New general systems of set-valued variational inclusions involving relative $(A,\eta )$-maximal monotone operators in Hilbert spaces

Abstract

The purpose of this paper is to introduce and study a class of new general systems of set-valued variational inclusions involving relative $(A,\eta )$-maximal monotone operators in Hilbert spaces. By using the generalized resolvent operator technique associated with relative $(A,\eta )$-maximal monotone operators, we also construct some new iterative algorithms for finding approximation solutions to the general systems of set-valued variational inclusions and prove the convergence of the sequences generated by the algorithms. The results presented in this paper improve and extend some known results in the literature.

1 Introduction

Recently, some systems of variational inequalities, variational inclusions, complementarity problems, and equilibrium problems have been studied by many authors because of their close relations to some problems arising in economics, mechanics, engineering science and other pure and applied sciences. Among these methods, the resolvent operator technique is very important. Huang and Fang [1] introduced a system of order complementarity problems and established some existence results for the system using fixed point theory. Verma [2] introduced and studied some systems of variational inequalities and developed some iterative algorithms for approximating the solutions of the systems of variational inequalities. Cho et al. [3] introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces. Further, the authors proved some existence and uniqueness theorems of solutions for the systems, and also constructed some iterative algorithms for approximating the solution of the systems of nonlinear variational inequalities, respectively.

Moreover, Fang et al. [4], Yan et al. [5], Fang and Huang [6] introduced and studied some new systems of variational inclusions involving H-monotone operators and $(H,\eta )$-monotone operators in Hilbert space, respectively. Using the corresponding resolvent operator technique associated with H-monotone operators, $(H,\eta )$-monotone operators, the authors proved the existence of solutions for the variational inclusion systems and constructed some algorithms for approximating the solutions of the systems and discussed convergence of the iteration sequences generated by the algorithms, respectively. Very recently, Lan et al. [7] introduced and studied a new system of nonlinear A-monotone multivalued variational inclusions in Hilbert spaces. By using the concept and properties of A-monotone operators, and the resolvent operator technique associated with A-monotone operators due to Verma [8], the authors constructed a new iterative algorithm for solving this system of nonlinear multivalued variational inclusions with A-monotone operators in Hilbert spaces and proved the existence of solutions for the nonlinear multivalued variational inclusion systems and the convergence of iterative sequences generated by the algorithm. For some related work, see, for example, [1–32] and the references therein.

On the other hand, Cao [33] introduced and studied a new system of generalized quasi-variational-like-inclusions applying the η-proximal mapping technique. Further, Agarwal and Verma [34] introduced and studied relative $(A,\eta )$-maximal monotone operators and discussed the approximation solvability of a new system of nonlinear (set-valued) variational inclusions involving $(A,\eta )$-maximal relaxed monotone and relative $(A,\eta )$-maximal monotone operators in Hilbert spaces based on a generalized hybrid iterative algorithm and the general $(A,\eta )$-resolvent operator method.

Inspired and motivated by the above works, the purpose of this paper is to consider the following new general system of set-valued variational inclusions involving relative $(A,\eta )$-maximal monotone operators in Hilbert spaces: Find $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })\in H_1\times H_2\times \cdots \times H_m$ and $u_{ij}^{\ast }\in U_{ij}(x_j^{\ast })$ for any $i,j=1,2,\dots ,m$ such that

$$0\in F_i(u_{i1}^{\ast },u_{i2}^{\ast },\dots ,u_{im}^{\ast })+M_i\left(g_i\left(x_i^{\ast }\right)\right),$$

(1.1)

where m is a given positive integer, $F_i:H_1\times H_2\times \cdots \times H_m\to H_i$, $A_i:H_i\to H_i$, $g_i:H_i\to H_i$ and ${\eta }_i:H_i\times H_i\to H_i$ are single-valued operators, $U_{ij}:H_j\to 2^{H_j}$ is a set-valued operator and $M_i:H_i\to 2^{H_i}$ is relative $(A_i,{\eta }_i)$-maximal monotone.

We note that for appropriate and suitable choices of positive integer m, the operators $F_i$, $g_i$, $A_i$, ${\eta }_i$, $M_i$, $U_{ij}$, and $H_i$ for $i,j=1,2,\dots ,m$, one can know that the problem (1.1) includes a number of known general problems of variational character, including variational inequality (system) problems, variational inclusion (system) problems as special cases. For more details, see [1–31, 35] and the following examples.

Example 1.1 For $i,j=1,2,\dots ,m$, if $U_{ij}=T_{ij}$ is single-valued operator, the problem (1.1) reduces to finding $x_j\in H_j$, such that

$$0\in F_i(T_{i1}{x}_1^{\ast },T_{i2}{x}_2^{\ast },\dots ,T_{im}{x}_m^{\ast })+M_i\left(g_i\left(x_i^{\ast }\right)\right).$$

(1.2)

Example 1.2 For $i=1,2,\dots ,m$, if $H_i=H$ and $A_i\equiv I$, an identity operator, and $M_i=\partial {\phi }_i$, where ${\phi }_i:H\to R\cup \{+\mathrm{\infty }\}$ is proper and lower semi-continuous ${\eta }_i$-subdifferentiable functional and $\partial {\phi }_i$ denotes ${\eta }_i$-subdifferential operator, then the problem (1.1) reduces to finding $x_i^{\ast }\in H$ and $u_{ij}^{\ast }\in U_{ij}(x_j^{\ast })$ for $j=1,2,\dots ,m$ such that

$$〈F_i(u_{i1}^{\ast },u_{i2}^{\ast },\dots ,u_{im}^{\ast }),{\eta }_i(x,g_i\left(x_i^{\ast }\right))〉\ge {\phi }_i\left(g_i\left(x_i^{\ast }\right)\right)-{\phi }_i(x),\mathrm{\forall }x\in H.$$

(1.3)

The problem (1.3) is called a set-valued nonlinear generalized quasi-variational-like-inclusion system, which was considered and studied by Cao [33].

Example 1.3 When $m=2$ and $g_i\equiv I$ for $i=1,2$, then the problem (1.1) is equivalent to the following nonlinear set-valued variational inclusion system problem: Find $(x_1^{\ast },x_2^{\ast })\in H_1\times H_2$ and $u_1^{\ast }\in U_1(x_1^{\ast })$, $u_2^{\ast }\in U_2(x_2^{\ast })$ such that

$$\begin{array}{r}0\in F_1(x_1^{\ast },u_2^{\ast })+M_1\left(x_1^{\ast }\right),\\ 0\in F_2(u_1^{\ast },x_2^{\ast })+M_2\left(x_2^{\ast }\right),\end{array}$$

(1.4)

which was studied by Agarwal and Verma [34].

Example 1.4 If $m=2$ and $M_i(x_i)=\partial {\phi }_i(x_i)$, where ${\phi }_i:H_i\to R\cup \{+\mathrm{\infty }\}$ is proper, convex, and lower semi-continuous functional and $\partial {\phi }_i$ denotes the subdifferential operator of ${\phi }_i$ for all $x_i\in H_i$, $i=1,2$, then the problem (1.4) reduces to the following system of set-valued mixed variational inequalities: Find $(x_1^{\ast },x_2^{\ast })\in H_1\times H_2$, $u_1^{\ast }\in U_1(x_1^{\ast })$ and $u_2^{\ast }\in U_2(x_2^{\ast })$ such that

$$\begin{array}{r}〈F_1(x_1^{\ast },u_2^{\ast }),x-x_1^{\ast }〉+{\phi }_1(x)-{\phi }_1\left(x_1^{\ast }\right)\ge 0,\mathrm{\forall }x\in H_1,\\ 〈F_2(u_1^{\ast },x_2^{\ast }),y-x_2^{\ast }〉+{\phi }_2(y)-{\phi }_2\left(x_2^{\ast }\right)\ge 0,\mathrm{\forall }y\in H_2.\end{array}$$

(1.5)

If $U_1=U_2\equiv I$, then the problem (1.5) reduces to finding $(x_1^{\ast },x_2^{\ast })\in H_1\times H_2$ such that

$$\begin{array}{r}〈F_1(x_1^{\ast },x_2^{\ast }),x-x_1^{\ast }〉+{\phi }_1(x)-{\phi }_1\left(x_1^{\ast }\right)\ge 0,\mathrm{\forall }x\in H_1,\\ 〈F_2(x_1^{\ast },x_2^{\ast }),y-x_2^{\ast }〉+{\phi }_2(y)-{\phi }_2\left(x_2^{\ast }\right)\ge 0,\mathrm{\forall }y\in H_2,\end{array}$$

(1.6)

which is called the system of nonlinear variational inequalities considered by Cho et al. [3]. Some specializations of the problem (1.6) are dealt by Kim and Kim [35].

Example 1.5 If $m=2$ and $U_1=U_2=g_1=g_2\equiv I$, then the problem (1.1) reduces to the problem of finding $(x_1^{\ast },x_2^{\ast })\in H_1\times H_2$ such that

$$\begin{array}{c}0\in F_1(x_1^{\ast },x_2^{\ast })+M_1\left(x_1^{\ast }\right),\hfill \\ 0\in F_2(x_1^{\ast },x_2^{\ast })+M_2\left(x_2^{\ast }\right),\hfill \end{array}$$

which was introduced and studied by Fang et al. [4].

Moreover, by using the generalized resolvent operator technique associated with relative $(A,\eta )$-maximal monotone operators, we also construct some new iterative algorithms for finding approximation solutions to the general systems of set-valued variational inclusions and prove convergence of the sequences generated by the algorithms.

2 Preliminaries

Throughout, let H and $H_i$ ($i=1,2,\dots ,m$) be real Hilbert spaces and endowed with the norm $\parallel \cdot \parallel$ and inner product $〈\cdot ,\cdot 〉$. Let $2^H$ and $C(H)$ denote the family of all the nonempty subsets of H and the family of all closed subsets of H, respectively.

Definition 2.1 Let $T:H\to H$ be a single-valued operator. Then the map T is said to be

• (i) r-strongly monotone, if there exists a constant $r>0$ such that

  $$〈T(x)-T(y),x-y〉\ge r{\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in H;$$

• (ii) β-Lipschitz continuous, if there exists a constant $\beta >0$ such that

  $$\parallel Tx-Ty\parallel \le \beta \parallel x-y\parallel ,\mathrm{\forall }x,y\in H.$$

Definition 2.2 Let $\eta :H\times H\to H$ and $A:H\to H$ be single-valued operators, $M:H\to 2^H$ be set-valued operator. Then

• (i) η is said to be t-strongly monotone, if there exists a constant $t>0$ such that

  $$〈\eta (x,y),x-y〉\ge t{\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in H;$$

• (ii) η is said to be τ-Lipschitz continuous, if there exists a constant $\tau >0$ such that

  $$\parallel \eta (x,y)\parallel \le \tau \parallel x-y\parallel ,\mathrm{\forall }x,y\in H;$$

• (iii) A is said to be η-monotone, if

  $$〈A(x)-A(y),\eta (x,y)〉\ge 0,\mathrm{\forall }x,y\in H;$$

• (iv) A is said to be strictly η-monotone, if A is η-monotone and

  $$〈A(x)-A(y),\eta (x,y)〉=0\text{if and only if}x=y;$$

• (v) A is said to be $(r,\eta )$-strongly monotone, if there exists a constant $r>0$ such that

  $$〈A(x)-A(y),\eta (x,y)〉\ge r{\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in H;$$

• (vi) M is said to be η-monotone with respect to A (or relative $(A,\eta )$-monotone) if

  $$〈u-v,\eta (A(x),A(y))〉\ge 0,\mathrm{\forall }x,y\in H,u\in M(x),v\in M(y);$$

• (vii) M is said to be relative $(A,\eta )$-maximal monotone, if M is η-monotone with respect to A (or relative $(A,\eta )$-monotone) and $(A+\lambda M)(H)=H$, where $\lambda >0$ is an arbitrary constant.

Definition 2.3 For $i,j=1,2,\dots ,m$, let $H_i$ be a Hilbert space, $A_j:H_j\to H_j$ be single-valued operator, $U_{ij}:H_j\to 2^{H_j}$ be set-valued operator. Then nonlinear operator $F_i:H_1\times H_2\times \cdots \times H_m\to H_i$ is said to be

• (i) $(U_{ij},c_j,{\mu }_j)$-relaxed cocoercive with respect to $A_j$ (or relative $(U_{ij},c_j,{\mu }_j)$-relaxed cocoercive) in the j th argument, if there exist constants $c_j,{\mu }_j>0$ such that for all $x_j^1,x_j^2\in H_j$, and for any $u_j^1\in U_{ij}(x_j^1)$, $u_j^2\in U_{ij}(x_j^2)$,

  $$\begin{array}{r}〈F_i(\dots ,u_j^1,\dots )-F_i(\dots ,u_j^2,\dots ),A_j\left(x_j^1\right)-A_j\left(x_j^2\right)〉\\ \ge (-c_j){\parallel F_i(\dots ,u_j^1,\dots )-F_i(\dots ,u_j^2,\dots )\parallel }^2+{\mu }_j{\parallel x_j^1-x_j^2\parallel }^2;\end{array}$$

• (ii) ${\zeta }_{ij}$-Lipschitz continuous in the j th argument, if there exists constant ${\zeta }_{ij}>0$ such that for all $x_j,y_j\in H_j$,

  $$\parallel F_i(x_1,\dots ,x_{j-1},x_j,x_{j+1},\dots ,x_m)-F_i(x_1,\dots ,x_{j-1},y_j,x_{j+1},\dots ,x_m)\parallel \le \parallel x_j-y_j\parallel .$$

Remark 2.1

• (i) When $m=1$ and $U=I$, then (i) and (ii) of Definition 2.3 reduce to corresponding concept of the relative relaxed cocoerciveness and Lipschitz continuity, respectively.

• (ii) If $U_{ij}=T_{ij}$ is single-valued operator for $i,j=1,2,\dots ,m$, then $F_i$ is $(U_{ij},c_j,{\mu }_j)$-relaxed cocoercive with respect to $A_j$ in the j th argument reduce to $(T_{ij},c_j,{\mu }_j)$-relaxed cocoercive with respect to $A_j$ in the j th argument, that is, if there exist constants $c_j,{\mu }_j>0$ such that for all $x_j^1,x_j^2\in H_j$,

  $$\begin{array}{c}〈F_i(\dots ,T_{ij}{x}_j^1,\dots )-F_i(\dots ,T_{ij}{x}_j^2,\dots ),A_j\left(x_j^1\right)-A_j\left(x_j^2\right)〉\hfill \\ \ge (-c_j){\parallel F_i(\dots ,T_{ij}{x}_j^1,\dots )-F_i(\dots ,T_{ij}{x}_j^2,\dots )\parallel }^2+{\mu }_j{\parallel x_j^1-x_j^2\parallel }^2.\hfill \end{array}$$

Lemma 2.1 ([34])

Let $\eta :H\times H\to H$ be a single-valued mapping, $A:H\to H$ be a strictly η-monotone mapping and $M:H\to 2^H$ be a relative $(A,\eta )$-maximal monotone mapping. Then the mapping $(A+\lambda M)$ is single-valued, where $\lambda >0$ is arbitrary constant.

Definition 2.4 Let $\eta :H\times H\to H$ be a single-valued mapping, $A:H\to H$ be a strictly η-monotone mapping and $M:H\to 2^H$ be a relative $(A,\eta )$-maximal monotone mapping. Then generalized resolvent operator $R_{M,\lambda }^{A,\eta }:H\to H$ is defined by

$$R_{M,\lambda }^{A,\eta }(z)={(A+\lambda M)}^{-1}(z),\mathrm{\forall }z\in H,$$

where $\lambda >0$ is a constant.

Lemma 2.2 ([34])

Let $\eta :H\times H\to H$ be a t-strongly monotone and τ-Lipschitz continuous mapping, $A:H\to H$ be an r-strongly monotone mapping, and $M:H\to 2^H$ be a relative $(A,\eta )$-maximal monotone mapping. Then generalized resolvent operator $R_{M,\lambda }^{A,\eta }:H\to H$ is $\frac{\tau }{rt}$-Lipschitz continuous, that is,

$$\parallel R_{M,\lambda }^{A,\eta }(x)-R_{M,\lambda }^{A,\eta }(y)\parallel \le \frac{\tau }{rt}\parallel x-y\parallel ,\mathrm{\forall }x,y\in H.$$

Definition 2.5 A set-valued operator $U:H\to 2^H$ is said to be D-γ-Lipschitz continuous, if there exists a constant $\gamma >0$ such that

$$D(U(x),U(y))\le \gamma \parallel x-y\parallel ,\mathrm{\forall }x,y\in H,$$

where $D:C(H)\times C(H)\to R\cup \{+\mathrm{\infty }\}$ is called the Hausdorff pseudo-metric defined as follows:

$$D(U,V)=max\{\underset{x\in U}{sup}\underset{y\in V}{inf}\parallel x-y\parallel ,\underset{y\in V}{sup}\underset{x\in U}{inf}\parallel x-y\parallel \},\mathrm{\forall }U,V\in C(H).$$

Furthermore, the Hausdorff pseudo-metric D reduces to the Hausdorff metric when $C(H)$ is restricted to closed bounded subsets of the family $\mathit{CB}(H)$.

Lemma 2.3 Let $\theta \in (0,1)$ be a constant. Then function $f(\lambda )=1-\lambda +\lambda \theta$ for $\lambda \in [0,1]$ is nonnegative and strictly decrease and $f(\lambda )\in [0,1]$. Further, if $\lambda \ne 0$, then $f(\lambda )\in (0,1)$.

Lemma 2.4 ([36])

Let $\{a_n\}$ and $\{b_n\}$ be two nonnegative real sequences satisfying

$$a_{n+1}\le \theta a_n+b_n$$

with $0<\theta <1$ and ${lim}_{n\to \mathrm{\infty }}{b}_n=0$. Then ${lim}_{n\to \mathrm{\infty }}{a}_n=0$.

3 Iterative algorithm and convergence analysis

In this section, we construct a class of new iterative algorithms for finding approximate solutions of the problems (1.1) and (1.2), respectively. Then the convergence criterion for the algorithms is also discussed.

Lemma 3.1 Let $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })\in H_1\times H_2\times \cdots \times H_m$ and $u_{ij}^{\ast }\in U_{ij}(x_j^{\ast })$ for $i,j=1,2,\dots ,m$, then $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast },u_{11}^{\ast },\dots ,u_{1m}^{\ast },\dots ,u_{m1}^{\ast },\dots ,u_{mm}^{\ast })$ (denoted by $(\ast )$) is a solution of the problem (1.1) if and only if $(\ast )$ satisfy

$$g_i\left(x_i^{\ast }\right)=R_{M_i,{\rho }_i}^{A_i,{\eta }_i}[A_i\left(g_i\left(x_i^{\ast }\right)\right)-{\rho }_i{F}_i(u_{i1}^{\ast },\dots ,u_{ii-1}^{\ast },u_{ii}^{\ast },u_{ii+1}^{\ast },\dots ,u_{im}^{\ast })],$$

(3.1)

where $R_{M_i,{\rho }_i}^{A_i,{\eta }_i}={(A_i+{\rho }_i{M}_i)}^{-1}$ and ${\rho }_i>0$ is a constant for $i=1,2,\dots ,m$.

Proof It follows from the definition of generalized resolvent operator $R_{M_i,{\rho }_i}^{A_i,{\eta }_i}$ that the proof can be obtained directly, and so it is omitted. □

Algorithm 3.1

Step 1. Setting $(x_1^0,x_2^0,\dots ,x_m^0)\in H_1\times H_2\times \cdots \times H_m$ and choose $u_{ij}^0\in U_{ij}(x_j^0)$ for $i,j=1,2,\dots ,m$.

Step 2. Let

$$\begin{array}{rcl}{x}_i^{n+1}& =& (1-\lambda )x_i^n+\lambda \{x_i^n-g_i\left(x_i^n\right)+R_{M_i,{\rho }_i}^{A_i,{\eta }_i}[A_i\left(g_i\left(x_i^n\right)\right)\\ -{\rho }_i{F}_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^n,u_{ii+1}^n,\dots ,u_{im}^n)\left]\right\}\end{array}$$

(3.2)

for all $i=1,2,\dots ,m$ and $n=0,1,2,\dots$ , where $\lambda \in (0,1]$ is a constant.

Step 3. By the results of Nadler [37], we can choose $u_{ij}^{n+1}\in U_{ij}(x_j^{n+1})$ such that

$$\parallel u_{ij}^{n+1}-u_{ij}^n\parallel \le (1+\frac{1}{n+1})D_j(U_{ij}\left(x_j^{n+1}\right),U_{ij}\left(x_j^n\right)),$$

(3.3)

where $D_j(\cdot ,\cdot )$ is the Hausdorff pseudo-metric on $C(H_j)$ and $i,j=1,2,\dots ,m$.

Step 4. If $x_i^{n+1}$ and $u_{ij}^{n+1}$ for $i,j=1,2,\dots ,m$ satisfy (3.2) to sufficient accuracy, stop. Otherwise, set $n:=n+1$ and return to Step 2.

Remark 3.1 If $R_{M_i,{\rho }_i}^{A_i,{\eta }_i}$ reduces to $J_{\rho }^{{\phi }_i}={(I+\rho \partial {\phi }_i)}^{-1}$, where ${\phi }_i:H_i\to R\cup \{+\mathrm{\infty }\}$ is proper and lower semi-continuous ${\eta }_i$-subdifferentiable functional, $H_i\equiv H$ for $i=1,2,\dots ,m$ and $\lambda =1$, then Algorithm 3.1 reduces to Algorithm (I) of Cao [33].

When $\lambda =1$ and $U_{ij}=T_{ij}$ is single-valued operator for $i,j=1,2,\dots ,m$, then Algorithm 3.1 reduces to the following algorithm for the problem (1.2).

Algorithm 3.2 For any given $(x_1^0,x_2^0,\dots ,x_m^0)\in H_1\times H_2\times \cdots \times H_m$, we compute $x_i^n$ as follows:

$$\begin{array}{rcl}{x}_i^{n+1}& =& x_i^n-g_i\left(x_i^n\right)+R_{M_i,{\rho }_i}^{A_i,{\eta }_i}[A_i\left(g_i\left(x_i^n\right)\right)\\ -{\rho }_i{F}_i(T_{i1}{x}_1^n,\dots ,T_{ii-1}{x}_{i-1}^n,T_{ii}{x}_i^n,T_{ii+1}{x}_{i+1}^n,\dots ,T_{im}{x}_m^n)]+w_i^n\end{array}$$

(3.4)

for $n=0,1,2,\dots$ and $i=1,2,\dots ,m$, where $w_i^n\in H_i$ is error to take into account a possible inexact computation of the resolvent operator point satisfying conditions ${lim}_{n\to \mathrm{\infty }}\parallel w_i^n\parallel =0$.

Remark 3.2

• (i) Let $m=2$, $g_i\equiv I$, $U_{ii}\equiv I$ for $i=1,2$, then Algorithm 3.1 reduces to Algorithm 4.3 of Agarwal and Verma [34].

• (ii) If for appropriate and suitable choices of positive integer m and mappings $F_i$, $g_i$, $A_i$, ${\eta }_i$, M, $U_{ij}$, and $H_i$ for $i,j=1,2,\dots ,m$, one can know that Algorithms 3.1-3.2 are extending a number of known algorithms.

In the sequel, we provide main result concerning the problem (1.1) with respect to Algorithm 3.1.

Theorem 3.1 For $i=1,2,\dots ,m$, let ${\eta }_i:H_i\times H_i\to H_i$ be ${\tau }_i$-Lipschitz continuous and $t_i$-strongly monotone operator, $A_i:H_i\to H_i$ be ${\beta }_i$-Lipschitz continuous and $r_i$-strongly monotone operator, $g_i:H_i\to H_i$ be ${\xi }_i$-Lipschitz continuous and ${\delta }_i$-strongly monotone operator and $M_i:H_i\to 2^{H_i}$ be relative $(A_i,{\eta }_i)$-maximal monotone. Suppose that $U_{ij}:H_j\to C{H}_j$ is $D_j$-${\gamma }_{ij}$-Lipschitz continuous, $F_i:H_1\times H_2\times \cdots \times H_m\to H_i$ is $(U_{ii},c_i,{\mu }_i)$-relaxed cocoercive with respect to $A_i$ in the ith argument and ${\zeta }_{ij}$-Lipschitz continuous in the jth for $i,j=1,2,\dots ,m$. If there exists constant ${\rho }_i>0$ for such that

$$\begin{array}{rcl}{\theta }_j& =& \frac{{\tau }_j}{r_j{t}_j}\cdot \sqrt{{\beta }_j^2{\xi }_j^2-2{\rho }_j{\mu }_j{\delta }_j^2+2{\rho }_j{c}_j{\zeta }_{jj}^2{\gamma }_{jj}^2+{\rho }_j^2{\zeta }_{jj}^2{\gamma }_{jj}^2}\\ +\sqrt{1-2{\delta }_j+{\xi }_j^2}+\sum _{i=1,i\ne j}^m\frac{{\rho }_i{\tau }_i{\zeta }_{ij}{\gamma }_{ij}}{r_i{t}_i}<1\end{array}$$

(3.5)

for all $j=1,2,\dots ,m$, then the problem (1.1) admits a solution $(\ast )$, i.e. $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast },u_{11}^{\ast },\dots ,u_{1m}^{\ast },\dots ,u_{m1}^{\ast },\dots ,u_{mm}^{\ast })$, where $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })\in H_1\times H_2\times \cdots \times H_m$ and $u_{ij}^{\ast }\in U_{ij}(x_j^{\ast })$ for $i,j=1,2,\dots ,m$. Moreover, iterative sequences $\{x_j^n\}$ and $\{u_{ij}^n\}$ generated by Algorithm  3.1 strongly converge to $x_j^{\ast }$ and $u_{ij}^{\ast }$ for $i,j=1,2,\dots ,m$, respectively.

Proof For $i=1,2,\dots ,m$, applying Algorithm 3.1 and Lemma 2.2, we have

$$\begin{array}{c}\parallel x_i^{n+1}-x_i^n\parallel \hfill \\ \le (1-\lambda )\parallel x_i^n-x_i^{n-1}\parallel +\lambda \parallel x_i^n-x_i^{n-1}-(g_i\left(x_i^n\right)-g_i\left(x_i^{n-1}\right))\parallel \hfill \\ +\lambda \parallel R_{M_i,{\rho }_i}^{A_i,{\eta }_i}[A_i\left(g_i\left(x_i^n\right)\right)-{\rho }_i{F}_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^n,u_{ii+1}^n,\dots ,u_{im}^n)]\hfill \\ -R_{M_i,{\rho }_i}^{A_i,{\eta }_i}[A_i\left(g_i\left(x_i^{n-1}\right)\right)-{\rho }_i{F}_i(u_{i1}^{n-1},\dots ,u_{ii-1}^{n-1},u_{ii}^{n-1},u_{ii+1}^{n-1},\dots ,u_{im}^{n-1})]\parallel \hfill \\ \le (1-\lambda )\parallel x_i^n-x_i^{n-1}\parallel +\lambda \parallel x_i^n-x_i^{n-1}-(g_i\left(x_i^n\right)-g_i\left(x_i^{n-1}\right))\parallel \hfill \\ +\frac{\lambda {\tau }_i}{r_i{t}_i}\parallel A_i\left(g_i\left(x_i^n\right)\right)-A_i\left(g_i\left(x_i^{n-1}\right)\right)\hfill \\ -{\rho }_i[F_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^n,u_{ii+1}^n,\dots ,u_{im}^n)\hfill \\ -F_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^{n-1},u_{ii+1}^n,\dots ,u_{im}^n)]\parallel \hfill \\ +\frac{\lambda {\tau }_i{\rho }_i}{r_i{t}_i}\parallel F_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^{n-1},u_{ii+1}^n,\dots ,u_{im}^n)\hfill \\ -F_i(u_{i1}^{n-1},\dots ,u_{ii-1}^{n-1},u_{ii}^{n-1},u_{ii+1}^{n-1},\dots ,u_{im}^{n-1})\parallel .\hfill \end{array}$$

(3.6)

By ${\xi }_i$-Lipschitz continuity and ${\delta }_i$-strongly monotonicity of $g_i$, we get

$$\begin{array}{c}{\parallel x_i^n-x_i^{n-1}-(g_i\left(x_i^n\right)-g_i\left(x_i^{n-1}\right))\parallel }^2\hfill \\ ={\parallel x_i^n-x_i^{n-1}\parallel }^2-2〈g_i\left(x_i^n\right)-g_i\left(x_i^{n-1}\right),x_i^n-x_i^{n-1}〉\hfill \\ +{\parallel g_i\left(x_i^n\right)-g_i\left(x_i^{n-1}\right)\parallel }^2\hfill \\ \le (1-2{\delta }_i+{\xi }_i^2){\parallel x_i^n-x_i^{n-1}\parallel }^2.\hfill \end{array}$$

(3.7)

Since $A_i$ is ${\beta }_i$-Lipschitz continuous, $F_i$ is $(U_{ii},c_i,{\mu }_i)$-relaxed cocoercive with respect to $A_i$ in the i th argument and $F_i$ is ${\zeta }_{ij}$-Lipschitz continuous in the j th argument, then we have

$$\begin{array}{c}\parallel A_i\left(g_i\left(x_i^n\right)\right)-A_i\left(g_i\left(x_i^{n-1}\right)\right)-{\rho }_i[F_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^n,u_{ii+1}^n,\dots ,u_{im}^n)\hfill \\ {-F_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^{n-1},u_{ii+1}^n,\dots ,u_{im}^n)]\parallel }^2\hfill \\ ={\parallel A_i\left(g_i\left(x_i^n\right)\right)-A_i\left(g_i\left(x_i^{n-1}\right)\right)\parallel }^2\hfill \\ -2{\rho }_i〈F_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^n,u_{ii+1}^n,\dots ,u_{im}^n)\hfill \\ -F_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^{n-1},u_{ii+1}^n,\dots ,u_{im}^n),A_i\left(g_i\left(x_i^n\right)\right)-A_i\left(g_i\left(x_i^{n-1}\right)\right)〉\hfill \\ +{\rho }_i^2\parallel F_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^n,u_{ii+1}^n,\dots ,u_{im}^n)\hfill \\ {-F_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^{n-1},u_{ii+1}^n,\dots ,u_{im}^n)\parallel }^2\hfill \\ \le {\beta }_i^2{\parallel g_i\left(x_i^n\right)-g_i\left(x_i^{n-1}\right)\parallel }^2\hfill \\ -2{\rho }_i[(-c_i)\parallel F_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^n,u_{ii+1}^n,\dots ,u_{im}^n)\hfill \\ {-F_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^{n-1},u_{ii+1}^n,\dots ,u_{im}^n)\parallel }^2\hfill \\ +{\mu }_i{\parallel g_i\left(x_i^n\right)-g_i\left(x_i^{n-1}\right)\parallel }^2]+{\rho }_i^2{\zeta }_{ii}^2{\parallel u_{ii}^n-u_{ii}^{n-1}\parallel }^2\hfill \\ \le ({\beta }_i^2{\xi }_i^2-2{\rho }_i{\mu }_i{\delta }_i^2){\parallel x_i^n-x_i^{n-1}\parallel }^2+(2{\rho }_i{c}_i{\zeta }_{ii}^2+{\rho }_i^2{\zeta }_{ii}^2){\parallel u_{ii}^n-u_{ii}^{n-1}\parallel }^2.\hfill \end{array}$$

(3.8)

By $D_j$-${\gamma }_{ij}$-Lipschitz continuity of the $U_{ij}$ and (3.3), we get

$$\begin{array}{c}\parallel F_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^{n-1},u_{ii+1}^n,\dots ,u_{im}^n)\hfill \\ -F_i(u_{i1}^{n-1},\dots ,u_{ii-1}^{n-1},u_{ii}^{n-1},u_{ii+1}^{n-1},\dots ,u_{im}^{n-1})\parallel \hfill \\ \le \parallel F_i(u_{i1}^n,u_{i2}^n,\dots ,u_{ii-1}^n,u_{ii}^{n-1},u_{ii+1}^n,\dots ,u_{im}^n)\hfill \\ -F_i(u_{i1}^{n-1},u_{i2}^n,\dots ,u_{ii-1}^n,u_{ii}^{n-1},u_{ii+1}^n,\dots ,u_{im}^n)\parallel \hfill \\ +\cdots +\parallel F_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{ii-1}^n,u_{ii}^{n-1},u_{ii+1}^n,\dots ,u_{im}^n)\hfill \\ -F_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{ii-1}^{n-1},u_{ii}^{n-1},u_{ii+1}^n,\dots ,u_{im}^n)\parallel \hfill \\ +\parallel F_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{ii-1}^{n-1},u_{ii}^{n-1},u_{ii+1}^n,\dots ,u_{im}^n)\hfill \\ -F_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{ii-1}^{n-1},u_{ii}^{n-1},u_{ii+1}^{n-1},\dots ,u_{im}^n)\parallel \hfill \\ +\cdots +\parallel F_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{ii-1}^{n-1},u_{ii}^{n-1},u_{ii+1}^{n-1},\dots ,u_{im}^n)\hfill \\ -F_i(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{ii-1}^{n-1},u_{ii}^{n-1},u_{ii+1}^{n-1},\dots ,u_{im}^{n-1})\parallel \hfill \\ \le {\zeta }_{i1}\parallel u_{i1}^n-u_{i1}^{n-1}\parallel +\cdots +{\zeta }_{ii-1}\parallel u_{ii-1}^n-u_{ii-1}^{n-1}\parallel \hfill \\ +{\zeta }_{ii+1}\parallel u_{ii+1}^n-u_{ii+1}^{n-1}\parallel +\cdots +{\zeta }_{im}\parallel u_{im}^n-u_{im}^{n-1}\parallel \hfill \\ =\sum _{j=1,j\ne i}^m{\zeta }_{ij}\parallel u_{ij}^n-u_{ij}^{n-1}\parallel \hfill \\ \le \sum _{j=1,j\ne i}^m{\zeta }_{ij}(1+\frac{1}{n})D_j(U_{ij}\left(x_j^n\right),U_{ij}\left(x_j^{n-1}\right))\hfill \\ \le (1+\frac{1}{n})\sum _{j=1,j\ne i}^m{\zeta }_{ij}{\gamma }_{ij}\parallel x_j^n-x_j^{n-1}\parallel \hfill \end{array}$$

(3.9)

and

$$\begin{array}{rcl}\parallel u_{ii}^n-u_{ii}^{n-1}\parallel & \le & (1+\frac{1}{n})D_i(U_{ii}\left(x_i^n\right),U_{ii}\left(x_i^{n-1}\right))\\ \le & (1+\frac{1}{n}){\gamma }_{ii}\parallel x_i^n-x_i^{n-1}\parallel .\end{array}$$

(3.10)

Combining (3.8) and (3.10), we have

$$\begin{array}{c}\parallel A_i\left(g_i\left(x_i^n\right)\right)-A_i\left(g_i\left(x_i^{n-1}\right)\right)-{\rho }_i[F_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^n,u_{ii+1}^n,\dots ,u_{im}^n)\hfill \\ {-F_i(u_{i1}^n,\dots ,u_{ii-1}^n,u_{ii}^{n-1},u_{ii+1}^n,\dots ,u_{im}^n)]\parallel }^2\hfill \\ \le [{\beta }_i^2{\xi }_i^2-2{\rho }_i{\mu }_i{\delta }_i^2\hfill \\ +{(1+\frac{1}{n})}^2{\gamma }_{ii}^2(2{\rho }_i{c}_i{\zeta }_{ii}^2+{\rho }_i^2{\zeta }_{ii}^2)]{\parallel x_i^n-x_i^{n-1}\parallel }^2.\hfill \end{array}$$

(3.11)

It follows from (3.6)-(3.9), and (3.11), that

$$\begin{array}{c}\parallel x_i^{n+1}-x_i^n\parallel \hfill \\ \le (1-\lambda +\lambda \sqrt{1-2{\delta }_i+{\xi }_i^2})\parallel x_i^n-x_i^{n-1}\parallel \hfill \\ +\frac{\lambda {\tau }_i}{r_i{t}_i}[\sqrt{{\beta }_i^2{\xi }_i^2-2{\rho }_i{\mu }_i{\delta }_i^2+{(1+n^{-1})}^2{\gamma }_{ii}^2(2{\rho }_i{c}_i{\zeta }_{ii}^2+{\rho }_i^2{\zeta }_{ii}^2)}\parallel x_i^n-x_i^{n-1}\parallel \hfill \\ +(1+\frac{1}{n}){\rho }_i\sum _{j=1,j\ne i}^m{\zeta }_{ij}{\gamma }_{ij}\parallel x_j^n-x_j^{n-1}\parallel ],\hfill \end{array}$$

which implies that

$$\begin{array}{rcl}\sum _{j=1}^m\parallel x_j^{n+1}-x_j^n\parallel & =& \sum _{i=1}^m\parallel x_i^{n+1}-x_i^n\parallel \\ \le & \sum _{i=1}^m[(1-\lambda +\lambda \sqrt{1-2{\delta }_i+{\xi }_i^2})\parallel x_i^n-x_i^{n-1}\parallel \\ +\frac{\lambda {\tau }_i}{r_i{t}_i}(\sqrt{{\beta }_i^2{\xi }_i^2-2{\rho }_i{\mu }_i{\delta }_i^2+{(1+\frac{1}{n})}^2{\gamma }_{ii}^2(2{\rho }_i{c}_i{\zeta }_{ii}^2+{\rho }_i^2{\zeta }_{ii}^2)}\parallel x_i^n-x_i^{n-1}\parallel \\ +(1+\frac{1}{n}){\rho }_i\sum _{j=1,j\ne i}^m{\zeta }_{ij}{\gamma }_{ij}\parallel x_j^n-x_j^{n-1}\parallel \left)\right]\\ =& \sum _{i=1}^m[(1-\lambda +\lambda \sqrt{1-2{\delta }_i+{\xi }_i^2})\\ +\frac{\lambda {\tau }_i}{r_i{t}_i}\sqrt{{\beta }_i^2{\xi }_i^2-2{\rho }_i{\mu }_i{\delta }_i^2+{(1+\frac{1}{n})}^2{\gamma }_{ii}^2(2{\rho }_i{c}_i{\zeta }_{ii}^2+{\rho }_i^2{\zeta }_{ii}^2)}]\parallel x_i^n-x_i^{n-1}\parallel \\ +(1+\frac{1}{n})\lambda \sum _{i=1}^m\sum _{j=1,j\ne i}^m\frac{{\rho }_i{\tau }_i{\zeta }_{ij}{\gamma }_{ij}}{r_i{t}_i}\parallel x_j^n-x_j^{n-1}\parallel \\ =& \sum _{j=1}^m[(1-\lambda +\lambda \sqrt{1-2{\delta }_j+{\xi }_j^2})\\ +\frac{\lambda {\tau }_j}{r_j{t}_j}\sqrt{{\beta }_j^2{\xi }_j^2-2{\rho }_j{\mu }_j{\delta }_j^2+{(1+\frac{1}{n})}^2{\gamma }_{jj}^2(2{\rho }_j{c}_j{\zeta }_{jj}^2+{\rho }_j^2{\zeta }_{jj}^2)}]\parallel x_j^n-x_j^{n-1}\parallel \\ +(1+\frac{1}{n})\lambda \sum _{j=1}^m\sum _{i=1,i\ne j}^m\frac{{\rho }_i{\tau }_i{\zeta }_{ij}{\gamma }_{ij}}{r_i{t}_i}\parallel x_j^n-x_j^{n-1}\parallel \\ =& \sum _{j=1}^m[(1-\lambda )+\lambda (\sqrt{1-2{\delta }_j+{\xi }_j^2}\\ +\frac{{\tau }_j}{r_j{t}_j}\sqrt{{\beta }_j^2{\xi }_j^2-2{\rho }_j{\mu }_j{\delta }_j^2+{(1+\frac{1}{n})}^2{\gamma }_{jj}^2(2{\rho }_j{c}_j{\zeta }_{jj}^2+{\rho }_j^2{\zeta }_{jj}^2)})\\ +(1+\frac{1}{n})\sum _{i=1,i\ne j}^m\frac{{\rho }_i{\tau }_i{\zeta }_{ij}{\gamma }_{ij}}{r_i{t}_i}]\parallel x_j^n-x_j^{n-1}\parallel \\ =& \sum _{j=1}^m[1-\lambda +\lambda {\theta }_j^n]\parallel x_j^n-x_j^{n-1}\parallel \le f_n(\lambda )\sum _{j=1}^m\parallel x_j^n-x_j^{n-1}\parallel ,\end{array}$$

(3.12)

where

$$\begin{array}{rcl}{\theta }_j^n& =& \frac{{\tau }_j}{r_j{t}_j}\sqrt{{\beta }_j^2{\xi }_j^2-2{\rho }_j{\mu }_j{\delta }_j^2+{(1+\frac{1}{n})}^2{\gamma }_{jj}^2(2{\rho }_j{c}_j{\zeta }_{jj}^2+{\rho }_j^2{\zeta }_{jj}^2)}\\ +\sqrt{1-2{\delta }_j+{\xi }_j^2}+(1+\frac{1}{n})\sum _{i=1,i\ne j}^m\frac{{\rho }_i{\tau }_i{\zeta }_{ij}{\gamma }_{ij}}{r_i{t}_i}\end{array}$$

and

$$f_n(\lambda )=\underset{1\le j\le m}{max}\{1-\lambda +\lambda {\theta }_j^n\}.$$

By condition (3.5), we know that sequence $\{{\theta }_j^n\}$ is monotone decreasing and ${\theta }_j^n\to {\theta }_j$ as $n\to \mathrm{\infty }$. Thus,

$$f(\lambda )=\underset{n\to \mathrm{\infty }}{lim}{f}_n(\lambda )=\underset{1\le j\le m}{max}\{1-\lambda +\lambda {\theta }_j\}.$$

Since $0<{\theta }_j<1$ for $j=1,2,\dots ,m$, we get $\theta ={max}_{1\le j\le m}\{{\theta }_j\}\in (0,1)$, by Lemma 2.3, we have $f(\lambda )=1-\lambda +\lambda \theta \in (0,1)$. From (3.12), it follows that $\{x_j^n\}$ is a Cauchy sequence and there exists $x_j^{\ast }\in H_j$ such that $x_j^n\to x_j^{\ast }$ as $n\to \mathrm{\infty }$ for $j=1,2,\dots ,m$.

Next, we show that $u_{ij}^n\to u_{ij}^{\ast }\in U_{ij}(x_j^{\ast })$ as $n\to \mathrm{\infty }$ for $i,j=1,2,\dots ,m$.

It follows from (3.9) and (3.10) that $\{u_{ij}^n\}$ are also Cauchy sequences. Hence, there exists $u_{ij}^{\ast }\in H_j$ such that $u_{ij}^n\to u_{ij}^{\ast }$ as $n\to \mathrm{\infty }$ for $i,j=1,2,\dots ,m$. Furthermore,

$$\begin{array}{rcl}d(u_{ij}^{\ast },U_{ij}\left(x_j^{\ast }\right))& =& inf\{\parallel u_{ij}^{\ast }-t\parallel :t\in U_{ij}\left(x_j^{\ast }\right)\}\\ \le & \parallel u_{ij}^{\ast }-u_{ij}^n\parallel +d(u_{ij}^n,U_{ij}\left(x_j^{\ast }\right))\\ \le & \parallel u_{ij}^{\ast }-u_{ij}^n\parallel +D_j(U_{ij}\left(x_j^n\right),U_{ij}\left(x_j^{\ast }\right))\\ \le & \parallel u_{ij}^{\ast }-u_{ij}^n\parallel +{\gamma }_{ij}\parallel x_j^n-x_j^{\ast }\parallel \to 0(n\to \mathrm{\infty }).\end{array}$$

Since $U_{ij}(x_j^{\ast })$ is closed for $i,j=1,2,\dots ,m$, we have $u_{ij}^{\ast }\in U_{ij}(x_j^{\ast })$ for $i,j=1,2,\dots ,m$. Using continuity, $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })\in H_1\times H_2\times \cdots \times H_m$ and $u_{ij}^{\ast }\in U_{ij}(x_j^{\ast })$ for $i,j=1,2,\dots ,m$ satisfy (3.1) and so in light of Lemma 3.1, $(\ast )$ is a solution to the problem (1.1). This completes the proof. □

Remark 3.3 If the generalized resolvent operator $R_{M_i,{\rho }_i}^{A_i,{\eta }_i}$ reduces to $J_{\rho }^{{\phi }_i}={(I+\rho \partial {\phi }_i)}^{-1}$, where ${\phi }_i:H_i\to R\cup \{+\mathrm{\infty }\}$ is proper and lower semi-continuous ${\eta }_i$-subdifferentiable functional, $H_i=H$ for $i=1,2,\dots ,m$, $\lambda =1$ and $(U_{ii},c_i,{\mu }_i)$-relaxed cocoerciveness with respect to $A_i$ in the i th argument of $F_i$ reduces to ${\mu }_i$-$(U_{ii},A_i)$-strongly monotonicity (right now, $c_i=0$, $A_i\equiv g_i$), then Theorem 3.1 reduces to Theorem 3.1 of Cao [33].

Theorem 3.2 Assume that ${\eta }_i$, $A_i$, $g_i$, $M_i$ are the same as in the Theorem  3.1 for $i=1,2,\dots ,m$. Suppose that $T_{ij}:H_j\to H_j$ is ${\gamma }_{ij}$-Lipschitz continuous, $F_i:H_1\times H_2\times \cdots \times H_m\to H_i$ is $(T_{ii},c_i,{\mu }_i)$-relaxed cocoercive with respect to $A_i$ in the ith argument and ${\zeta }_{ij}$-Lipschitz continuous in the jth for $i,j=1,2,\dots ,m$. If there exists constant ${\rho }_i>0$ for such that

$$\begin{array}{rcl}{\theta }_j& =& \frac{{\tau }_j}{r_j{t}_j}\cdot \sqrt{{\beta }_j^2{\xi }_j^2-2{\rho }_j{\mu }_j{\delta }_j^2+2{\rho }_j{c}_j{\zeta }_{jj}^2{\gamma }_{jj}^2+{\rho }_j^2{\zeta }_{jj}^2{\gamma }_{jj}^2}\\ +\sqrt{1-2{\delta }_j+{\xi }_j^2}+\sum _{i=1,i\ne j}^m\frac{{\rho }_i{\tau }_i{\zeta }_{ij}{\gamma }_{ij}}{r_i{t}_i}<1\end{array}$$

for $j=1,2,\dots ,m$, then the problem (1.2) has a unique solution $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })\in H_1\times H_2\times \cdots \times H_m$. Moreover, the iterative sequences $\{x_j^n\}$ generated by Algorithm  3.2 strongly converge to $x_j^{\ast }$ for $j=1,2,\dots ,m$.

Proof Define the norm ${\parallel \cdot \parallel }_{\ast }$ on product space $H_1\times H_2\times \cdots \times H_m$ by

$${\parallel (x_1,x_2,\dots ,x_m)\parallel }_{\ast }=\sum _{j=1}^m\parallel x_j\parallel ,\mathrm{\forall }(x_1,x_2,\dots ,x_m)\in H_1\times H_2\times \cdots \times H_m.$$

It is easy to see that $(H_1\times H_2\times \cdots \times H_m,{\parallel \cdot \parallel }_{\ast })$ is a Banach space. Set

$$\begin{array}{rcl}{y}_i& =& x_i-g_i(x_i)+R_{M_i,{\rho }_i}^{A_i,{\eta }_i}[A_i(g_i(x_i))\\ -{\rho }_i{F}_i(T_{i1}{x}_1,\dots ,T_{ii-1}{x}_{i-1},T_{ii}{x}_i,T_{ii+1}{x}_{i+1},\dots ,T_{im}{x}_m)].\end{array}$$

Let $G:H_1\times H_2\times \cdots \times H_m\to H_1\times H_2\times \cdots \times H_m$ be defined by

$$G(x_1,x_2,\dots ,x_m)=(y_1,y_2,\dots ,y_m),\mathrm{\forall }(x_1,x_2,\dots ,x_m)\in H_1\times H_2\times \cdots \times H_m.$$

For any $(x_1^1,x_2^1,\dots ,x_m^1),(x_1^2,x_2^2,\dots ,x_m^2)\in H_1\times H_2\times \cdots \times H_m$, it follows from Lemma 2.2 that

$$\begin{array}{c}{\parallel G(x_1^1,x_2^1,\dots ,x_m^1)-G(x_1^2,x_2^2,\dots ,x_m^2)\parallel }_{\ast }\hfill \\ =\sum _{i=1}^m\parallel y_i^1-y_i^2\parallel \hfill \\ \le \sum _{i=1}^m\{\parallel x_i^1-x_i^2-(g_i\left(x_i^1\right)-g_i\left(x_i^2\right))\parallel +\parallel R_{M_i,{\rho }_i}^{A_i,{\eta }_i}[A_i\left(g_i\left(x_i^1\right)\right)\hfill \\ -{\rho }_i{F}_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)]\hfill \\ -R_{M_i,{\rho }_i}^{A_i,{\eta }_i}[A_i\left(g_i\left(x_i^2\right)\right)\hfill \\ -{\rho }_i{F}_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)]\parallel \}\hfill \\ \le \sum _{i=1}^m\{\parallel x_i^1-x_i^2-(g_i\left(x_i^1\right)-g_i\left(x_i^2\right))\parallel \hfill \\ +\frac{{\tau }_i}{r_i{t}_i}\parallel A_i\left(g_i\left(x_i^1\right)\right)-A_i\left(g_i\left(x_i^2\right)\right)\hfill \\ -{\rho }_i[F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\hfill \\ -F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)]\parallel \hfill \\ +\frac{{\tau }_i{\rho }_i}{r_i{t}_i}\parallel F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\hfill \\ -F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\parallel \}.\hfill \end{array}$$

(3.13)

By ${\xi }_i$-Lipschitz continuity and ${\delta }_i$-strongly monotonicity of $g_i$, we get

$$\parallel x_i^1-x_i^2-(g_i\left(x_i^1\right)-g_i\left(x_i^2\right))\parallel \le \sqrt{1-2{\delta }_i+{\xi }_i^2}\parallel x_i^1-x_i^2\parallel .$$

(3.14)

Since $A_i$ is ${\beta }_i$-Lipschitz continuous, $F_i$ is $(T_{ii},c_i,{\mu }_i)$-relaxed cocoercive with respect to $A_i$ in the i th argument and $F_i$ is ${\zeta }_{ij}$-Lipschitz continuous in the j th argument and $T_{ij}:H_j\to H_j$ is ${\gamma }_{ij}$-Lipschitz continuous, then we have

$$\begin{array}{c}\parallel A_i\left(g_i\left(x_i^1\right)\right)-A_i\left(g_i\left(x_i^2\right)\right)\hfill \\ -{\rho }_i[F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\hfill \\ {-F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)]\parallel }^2\hfill \\ \le {\beta }_i^2{\parallel g_i\left(x_i^1\right)-g_i\left(x_i^2\right)\parallel }^2\hfill \\ -2{\rho }_i[(-c_i)\parallel F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^1,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\hfill \\ {-F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\parallel }^2\hfill \\ +{\mu }_i{\parallel g_i\left(x_i^1\right)-g_i\left(x_i^2\right)\parallel }^2]+{\rho }_i^2{\zeta }_{ii}^2{\parallel T_{ii}{x}_i^1-T_{ii}{x}_i^2\parallel }^2\hfill \\ \le ({\beta }_i^2{\xi }_i^2-2{\rho }_i{\mu }_i{\delta }_i^2){\parallel x_i^1-x_i^2\parallel }^2+(2{\rho }_i{c}_i{\zeta }_{ii}^2+{\rho }_i^2{\zeta }_{ii}^2){\parallel T_{ii}{x}_i^1-T_{ii}{x}_i^2\parallel }^2\hfill \\ \le ({\beta }_i^2{\xi }_i^2-2{\rho }_i{\mu }_i{\delta }_i^2+2{\rho }_i{c}_i{\zeta }_{ii}^2{\gamma }_{ii}+{\rho }_i^2{\zeta }_{ii}^2{\gamma }_{ii}){\parallel x_i^1-x_i^2\parallel }^2\hfill \end{array}$$

(3.15)

and

$$\begin{array}{c}\parallel F_i(T_{i1}{x}_1^1,\dots ,T_{ii-1}{x}_{i-1}^1,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^1,\dots ,T_{im}{x}_m^1)\hfill \\ -F_i(T_{i1}{x}_1^2,\dots ,T_{ii-1}{x}_{i-1}^2,T_{ii}{x}_i^2,T_{ii+1}{x}_{i+1}^2,\dots ,T_{im}{x}_m^2)\parallel \hfill \\ \le {\zeta }_{i1}\parallel T_{i1}{x}_1^1-T_{i1}{x}_1^2\parallel +\cdots +{\zeta }_{ii-1}\parallel T_{ii-1}{x}_{i-1}^1-T_{ii-1}{x}_{i-1}^2\parallel \hfill \\ +{\zeta }_{ii+1}\parallel T_{ii+1}{x}_{i+1}^1-T_{ii+1}{x}_{ii+1}^2\parallel +\cdots +{\zeta }_{im}\parallel T_{im}{x}_m^1-T_{im}{x}_m^2\parallel \hfill \\ =\sum _{j=1,j\ne i}^m{\zeta }_{ij}\parallel T_{ij}{x}_j^1-T_{ij}{x}_j^2\parallel \hfill \\ \le \sum _{j=1,j\ne i}^m{\zeta }_{ij}{\gamma }_{ij}\parallel x_j^1-x_j^2\parallel .\hfill \end{array}$$

(3.16)

From (3.13)-(3.16), we have

$$\begin{array}{c}{\parallel G(x_1^1,x_2^1,\dots ,x_m^1)-G(x_1^2,x_2^2,\dots ,x_m^2)\parallel }_{\ast }\hfill \\ \le \sum _{i=1}^m(\sqrt{1-2{\delta }_i+{\xi }_i^2}\hfill \\ +\frac{{\tau }_i}{r_i{t}_i}\sqrt{{\beta }_i^2{\xi }_i^2-2{\rho }_i{\mu }_i{\delta }_i^2+2{\rho }_i{c}_i{\zeta }_{ii}^2{\gamma }_{ii}^2+{\rho }_i^2{\zeta }_{ii}^2{\gamma }_{ii}^2})\parallel x_i^1-x_i^2\parallel \hfill \\ +\sum _{j=1,j\ne i}^m\frac{{\rho }_j{\tau }_j{\zeta }_{ij}{\gamma }_{ij}}{r_j{t}_j}\parallel x_j^1-x_j^2\parallel \hfill \\ =\sum _{j=1}^m{\theta }_j\parallel x_j^1-x_j^2\parallel \hfill \\ \le \theta \sum _{j=1}^m\parallel x_j^1-x_j^1\parallel \hfill \\ =\theta {\parallel (x_1^1,x_2^1,\dots ,x_m^1)-(x_1^2,x_2^2,\dots ,x_m^2)\parallel }_{\ast },\hfill \end{array}$$

where $\theta ={max}_{1\le j\le m}{\theta }_j$. It follows from assumption (3.5) that $0<\theta <1$. This shows that $G:H_1\times H_2\times \cdots \times H_m\to H_1\times H_2\times \cdots \times H_m$ is a contractive operator, and so there exists a unique $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })\in H_1\times H_2\times \cdots \times H_m$ such that $G(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })=(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })$. Thus, $(x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast })$ is the unique solution of the problem (1.2).

Now we prove that $x_i^n\to x_i^{\ast }$ as $n\to \mathrm{\infty }$ for $i=1,2,\dots ,m$. In fact, it follows from (3.4) and Lemma 2.2 that

$$\begin{array}{c}\parallel x_i^{n+1}-x^{\ast }\parallel \hfill \\ \le \parallel x_i^n-x_i^{\ast }-(g_i\left(x_i^n\right)-g_i\left(x_i^{\ast }\right))\parallel \hfill \\ +\parallel R_{M_i,{\rho }_i}^{A_i,{\eta }_i}[A_i\left(g_i\left(x_i^n\right)\right)\hfill \\ -{\rho }_i{F}_i(T_{i1}{x}_1^n,\dots ,T_{ii-1}{x}_{i-1}^n,T_{ii}{x}_i^n,T_{ii+1}{x}_{i+1}^n,\dots ,T_{im}{x}_m^n)]\hfill \\ -R_{M_i,{\rho }_i}^{A_i,{\eta }_i}[A_i\left(g_i\left(x_i^{\ast }\right)\right)\hfill \\ -{\rho }_i{F}_i(T_{i1}{x}_1^{\ast },\dots ,T_{ii-1}{x}_{i-1}^{\ast },T_{ii}{x}_i^{\ast },T_{ii+1}{x}_{i+1}^{\ast },\dots ,T_{im}{x}_m^{\ast })]\parallel +\parallel w_i^n\parallel \hfill \\ \le \parallel x_i^n-x_i^{\ast }-(g_i\left(x_i^n\right)-g_i\left(x_i^{\ast }\right))\parallel +\parallel w_i^n\parallel \hfill \\ +\frac{{\tau }_i}{r_i{t}_i}\parallel A_i\left(g_i\left(x_i^n\right)\right)-A_i\left(g_i\left(x_i^{\ast }\right)\right)\hfill \\ -{\rho }_i[F_i(T_{i1}{x}_1^n,\dots ,T_{ii-1}{x}_{i-1}^n,T_{ii}{x}_i^n,T_{ii+1}{x}_{i+1}^n,\dots ,T_{im}{x}_m^n)\hfill \\ -F_i(T_{i1}{x}_1^n,\dots ,T_{ii-1}{x}_{i-1}^n,T_{ii}{x}_i^{\ast },T_{ii+1}{x}_{i+1}^n,\dots ,T_{im}{x}_m^n)]\parallel \hfill \\ +\frac{{\tau }_i{\rho }_i}{r_i{t}_i}\parallel F_i(T_{i1}{x}_1^n,\dots ,T_{ii-1}{x}_{i-1}^n,T_{ii}{x}_i^{\ast },T_{ii+1}{x}_{i+1}^n,\dots ,T_{im}{x}_m^n)\hfill \\ -F_i(T_{i1}{x}_1^{\ast },\dots ,T_{ii-1}{x}_{i-1}^{\ast },T_{ii}{x}_i^{\ast },T_{ii+1}{x}_{i+1}^{\ast },\dots ,T_{im}{x}_m^{\ast })\parallel .\hfill \end{array}$$

(3.17)

Following very similar arguments from (3.14)-(3.16), we have

$$\begin{array}{c}\parallel x_i^{n+1}-x_i^{\ast }\parallel \hfill \\ \le \sqrt{1-2{\delta }_i+{\xi }_i^2}\parallel x_i^n-x_i^{\ast }\parallel \hfill \\ +\frac{{\tau }_i}{r_i{t}_i}[\sqrt{{\beta }_i^2{\xi }_i^2-2{\rho }_i{\mu }_i{\delta }_i^2+2{\rho }_i{c}_i{\zeta }_{ii}^2{\gamma }_{ii}^2+{\rho }_i^2{\zeta }_{ii}^2{\gamma }_{ii}^2}\parallel x_i^n-x_i^{\ast }\parallel \hfill \\ +{\rho }_i\sum _{j=1,j\ne i}^m{\zeta }_{ij}{\gamma }_{ij}\parallel x_j^n-x_j^{\ast }\parallel ]+\parallel w_i^n\parallel ,\hfill \end{array}$$

(3.18)

which implies that

$$\begin{array}{rcl}\sum _{j=1}^m\parallel x_j^{n+1}-x_j^{\ast }\parallel & =& \sum _{j=1}^m{\theta }_j\parallel x_j^n-x_j^{\ast }\parallel +\sum _{j=1}^m\parallel w_j^n\parallel \\ \le & \theta \sum _{j=1}^m\parallel x_j^n-x_j^{\ast }\parallel +\sum _{j=1}^m\parallel w_j^n\parallel ,\end{array}$$

where $a_n={\sum }_{j=1}^m\parallel x_j^n-x_j^{\ast }\parallel$, $b_n={\sum }_{j=1}^m\parallel w_j^n\parallel$. The condition of Algorithm 3.2 yields ${lim}_{n\to \mathrm{\infty }}{b}_n=0$. Now Lemma 2.4 implies that ${lim}_{n\to \mathrm{\infty }}{a}_n=0$, and so $x_j^n\to x_j^{\ast }$ as $n\to \mathrm{\infty }$ for $j=1,2,\dots ,m$. This completes the proof. □

Remark 3.4 If $m=2$, $g_1=g_2=U_{11}=U_{22}\equiv I$ (right now, ${\delta }_i={\xi }_i={\zeta }_{ii}=1$ for $i=1,2$), then Theorem 3.1 reduces to Theorem 4.5 based on Algorithm 4.3 of Agarwal and Verma [34]. Our presented results improve and extend some known results in the literature.