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# Solvability and iterative algorithms for a system of generalized nonlinear mixed quasivariational inclusions with $({H}_{i},{\eta}_{i})$-monotone operators

- Zeqing Liu
^{1}, - Lili Wang
^{1}, - Jeong Sheok Ume
^{2}Email author and - Shin Min Kang
^{3}

**2012**:235

https://doi.org/10.1186/1029-242X-2012-235

© Liu et al.; licensee Springer 2012

**Received:**12 May 2012**Accepted:**28 September 2012**Published:**17 October 2012

## Abstract

In this paper, we introduce and discuss a new system of generalized nonlinear mixed quasivariational inclusions with $({H}_{i},{\eta}_{i})$-monotone operators in Hilbert spaces, which includes several systems of variational inequalities and variational inclusions as special cases. By employing the resolvent operator technique associated with $({H}_{i},{\eta}_{i})$-monotone operators, we suggest two iterative algorithms for computing the approximate solutions of the system of generalized nonlinear mixed quasivariational inclusions. Under certain conditions, we obtain the existence of solutions for the system of generalized nonlinear mixed quasivariational inclusions and prove the convergence of the iterative sequences generated by the iterative algorithms. The results presented in this paper extend, improve and unify many known results in recent literature.

**MSC:**47J20, 49J40.

## Keywords

- $({H}_{i},{\eta}_{i})$-monotone operators
- resolvent operator technique
- iterative algorithm
- Mann perturbed iterative algorithm with mixed errors
- set-valued mapping
- convergence
- system of generalized nonlinear mixed quasivariational inclusions
- Hilbert space

## 1 Introduction

Variational inclusions, as important extensions of the classical variational inequalities, provide us with simple, natural, general and unified frameworks in the study of many fields including mechanics, physics, optimization and control, nonlinear programming, economics and transportation equilibrium, as well as engineering sciences. Owing to their wide applications, a lot of existence results and iterative algorithms of solutions for various variational inclusions have been studied in recent years. For details, we refer to [1–20] and the references therein. Among these methods, the resolvent operator techniques to solve various variational inclusions are interesting and important.

In the past years, Agarwal-Huang-Tan [1] introduced a system of generalized nonlinear mixed quasivariational inclusions and investigated the sensitivity analysis of solutions for the system of generalized nonlinear mixed quasivariational inclusions in Hilbert spaces. Kazmi-Bhat [8] introduced a system of nonlinear variational-like inclusions and gave an iterative algorithm for finding its approximate solution. Fang-Huang [4], Huang-Fang [6] and Liu-Kang-Ume [13] introduced some new systems of variational inclusions in Hilbert spaces using the resolvent operator associated with *H*-monotone operators, maximal *η*-monotone operators, maximal monotone operators and *A*-monotone operators, respectively. Afterward, Fang-Huang-Thompson [5] studied a class of variational inclusions with $(H,\eta )$-monotone operators, which provides a unifying framework for the classes of maximal monotone operators, maximal *η*-monotone operators and *H*-monotone operators. Jin-Liu-Lee [7] studied a class of generalized nonlinear mixed quasivariational inclusions including relaxed Lipschitz and relaxed monotone operators.

Motivated and inspired by the above works, the goal of this paper is as follows. First, a new system of generalized nonlinear mixed quasivariational inclusions with $({H}_{i},{\eta}_{i})$-monotone operators is introduced and studied in Hilbert spaces. Secondly, by utilizing some properties of the resolvent operators with $({H}_{i},{\eta}_{i})$-monotone operators, two new iterative algorithms for approximating solutions of the system of generalized nonlinear mixed quasivariational inclusions are constructed. Finally, we prove the existence of solutions for the system of generalized nonlinear mixed quasivariational inclusions and show the convergence of the iterative sequences generated by the iterative algorithms in Hilbert spaces. Our results generalize, improve and unify a lot of previously known results in this area.

## 2 Preliminaries

where $d(a,B)={inf}_{b\in B}\parallel a-b\parallel $, $d(A,b)={inf}_{a\in A}\parallel a-b\parallel $. Set
, *ω* and
denote the sets of all nonnegative and positive integers, respectively.

In what follows, we recall some basic concepts, assumptions and results, which will be used in the sequel.

**Definition 2.1**Let $\eta :\mathcal{H}\times \mathcal{H}\to \mathcal{H}$, $H:\mathcal{H}\to \mathcal{H}$ be two operators and $M:\mathcal{H}\to {2}^{\mathcal{H}}$ be a set-valued operator.

*M*is said to be

- (1)
*η*-*monotone*if $\u3008x-y,\eta (u,v)\u3009\ge 0$, $\mathrm{\forall}u,v\in \mathcal{H}$, $x\in Mu$, $y\in Mv$; - (2)
$(H,\eta )$-

*monotone*if*M*is*η*-monotone and $(H+\lambda M)(\mathcal{H})=\mathcal{H}$ for all $\lambda >0$.

**Remark 2.1** The class of $(H,\eta )$-monotone operators is much more universal than all maximal *η*-monotone, maximal monotone, *H*-monotone and *η*-monotone operators.

**Definition 2.2**An operator $\eta :\mathcal{H}\times \mathcal{H}\to \mathcal{H}$ is said to be Lipschitz continuous if there exists a constant $\tau >0$ such that

**Definition 2.3**Let $g,m:\mathcal{H}\to \mathcal{H}$, $\eta :\mathcal{H}\times \mathcal{H}\to \mathcal{H}$ be operators. The operator

*g*is said to be

- (1)
*Lipschitz continuous*if there exists a constant $s>0$ such that$\parallel g(u)-g(v)\parallel \le s\parallel u-v\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}u,v\in \mathcal{H};$ - (2)
*strongly monotone*if there exists a constant $r>0$ satisfying$\u3008g(u)-g(v),u-v\u3009\ge r{\parallel u-v\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}u,v\in \mathcal{H};$ - (3)
*η*-*monotone*if $\u3008g(u)-g(v),\eta (u,v)\u3009\ge 0$, $\mathrm{\forall}u,v\in \mathcal{H}$; - (4)
*strictly**η-monotone*if*g*is*η*-monotone and $\u3008g(u)-g(v),\eta (u,v)\u3009=0$ if and only if $u=v$; - (5)
*strongly**η-monotone*if there exists a constant $\sigma >0$ fulfilling$\u3008g(u)-g(v),\eta (u,v)\u3009\ge \sigma {\parallel u-v\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}u,v\in \mathcal{H};$ - (6)
*m*-*relaxed monotone*if there exists a constant $k>0$ such that$\u3008g(u)-g(v),m(u)-m(v)\u3009\ge -k{\parallel u-v\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}u,v\in \mathcal{H}.$

**Definition 2.4**A set-valued operator $A:\mathcal{H}\to CB(\mathcal{H})$ is said to be $\stackrel{\u02c6}{\mathcal{H}}$-

*Lipschitz continuous*if there exists a constant ${l}_{A}>0$ such that

**Definition 2.5** ([5])

*η*-monotone operator and $M:\mathcal{H}\to {2}^{\mathcal{H}}$ be an $(H,\eta )$-monotone operator. Then the

*resolvent operator*${J}_{M,\lambda}^{H,\eta}:\mathcal{H}\to \mathcal{H}$ is defined by

**Lemma 2.1** ([5])

*Let*$\eta :\mathcal{H}\times \mathcal{H}\to \mathcal{H}$

*be a Lipschitz continuous operator with a constant*$\tau >0$, $H:\mathcal{H}\to \mathcal{H}$

*be a strongly*

*η*-

*monotone operator with a constant*$\sigma >0$

*and*$M:\mathcal{H}\to {2}^{\mathcal{H}}$

*be an*$(H,\eta )$-

*monotone operator*.

*Then the resolvent operator*${J}_{M,\lambda}^{H,\eta}:\mathcal{H}\to \mathcal{H}$

*is Lipschitz continuous with a constant*$\frac{\tau}{\sigma}$,

*that is*,

**Lemma 2.2** ([21])

*Let*$\{{a}_{n}\}$, $\{{b}_{n}\}$, $\{{c}_{n}\}$

*be three nonnegative real sequences satisfying*

*where* ${\{{t}_{n}\}}_{n\in \omega}\subset [0,1]$, ${\sum}_{n=0}^{\mathrm{\infty}}{t}_{n}=+\mathrm{\infty}$, ${lim}_{n\to \mathrm{\infty}}{b}_{n}=0$ *and* ${\sum}_{n=0}^{\mathrm{\infty}}{c}_{n}<+\mathrm{\infty}$. *Then* ${lim}_{n\to \mathrm{\infty}}{a}_{n}=0$.

**Lemma 2.3** ([22])

*Let*$(E,d)$

*be a metric space and*$T:E\to CB(E)$

*be a set*-

*valued mapping*.

*Then for any given*$\u03f5>0$

*and any given*$x,y\in E$, $u\in Tx$,

*there exists*$v\in Ty$

*such that*

*where* *H* *is the Hausdorff metric on* $CB(E)$.

In what follows, unless specified otherwise, we always assume that for each $i\in \{1,2\}$, ${\mathcal{H}}_{i}$ is a real Hilbert space with norm ${\parallel \cdot \parallel}_{i}$, inner product ${\u3008\cdot ,\cdot \u3009}_{i}$ and Hausdorff metric $\stackrel{\u02c6}{{\mathcal{H}}_{i}}$.

**Definition 2.6**Let $H:{\mathcal{H}}_{1}\to {\mathcal{H}}_{1}$, $F:{\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\to {\mathcal{H}}_{1}$ be two operators. The operator

*F*is said to be

- (1)
*Lipschitz continuous*in the first argument if there exists a constant $\beta >0$ such that${\parallel F(u,z)-F(v,z)\parallel}_{1}\le \beta {\parallel u-v\parallel}_{1},\phantom{\rule{1em}{0ex}}\mathrm{\forall}u,v\in {\mathcal{H}}_{1},z\in {\mathcal{H}}_{2};$ - (2)
*H*-*relaxed monotone*in the first argument if there exists a constant $\alpha >0$ satisfying${\u3008F(u,z)-F(v,z),H(u)-H(v)\u3009}_{1}\ge -\alpha {\parallel u-v\parallel}_{1}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}u,v\in {\mathcal{H}}_{1},z\in {\mathcal{H}}_{2}.$

In a similar way, we can define the corresponding concepts for the operator *F* in the second argument.

## 3 A system of generalized nonlinear mixed quasivariational inclusions and iterative algorithms

In this section, we shall introduce a new system of generalized nonlinear mixed quasivariational inclusions including $({H}_{i},{\eta}_{i})$-monotone operators in Hilbert spaces, and construct a new iterative algorithm for solving the system of generalized nonlinear mixed quasivariational inclusions. Furthermore, a more general and unified iterative algorithm called the Mann perturbed iterative algorithm with mixed errors is constructed as well in Hilbert spaces.

For each $i\in \{1,2\}$, let ${H}_{i},{g}_{i},{m}_{i}:{\mathcal{H}}_{i}\to {\mathcal{H}}_{i}$, ${\eta}_{i}:{\mathcal{H}}_{i}\times {\mathcal{H}}_{i}\to {\mathcal{H}}_{i}$ be operators, ${M}_{i}:{\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\to {2}^{{\mathcal{H}}_{i}}$ be set-valued operator such that for each fixed $(x,y)\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}$, ${M}_{1}(\cdot ,y):{\mathcal{H}}_{1}\to {2}^{{\mathcal{H}}_{1}}$ is $({H}_{1},{\eta}_{1})$-monotone and ${M}_{2}(x,\cdot ):{\mathcal{H}}_{2}\to {2}^{{\mathcal{H}}_{2}}$ is $({H}_{2},{\eta}_{2})$-monotone, $F,{N}_{1}:{\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\to {\mathcal{H}}_{1}$, $G,{N}_{2}:{\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\to {\mathcal{H}}_{2}$ be operators and $A,C:{\mathcal{H}}_{1}\to CB({\mathcal{H}}_{1})$, $B,D:{\mathcal{H}}_{2}\to CB({\mathcal{H}}_{2})$ be four set-valued operators. Let *ρ* and *ν* be two positive constants. For any $(f,g)\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}$, we consider the following problem:

where $({g}_{i}-{m}_{i})({x}^{\prime})={g}_{i}({x}^{\prime})-{m}_{i}({x}^{\prime})$, $\mathrm{\forall}{x}^{\prime}\in {\mathcal{H}}_{i}$, $i\in \{1,2\}$. The problem (3.1) is called *a system of generalized nonlinear mixed quasivariational inclusions*.

- (A)If $\rho ,\nu =1$, $f=g=0$, ${M}_{1}(({g}_{1}-{m}_{1})(x),y)={M}_{1}({g}_{1}(x))$ and ${M}_{2}(x,({g}_{2}-{m}_{2})(y))={M}_{2}({g}_{2}(y))$ for all $(x,y)\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}$, then the problem (3.1) is equivalent to finding $(x,y,u,v,w,z)$ such that $(x,y)\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}$, $u\in A(x)$, $v\in B(y)$, $w\in C(x)$, $z\in D(y)$, ${g}_{1}(x)\in dom({M}_{1})$, ${g}_{2}(y)\in dom({M}_{2})$ and$\begin{array}{r}0\in {N}_{1}(u,v)+F(x,y)+{M}_{1}({g}_{1}(x)),\\ 0\in {N}_{2}(w,z)+G(x,y)+{M}_{2}({g}_{2}(y)),\end{array}$(3.2)

- (B)If ${N}_{1}={N}_{2}=0$, ${g}_{1}={I}_{1}$ (the identity map on ${\mathcal{H}}_{1}$), ${g}_{2}={I}_{2}$ (the identity map on ${\mathcal{H}}_{2}$), ${\eta}_{1}(a,x)=a-x$ for every $a,x\in {\mathcal{H}}_{1}$, ${\eta}_{2}(b,y)=b-y$ for any $b,y\in {\mathcal{H}}_{2}$, ${M}_{1}(x)=\partial \phi (x)$ and ${M}_{2}(y)=\partial \psi (y)$, where and are two proper, convex, lower semi-continuous functionals, $\partial \phi (x)$ is the subdifferential of
*φ*at*x*, $\partial \psi (y)$ is the subdifferential of*ψ*at*y*, then the problem (3.2) reduces to the following system of nonlinear variational inequalities introduced by Cho-Fang-Huang-Hwang [2]: seek $(x,y)\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}$ such that$\begin{array}{r}{\u3008F(x,y),a-x\u3009}_{1}+\phi (a)-\phi (x)\ge 0,\\ {\u3008G(x,y),b-y\u3009}_{2}+\psi (b)-\psi (y)\ge 0.\end{array}$(3.3)

In brief, for suitable choices of the mappings presented in the problem (3.1) and the constants *ρ*, *ν*, one can obtain various new and previously known systems of variational inequalities and variational inclusions as special cases of the problem (3.1).

**Lemma 3.1**

*Let*

*λ*

*and*

*μ*

*be two positive constants*.

*For each*$i\in \{1,2\}$,

*let*${\eta}_{i}:{\mathcal{H}}_{i}\times {\mathcal{H}}_{i}\to {\mathcal{H}}_{i}$

*be an operator*, ${H}_{i}:{\mathcal{H}}_{i}\to {\mathcal{H}}_{i}$

*be a strictly*${\eta}_{i}$-

*monotone operator*, ${M}_{i}:{\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\to {2}^{{\mathcal{H}}_{i}}$

*be a set*-

*valued operator such that for each*$(x,y)\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}$, ${M}_{1}(\cdot ,y):{\mathcal{H}}_{1}\to {2}^{{\mathcal{H}}_{1}}$

*is*$({H}_{1},{\eta}_{1})$-

*monotone and*${M}_{2}(x,\cdot ):{\mathcal{H}}_{2}\to {2}^{{\mathcal{H}}_{2}}$

*is*$({H}_{2},{\eta}_{2})$-

*monotone*.

*Then*$(x,y,u,v,w,z)$

*with*$(x,y)\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}$, $u\in A(x)$, $v\in B(y)$, $w\in C(x)$, $z\in D(y)$

*is a solution of the problem*(3.1)

*if and only if*

*where* ${J}_{{M}_{1}(\cdot ,y),\lambda \rho}^{{H}_{1},{\eta}_{1}}={({H}_{1}+\lambda \rho {M}_{1}(\cdot ,y))}^{-1}$, ${J}_{{M}_{2}(x,\cdot ),\mu \nu}^{{H}_{2},{\eta}_{2}}={({H}_{2}+\mu \nu {M}_{2}(x,\cdot ))}^{-1}$ *and* *λ*, *μ* *are both positive constants*.

This completes the proof. □

Based on Lemma 2.3 and Lemma 3.1, we suggest the following two sorts of iterative algorithms for the system of generalized nonlinear mixed quasivariational inclusions (3.1).

**Algorithm 3.1** Let *λ* and *μ* be two positive constants. For each $i\in \{1,2\}$, let ${\eta}_{i}:{\mathcal{H}}_{i}\times {\mathcal{H}}_{i}\to {\mathcal{H}}_{i}$ be an operator, ${H}_{i}:{\mathcal{H}}_{i}\to {\mathcal{H}}_{i}$ be a strictly ${\eta}_{i}$-monotone operator, ${M}_{i}:{\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\to {2}^{{\mathcal{H}}_{i}}$ be a set-valued operator such that for each $(x,y)\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}$, ${M}_{1}(\cdot ,y):{\mathcal{H}}_{1}\to {2}^{{\mathcal{H}}_{1}}$ is $({H}_{1},{\eta}_{1})$-monotone and ${M}_{2}(x,\cdot ):{\mathcal{H}}_{2}\to {2}^{{\mathcal{H}}_{2}}$ is $({H}_{2},{\eta}_{2})$-monotone.

Step 0: Choose $({x}_{0},{y}_{0})\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}$ and take ${u}_{0}\in A({x}_{0})$, ${v}_{0}\in B({y}_{0})$, ${w}_{0}\in C({x}_{0})$, ${z}_{0}\in D({y}_{0})$. Set $n=0$.

Step 2: If ${x}_{n+1}$, ${y}_{n+1}$, ${u}_{n+1}$, ${v}_{n+1}$, ${w}_{n+1}$, ${z}_{n+1}$ satisfy Lemma 3.1 to sufficient accuracy, stop; otherwise, set $n:=n+1$ and return to Step 1.

**Algorithm 3.2**Let

*λ*and

*μ*be two positive constants. For each $i\in \{1,2\}$ and $n\in \omega $, let ${\eta}_{i}:{\mathcal{H}}_{i}\times {\mathcal{H}}_{i}\to {\mathcal{H}}_{i}$ be an operator, ${H}_{i}:{\mathcal{H}}_{i}\to {\mathcal{H}}_{i}$ be a strictly ${\eta}_{i}$-monotone operator, ${M}_{i},{M}_{i}^{n}:{\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\to {2}^{{\mathcal{H}}_{i}}$ be set-valued operators such that for each $(x,y)\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}$, ${M}_{1}(\cdot ,y),{M}_{1}^{n}(\cdot ,y):{\mathcal{H}}_{1}\to {2}^{{\mathcal{H}}_{1}}$ is $({H}_{1},{\eta}_{1})$-monotone and ${M}_{2}(x,\cdot ),{M}_{2}^{n}(x,\cdot ):{\mathcal{H}}_{2}\to {2}^{{\mathcal{H}}_{2}}$ is $({H}_{2},{\eta}_{2})$-monotone. Given $({x}_{0},{y}_{0})\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}$, compute ${\{({x}_{n},{y}_{n},{u}_{n},{v}_{n},{w}_{n},{z}_{n})\}}_{n\in \omega}$ by the iterative schemes

- (a)
${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}^{\ast}=+\mathrm{\infty}$;

- (b)
${e}_{n}={e}_{n}^{\prime}+{e}_{n}^{\u2033}$, $\parallel {e}_{n}^{\u2033}\parallel ={\xi}_{n}^{\ast}{\alpha}_{n}^{\ast}$, $\mathrm{\forall}n\in \omega $, ${\sum}_{n=0}^{\mathrm{\infty}}\parallel {e}_{n}^{\prime}\parallel <+\mathrm{\infty}$, ${lim}_{n\to \mathrm{\infty}}{\xi}_{n}^{\ast}=0$;

- (c)
${f}_{n}={f}_{n}^{\prime}+{f}_{n}^{\u2033}$, $\parallel {f}_{n}^{\u2033}\parallel ={\varsigma}_{n}^{\ast}{\alpha}_{n}^{\ast}$, $\mathrm{\forall}n\in \omega $, ${\sum}_{n=0}^{\mathrm{\infty}}\parallel {f}_{n}^{\prime}\parallel <+\mathrm{\infty}$, ${lim}_{n\to \mathrm{\infty}}{\varsigma}_{n}^{\ast}=0$.

**Remark 3.1** Algorithm 3.2 includes several known algorithms in [2–6, 13, 15–18] as special cases.

## 4 Existence and convergence theorems

At present, we seek those conditions which ensure the existence of solutions for the problem (3.1) and the convergence of the iterative sequence generated by Algorithm 3.1. Furthermore, based on the existence of the solutions for the problem (3.1), the convergence of the Mann perturbed iterative sequence generated by Algorithm 3.2 is discussed.

**Theorem 4.1**

*Let*

*ρ*

*and*

*ν*

*be two positive constants*.

*For*$i\in \{1,2\}$,

*let*${\eta}_{i}:{\mathcal{H}}_{i}\times {\mathcal{H}}_{i}\to {\mathcal{H}}_{i}$

*be Lipschitz continuous with a constant*${\tau}_{i}$, ${H}_{i}:{\mathcal{H}}_{i}\to {\mathcal{H}}_{i}$

*be strongly*${\eta}_{i}$-

*monotone and Lipschitz continuous with constants*${\sigma}_{i}$

*and*${\delta}_{i}$,

*respectively*, ${g}_{i},{m}_{i}:{\mathcal{H}}_{i}\to {\mathcal{H}}_{i}$

*be Lipschitz continuous with constants*${s}_{i}$

*and*${t}_{i}$,

*respectively*, ${g}_{i}$

*be*${m}_{i}$-

*relaxed monotone with a constant*${k}_{i}$, ${g}_{i}-{m}_{i}$

*be strongly monotone with a constant*${r}_{i}$, ${M}_{i}:{\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\to {2}^{{\mathcal{H}}_{i}}$

*be set*-

*valued operators such that for each*$(x,y)\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}$, ${M}_{1}(\cdot ,y):{\mathcal{H}}_{1}\to {2}^{{\mathcal{H}}_{1}}$

*is*$({H}_{1},{\eta}_{1})$-

*monotone and*${M}_{2}(x,\cdot ):{\mathcal{H}}_{2}\to {2}^{{\mathcal{H}}_{2}}$

*is*$({H}_{2},{\eta}_{2})$-

*monotone*.

*Let*$A,C:{\mathcal{H}}_{1}\to CB({\mathcal{H}}_{1})$

*be*$\stackrel{\u02c6}{{\mathcal{H}}_{1}}$-

*Lipschitz continuous with constants*${l}_{A}$, ${l}_{C}$,

*respectively*,

*and*$B,D:{\mathcal{H}}_{2}\to CB({\mathcal{H}}_{2})$

*be*-

*Lipschitz continuous with constants*${l}_{B}$, ${l}_{D}$,

*respectively*.

*Let*$F:{\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\to {\mathcal{H}}_{1}$

*be*${H}_{1}({g}_{1}-{m}_{1})$-

*relaxed monotone in the first argument with a constant*${\alpha}_{1}$,

*Lipschitz continuous in the first and second arguments with constants*${\beta}_{1}$

*and*${\gamma}_{1}$,

*respectively*.

*Let*$G:{\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\to {\mathcal{H}}_{2}$

*be*${H}_{2}({g}_{2}-{m}_{2})$-

*relaxed monotone in the second argument with a constant*${\alpha}_{2}$,

*Lipschitz continuous in the first and second arguments with constants*${\beta}_{2}$

*and*${\gamma}_{2}$,

*respectively*.

*Assume that*${N}_{1}:{\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\to {\mathcal{H}}_{1}$

*is Lipschitz continuous in the first and second arguments with constants*${b}_{1}$

*and*${c}_{1}$,

*respectively*, ${N}_{2}:{\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\to {\mathcal{H}}_{2}$

*is Lipschitz continuous in the first and second arguments with constants*${b}_{2}$

*and*${c}_{2}$,

*respectively*.

*Assume that there exist positive constants*${\lambda}_{1}$, ${\lambda}_{2}$,

*λ*

*and*

*μ*

*satisfying*

*where*

*Then the problem* (3.1) *admits a solution* $(x,y,u,v,w,z)\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\times A(x)\times B(y)\times C(x)\times D(y)$ *and the sequence* ${\{({x}_{n},{y}_{n},{u}_{n},{v}_{n},{w}_{n},{z}_{n})\}}_{n\in \omega}$ *generated by Algorithm* 3.1 *converges to* $(x,y,u,v,w,z)$ *as* $n\to \mathrm{\infty}$.

*Proof*Set

*F*is relaxed monotone with respect to ${H}_{1}({g}_{1}-{m}_{1})$ in the first argument and Lipschitz continuous in the first argument, respectively, ${g}_{1}$, ${m}_{1}$ are both Lipschitz continuous, and ${g}_{1}$ is ${m}_{1}$-relaxed monotone, we derive that

*F*is Lipschitz continuous in the second argument, we get that

*A*and

*B*, we arrive at

In view of (4.2) and (4.14), we know that $0<\zeta <1$ and so (4.13) implies that ${\{{x}_{n}\}}_{n\in \omega}$, ${\{{y}_{n}\}}_{n\in \omega}$ are both Cauchy sequences. Thus, there exist $x\in {\mathcal{H}}_{1}$ and $y\in {\mathcal{H}}_{2}$ satisfying ${x}_{n}\to x$ and ${y}_{n}\to y$ as $n\to \mathrm{\infty}$.

*A*and (3.5) that

*F*,

*G*, ${N}_{i}$, ${J}_{{M}_{1}(\cdot ,y),\lambda \rho}^{{H}_{1},{\eta}_{1}}$, ${J}_{{M}_{2}(x,\cdot ),\mu \nu}^{{H}_{2},{\eta}_{2}}$ and Algorithm 3.1, we find that $(x,y,u,v,w,z)$ satisfy the following relations:

It follows that $(x,y,u,v,w,z)$ is a solution of the problem (3.1) from Lemma 3.1. This completes the proof. □

**Remark 4.1** Theorem 4.1 generalizes and unifies those results in [2–6, 13, 15–18] from the following aspects:

(d1) The problem (3.1) includes the variational inequalities and variational inclusions in [2, 15–17] as special cases;

(d2) For every $i\in \{1,2\}$, the $({H}_{i},{\eta}_{i})$-monotone operators and ${H}_{i}({g}_{i}-{m}_{i})$-relaxed monotone operators we utilized here are much more universal than those used in [3, 4, 6, 13, 18];

(d3) Algorithm 3.1 is very different from those in [2–6, 13, 15–18] for finding approximate solutions.

**Theorem 4.2**

*Let*

*ρ*,

*ν*, ${\eta}_{i}$, ${H}_{i}$, ${g}_{i}$, ${m}_{i}$,

*A*,

*B*,

*C*,

*D*,

*F*,

*G*, ${N}_{i}$, ${M}_{i}$, $i\in \{1,2\}$,

*be the same as in Theorem*4.1.

*For*$n\in \omega $,

*let*${M}_{1}^{n}:{\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\to {2}^{{\mathcal{H}}_{1}}$, ${M}_{2}^{n}:{\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\to {2}^{{\mathcal{H}}_{2}}$

*be set*-

*valued operators such that*${M}_{1}^{n}(\cdot ,y):{\mathcal{H}}_{1}\to {2}^{{\mathcal{H}}_{1}}$

*is*$({H}_{1},{\eta}_{1})$-

*monotone for each fixed*$y\in {\mathcal{H}}_{2}$, ${M}_{2}^{n}(x,\cdot ):{\mathcal{H}}_{2}\to {2}^{{\mathcal{H}}_{2}}$

*is*$({H}_{2},{\eta}_{2})$-

*monotone for each fixed*$x\in {\mathcal{H}}_{1}$.

*Assume that there exist positive constants*${\lambda}_{1}$, ${\lambda}_{2}$,

*λ*

*and*

*μ*

*satisfying*(4.2)-(4.3),

*and*

*Then the problem* (3.1) *admits a solution* $(x,y,u,v,w,z)\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\times A(x)\times B(y)\times C(x)\times D(y)$ *and the sequence* ${\{({x}_{n},{y}_{n},{u}_{n},{v}_{n},{w}_{n},{z}_{n})\}}_{n\in \omega}$ *generated by Algorithm* 3.2 *converges to* $(x,y,u,v,w,z)$ *as* $n\to \mathrm{\infty}$.

*Proof*It follows from Theorem 4.1 that the problem (3.1) possesses a solution $(x,y,u,v,w,z)\in {\mathcal{H}}_{1}\times {\mathcal{H}}_{2}\times A(x)\times B(y)\times C(x)\times D(y)$. Further, we get that by Lemma 3.1

*ζ*are defined by (4.14). Thus, ${\zeta}_{n}\to \zeta $ as $n\to \mathrm{\infty}$ and $0<\zeta <1$. Hence, for $\theta =\frac{1+\zeta}{2}$, there exist such that ${\zeta}_{n}<\theta $ for every $n\ge {n}_{0}$. By (b) and (c) in Algorithm 3.2 and (4.20), we gain that

On the grounds of (4.16), (a)-(c) in Algorithm 3.2 and Lemma 2.2, we obtain that ${x}_{n}\to x$ and ${y}_{n}\to y$ as $n\to \mathrm{\infty}$. The remainder of the proof is the same as that in Theorem 4.1 and is omitted. This completes the proof. □

**Remark 4.2** Theorem 4.2 improves and extends those corresponding results in [1, 2, 4, 5, 15–18, 21].

## Declarations

### Acknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2002165).

## Authors’ Affiliations

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