }_{m}\bigl(z_{t},{ \varDelta }^{m}_{t_{n}}y_{t_{n}}\bigr) \\ &\quad=\varphi_{m}\bigl({ \varDelta }^{m}_{t_{n}}y_{t_{n}} \bigr)-\varphi_{m}(z_{t})+\bigl\langle z_{t}-{ \varDelta }^{m}_{t_{n}}y_{t_{n}},{\mathcal{A}}_{m}z_{t}-{ \mathcal{A}}_{m}{ \varDelta }^{m}_{t_{n}}y_{t_{n}} \bigr\rangle \\ &\qquad{} +\bigl\langle z_{t}-{ \varDelta }^{m}_{t_{n}}y_{t_{n}},{ \mathcal{A}}_{m}{ \varDelta }^{m}_{t_{n}}y_{t_{n}}-{ \mathcal{A}}_{m}{ \varDelta }^{m-1}_{t_{n}}y_{t_{n}} \bigr\rangle \\ &\qquad{} -\biggl\langle z_{t}-{ \varDelta }^{m}_{t_{n}}y_{t_{n}}, \frac{{ \varDelta }^{m}_{t_{n}}y_{t_{n}}-{ \varDelta }^{m-1}_{t_{n}}y_{t_{n}}}{r_{m,t_{n}}}\biggr\rangle +{ \varTheta }_{m}\bigl(z_{t},{ \varDelta }^{m}_{t_{n}}y_{t_{n}}\bigr). \end{aligned}$$
By (3.16), we have \(\|{\mathcal{A}}_{m}{ \varDelta }^{m}_{t_{n}}y_{t_{n}}-{\mathcal{A}}_{m}{ \varDelta }^{m-1}_{t_{n}}y_{t_{n}}\|\to0\) as \(n\to\infty\). Furthermore, by the monotonicity of \({\mathcal{A}}_{m}\), we obtain \(\langle z_{t}-{ \varDelta }^{m}_{t_{n}}y_{t_{n}},{\mathcal{A}}_{m}z_{t}-{\mathcal{A}}_{m}{ \varDelta }^{m}_{t_{n}}y_{t_{n}}\rangle\geq0\). Then, by (A4), we obtain
$$\bigl\langle z_{t}-x^{*},{\mathcal{A}}_{m}z_{t} \bigr\rangle \geq\varphi_{m}\bigl(x^{*}\bigr)-\varphi _{m}(z_{t})+{ \varTheta }_{m}\bigl(z_{t},x^{*}\bigr). $$
Utilizing (A1), (A4), and the last inequality, we obtain
$$\begin{aligned} 0&={ \varTheta }_{m}(z_{t},z_{t})+ \varphi_{m}(z_{t})-\varphi _{m}(z_{t}) \\ &\leq t{ \varTheta }_{m}(z_{t},y)+(1-t){ \varTheta }_{m} \bigl(z_{t},x^{*}\bigr)+t\varphi _{m}(y)+(1-t) \varphi_{m}\bigl(x^{*}\bigr)-\varphi_{m}(z_{t}) \\ &\leq t\bigl[{ \varTheta }_{m}(z_{t},y)+\varphi_{m}(y)- \varphi_{m}(z_{t})\bigr]+(1-t)\bigl\langle z_{t}-x^{*},{\mathcal{A}}_{m}z_{t}\bigr\rangle \\ &=t\bigl[{ \varTheta }_{m}(z_{t},y)+\varphi_{m}(y)- \varphi_{m}(z_{t})\bigr]+(1-t)t\bigl\langle y-x^{*},{ \mathcal{A}}_{m}z_{t}\bigr\rangle , \end{aligned}$$
and hence
$$0\leq{ \varTheta }_{m}(z_{t},y)+\varphi_{m}(y)- \varphi_{m}(z_{t})+(1-t)\bigl\langle y-x^{*},{ \mathcal{A}}_{m}z_{t}\bigr\rangle . $$
Letting \(t\to0\), we have, for each \(y\in C\),
$$0\leq{ \varTheta }_{m}\bigl(x^{*},y\bigr)+\varphi_{m}(y)- \varphi_{m}\bigl(x^{*}\bigr)+\bigl\langle y-x^{*},{\mathcal{A}}_{m}x^{*} \bigr\rangle . $$
This implies that \(x^{*}\in\operatorname{GMEP}({ \varTheta }_{m},\varphi_{m},{\mathcal {A}}_{m})\) and hence \(x^{*}\in\bigcap^{N}_{m=1}\operatorname{GMEP}({ \varTheta }_{m},\varphi _{m},{\mathcal{A}}_{m})\). Thus, \(x^{*}\in\bigcap^{N}_{m=1}\operatorname{GMEP}({ \varTheta }_{m},\varphi_{m},{\mathcal{A}}_{m})\cap\operatorname{Fix}(T)\cap{ \varXi }\).
Next, we prove that \(x_{t}\to\tilde{x}\) as \(t\to0\). First, let us assert that \(x^{*}\) is a solution of the VIP (3.2). As a matter of fact, since
$$x_{t}=x_{t}-w_{t}+(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}+\theta_{t}\bigl(t \gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}_{t}Gx_{t} \bigr), $$
we have
$$x_{t}-S{ \varDelta }^{N}_{t}Gx_{t}=x_{t}-w_{t}+ \theta_{t}(I-A)S{ \varDelta }^{N}_{t}Gx_{t}+ \theta_{t}t\bigl(\gamma Vx_{t}-\mu FS{ \varDelta }^{N}_{t}Gx_{t} \bigr). $$
Since \({ \varDelta }^{N}_{t}\) is nonexpansive (due to (2.1) and Proposition 2.2(ii)), G is nonexpansive (due to Proposition 2.3) and S is nonexpansive (due to Lemma 2.4), \(I-S{ \varDelta }^{N}_{t}G\) is monotone. So, from the monotonicity of \(I-S{ \varDelta }^{N}_{t}G\), it follows that, for \(p\in{ \varOmega }\),
$$\begin{aligned} 0 \leq&\bigl\langle \bigl(I-S{ \varDelta }^{N}_{t}G \bigr)x_{t}-\bigl(I-S{ \varDelta }^{N}_{t}G \bigr)p,x_{t}-p\bigr\rangle =\bigl\langle \bigl(I-S{ \varDelta }^{N}_{t}G\bigr)x_{t},x_{t}-p \bigr\rangle \\ =&\langle x_{t}-w_{t},x_{t}-p\rangle+ \theta_{t}\bigl\langle (I-A)S{ \varDelta }^{N}_{t}Gx_{t},x_{t}-p \bigr\rangle \\ &{} +\theta_{t}t\bigl\langle \gamma Vx_{t}-\mu FS{ \varDelta }^{N}_{t}Gx_{t},x_{t}-p\bigr\rangle \\ \leq&\theta_{t}\bigl\langle (I-A)S{ \varDelta }^{N}_{t}Gx_{t},x_{t}-p \bigr\rangle +\theta _{t}t\bigl\langle \gamma Vx_{t}-\mu FS{ \varDelta }^{N}_{t}Gx_{t},x_{t}-p\bigr\rangle \\ =&\theta_{t}\bigl\langle (I-A)x_{t},x_{t}-p \bigr\rangle +\theta_{t}\bigl\langle (I-A) \bigl(S{ \varDelta }^{N}_{t}G-I\bigr)x_{t},x_{t}-p\bigr\rangle \\ &{} +\theta_{t}t\bigl\langle \gamma Vx_{t}-\mu FS{ \varDelta }^{N}_{t}Gx_{t},x_{t}-p\bigr\rangle . \end{aligned}$$
This implies that
$$ \bigl\langle (A-I)x_{t},x_{t}-p\bigr\rangle \leq\bigl\langle (I-A) \bigl(S{ \varDelta }^{N}_{t}G-I \bigr)x_{t},x_{t}-p\bigr\rangle +t\bigl\langle \gamma Vx_{t}-\mu FS{ \varDelta }^{N}_{t}Gx_{t},x_{t}-p \bigr\rangle . $$
(3.24)
Now, replacing t in (3.24) with \(t_{n}\) and letting \(n\to\infty\), noticing the boundedness of \(\{\gamma Vx_{t_{n}}-\mu FS { \varDelta }^{N}_{t_{n}}Gx_{t_{n}}\}\) and the fact that \((I-A)(S{ \varDelta }^{N}_{t_{n}}G-I)x_{t_{n}}\to0\) as \(n\to\infty\) (due to (3.4)), we obtain
$$\bigl\langle (A-I)x^{*},x^{*}-p\bigr\rangle \leq0. $$
That is, \(x^{*}\in{ \varOmega }\) is a solution of the VIP (3.2); hence \(x^{*}=\tilde{x}\) by uniqueness. In summary, we have proven that each cluster point of \(\{x_{t}\}\) (as \(t\to0\)) equals . Consequently, \(x_{t}\to\tilde{x}\) as \(t\to0\).
The VIP (3.2) can be rewritten as
$$\bigl\langle (2I-A)\tilde{x}-\tilde{x},\tilde{x}-p\bigr\rangle \geq0,\quad \forall p \in { \varOmega }. $$
Recalling Proposition 2.1(i), the last inequality is equivalent to the fixed point equation
$$P_{{ \varOmega }}(2I-A)\tilde{x}=\tilde{x}. $$
 □

Taking \(F=\frac{1}{2}I\), \(\mu=2\), and \(\gamma=1\) in Theorem 3.1, we get

Corollary 3.1

Let \(\{x_{t}\}\) be defined by
$$x_{t}=P_{C}\bigl[(I-\theta_{t}A)S{ \varDelta }^{N}_{t}Gx_{t}+\theta_{t} \bigl(tVx_{t}+(1-t)S{ \varDelta }^{N}_{t}Gx_{t} \bigr)\bigr]. $$
If \(\lim_{t\to0}\theta_{t}=0\), then \(\{x_{t}\}\) converges strongly as \(t\to 0\) to a point \(\tilde{x}\in{ \varOmega }\), which is the unique solution of the VIP (3.2).

First, we prove the following result in order to establish the strong convergence of the sequence \(\{x_{n}\}\) generated by the composite explicit relaxed extragradient-like scheme (3.3).

Theorem 3.2

Let \(\{x_{n}\}\) be the sequence generated by the explicit scheme (3.3), where \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) satisfy the following condition:
  1. (C1)

    \(\{\alpha_{n}\}\subset[0,1]\), \(\{\beta_{n}\}\subset(0,1]\) and \(\alpha _{n}\to0\), \(\beta_{n}\to0\) as \(n\to\infty\).

     
Let LIM be a Banach limit. Then
$$\operatorname{LIM}_{n}\bigl\langle (A-I)\tilde{x},\tilde{x}-x_{n}\bigr\rangle \leq0, $$
where \(\tilde{x}=\lim_{t\to0^{+}}x_{t}\) with \(x_{t}\) being defined by
$$ x_{t}=P_{C}\bigl[(I-\theta_{t}A)S{ \varDelta }^{N}Gx_{t}+\theta_{t}\bigl(t\gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}Gx_{t}\bigr) \bigr], $$
(3.25)
where \(S,G,{ \varDelta }^{N}:C\to C\) are defined by \(Sx=\lambda x+(1-\lambda )Tx\), \(Gx=T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu_{1}F_{1})T^{{ \varPhi } _{2}}_{\nu_{2}}(I-\nu_{2}F_{2})x\) and \({ \varDelta }^{N}x=T^{({ \varTheta }_{N},\varphi _{N})}_{r_{N}}(I-r_{N}{\mathcal{A}}_{N})\cdots T^{({ \varTheta }_{1} ,\varphi_{1})}_{r_{1}}(I-r_{1}{\mathcal{A}}_{1})x\) with \(0\leq k\leq\lambda<1\) and \(r_{i}\in[a_{i},b_{i}]\subset(0,2\eta_{i})\) for each \(i=1,\ldots,N\).

Proof

First, note that from the condition (C1), without loss of generality, we may assume that \(0<\beta_{n}\leq\|A\|^{-1}\) for all \(n\geq0\).

Let \(\{x_{t}\}\) be the net generated by (3.25). Since \({ \varDelta }^{N}\) is a nonexpansive self-mapping on C, by Theorem 3.1 with \({ \varDelta }^{N}_{t}={ \varDelta }^{N}\), there exists \(\lim_{t\to0}x_{t}\in { \varOmega }\). Denote it by . Moreover, is the unique solution of the VIP (3.2). From Proposition 3.1(i) with \({ \varDelta }^{N}_{t}={ \varDelta }^{N}\), we know that \(\{x_{t}\}\) is bounded and so are the nets \(\{Vx_{t}\}\), \(\{{ \varDelta }^{N}Gx_{t}\}\), and \(\{FS{ \varDelta }^{N}Gx_{t}\}\).

First of all, let us show that \(\{x_{n}\}\) is bounded. To this end, take \(p\in{ \varOmega }\). Then we get
$$\begin{aligned} \|y_{n}-p\|&=\bigl\| \alpha_{n}\gamma Vx_{n}+(I- \alpha_{n}\mu F)S{ \varDelta }^{N}_{n}Gx_{n}-p \bigr\| \\ &=\bigl\| \alpha_{n}(\gamma Vx_{n}-\mu Fp)+(I-\alpha_{n} \mu F)S{ \varDelta }^{N}_{n}Gx_{n}-(I- \alpha_{n}\mu F)S{ \varDelta }^{N}_{n}Gp\bigr\| \\ &\leq\alpha_{n}\gamma l\|x_{n}-p\|+\alpha_{n}\bigl\| ( \gamma V-\mu F)p\bigr\| +(1-\alpha _{n}\tau)\|x_{n}-p\| \\ &=\bigl(1-\alpha_{n}(\tau-\gamma l)\bigr)\|x_{n}-p\|+ \alpha_{n}\bigl\| (\gamma V-\mu F)p\bigr\| , \end{aligned}$$
together with Lemma 2.9, implies that
$$\begin{aligned} \|x_{n+1}-p\| =&\bigl\| P_{C}\bigl[(I-\beta_{n}A)S{ \varDelta }^{N}_{n}Gx_{n}+\beta_{n}y_{n} \bigr]-p\bigr\| \\ \leq&\bigl\| (I-\beta_{n}A)S{ \varDelta }^{N}_{n}Gx_{n}+ \beta_{n}y_{n}-p\bigr\| \\ =&\bigl\| (I-\beta_{n}A)S{ \varDelta }^{N}_{n}Gx_{n}-(I- \beta_{n}A)S{ \varDelta }^{N}_{n}Gp \\ &{} +\beta_{n}(y_{n}-p)+\beta_{n}(I-A)p\bigr\| \\ \leq&\bigl\| (I-\beta_{n}A)S{ \varDelta }^{N}_{n}Gx_{n}-(I- \beta_{n}A)S{ \varDelta }^{N}_{n}Gp\bigr\| \\ &{} +\beta_{n}\|y_{n}-p\|+\beta_{n}\|I-A\|\|p\| \\ \leq&(1-\beta_{n}\bar{\gamma})\|x_{n}-p\|+\beta_{n} \bigl[\bigl(1-\alpha_{n}(\tau-\gamma l)\bigr)\|x_{n}-p\| \\ &{} +\alpha_{n}\bigl\| (\gamma V-\mu F)p\bigr\| \bigr]+\beta_{n}\|I-A\| \|p\| \\ \leq&\bigl(1-\beta_{n}(\bar{\gamma}-1)\bigr)\|x_{n}-p\|+ \beta_{n}\bigl(\bigl\| (\gamma V-\mu F)p\bigr\| +\|I-A\|\|p\|\bigr) \\ =&\bigl(1-\beta_{n}(\bar{\gamma}-1)\bigr)\|x_{n}-p\|+ \beta_{n}(\bar{\gamma}-1)\frac{\| (\gamma V-\mu F)p\|+\|I-A\|\|p\|}{\bar{\gamma}-1} \\ \leq&\max\biggl\{ \|x_{n}-p\|,\frac{\|(\gamma V-\mu F)p\|+\|I-A\|\|p\|}{\bar{\gamma}-1}\biggr\} . \end{aligned}$$
By induction
$$\|x_{n}-p\|\leq\max\biggl\{ \|x_{0}-p\|,\frac{\|(\gamma V-\mu F)p\|+\|I-A\|\|p\| }{\bar{\gamma}-1} \biggr\} ,\quad \forall n\geq0. $$
This implies that \(\{x_{n}\}\) is bounded and so are \(\{Vx_{n}\}\), \(\{{ \varDelta }^{N}_{n}Gx_{n}\}\), \(\{FS{ \varDelta }^{N}_{n}Gx_{n}\}\), and \(\{y_{n}\}\). Thus, utilizing the control condition (C1), we get
$$\begin{aligned} \bigl\| x_{n+1}-S{ \varDelta }^{N}_{n}Gx_{n}\bigr\| &= \bigl\| P_{C}\bigl[(I-\beta _{n}A)S{ \varDelta }^{N}_{n}Gx_{n}+ \beta_{n}y_{n}\bigr]-S{ \varDelta }^{N}_{n}Gx_{n} \bigr\| \\ &\leq\bigl\| (I-\beta_{n}A)S{ \varDelta }^{N}_{n}Gx_{n}+ \beta_{n}y_{n}-S{ \varDelta }^{N}_{n}Gx_{n} \bigr\| \\ &=\beta_{n}\bigl\| y_{n}-AS{ \varDelta }^{N}_{n}Gx_{n} \bigr\| \to0 \quad\mbox{as } n\to\infty. \end{aligned}$$
Utilizing similar arguments to those of (3.20), we have
$$\bigl\| { \varDelta }^{N}_{n}Gx_{n}-{ \varDelta }^{N}Gx_{n} \bigr\| \leq\widetilde{M}_{1}{ \sum^{N}_{i=1}}|r_{i,n}-r_{i}|, $$
where \(\sup_{n\geq0}\{\sum^{N}_{i=1}[\|{\mathcal{A}}_{i}{ \varDelta }^{i-1}_{n}Gx_{n}\| +\frac{1}{r_{i,n}}\|T^{({ \varTheta }_{i},\varphi _{i})}_{r_{i,n}}(I-r_{i,n}{\mathcal{A}}_{i}){ \varDelta }^{i-1}_{n}Gx_{n} -(I-r_{i,n}{\mathcal{A}}_{i}){ \varDelta }^{i-1}_{n}Gx_{n}\|]\}\leq\widetilde{M}_{1}\) for some \(\widetilde{M}_{1}>0\). Consequently, it is not hard to find that
$$\begin{aligned} &\bigl\| S{ \varDelta }^{N}Gx_{t}-x_{n+1}\bigr\| \\ &\quad\leq\bigl\| S{ \varDelta }^{N}Gx_{t}-S{ \varDelta }^{N}Gx_{n} \bigr\| +\bigl\| S{ \varDelta }^{N}Gx_{n}-S{ \varDelta }^{N}_{n}Gx_{n} \bigr\| +\bigl\| S{ \varDelta }^{N}_{n}Gx_{n}-x_{n+1}\bigr\| \\ &\quad\leq\|x_{t}-x_{n}\|+\bigl\| { \varDelta }^{N}Gx_{n}-{ \varDelta }^{N}_{n}Gx_{n}\bigr\| +\bigl\| S{ \varDelta }^{N}_{n}Gx_{n}-x_{n+1} \bigr\| \\ &\quad\leq\|x_{t}-x_{n}\|+\widetilde{M}_{1}{ \sum ^{N}_{i=1}}|r_{i,n}-r_{i}|+ \bigl\| S{ \varDelta }^{N}_{n}Gx_{n}-x_{n+1}\bigr\| \\ &\quad=\|x_{t}-x_{n}\|+\epsilon_{n}, \end{aligned}$$
(3.26)
where \(\epsilon_{n}=\widetilde{M}_{1}\sum^{N}_{i=1}|r_{i,n}-r_{i}|+\| x_{n+1}-S{ \varDelta }^{N}_{n}Gx_{n}\|\to0\) as \(n\to\infty\). Also, by observing that A is strongly positive, we have
$$ \langle Ax_{t}-Ax_{n},x_{t}-x_{n} \rangle=\bigl\langle A(x_{t}-x_{n}),x_{t}-x_{n} \bigr\rangle \geq \bar{\gamma}\|x_{t}-x_{n}\|^{2}. $$
(3.27)
For simplicity, we write \(w_{t}=(I-\theta_{t}A)S{ \varDelta }^{N}Gx_{t}+\theta _{t}(t\gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}Gx_{t})\). Then we obtain \(x_{t}=P_{C}w_{t}\) and
$$\begin{aligned} x_{t}-x_{n+1} ={}&x_{t}-w_{t}+(I-\theta_{t}A)S{ \varDelta }^{N}Gx_{t}+\theta_{t}\bigl(t\gamma Vx_{t}+(I-t\mu F)S{ \varDelta }^{N}Gx_{t} \bigr)-x_{n+1} \\ ={}&(I-\theta_{t}A)S{ \varDelta }^{N}Gx_{t}-(I- \theta_{t}A)x_{n+1}+\theta _{t}\bigl(t\gamma Vx_{t} \\ &{} +(I-t\mu F)S{ \varDelta }^{N}Gx_{t}-Ax_{n+1} \bigr)+x_{t}-w_{t}. \end{aligned}$$
Applying Lemma 2.1, we have
$$\begin{aligned} &\|x_{t}-x_{n+1}\|^{2} \\ &\quad\leq\bigl\| (I-\theta_{t}A)S{ \varDelta }^{N}Gx_{t}-(I- \theta_{t}A)x_{n+1}\bigr\| ^{2} \\ &\qquad{} +2\theta_{t}\bigl\langle S{ \varDelta }^{N}Gx_{t}-t \bigl(\mu FS{ \varDelta }^{N}Gx_{t}-\gamma Vx_{t} \bigr)-Ax_{n+1},x_{t}-x_{n+1}\bigr\rangle +2\langle x_{t}-w_{t},x_{t}-x_{n+1} \rangle \\ &\quad\leq\bigl\| (I-\theta_{t}A)S{ \varDelta }^{N}Gx_{t}-(I- \theta_{t}A)x_{n+1}\bigr\| ^{2} \\ &\qquad{} +2\theta_{t}\bigl\langle S{ \varDelta }^{N}Gx_{t}-t \bigl(\mu FS{ \varDelta }^{N}Gx_{t}-\gamma Vx_{t} \bigr)-Ax_{n+1},x_{t}-x_{n+1}\bigr\rangle \\ &\quad\leq(1-\theta_{t}\bar{\gamma})^{2}\bigl\| S{ \varDelta }^{N}Gx_{t}-x_{n+1}\bigr\| ^{2}+2\theta _{t}\bigl\langle S{ \varDelta }^{N}Gx_{t}-x_{t},x_{t}-x_{n+1} \bigr\rangle \\ &\qquad{} -2\theta_{t}t\bigl\langle \mu FS{ \varDelta }^{N}Gx_{t}- \gamma Vx_{t},x_{t}-x_{n+1}\bigr\rangle +2 \theta_{t}\langle x_{t}-Ax_{n+1},x_{t}-x_{n+1} \rangle. \end{aligned}$$
(3.28)
Using (3.26) and (3.27) in (3.28), we obtain
$$\begin{aligned} &\|x_{t}-x_{n+1}\|^{2} \\ &\quad\leq(1-\theta_{t}\bar{\gamma})^{2}\bigl\| S{ \varDelta }^{N}Gx_{t}-x_{n+1}\bigr\| ^{2}+2\theta _{t}\bigl\langle S{ \varDelta }^{N}Gx_{t}-x_{t},x_{t}-x_{n+1} \bigr\rangle \\ &\qquad{} +2\theta_{t}t\bigl\langle \gamma Vx_{t}-\mu FS{ \varDelta }^{N}Gx_{t},x_{t}-x_{n+1}\bigr\rangle +2\theta_{t}\langle x_{t}-Ax_{n+1},x_{t}-x_{n+1} \rangle \\ &\quad\leq(1-\theta_{t}\bar{\gamma})^{2}\bigl( \|x_{t}-x_{n}\|+\epsilon_{n}\bigr)^{2}+2 \theta_{t}\bigl\| S{ \varDelta }^{N}Gx_{t}-x_{t} \bigr\| \|x_{t}-x_{n+1}\| \\ &\qquad{} +2\theta_{t}t\bigl\| \gamma Vx_{t}-\mu FS{ \varDelta }^{N}Gx_{t}\bigr\| \|x_{t}-x_{n+1}\| +2 \theta_{t}\langle x_{t}-Ax_{n+1},x_{t}-x_{n+1} \rangle \\ &\quad=\bigl(\theta^{2}_{t}\bar{\gamma}-2 \theta_{t}\bigr)\bar{\gamma}\|x_{t}-x_{n} \|^{2}+\|x_{t}-x_{n}\| ^{2}+(1- \theta_{t}\bar{\gamma})^{2}\bigl(2\|x_{t}-x_{n} \|\epsilon_{n}+\epsilon^{2}_{n}\bigr) \\ &\qquad{} +2\theta_{t}\bigl\| S{ \varDelta }^{N}Gx_{t}-x_{t} \bigr\| \|x_{t}-x_{n+1}\|+2\theta_{t}t\bigl\| \gamma Vx_{t}-\mu FS{ \varDelta }^{N}Gx_{t}\bigr\| \|x_{t}-x_{n+1}\| \\ &\qquad{} +2\theta_{t}\langle x_{t}-Ax_{n+1},x_{t}-x_{n+1} \rangle \\ &\quad\leq\bigl(\theta^{2}_{t}\bar{\gamma}-2 \theta_{t}\bigr)\langle Ax_{t}-Ax_{n},x_{t}-x_{n} \rangle +\|x_{t}-x_{n}\|^{2} +(1- \theta_{t}\bar{\gamma})^{2}\bigl(2\|x_{t}-x_{n} \|\epsilon_{n}+\epsilon^{2}_{n}\bigr) \\ &\qquad{} +2\theta_{t}\bigl\| S{ \varDelta }^{N}Gx_{t}-x_{t} \bigr\| \|x_{t}-x_{n+1}\|+2\theta_{t}t\bigl\| \gamma Vx_{t}-\mu FS{ \varDelta }^{N}Gx_{t}\bigr\| \|x_{t}-x_{n+1}\| \\ &\qquad{} +2\theta_{t}\langle x_{t}-Ax_{n+1},x_{t}-x_{n+1} \rangle \\ &\quad=\theta^{2}_{t}\bar{\gamma}\langle Ax_{t}-Ax_{n},x_{t}-x_{n} \rangle+\|x_{t}-x_{n}\| ^{2}+(1- \theta_{t}\bar{\gamma})^{2}\bigl(2\|x_{t}-x_{n} \|\epsilon_{n}+\epsilon^{2}_{n}\bigr) \\ &\qquad{} +2\theta_{t}\bigl\| S{ \varDelta }^{N}Gx_{t}-x_{t} \bigr\| \|x_{t}-x_{n+1}\|+2\theta_{t}t\bigl\| \gamma Vx_{t}-\mu FS{ \varDelta }^{N}Gx_{t}\bigr\| \|x_{t}-x_{n+1}\| \\ &\qquad{} +2\theta_{t}\bigl[\langle x_{t}-Ax_{n+1},x_{t}-x_{n+1} \rangle-\langle Ax_{t}-Ax_{n},x_{t}-x_{n} \rangle\bigr] \\ &=\theta^{2}_{t}\bar{\gamma}\bigl\langle A(x_{t}-x_{n}),x_{t}-x_{n} \bigr\rangle +\|x_{t}-x_{n}\| ^{2}+(1- \theta_{t}\bar{\gamma})^{2}\bigl(2\|x_{t}-x_{n} \|\epsilon_{n}+\epsilon^{2}_{n}\bigr) \\ &\qquad{} +2\theta_{t}\bigl\| S{ \varDelta }^{N}Gx_{t}-x_{t} \bigr\| \|x_{t}-x_{n+1}\|+2\theta_{t}t\bigl\| \gamma Vx_{t}-\mu FS{ \varDelta }^{N}Gx_{t}\bigr\| \|x_{t}-x_{n+1}\| \\ &\qquad{} +2\theta_{t}\bigl[\bigl\langle (I-A)x_{t},x_{t}-x_{n+1} \bigr\rangle +\bigl\langle A(x_{t}-x_{n+1}),x_{t}-x_{n+1} \bigr\rangle \\ &\qquad{} -\bigl\langle A(x_{t}-x_{n}),x_{t}-x_{n} \bigr\rangle \bigr]. \end{aligned}$$
(3.29)
Applying the Banach limit LIM to (3.29), from \(\epsilon_{n}\to0\) we have
$$\begin{aligned} &\operatorname{LIM}_{n}\|x_{t}-x_{n+1} \|^{2} \\ &\quad\leq\theta^{2}_{t}\bar{\gamma}\operatorname{LIM}_{n} \bigl\langle A(x_{t}-x_{n}),x_{t}-x_{n} \bigr\rangle +\operatorname{LIM}_{n}\|x_{t}-x_{n} \|^{2} \\ &\qquad{} +2\theta_{t}\bigl\| S{ \varDelta }^{N}Gx_{t}-x_{t} \bigr\| \operatorname{LIM}_{n}\|x_{t}-x_{n+1}\| \\ &\qquad{} +2\theta_{t}t\bigl\| \gamma Vx_{t}-\mu FS{ \varDelta }^{N}Gx_{t}\bigr\| \operatorname{LIM}_{n}\| x_{t}-x_{n+1}\| \\ &\qquad{} +2\theta_{t}\bigl[\operatorname{LIM}_{n}\bigl\langle (I-A)x_{t},x_{t}-x_{n+1} \bigr\rangle + \operatorname{LIM}_{n}\bigl\langle A(x_{t}-x_{n+1}),x_{t}-x_{n+1} \bigr\rangle \\ &\qquad{} -\operatorname{LIM}_{n}\bigl\langle A(x_{t}-x_{n}),x_{t}-x_{n} \bigr\rangle \bigr]. \end{aligned}$$
(3.30)
Utilizing the property \(\operatorname{LIM}_{n}a_{n}=\operatorname{LIM}_{n}a_{n+1}\) of the Banach limit in (3.30), we obtain
$$\begin{aligned} &\operatorname{LIM}_{n}\bigl\langle (A-I)x_{t},x_{t}-x_{n} \bigr\rangle \\ &\quad=\operatorname{LIM}_{n}\bigl\langle (A-I)x_{t},x_{t}-x_{n+1} \bigr\rangle \\ &\quad\leq\frac{\theta_{t}\bar{\gamma}}{2}\operatorname{LIM}_{n}\bigl\langle A(x_{t}-x_{n}),x_{t}-x_{n}\bigr\rangle +\frac{1}{2\theta_{t}}\bigl[\operatorname{LIM}_{n}\|x_{t}-x_{n} \|^{2}-\operatorname{LIM}_{n}\| x_{t}-x_{n+1} \|^{2}\bigr] \\ &\qquad{} +\bigl\| S{ \varDelta }^{N}Gx_{t}-x_{t}\bigr\| \operatorname{LIM}_{n}\|x_{t}-x_{n}\|+t\bigl\| \gamma Vx_{t}-\mu FS{ \varDelta }^{N}Gx_{t}\bigr\| \operatorname{LIM}_{n}\|x_{t}-x_{n}\| \\ &\qquad{} +\operatorname{LIM}_{n}\bigl\langle A(x_{t}-x_{n+1}),x_{t}-x_{n+1} \bigr\rangle -\operatorname{LIM}_{n}\bigl\langle A(x_{t}-x_{n}),x_{t}-x_{n} \bigr\rangle \\ &\quad\leq\frac{\theta_{t}\bar{\gamma}}{2}\operatorname{LIM}_{n}\bigl\langle A(x_{t}-x_{n}),x_{t}-x_{n}\bigr\rangle + \bigl\| S{ \varDelta }^{N}Gx_{t}-x_{t}\bigr\| \operatorname{LIM}_{n}\| x_{t}-x_{n}\| \\ &\qquad{} +t\bigl\| \gamma Vx_{t}-\mu FS{ \varDelta }^{N}Gx_{t} \bigr\| \operatorname{LIM}_{n}\|x_{t}-x_{n}\|. \end{aligned}$$
(3.31)
Since as \(t\to0\),
$$ \theta_{t}\bigl\langle A(x_{t}-x_{n}),x_{t}-x_{n} \bigr\rangle \leq\theta_{t}\|A\|\|x_{t}-x_{n}\| ^{2}\leq\theta_{t}K\to0, $$
(3.32)
where \(\|A\|\|x_{t}-x_{n}\|^{2}\leq K\),
$$ \begin{aligned} &\bigl\| S{ \varDelta }^{N}Gx_{t}-x_{t}\bigr\| \to0 \quad\bigl( \mbox{see (3.4)}\bigr)\quad \mbox{and}\\ & t\bigl\| \gamma Vx_{t}-\mu FS{ \varDelta }^{N}Gx_{t}\bigr\| \to0 \quad \mbox{as } t\to0, \end{aligned} $$
(3.33)
we conclude from (3.31)-(3.33) that
$$\begin{aligned} &\operatorname{LIM}_{n}\bigl\langle (A-I)\tilde{x},\tilde{x}-x_{n}\bigr\rangle \\ &\quad\leq \limsup_{t\to0}\operatorname{LIM}_{n} \bigl\langle (A-I)x_{t},x_{t}-x_{n}\bigr\rangle \\ &\quad\leq \limsup_{t\to0}\frac{\theta_{t}\bar{\gamma}}{2} \operatorname{LIM}_{n}\bigl\langle A(x_{t}-x_{n}),x_{t}-x_{n} \bigr\rangle + \limsup_{t\to0}\bigl\| S{ \varDelta }^{N}Gx_{t}-x_{t} \bigr\| \operatorname{LIM}_{n}\| x_{t}-x_{n}\| \\ &\qquad{} + \limsup_{t\to0}t\bigl\| \gamma Vx_{t}-\mu FS{ \varDelta }^{N}Gx_{t}\bigr\| \operatorname{LIM}_{n} \|x_{t}-x_{n}\| \\ &\quad=0. \end{aligned}$$
This completes the proof. □

Now, using Theorem 3.2, we establish the strong convergence of the sequence \(\{x_{n}\}\) generated by the composite explicit relaxed extragradient-like scheme (3.3) to a point \(\tilde{x}\in{ \varOmega }\), which is also the unique solution of the VIP (3.2).

Theorem 3.3

Let \(\{x_{n}\}\) be the sequence generated by the explicit scheme (3.3), where \(\{\alpha_{n}\}\) and \(\{\beta _{n}\}\) satisfy the following conditions:
  1. (C1)

    \(\{\alpha_{n}\}\subset[0,1]\), \(\{\beta_{n}\}\subset(0,1]\), and \(\alpha _{n}\to0\), \(\beta_{n}\to0\) as \(n\to\infty\);

     
  2. (C2)

    \(\sum^{\infty}_{n=0}\beta_{n}=\infty\).

     
If \(\{x_{n}\}\) is weakly asymptotically regular (i.e., \(x_{n+1}-x_{n}\rightharpoonup0\)), then \(x_{n}\) converges strongly to a point \(\tilde{x}\in{ \varOmega }\), which is the unique solution of the VIP (3.2).

Proof

First, note that from the condition (C1), without loss of generality, we may assume that \(\alpha_{n}\tau<1\) and \(\frac{2\beta_{n}(\bar{\gamma}-1)}{1-\beta_{n}}<1\) for all \(n\geq0\).

Let \(x_{t}\) be defined by (3.25), that is,
$$x_{t}=P_{C}\bigl[(I-\theta_{t}A)S{ \varDelta }^{N}Gx_{t}+\theta_{t}(S{ \varDelta }^{N}Gx_{t}-t\bigl(\mu FS{ \varDelta }^{N}Gx_{t}- \gamma Vx_{t}\bigr)\bigr], $$
for \(t\in(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})\), where \(Sx=\lambda x+(1-\lambda)Tx\) for \(0\leq k\leq\lambda<1\), \(Gx=T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu_{1}F_{1})T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu _{2}F_{2})x\) for \(\nu_{j}\in(0,2\zeta_{j})\), \(j=1,2\), \({ \varDelta }^{N}x =T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N}}(I-r_{N}{\mathcal{A}}_{N})\cdots T^{({ \varTheta }_{1},\varphi_{1})}_{r_{1}}(I-r_{1}{\mathcal{A}}_{1})x\) for \(r_{i}\in[a_{i},b_{i}]\subset(0,2\eta_{i})\), \(i=1,\ldots,N\), and \(\lim_{t\to 0}x_{t}:=\tilde{x}\in{ \varOmega }\) (due to Theorem 3.1). Then is the unique solution of the VIP (3.2).

We divide the rest of the proof into several steps.

Step 1. We see that
$$\|x_{n}-p\|\leq\max\biggl\{ \|x_{0}-p\|,\frac{\|(\gamma V-\mu F)p\|+\|I-A\|\|p\| }{\bar{\gamma}-1} \biggr\} , \quad\forall n\geq0, $$
for all \(p\in{ \varOmega }\) as in the proof of Theorem 3.2. Hence \(\{x_{n}\} \) is bounded and so are \(\{Vx_{n}\}\), \(\{{ \varDelta }^{N}Gx_{n}\}\), \(\{{ \varDelta }^{N}_{n}Gx_{n}\}\), \(\{FS{ \varDelta }^{N}_{n}Gx_{n}\}\), and \(\{y_{n}\}\).
Step 2. We show that \(\limsup_{n\to\infty}\langle(I-A)\tilde{x},x_{n}-\tilde{x}\rangle\leq0\). To this end, put
$$a_{n}:=\bigl\langle (A-I)\tilde{x},\tilde{x}-x_{n}\bigr\rangle , \quad\forall n\geq0. $$
Then, by Theorem 3.2, we get \(\operatorname{LIM}_{n}a_{n}\leq0\) for any Banach limit LIM. Since \(\{x_{n}\}\) is bounded, there exists a subsequence \(\{x_{n_{j}}\}\) of \(\{x_{n}\}\) such that
$$\limsup_{n\to\infty}(a_{n+1}-a_{n})=\limsup _{j\to\infty}(a_{n_{j}+1}-a_{n_{j}}) $$
and \(x_{n_{j}}\rightharpoonup v\in H\). This implies that \(x_{n_{j}+1}\rightharpoonup v\) since \(\{x_{n}\}\) is weakly asymptotically regular. Therefore, we have
$$w-\lim_{j\to\infty}(\tilde{x}-x_{n_{j}+1})=w-\lim _{j\to\infty}(\tilde{x}-x_{n_{j}})=(\tilde{x}-v), $$
and so
$$\limsup_{n\to\infty}(a_{n+1}-a_{n})=\lim _{j\to\infty}\bigl\langle (A-I)\tilde{x},(\tilde{x}-x_{n_{j}+1})-( \tilde{x}-x_{n_{j}})\bigr\rangle =0. $$
Then, by Lemma 2.10, we obtain \(\limsup_{n\to\infty}a_{n}\leq0\), that is,
$$\limsup_{n\to\infty}\bigl\langle (I-A)\tilde{x},x_{n}-\tilde{x}\bigr\rangle =\limsup_{n\to\infty}\bigl\langle (A-I)\tilde{x},\tilde{x}-x_{n}\bigr\rangle \leq0. $$
Step 3. We show that \(\lim_{n\to\infty}\|x_{n}-\tilde{x}\|=0\). Indeed, for simplicity, we write \(w_{n}=(I-\beta_{n}A)S{ \varDelta }^{N}_{n} Gx_{n}+\beta_{n}y_{n}\) for all \(n\geq0\). Then \(x_{n+1}=P_{C}w_{n}\). Utilizing (3.3) and \(S{ \varDelta }^{N}_{n}G\tilde{x}=\tilde{x}\), we have
$$y_{n}-\tilde{x}=(I-\alpha_{n}\mu F)S{ \varDelta }^{N}_{n}Gx_{n}-(I- \alpha_{n}\mu F)S{ \varDelta }^{N}_{n}G\tilde{x}+ \alpha_{n}(\gamma Vx_{n}-\mu F\tilde{x}) $$
and
$$x_{n+1}-\tilde{x}=x_{n+1}-w_{n}+(I- \beta_{n}A) \bigl(S{ \varDelta }^{N}_{n}Gx_{n}-S{ \varDelta }^{N}_{n}G\tilde{x}\bigr)+\beta_{n}(y_{n}- \tilde{x})+\beta_{n}(I-A)\tilde{x}. $$
Applying Lemmas 2.1, 2.7, and 2.9, we obtain
$$\begin{aligned} \|y_{n}-\tilde{x}\|^{2} =&\bigl\| (I-\alpha_{n}\mu F)S{ \varDelta }^{N}_{n}Gx_{n}-(I-\alpha_{n}\mu F)S{ \varDelta }^{N}_{n}G\tilde{x} +\alpha_{n}(\gamma Vx_{n}-\mu F\tilde{x})\bigr\| ^{2} \\ \leq&\bigl\| (I-\alpha_{n}\mu F)S{ \varDelta }^{N}_{n}Gx_{n}-(I- \alpha_{n}\mu F)S{ \varDelta }^{N}_{n}G\tilde{x} \bigr\| ^{2} \\ &{} +2\alpha_{n}\langle\gamma Vx_{n}-\mu F\tilde{x},y_{n}-\tilde{x}\rangle \\ \leq&(1-\alpha_{n}\tau)^{2}\|x_{n}-\tilde{x} \|^{2}+2\alpha_{n}\|\gamma Vx_{n}-\mu F\tilde{x}\| \|y_{n}-\tilde{x}\| \\ \leq&\|x_{n}-\tilde{x}\|^{2}+2\alpha_{n}\|\gamma Vx_{n}-\mu F\tilde{x}\|\| y_{n}-\tilde{x}\|, \end{aligned}$$
and hence
$$\begin{aligned} &\|x_{n+1}-\tilde{x}\|^{2} \\ &\quad=\bigl\| (I-\beta_{n}A) \bigl(S{ \varDelta }^{N}_{n}Gx_{n}-S{ \varDelta }^{N}_{n}G\tilde{x}\bigr)+\beta _{n}(y_{n}- \tilde{x})+\beta_{n}(I-A)\tilde{x}+x_{n+1}-w_{n} \bigr\| ^{2} \\ &\quad\leq\bigl\| (I-\beta_{n}A) \bigl(S{ \varDelta }^{N}_{n}Gx_{n}-S{ \varDelta }^{N}_{n}G\tilde{x}\bigr)\bigr\| ^{2}+2 \beta_{n}\langle y_{n}-\tilde{x},x_{n+1}-\tilde{x} \rangle \\ &\qquad{} +2\beta_{n}\bigl\langle (I-A)\tilde{x},x_{n+1}-\tilde{x}\bigr\rangle +2\langle x_{n+1}-w_{n},x_{n+1}-\tilde{x}\rangle \\ &\quad\leq\bigl\| (I-\beta_{n}A) \bigl(S{ \varDelta }^{N}_{n}Gx_{n}-S{ \varDelta }^{N}_{n}G\tilde{x}\bigr)\bigr\| ^{2}+2 \beta_{n}\langle y_{n}-\tilde{x},x_{n+1}-\tilde{x} \rangle \\ &\qquad{} +2\beta_{n}\bigl\langle (I-A)\tilde{x},x_{n+1}-\tilde{x}\bigr\rangle \\ &\quad\leq(1-\beta_{n}\bar{\gamma})^{2}\|x_{n}- \tilde{x}\|^{2}+2\beta_{n}\|y_{n}-\tilde{x}\| \|x_{n+1}-\tilde{x}\| +2\beta_{n}\bigl\langle (I-A)\tilde{x},x_{n+1}-\tilde{x}\bigr\rangle \\ &\quad\leq(1-\beta_{n}\bar{\gamma})^{2}\|x_{n}- \tilde{x}\|^{2}+\beta_{n}\bigl(\|y_{n}-\tilde{x}\| ^{2}+\|x_{n+1}-\tilde{x}\|^{2}\bigr) +2\beta_{n}\bigl\langle (I-A)\tilde{x},x_{n+1}-\tilde{x}\bigr\rangle \\ &\quad\leq(1-\beta_{n}\bar{\gamma})^{2}\|x_{n}- \tilde{x}\|^{2}+\beta_{n}\bigl[\|x_{n}-\tilde{x}\| ^{2}+2\alpha_{n}\|\gamma Vx_{n}-\mu F\tilde{x}\| \|y_{n}-\tilde{x}\|\bigr] \\ &\qquad{} +\beta_{n}\|x_{n+1}-\tilde{x}\|^{2}+2 \beta_{n}\bigl\langle (I-A)\tilde{x},x_{n+1}-\tilde{x}\bigr\rangle \\ &\quad=\bigl[(1-\beta_{n}\bar{\gamma})^{2}+\beta_{n} \bigr]\|x_{n}-\tilde{x}\|^{2}+2\alpha_{n}\beta _{n}\|\gamma Vx_{n}-\mu F\tilde{x}\|\|y_{n}-\tilde{x} \| \\ &\qquad{} +\beta_{n}\|x_{n+1}-\tilde{x}\|^{2}+2 \beta_{n}\bigl\langle (I-A)\tilde{x},x_{n+1}-\tilde{x}\bigr\rangle . \end{aligned}$$
(3.34)
It then follows from (3.34) that
$$\begin{aligned} \|x_{n+1}-\tilde{x}\|^{2} \leq&\frac{(1-\beta_{n}\bar{\gamma})^{2}+\beta_{n}}{1-\beta_{n}} \|x_{n}-\tilde{x}\|^{2} +\frac{\beta_{n}}{1-\beta_{n}}\bigl[2 \alpha_{n}\|\gamma Vx_{n}-\mu F\tilde{x}\|\| y_{n}- \tilde{x}\| \\ &{} +2\bigl\langle (I-A)\tilde{x},x_{n+1}-\tilde{x}\bigr\rangle \bigr] \\ =&\biggl(1-\frac{2\beta_{n}(\bar{\gamma}-1)}{1-\beta_{n}}\biggr)\|x_{n}-\tilde{x}\|^{2} \\ &{} +\frac{2\beta_{n}(\bar{\gamma}-1)}{1-\beta_{n}}\cdot\frac{1}{2(\bar{\gamma}-1)}\bigl[2\alpha_{n}\| \gamma Vx_{n}-\mu F\tilde{x}\|\|y_{n}-\tilde{x}\| \\ &{} +\beta_{n}\bar{\gamma}^{2}\|x_{n}-\tilde{x} \|^{2}+2\bigl\langle (I-A)\tilde{x},x_{n+1}-\tilde{x}\bigr\rangle \bigr] \\ =&(1-\omega_{n})\|x_{n}-\tilde{x}\|^{2}+ \omega_{n}\delta_{n}, \end{aligned}$$
where \(\omega_{n}=\frac{2\beta_{n}(\bar{\gamma}-1)}{1-\beta_{n}}\) and
$$\delta_{n}=\frac{1}{2(\bar{\gamma}-1)}\bigl[2\alpha_{n}\|\gamma Vx_{n}-\mu F\tilde{x}\|\|y_{n}-\tilde{x}\| +\beta_{n} \bar{\gamma}^{2}\|x_{n}-\tilde{x}\|^{2}+2\bigl\langle (I-A)\tilde{x},x_{n+1}-\tilde{x}\bigr\rangle \bigr]. $$
It can be readily seen from Step 2 and conditions (C1) and (C2) that \(\omega_{n}\to0\), \(\sum^{\infty}_{n=0}\omega_{n}=\infty\) and \(\limsup_{n\to\infty}\delta_{n}\leq0\). By Lemma 2.8 with \(r_{n}=0\), we conclude that \(\lim_{n\to\infty}\|x_{n}-\tilde{x}\|=0\). This completes the proof. □

Corollary 3.2

Let \(\{x_{n}\}\) be the sequence generated by the explicit scheme (3.3). Assume that the sequences \(\{ \alpha_{n}\}\) and \(\{\beta_{n}\}\) satisfy the conditions (C1) and (C2) in Theorem 3.3. If \(\{x_{n}\}\) is asymptotically regular (i.e., \(x_{n+1}-x_{n}\to0\)), then \(\{x_{n}\}\) converges strongly to a point \(\tilde{x}\in{ \varOmega }\), which is the unique solution of the VIP (3.2).

Putting \(\mu=2\), \(F=\frac{1}{2}I\), and \(\gamma=1\) in Theorem 3.3, we obtain the following.

Corollary 3.3

Let \(\{x_{n}\}\) be generated by the following iterative scheme:
$$\left \{ \textstyle\begin{array}{@{}l} y_{n}=\alpha_{n}Vx_{n}+(1-\alpha_{n})S{ \varDelta }^{N}_{n}Gx_{n},\\ x_{n+1}=P_{C}[(I-\beta_{n}A)S{ \varDelta }^{N}_{n}Gx_{n}+\beta_{n}y_{n}],\quad \forall n\geq0. \end{array}\displaystyle \right . $$
Assume that the sequences \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) satisfy the conditions (C1) and (C2) in Theorem  3.3. If \(\{x_{n}\}\) is weakly asymptotically regular (i.e., \(x_{n+1}-x_{n}\rightharpoonup0\)), then \(\{x_{n}\}\) converges strongly to a point \(\tilde{x}\in{ \varOmega }\), which is the unique solution of the VIP (3.2).

Putting \(\alpha_{n}=0\), \(\forall n\geq0\) in Corollary 3.3, we get the following.

Corollary 3.4

Let \(\{x_{n}\}\) be generated by the following iterative scheme:
$$x_{n+1}=P_{C}\bigl[\bigl(I-\beta_{n}(A-I)\bigr)S{ \varDelta }^{N}_{n}Gx_{n}\bigr],\quad \forall n\geq0. $$
Assume that the sequence \(\{\beta_{n}\}\) satisfies the conditions (C1) and (C2) in Theorem 3.3 with \(\alpha_{n}=0\), \(\forall n\geq0\). If \(\{x_{n}\}\) is weakly asymptotically regular (i.e., \(x_{n+1}-x_{n}\rightharpoonup0\)), then \(\{x_{n}\}\) converges strongly to a point \(\tilde{x}\in{ \varOmega }\), which is the unique solution of the VIP (3.2).

Remark 3.1

If \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) in Corollary 3.2 and \(\{r_{i,n}\}^{N}_{i=1}\) in \({ \varDelta }^{N}_{n}\) satisfy conditions (C2) and
  1. (C3)

    \(\sum^{\infty}_{n=0}|\alpha_{n+1}-\alpha_{n}|<\infty\) and \(\sum^{\infty}_{n=0}|\beta_{n+1}-\beta_{n}|<\infty\); or

     
  2. (C4)

    \(\sum^{\infty}_{n=0}|\alpha_{n+1}-\alpha_{n}|<\infty\) and \(\lim_{n\to \infty}\frac{\beta_{n}}{\beta_{n+1}}=1\) or, equivalently, \(\lim_{n\to\infty}\frac{\alpha_{n}-\alpha_{n+1}}{\alpha_{n+1}}=0\) and \(\lim_{n\to\infty}\frac{\beta_{n}-\beta_{n+1}}{\beta_{n+1}}=0\); or,

     
  3. (C5)

    \(\sum^{\infty}_{n=0}|\alpha_{n+1}-\alpha_{n}|<\infty\) and \(|\beta _{n+1}-\beta_{n}|\leq o(\beta_{n+1})+\sigma_{n}\), \(\sum^{\infty}_{n=0}\sigma_{n}<\infty\) (the perturbed control condition);

     
  4. (C6)

    \(\sum^{\infty}_{n=0}|r_{i,n+1}-r_{i,n}|<\infty\) for each \(i=1,\ldots,N\),

     
then the sequence \(\{x_{n}\}\) generated by (3.3) is asymptotically regular. Now we give only the proof in the case when \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), and \(\{r_{i,n}\}^{N}_{i=1}\) satisfy the conditions (C2), (C5), and (C6). By Step 1 in the proof of Theorem 3.3, there exists a constant \(\widetilde{M}_{2}>0\) such that for all \(n\geq0\), \(\mu\|FS{ \varDelta }^{N}_{n}Gx_{n}\|+\gamma\|Vx_{n}\| \leq\widetilde{M}_{2}\), \(\|A\|\|S{ \varDelta }^{N}_{n}Gx_{n}\|+\|y_{n}\|\leq \widetilde{M}_{2}\), and
$$\sum^{N}_{i=1}\biggl[\bigl\| { \mathcal{A}}_{i}{ \varDelta }^{i-1}_{n}Gx_{n}\bigr\| +\frac{1}{r_{i,n}}\bigl\| T^{({ \varTheta }_{i},\varphi _{i})}_{r_{i,n}}(I-r_{i,n}{ \mathcal{A}}_{i}){ \varDelta }^{i-1}_{n}Gx_{n} -(I-r_{i,n}{\mathcal{A}}_{i}){ \varDelta }^{i-1}_{n}Gx_{n} \bigr\| \biggr]\leq\widetilde{M}_{2}. $$
Utilizing similar arguments to those of (3.20), we obtain
$$\bigl\| { \varDelta }^{N}_{n}Gx_{n}-{ \varDelta }^{N}_{n-1}Gx_{n-1} \bigr\| \leq\widetilde{M}_{2}{ \sum^{N}_{i=1}}|r_{i,n}-r_{i,n-1}|+ \| x_{n}-x_{n-1}\|. $$
So, we obtain, for all \(n\geq0\),
$$\begin{aligned} &\|y_{n}-y_{n-1}\| \\ &\quad=\bigl\| \alpha_{n}\gamma(Vx_{n}-Vx_{n-1})+\gamma( \alpha_{n}-\alpha _{n-1})Vx_{n-1} \\ &\qquad{} +(I-\alpha_{n}\mu F)S{ \varDelta }^{N}_{n}Gx_{n}-(I- \alpha_{n}\mu F)S{ \varDelta }^{N}_{n-1}Gx_{n-1} \\ &\qquad{} +\mu(\alpha_{n}-\alpha_{n-1})FS{ \varDelta }^{N}_{n-1}Gx_{n-1}\bigr\| \\ &\quad\leq\alpha_{n}\gamma l\|x_{n}-x_{n-1}\|+(1- \alpha_{n}\tau)\bigl\| { \varDelta }^{N}_{n}Gx_{n}-{ \varDelta }^{N}_{n-1}Gx_{n-1}\bigr\| \\ &\qquad{} +|\alpha_{n}-\alpha_{n-1}|\bigl(\gamma \|Vx_{n-1}\|+\mu\bigl\| FS{ \varDelta }^{N}_{n-1}Gx_{n-1} \bigr\| \bigr) \\ &\quad\leq\alpha_{n}\gamma l\|x_{n}-x_{n-1}\|+(1- \alpha_{n}\tau)\Biggl[\|x_{n}-x_{n-1}\| +\widetilde{M}_{2}{ \sum^{N}_{i=1}}|r_{i,n}-r_{i,n-1}| \Biggr] \\ &\qquad{} +|\alpha_{n}-\alpha_{n-1}|\bigl(\gamma \|Vx_{n-1}\|+\mu\bigl\| FS{ \varDelta }^{N}_{n-1}Gx_{n-1} \bigr\| \bigr) \\ &\quad\leq\bigl(1-\alpha_{n}(\tau-\gamma l)\bigr)\|x_{n}-x_{n-1} \|+\Biggl({ \sum^{N}_{i=1}}|r_{i,n}-r_{i,n-1}| +|\alpha_{n}-\alpha_{n-1}|\Biggr)\widetilde{M}_{2}, \end{aligned}$$
and hence
$$\begin{aligned} &\|x_{n+1}-x_{n}\| \\ &\quad\leq\bigl\| (I-\beta_{n}A)S{ \varDelta }^{N}_{n}Gx_{n}+ \beta_{n}y_{n}-(I-\beta _{n-1}A)S{ \varDelta }^{N}_{n-1}Gx_{n-1}-\beta_{n-1}y_{n-1} \bigr\| \\ &\quad\leq\bigl\| (I-\beta_{n}A) \bigl(S{ \varDelta }^{N}_{n}Gx_{n}-S{ \varDelta }^{N}_{n-1}Gx_{n-1}\bigr)\bigr\| \\ &\qquad{} +|\beta_{n}-\beta_{n-1}|\|A\|\bigl\| S{ \varDelta }^{N}_{n-1}Gx_{n-1}\bigr\| +\beta _{n} \|y_{n}-y_{n-1}\|+|\beta_{n}-\beta_{n-1}| \|y_{n-1}\| \\ &\quad\leq(1-\beta_{n}\bar{\gamma})\bigl\| S{ \varDelta }^{N}_{n}Gx_{n}-S{ \varDelta }^{N}_{n-1}Gx_{n-1}\bigr\| \\ &\qquad{} +\beta_{n}\Biggl[\bigl(1-\alpha_{n}(\tau-\gamma l) \bigr)\|x_{n}-x_{n-1}\|+\Biggl({ \sum ^{N}_{i=1}}|r_{i,n}-r_{i,n-1}| +| \alpha_{n}-\alpha_{n-1}|\Biggr)\widetilde{M}_{2} \Biggr] \\ &\qquad{} +|\beta_{n}-\beta_{n-1}|\bigl(\|A\|\bigl\| S{ \varDelta }^{N}_{n-1}Gx_{n-1}\bigr\| +\| y_{n-1}\| \bigr) \\ &\quad\leq(1-\beta_{n}\bar{\gamma})\Biggl[\|x_{n}-x_{n-1} \|+\widetilde{M}_{2}{ \sum^{N}_{i=1}}|r_{i,n}-r_{i,n-1}| \Biggr] \\ &\qquad{} +\beta_{n}\Biggl[\bigl(1-\alpha_{n}(\tau-\gamma l) \bigr)\|x_{n}-x_{n-1}\| \\ &\qquad{} +\Biggl({ \sum^{N}_{i=1}}|r_{i,n}-r_{i,n-1}|+| \alpha_{n}-\alpha _{n-1}|\Biggr)\widetilde{M}_{2} \Biggr]+|\beta_{n}-\beta_{n-1}|\widetilde{M}_{2} \\ &\quad\leq(1-\beta_{n}\bar{\gamma})\Biggl[\|x_{n}-x_{n-1} \|+\Biggl({ \sum^{N}_{i=1}}|r_{i,n}-r_{i,n-1}|+| \alpha_{n}-\alpha_{n-1}|\Biggr)\widetilde{M}_{2} \Biggr] \\ &\qquad{} +\beta_{n}\Biggl[\|x_{n}-x_{n-1}\| + \Biggl({ \sum^{N}_{i=1}}|r_{i,n}-r_{i,n-1}|+| \alpha_{n}-\alpha _{n-1}|\Biggr)\widetilde{M}_{2} \Biggr]+|\beta_{n}-\beta_{n-1}|\widetilde{M}_{2} \\ &\quad=\bigl(1-\beta_{n}(\bar{\gamma}-1)\bigr)\Biggl[ \|x_{n}-x_{n-1}\|+\Biggl({ \sum^{N}_{i=1}}|r_{i,n}-r_{i,n-1}|+| \alpha_{n}-\alpha_{n-1}|\Biggr)\widetilde{M}_{2} \Biggr] \\ &\qquad{} +|\beta_{n}-\beta_{n-1}|\widetilde{M}_{2} \\ &\quad\leq\bigl(1-\beta_{n}(\bar{\gamma}-1)\bigr)\|x_{n}-x_{n-1} \|+\Biggl({ \sum^{N}_{i=1}}|r_{i,n}-r_{i,n-1}|+| \alpha_{n}-\alpha_{n-1}|\Biggr)\widetilde{M}_{2} \\ &\qquad{} +\bigl(o(\beta_{n})+\sigma_{n-1}\bigr)\widetilde{M}_{2}. \end{aligned}$$
(3.35)
By taking \(a_{n+1}=\|x_{n+1}-x_{n}\|\), \(\omega_{n}=\beta_{n}(\bar{\gamma}-1)\), \(\omega_{n}\delta_{n}=\widetilde{M}_{2}o(\beta_{n})\) and \(r_{n}=({ \sum^{N}_{i=1}}|r_{i,n}-r_{i,n-1}|+|\alpha _{n}-\alpha_{n-1}|+\sigma_{n-1})\widetilde{M}_{2}\), from (3.35) we have
$$a_{n+1}\leq(1-\omega_{n})a_{n}+\omega_{n} \delta_{n}+r_{n}. $$
Consequently, utilizing the conditions (C2), (C5), (C6), and Lemma 2.8, we obtain
$$\lim_{n\to\infty}\|x_{n+1}-x_{n}\|=0. $$

In view of this observation, we have the following.

Corollary 3.5

Let \(\{x_{n}\}\) be the sequence generated by the explicit scheme (3.3), where the sequences \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), and \(\{r_{i,n}\}^{N}_{i=1}\) satisfy the conditions (C1), (C2), (C5), and (C6) (or the conditions (C1), (C2), (C3), and (C6), or the conditions (C1), (C2), (C4), and (C6)). Then \(\{ x_{n}\}\) converges strongly to a point \(\tilde{x}\in{ \varOmega }\), which is the unique solution of the VIP (3.2).

4 Applications

Let C be a nonempty, closed, and convex subset of a real Hilbert space H. For a given nonlinear mapping \({\mathcal{A}}:C\to H\), we consider the variational inequality problem (VIP) of finding \(x^{*}\in C\) such that
$$ \bigl\langle {\mathcal{A}}x^{*},y-x^{*}\bigr\rangle \geq0, \quad\forall y\in C. $$
(4.1)
We will denote by \(\operatorname{VI}(C,{\mathcal{A}})\) the set of solutions of the VIP (4.1).
Recall that if u is a point in C, then the following relation holds:
$$u\in\operatorname{VI}(C,{\mathcal{A}}) \quad\Leftrightarrow\quad u=P_{C}(I- \lambda{\mathcal {A}})u,\quad \lambda>0. $$
In the meantime, it is easy to see that the following relation holds:
$$ \operatorname{SVI}\mbox{ (1.13) with }F_{2}=0 \quad \Leftrightarrow\quad \operatorname{VIP}\mbox{ (4.1) with } { \mathcal{A}}=F_{1}. $$
(4.2)
An operator \({\mathcal{A}}:C\to H\) is said to be an α-inverse strongly monotone operator if there exists a constant \(\alpha>0\) such that
$$\langle{\mathcal{A}}x-{\mathcal{A}}y,x-y\rangle\geq\alpha\|{\mathcal {A}}x-{ \mathcal{A}}y\|^{2}, \quad\forall x,y\in C. $$

As an example, we recall that the α-inverse strongly monotone operators are firmly nonexpansive mappings if \(\alpha \geq1\) and that every α-inverse strongly monotone operator is also \(\frac{1}{\alpha}\)-Lipschitz-continuous (see [17]).

Let us observe also that, if \({\mathcal{A}}\) is α-inverse strongly monotone, the mappings \(P_{C}(I-\lambda{\mathcal{A}})\) are nonexpansive for all \(\lambda\in(0,2\alpha]\) since they are compositions of nonexpansive mappings.

Throughout the rest of this paper, we always assume the following:
  • \(F:C\to H\) is a κ-Lipschitzian and η-strongly monotone operator with positive constants \(\kappa,\eta>0\);

  • \(F_{j}:C\to H\) is \(\zeta_{j}\)-inverse strongly monotone for \(j=1,2\) and \(T_{i}:C\to C\) is a \(k_{i}\)-strictly pseudocontractive mapping for each \(i=1,\ldots,N\);

  • A is a γ̄-strongly positive bounded linear operator on H with \(\bar{\gamma}\in(1,2)\) and \(V:C\to H\) is an l-Lipschitzian mapping with \(l\geq0\);

  • \(0<\mu<\frac{2\eta}{\kappa^{2}}\) and \(0\leq\gamma l<\tau\) with \(\tau =1-\sqrt{1-\mu(2\eta-\mu\kappa^{2})}\);

  • \(G:C\to C\) is a mapping defined by \(Gx=P_{C}(I-\nu_{1}F_{1})P_{C}(I-\nu _{2}F_{2})x\) with \(0<\nu_{j}<2\zeta_{j}\) for \(j=1,2\), and the fixed point set of G is denoted by Ξ;

  • \({ \varDelta }^{N}_{t}:C\to C\) is a mapping defined by \({ \varDelta }^{N}_{t}x=(I-r_{N,t}{\mathcal{A}}_{N})\cdots(I-r_{1,t}{\mathcal{A}}_{1})x\), \(t\in(0,1)\) with \({\mathcal{A}}_{i}=I-T_{i}\) and \(\{r_{i,t}\}\subset [a_{i},b_{i}]\subset(0,1-k_{i})\) for each \(i=1,\ldots,N\);

  • \({ \varDelta }^{N}_{n}:C\to C\) is a mapping defined by \({ \varDelta }^{N}_{n}x=(I-r_{N,n}{\mathcal{A}}_{N})\cdots(I-r_{1,n}{\mathcal{A}}_{1})x\) with \(\{r_{i,n}\}\subset[a_{i},b_{i}]\subset(0,1-k_{i})\) and \(\lim_{n\to \infty}r_{i,n}=r_{i}\), for each \(i=1,\ldots,N\);

  • \({ \varOmega }=\bigcap^{N}_{i=1}\operatorname{Fix}(T_{i})\cap{ \varXi }\neq\emptyset\) and \(P_{{ \varOmega }}\) is the metric projection of H onto Ω;

  • \(\{\alpha_{n}\}\subset[0,1]\), \(\{\beta_{n}\}\subset(0,1]\) and \(\{\theta_{t}\} _{t\in(0,\min\{1,\frac{2-\bar{\gamma}}{\tau-\gamma l}\})} \subset(0,1)\).

We now introduce the following composite implicit relaxed extragradient-like scheme that generates a net \(\{x_{t}\}_{t\in(0,\min\{1, \frac{2-\bar{\gamma}}{\tau-\gamma l}\})}\) in one implicit manner:
$$ x_{t}=P_{C}\bigl[(I-\theta_{t}A){ \varDelta }^{N}_{t}Gx_{t}+\theta_{t}\bigl(t \gamma Vx_{t}+(I-t\mu F){ \varDelta }^{N}_{t}Gx_{t} \bigr)\bigr]. $$
(4.3)
Moreover, we also propose the following composite explicit relaxed extragradient-like scheme, which generates a sequence in another explicit way:
$$ \left \{ \textstyle\begin{array}{@{}l} y_{n}=\alpha_{n}\gamma Vx_{n}+(I-\alpha_{n}\mu F){ \varDelta }^{N}_{n}Gx_{n},\\ x_{n+1}=P_{C}[(I-\beta_{n}A){ \varDelta }^{N}_{n}Gx_{n}+\beta_{n}y_{n}],\quad \forall n\geq0, \end{array}\displaystyle \right . $$
(4.4)
where \(x_{0}\in C\) is an arbitrary initial guess.

Theorem 4.1

Let the net \(\{x_{t}\}\) be defined via (4.3). If \(\lim_{t\to0}\theta_{t}=0\), then \(x_{t}\) converges strongly to a point \(\tilde{x}\in{ \varOmega }\) as \(t\to0\), which solves the VIP
$$ \bigl\langle (A-I)\tilde{x},\tilde{x}-p\bigr\rangle \leq0,\quad \forall p\in{ \varOmega }. $$
(4.5)
Equivalently, we have \(P_{{ \varOmega }}(2I-A)\tilde{x}=\tilde{x}\).

Proof

First of all, it is easy to see that \(T_{i}:C\to C\) is \(k_{i}\)-strictly pseudocontractive if and only if
$$\langle T_{i}x-T_{i}y,x-y\rangle\leq\|x-y\|^{2}- \frac{1-k_{i}}{2}\bigl\| (I-T_{i})x-(I-T_{i})y\bigr\| ^{2}, \quad \forall x,y\in C. $$
It is clear that in this case the mapping \(I-T_{i}\) is \(\frac {1-k_{i}}{2}\)-inverse strongly monotone. Moreover, by Lemma 2.3(i) we know that if \(T_{i}:C\to C\) is \(k_{i}\)-strictly pseudocontractive, then \(T_{i}\) is Lipschitz-continuous with constant \(\frac{1 +k_{i}}{1-k_{i}}\), i.e., \(\|T_{i}x-T_{i}y\|\leq\frac{1+k_{i}}{1-k_{i}}\| x-y\|\) for all \(x,y\in C\).
In Theorem 3.1, we put \({ \varTheta }_{i}=0\), \(\varphi_{i}=0\), \({ \varPhi }_{1}={ \varPhi }_{2}=0\), \(T=I\), \(\lambda=0\), \({\mathcal{A}}_{i}=I-T_{i}\) and \(\eta_{i}=\frac{1-k_{i}}{2}\) for each \(i=1,\ldots,N\). Then \(T:C\to C\) is a k-strictly pseudocontractive mapping with \(k=0\) and \({\mathcal{A}}_{i}\) is \(\eta_{i}\)-inverse strongly monotone with \(\eta_{i}=\frac{1-k_{i}}{2}\) for each \(i=1,\ldots,N\). In this case, \(S=I\) and \(\operatorname{GMEP}({ \varTheta }_{i},\varphi_{i}, {\mathcal{A}}_{i})=\operatorname{VI}(C,{\mathcal{A}}_{i})\) for each \(i=1,\ldots,N\). Next let us show \(\operatorname{VI}(C,{\mathcal{A}}_{i})=\operatorname{Fix} (T_{i})\) for each \(i=1,\ldots,N\). Indeed, we have, for \(\nu>0\),
$$\begin{aligned} u\in\operatorname{VI}(C,{\mathcal{A}}_{i})&\quad\Leftrightarrow\quad \langle {\mathcal{A}}_{i}u,y-u\rangle\geq0,\quad\forall y\in C \\ &\quad\Leftrightarrow\quad \langle u-\nu{\mathcal{A}}_{i}u-u,u-y \rangle\geq 0,\quad\forall y\in C \\ &\quad\Leftrightarrow\quad u=P_{C}(u-\nu{\mathcal{A}}_{i}u) \\ &\quad\Leftrightarrow\quad u=P_{C}(u-\nu u+\nu T_{i}u) \\ &\quad\Leftrightarrow\quad \langle u-\nu u+\nu T_{i}u-u,u-y\rangle \geq0,\quad\forall y\in C \\ &\quad\Leftrightarrow\quad \langle u-T_{i}u,u-y\rangle\leq0,\quad \forall y\in C \\ &\quad\Leftrightarrow\quad u=T_{i}u \\ &\quad\Leftrightarrow\quad u\in\operatorname{Fix}(T_{i}). \end{aligned}$$
Hence, we conclude that
$${ \varOmega }=\bigcap^{N}_{i=1} \operatorname{GMEP}({ \varTheta }_{i},\varphi_{i},{\mathcal {A}}_{i})\cap\operatorname{Fix}(T)\cap{ \varXi } =\bigcap ^{N}_{i=1}\operatorname{VI}(C,{\mathcal{A}}_{i}) \cap\operatorname{Fix}(I)\cap{ \varXi }=\bigcap^{N}_{i=1} \operatorname{Fix}(T_{i})\cap{ \varXi }. $$
Also, observe that
$$\begin{aligned} S{ \varDelta }^{N}_{t}Gx_{t}&=ST^{({ \varTheta }_{N},\varphi _{N})}_{r_{N,t}}(I-r_{N,t}{ \mathcal{A}}_{N})\cdots T^{({ \varTheta }_{1},\varphi_{1})}_{r_{1,t}}(I-r_{1,t}{ \mathcal{A}}_{1})Gx_{t} \\ &=T^{({ \varTheta }_{N},\varphi_{N})}_{r_{N,t}}(I-r_{N,t}{\mathcal {A}}_{N})\cdots T^{({ \varTheta }_{1},\varphi_{1})}_{r_{1,t}}(I-r_{1,t}{ \mathcal{A}}_{1})Gx_{t} \\ &=P_{C}(I-r_{N,t}{\mathcal{A}}_{N})\cdots P_{C}(I-r_{2,t}{\mathcal {A}}_{2})P_{C} \bigl[(1-r_{1,t})Gx_{t}+r_{1,t}T_{1}Gx_{t} \bigr] \\ &=P_{C}(I-r_{N,t}{\mathcal{A}}_{N})\cdots P_{C}(I-r_{2,t}{\mathcal {A}}_{2}) (I-r_{1,t}{\mathcal{A}}_{1})Gx_{t} \\ &=\cdots \\ &=(I-r_{N,t}{\mathcal{A}}_{N})\cdots(I-r_{2,t}{ \mathcal {A}}_{2}) (I-r_{1,t}{\mathcal{A}}_{1})Gx_{t}. \end{aligned}$$
In this case, the implicit scheme (3.1) reduces to (4.3). Consequently, utilizing Theorem 3.1 we obtain the desired result. □

Remark 4.1

Theorem 4.1 extends and improves Ceng et al.’s hierarchical fixed point problem (1.7) for a nonexpansive mapping (see [32], Theorem 3.1) and Jung’s hierarchical fixed point problem (1.10) for a strict pseudocontraction (see [1], Theorem 3.1) to the hierarchical fixed point problem (4.5) with the constraint of SVI (1.13).

Taking \(F=\frac{1}{2}I\), \(\mu=2\), and \(\gamma=1\) in Theorem 4.1, we get

Corollary 4.1

Let \(\{x_{t}\}\) be defined by
$$x_{t}=P_{C}\bigl[(I-\theta_{t}A){ \varDelta }^{N}_{t}Gx_{t}+\theta_{t} \bigl(tVx_{t}+(1-t){ \varDelta }^{N}_{t}Gx_{t} \bigr)\bigr]. $$
If \(\lim_{t\to0}\theta_{t}=0\), then \(\{x_{t}\}\) converges strongly as \(t\to 0\) to a point \(\tilde{x}\in{ \varOmega }\), which is the unique solution of the VIP (4.5).

Next, by utilizing Theorem 3.2, we prove the following result in order to establish the strong convergence of the sequence \(\{x_{n}\}\) generated by the composite explicit relaxed extragradient-like scheme (4.4).

Theorem 4.2

Let \(\{x_{n}\}\) be the sequence generated by the explicit scheme (4.4), where \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) satisfy the following condition:
  1. (C1)

    \(\{\alpha_{n}\}\subset[0,1]\), \(\{\beta_{n}\}\subset(0,1]\), and \(\alpha _{n}\to0\), \(\beta_{n}\to0\) as \(n\to\infty\).

     
Let LIM be a Banach limit. Then
$$\operatorname{LIM}_{n}\bigl\langle (A-I)\tilde{x},\tilde{x}-x_{n}\bigr\rangle \leq0, $$
where \(\tilde{x}=\lim_{t\to0^{+}}x_{t}\) with \(x_{t}\) being defined by
$$ x_{t}=P_{C}\bigl[(I-\theta_{t}A){ \varDelta }^{N}Gx_{t}+\theta_{t}\bigl(t\gamma Vx_{t}+(I-t\mu F){ \varDelta }^{N}Gx_{t}\bigr) \bigr], $$
(4.6)
where \(G,{ \varDelta }^{N}:C\to C\) are defined by \(Gx=P_{C}(I-\nu _{1}F_{1})P_{C}(I-\nu_{2}F_{2})x\) and \({ \varDelta }^{N}x=(I-r_{N}{\mathcal{A}}_{N})\cdots(I-r _{1}{\mathcal{A}}_{1})x\) with \(\nu_{j}\in(0,2\zeta_{j})\), \(j=1,2\) and \(r_{i}\in [a_{i},b_{i}]\subset(0,1-k_{i})\) for each \(i=1,\ldots,N\).

Proof

In Theorem 3.2, we put \({ \varTheta }_{i}=0\), \(\varphi_{i}=0\), \({ \varPhi }_{1}={ \varPhi }_{2}=0\), \(T=I\), \(\lambda=0\), \({\mathcal{A}} _{i}=I-T_{i}\) and \(\eta_{i}=\frac{1-k_{i}}{2}\) for each \(i=1,\ldots,N\). Then \({\mathcal{A}}_{i}\) is \(\eta_{i}\)-inverse strongly monotone with \(\eta_{i}=\frac{1-k_{i}}{2}\) for each \(i=1,\ldots,N\). Utilizing similar arguments to those in the proof of Theorem 4.1, we get \(\operatorname{GMEP}({ \varTheta }_{i},\varphi_{i},{\mathcal{A}}_{i})=\operatorname{VI} (C,{\mathcal{A}}_{i})=\operatorname{Fix}(T_{i})\), \({ \varDelta }^{N}_{n}Gx_{n}=(I- r_{N,n}{\mathcal{A}}_{N})\cdots(I-r_{1,n}{\mathcal{A}}_{1})Gx_{n}\) and \({ \varDelta }^{N}Gx_{t}=(I-r_{N}{\mathcal{A}}_{N})\cdots(I-r_{1}{\mathcal {A}}_{1})Gx_{t}\). Thus, it is easy to see that the schemes (3.3) and (3.25) reduce to the ones (4.4) and (4.6), respectively. Consequently, by utilizing Theorem 3.2, we derive the desired result. □

Remark 4.2

Theorem 4.2 extends and improves Jung’s hierarchical fixed point problem (1.10) for a strict pseudocontraction (see [1], Theorem 3.2) to the hierarchical fixed point problem (4.5) with the constraint of SVI (1.13).

Now, using Theorem 4.2, we establish the strong convergence of the sequence \(\{x_{n}\}\) generated by the composite explicit relaxed extragradient-like scheme (4.4) to a point \(\tilde{x}\in{ \varOmega }\), which is also the unique solution of the VIP (4.5).

Theorem 4.3

Let \(\{x_{n}\}\) be the sequence generated by the explicit scheme (4.4), where \(\{\alpha_{n}\}\) and \(\{\beta _{n}\}\) satisfy the following conditions:
  1. (C1)

    \(\{\alpha_{n}\}\subset[0,1]\), \(\{\beta_{n}\}\subset(0,1]\), and \(\alpha _{n}\to0\), \(\beta_{n}\to0\) as \(n\to\infty\);

     
  2. (C2)

    \(\sum^{\infty}_{n=0}\beta_{n}=\infty\).

     
If \(\{x_{n}\}\) is weakly asymptotically regular (i.e., \(x_{n+1}-x_{n}\rightharpoonup0\)), then \(x_{n}\) converges strongly to a point \(\tilde{x}\in{ \varOmega }\), which is the unique solution of the VIP (4.5).

Proof

In Theorem 3.3, we put \({ \varTheta }_{i}=0\), \(\varphi_{i}=0\), \({ \varPhi }_{1}={ \varPhi }_{2}=0\), \(T=I\), \(\lambda=0\), \({\mathcal{A}}_{i}=I-T_{i}\), and \(\eta_{i}=\frac{1-k_{i}}{2}\) for each \(i=1,\ldots,N\). Then \({\mathcal{A}}_{i}\) is \(\eta_{i}\)-inverse strongly monotone with \(\eta_{i}=\frac{1-k_{i}}{2}\) for each \(i=1,\ldots,N\). Utilizing similar arguments to those in the proof of Theorem 4.1, we get \({ \varOmega }=\bigcap^{N}_{i=1}\operatorname{Fix}(T_{i})\cap{ \varXi }\) and \({ \varDelta }^{N}_{n}Gx_{n}=(I-r_{N,n}{\mathcal{A}}_{N})\cdots(I-r_{1,n} {\mathcal{A}}_{1})Gx_{n}\). So, it is easy to see that the scheme (3.3) reduces to the ones (4.4). Thus, in terms of Theorem 3.3 we derive the desired result. □

Remark 4.3

Theorem 4.3 extends and improves Ceng et al.’s hierarchical fixed point problem (1.7) for a nonexpansive mapping (see [32], Theorem 3.2) and Jung’s hierarchical fixed point problem (1.10) for a strict pseudocontraction (see [1], Theorem 3.3) to the hierarchical fixed point problem (4.5) with the constraint of SVI (1.13).

Corollary 4.2

Let \(\{x_{n}\}\) be the sequence generated by the explicit scheme (4.4). Assume that the sequences \(\{ \alpha_{n}\}\) and \(\{\beta_{n}\}\) satisfy the conditions (C1) and (C2) in Theorem 4.3. If \(\{x_{n}\}\) is asymptotically regular (i.e., \(x_{n+1}-x_{n}\to0\)), then \(\{x_{n}\}\) converges strongly to a point \(\tilde{x}\in{ \varOmega }\), which is the unique solution of the VIP (4.5).

Putting \(\mu=2\), \(F=\frac{1}{2}I\), and \(\gamma=1\) in Theorem 4.3, we obtain the following.

Corollary 4.3

Let \(\{x_{n}\}\) be generated by the following iterative scheme:
$$\left \{ \textstyle\begin{array}{@{}l} y_{n}=\alpha_{n}Vx_{n}+(1-\alpha_{n}){ \varDelta }^{N}_{n}Gx_{n},\\ x_{n+1}=P_{C}[(I-\beta_{n}A){ \varDelta }^{N}_{n}Gx_{n}+\beta_{n}y_{n}], \quad\forall n\geq0. \end{array}\displaystyle \right . $$
Assume that the sequences \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) satisfy the conditions (C1) and (C2) in Theorem  4.3. If \(\{x_{n}\}\) is weakly asymptotically regular (i.e., \(x_{n+1}-x_{n}\rightharpoonup0\)), then \(\{x_{n}\}\) converges strongly to a point \(\tilde{x}\in{ \varOmega }\), which is the unique solution of the VIP (4.5).

Putting \(\alpha_{n}=0\), \(\forall n\geq0\) in Corollary 4.3, we get the following.

Corollary 4.4

Let \(\{x_{n}\}\) be generated by the following iterative scheme:
$$x_{n+1}=P_{C}\bigl[\bigl(I-\beta_{n}(A-I)\bigr){ \varDelta }^{N}_{n}Gx_{n}\bigr], \quad\forall n\geq0. $$
Assume that the sequence \(\{\beta_{n}\}\) satisfies the conditions (C1) and (C2) in Theorem 4.3 with \(\alpha_{n}=0\), \(\forall n\geq0\). If \(\{x_{n}\}\) is weakly asymptotically regular (i.e., \(x_{n+1}-x_{n}\rightharpoonup0\)), then \(\{x_{n}\}\) converges strongly to a point \(\tilde{x}\in{ \varOmega }\), which is the unique solution of the VIP (4.5).

Remark 4.4

If \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) in Corollary 4.2 and \(\{r_{i,n}\}^{N}_{i=1}\) in \({ \varDelta }^{N}_{n}\) satisfy conditions (C2) and
  1. (C3)

    \(\sum^{\infty}_{n=0}|\alpha_{n+1}-\alpha_{n}|<\infty\) and \(\sum^{\infty}_{n=0}|\beta_{n+1}-\beta_{n}|<\infty\); or

     
  2. (C4)

    \(\sum^{\infty}_{n=0}|\alpha_{n+1}-\alpha_{n}|<\infty\) and \(\lim_{n\to \infty}\frac{\beta_{n}}{\beta_{n+1}}=1\); or, equivalently, \(\lim_{n\to\infty}\frac{\alpha_{n}-\alpha_{n+1}}{\alpha_{n+1}}=0\) and \(\lim_{n\to\infty}\frac{\beta_{n}-\beta_{n+1}}{\beta_{n+1}}=0\); or,

     
  3. (C5)

    \(\sum^{\infty}_{n=0}|\alpha_{n+1}-\alpha_{n}|<\infty\) and \(|\beta _{n+1}-\beta_{n}|\leq o(\beta_{n+1})+\sigma_{n}\), \(\sum^{\infty}_{n=0}\sigma_{n}<\infty\) (the perturbed control condition);

     
  4. (C6)

    \(\sum^{\infty}_{n=0}|r_{i,n+1}-r_{i,n}|<\infty\) for each \(i=1,\ldots,N\),

     
then the sequence \(\{x_{n}\}\) generated by (4.4) is asymptotically regular. Now we give only the proof in the case when \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), and \(\{r_{i,n}\}^{N}_{i=1}\) satisfy the conditions (C2), (C5), and (C6). Indeed, in Remark 3.1, we put \({ \varTheta }_{i}=0\), \(\varphi_{i}=0\), \({ \varPhi }_{1}={ \varPhi }_{2}=0\), \(T=I\), \(\lambda=0\), \({\mathcal{A}}_{i}=I-T_{i}\), and \(\eta_{i}=\frac{1- k_{i}}{2}\) for each \(i=1,\ldots,N\). Utilizing Remark 3.1, we derive the claim.

In view of Remark 4.4, we have the following.

Corollary 4.5

Let \(\{x_{n}\}\) be the sequence generated by the explicit scheme (4.4), where the sequences \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), and \(\{r_{i,n}\}^{N}_{i=1}\) satisfy Remark 4.4(C1), (C2), (C5), and (C6) (or Remark  4.4(C1), (C2), (C3), and (C6), or Remark 4.4(C1), (C2), (C4), and (C6)). Then \(\{x_{n}\}\) converges strongly to a point \(\tilde{x}\in{ \varOmega }\), which is the unique solution of the VIP (4.5).

Remark 4.5

Corollary 4.5 extends and improves Jung’s hierarchical fixed point problem (1.10) for a strict pseudocontraction (see [1], Corollary 4.5) to the hierarchical fixed point problem (4.5) with the constraint of SVI (1.13).

5 Concluding remarks

We introduced and analyzed one composite implicit relaxed extragradient-like scheme and another composite explicit relaxed extragradient-like scheme for finding a common solution of a finite family of generalized mixed equilibrium problems (GMEPs) with the constraints of a system of generalized equilibrium problems (SGEP) and the hierarchical fixed point problem (HFPP) for a strictly pseudocontractive mapping by virtue of the general composite implicit and explicit schemes for a nonexpansive mapping \(T:H\to H\) (see [32]) and the general composite implicit and explicit ones for a strict pseudocontraction \(T:H\to H\) (see [1]). Our Theorems 3.1-3.3 and Corollary 3.5 improve, extend, supplement, and develop Theorems 3.1 and 3.2 of [32], Theorems 3.1-3.3 and Corollary 3.5 of [1] and Theorem 3.1 of [23] in the following aspects.

(i) Ceng et al.’s general composite implicit scheme for a nonexpansive mapping \(T:H\to H\) (see (3.1) in [32]) and Jung’s general composite implicit one for a strict pseudocontraction \(T:H\to H\) (see (3.1) in [1]) extends to developing the composite implicit relaxed extragradient-like scheme (3.1) for a finite family of GMEPs with constraints of SGEP (1.12) and the HFPP for a strict pseudocontraction. Moreover, Ceng et al.’s general composite explicit scheme for a nonexpansive mapping \(T:H\to H\) (see (3.5) in [32]) and Jung’s general composite explicit one for a strict pseudocontraction (see (3.3) in [1]) extends to developing the composite explicit relaxed extragradient-like one (3.3) for a finite family of GMEPs with constraints of SGEP (1.12) and the HFPP for a strict pseudocontraction.

(ii) The argument techniques in our Theorems 3.1-3.3 and Corollary 3.5 are very different from those techniques in [32] Theorems 3.1-3.2 and [1] Theorems 3.1-3.3 and Corollary 3.5 because we make use of the properties of the resolvent \(T^{({ \varTheta }, \varphi)}_{r}\) (see, e.g., Proposition 2.2 and the argument of (3.9), (3.14), (3.16), and (3.20)), the ones of the strong positive bounded linear operators (see Lemma 2.9), the ones of the Banach limit LIM (see Lemma 2.10), the equivalence of the fixed point equation \(x^{*}=T^{{ \varPhi }_{1}}_{\nu_{1}}(I-\nu _{1}F_{1})T^{{ \varPhi }_{2}}_{\nu_{2}}(I-\nu_{2}F_{2})x^{*}\) to the SGEP (1.12) for \(\zeta_{j}\)-inverse strongly monotone mappings \(F_{j}:C\to H\), \(j=1,2\) (see Proposition 2.3) and the contractive coefficient estimates for the contractions \(T^{\lambda}\) associating with nonexpansive mappings (see Lemma 2.7).

(iii) The problem of finding a common solution \(\tilde{x}\in\bigcap^{N}_{i=1}\operatorname{GMEP}({ \varTheta }_{i},\varphi_{i},{\mathcal{A}}_{i}) \cap\operatorname{Fix}(T)\cap{ \varXi }\) of SGEP (1.12), the fixed point problem of a k-strict pseudocontraction T and a finite family of GMEPs in our Theorems 3.1-3.3 and Corollary 3.5 is more general and more flexible than the one of finding a fixed point of a nonexpansive mapping \(T:H\to H\) in [32] Theorems 3.1 and 3.2, the one of finding a fixed point of a strictly pseudocontractive mapping \(T:H\to H\) in [1] Theorems 3.1-3.3 and Corollary 3.5, and the one of finding a common solution of GMEP (1.11), SGEP (1.12), and the fixed point problem of a k-strict pseudocontraction T in [17] Theorem 3.1. It is worth pointing out that the problem of finding \(\tilde{x}\in(\bigcap^{N}_{i=1}\operatorname{GMEP}({ \varTheta }_{i},\varphi_{i},{\mathcal{A}}_{i}))\cap\operatorname{Fix}(T)\cap{ \varXi }\) extends the fixed point problems in [1, 32] from the domain H of the mapping T to the domain C for the one of finding \(\tilde{x}\in(\bigcap^{N}_{i=1}\operatorname{GMEP}({ \varTheta }_{i},\varphi_{i},{\mathcal {A}}_{i}))\cap\operatorname{Fix}(T)\cap{ \varXi }\) and generalizes the fixed point problems in [1, 32] to the setting of SGEP (1.12) and a finite family of GMEPs. In the meantime, the problem of finding \(\tilde{x} \in(\bigcap^{N}_{i=1}\operatorname{GMEP}({ \varTheta }_{i},\varphi_{i},{\mathcal {A}}_{i}))\cap\operatorname{Fix}(T)\cap{ \varXi }\) extends the problem of finding \(\tilde{x} \in\operatorname{GMEP}({ \varTheta },\varphi, {\mathcal{A}})\cap\operatorname{Fix}(T)\cap{ \varXi }\) in [23] from one GMEP to a finite family of GMEPs.

(iv) Our Theorems 3.1-3.3 and Corollary 3.5 generalize [32] Theorems 3.1 and 3.2 from a nonexpansive mapping \(T:H\to H\) to a k-strict pseudocontraction \(T:C\to C\) and extend [32] Theorems 3.1 and 3.2 to the setting of SGEP (1.12) and a finite family of GMEPs. Moreover, Theorems 3.1-3.3 and Corollary 3.5 generalize Theorems 3.1-3.3 and Corollary 3.5 of [1] from a strict pseudocontraction \(T:H\to H\) to the setting of SGEP (1.12) and a finite family of GMEPs. In the meantime, the operators \(T_{t}\) in the implicit scheme (3.1) of Jung [1] are replaced by the composite ones \(S{ \varDelta }^{N}_{t}G\) in our implicit scheme (3.1) and the operators \(T_{n}\) in the explicit scheme (3.3) of Jung [1] are replaced by the composite ones \(S{\varDelta }^{N}_{n}G\) in our explicit scheme (3.3).

Declarations

Acknowledgements

In this research, the first author was partially supported by the Innovation Program of Shanghai Municipal Education Commission (15ZZ068), the Ph.D. Program Foundation of Ministry of Education of China (20123127110002), and the Program for Outstanding Academic Leaders in Shanghai City (15XD1503100). The second author and third author were supported partly by the National Science Council of the Republic of China.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities
(2)
Department of Information Management, and Innovation Center for Big Data and Digital Convergence, Yuan Ze University
(3)
Center for Fundamental Science, Kaohsiung Medical University

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