Open Access

Auxiliary principle and iterative algorithm for a system of generalized set-valued strongly nonlinear mixed implicit quasi-variational-like inequalities

Journal of Inequalities and Applications20162016:38

https://doi.org/10.1186/s13660-016-0978-3

Received: 21 September 2015

Accepted: 18 January 2016

Published: 1 February 2016

Abstract

In this paper, the auxiliary principle technique is extended to study a system of generalized set-valued strongly nonlinear mixed implicit quasi-variational-like inequalities problem in Hilbert spaces. First, we establish the existence of solutions of the corresponding system of auxiliary variational inequalities problem. Then, using the existence result, we construct a new iterative algorithm. Finally, both the existence of solutions of the original problem and the convergence of iterative sequences generated by the algorithm are proved. We give an affirmative answer to the open problem raised by Noor et al. (Korean J. Comput. Appl. Math. 1:73-89, 1998; J. Comput. Appl. Math. 47:285-312, 1993). Our results improve and extend some known results.

Keywords

system of generalized mixed implicit quasi-variational-like inequalitiesauxiliary principleiterative algorithmconvergence

MSC

47H1049J30

1 Introduction

Throughout the paper, let \(I=\{1,2\}\) be an index set, let, for each \(i \in I\), \(H_{i}\) be a Hilbert space with inner product \(\langle\cdot,\cdot\rangle_{i}\) and norm \(\Vert \cdot \Vert _{i}\), and \(CB(H_{i})\) be the family of all nonempty bounded closed subsets of \(H_{i}\). Denoting by \(2^{H_{i}}\) the family of all nonempty subsets of \(H_{i}\), let \(K_{1}:H_{1}\rightarrow2^{H_{1}}\) and \(K_{2}:H_{2}\rightarrow 2^{H_{2}}\) be two set-valued mappings such that for each \(x\in H_{1}\), \(K_{1}(x)\) is a closed convex subset of \(H_{1}\) and for each \(y\in H_{2}\), \(K_{2}(y)\) is a closed convex subset of \(H_{2}\). Let \(N_{i}, \eta_{i}:H_{i}\times H_{i}\rightarrow H_{i}\) be nonlinear single-valued mappings, and let \(A_{i}, B_{i}:H_{i}\rightarrow CB(H_{i})\) and \(F_{i}:H_{1}\times H_{2}\rightarrow CB(H_{i})\) be set-valued mappings. We consider the following generalized set-valued strongly nonlinear mixed implicit quasi-variational-like inequalities problem: Find \((x,y)\in K_{1}(x)\times K_{2}(y)\), \(u_{1}\in A_{1}x\), \(v_{1}\in B_{1}x\), \(u_{2}\in A_{2}y\), \(v_{2}\in B_{2}y\), and \(w_{i}\in F_{i}(x,y)\) such that
$$ \textstyle\begin{cases} \langle N_{1}(u_{1},v_{1})+w_{1},\eta_{1}(h_{1},x)\rangle _{1}+b_{1}(x,h_{1})-b_{1}(x,x)\geq 0,& \forall h_{1}\in K_{1}(x),\\ \langle N_{2}(u_{2},v_{2})+w_{2},\eta_{2}(h_{2},y)\rangle _{2}+b_{2}(y,h_{2})-b_{2}(y,y)\geq 0,& \forall h_{2}\in K_{2}(y), \end{cases} $$
(1.1)
where \(b_{i}:H_{i}\times H_{i}\rightarrow R\), which is not necessarily differentiable, possesses the following properties:
  1. (i)

    \(b_{i}(\cdot,\cdot)\) is linear in the first argument;

     
  2. (ii)
    \(b_{i}(\cdot,\cdot)\) is bounded, that is, there exists a constant \(\gamma_{i}>0\) such that
    $$b_{i}(u_{i},v_{i})\leq \gamma_{i}\Vert u_{i}\Vert _{i}\Vert v_{i}\Vert _{i},\quad \forall u_{i},v_{i}\in H_{i}; $$
     
  3. (iii)

    \(b_{i}(u_{i},v_{i})-b_{i}(u_{i},w_{i})\leq b_{i}(u_{i},v_{i}-w_{i})\), \(\forall u_{i},v_{i},w_{i}\in H_{i}\);

     
  4. (iv)

    \(b_{i}(\cdot,\cdot)\) is convex in the second argument.

     
In many important applications, \(K_{1}(x)\) and \(K_{2}(y)\) have the following forms [1, 3]:
$$ \textstyle\begin{cases} K_{1}(x)=m_{1}(x)+K_{1}, & \forall x \in H_{1},\\ K_{2}(y)=m_{2}(y)+K_{2}, & \forall y \in H_{2}, \end{cases} $$
(1.2)
where \(m_{i}:H_{i}\rightarrow H_{i}\) are single-valued mappings, and \(K_{i}\) are closed convex subsets of \(H_{i}\).

Choosing different mappings in problem (1.1), we have the following special cases.

(I) If \(F_{1}(\cdot,\cdot)=F_{2}(\cdot,\cdot)=0\), then problem (1.1) reduces to the problem of finding \((x,y)\in H_{1}\times H_{2}\), \(u_{1}\in A_{1}x\), \(v_{1}\in B_{1}x\), \(u_{2}\in A_{2}y\), and \(v_{2}\in B_{2}y\) such that
$$ \textstyle\begin{cases} \langle N_{1}(u_{1},v_{1}),\eta_{1}(h_{1},x)\rangle _{1}+b_{1}(x,h_{1})-b_{1}(x,x)\geq 0,& \forall h_{1}\in H_{1}, \\ \langle N_{2}(u_{2},v_{2}),\eta_{2}(h_{2},y)\rangle _{2}+b_{2}(y,h_{2})-b_{2}(y,y)\geq 0, &\forall h_{2}\in H_{2}. \end{cases} $$
(1.3)
Problem (1.3) has been introduced and studied by Kazmi and Khan [4].
(II) If \(H_{1}=H_{2}=H\), \(b_{1}(\cdot,\cdot)=b_{2}(\cdot,\cdot)=0\), \(\eta_{1}(h_{1},x)=h_{1}-x\), \(\eta_{2}(h_{2},y)=h_{2}-y\), and \(A_{i}=B_{i}=I_{i}\), then problem (1.1) reduces to the problem of finding \((x,y)\in K_{1}(x)\times K_{2}(y)\) and \(w_{i}\in F_{i}(x,y)\) such that
$$ \textstyle\begin{cases} \langle N_{1}(x,y)+w_{1},h_{1}-x\rangle\geq0,& \forall h_{1}\in K_{1}(x), \\ \langle N_{2}(x,y)+w_{2},h_{2}-y\rangle\geq0, &\forall h_{2}\in K_{2}(y). \end{cases} $$
(1.4)
Problem (1.4) has been introduced and studied by Qiu et al. [5].
(III) If \(N_{1}=N_{2}=N\), \(A_{1}=A_{2}=A\), \(B_{1}=B_{2}=B\), \(K_{1}=K_{2}=K\), \(F_{1}=F_{2}=0\), \(\eta _{1}(h_{1},x)=\eta_{2}(h_{2},y)=h-x\), and \(b_{1}=b_{2}=b\), then problem (1.1) reduces to the problem of finding \(x\in K(x)\), \(u\in Ax\), and \(v\in Bx\) such that
$$ \bigl\langle N(u,v),h-x\bigr\rangle +b(x,h)-b(x,x)\geq0,\quad \forall h\in K(x). $$
(1.5)

The problem has been introduced by Ding [3].

(IV) If \(N_{1}(\cdot,\cdot)=N_{2}(\cdot,\cdot)=(V\circ A+B)(\cdot )\), \(F_{1}(\cdot,\cdot)=F_{2}(\cdot,\cdot)=F(\cdot)\), \(K_{1}=K_{2}=K\), and \(A_{i}=B_{i}=I_{i}\), then problem (1.1) is equivalent to finding \(x\in K(x)\) such that \(w\in F(x)\), \(u\in A(x)\), and
$$ \langle\omega+Vu+Bx,h-x\rangle\geq0,\quad \forall h\in K(x), $$
(1.6)
which is due to Noor [1].
(V) If \(N_{1}=N_{2}=N\), \(A_{1}=A_{2}=A\), \(B_{1}=B_{2}=B\), \(\eta_{1}=\eta_{2}=\eta \), \(F_{1}=F_{2}=0\), and \(b_{1}=b_{2}=b\), then problem (1.1) reduces to the problem of finding \(x\in H\), \(u\in Ax\), and \(v\in Bx\) such that
$$ \bigl\langle N(u,v),\eta(h,x)\bigr\rangle +b(x,h)-b(x,x)\geq0,\quad \forall h\in H. $$
(1.7)

The problem was introduced by Zeng et al. [6] and Huang et al. [7].

There are many special cases of problems (1.1) and (1.3)-(1.7), which can be also found in Cohen [8], Noor [2, 9, 10], and the references therein. In the system of the generalized set-valued strongly nonlinear mixed implicit quasi-variational-like inequalities problem (1.1), the two convex sets depend on the solution implicitly or explicitly, and \(b_{i}\) is a nonlinear mapping, so the projection method cannot be applied to it. This fact motivated many authors to develop the auxiliary principle technique to study the existence of solutions of generalized mixed type quasi-variational inequalities and also to develop a large number of numerical methods for solving various variational inequalities, complementarity problems, and optimization problems. The auxiliary principle technique was first introduced by Glowinski et al. [11]. Then Noor [12] extended it to study the existence and uniqueness of solutions when A and B are compact set-valued mappings. Regretfully, A is actually a single-valued mapping in his Theorem 4.1, as it was pointed out in Liu and Li [13]. Noor [1, 2] put forward that extending the projection methods and its variant forms for generalized set-valued mixed nonlinear variational inequalities involving the nonlinear form \(b(\cdot,\cdot)\) satisfying properties (i), (ii), and (iii) is still an open problem. The theory of quasi-variational inequalities is not developed to an extent providing a complete framework for studying these problems, and this is another direction for future research in this fascinating and elegant area. Also, extending the auxiliary principle technique for quasi-variational inequalities is still an open problem, and this needs further research efforts.

In this paper, we extend the auxiliary principle technique to study the generalized set-valued strongly nonlinear mixed implicit quasi-variational-like inequalities problem (1.1) in Hilbert spaces. First, we establish the existence of solutions of the corresponding system of auxiliary variational inequalities (2.1). Then, using the existence result, we construct a new iterative algorithm. Finally, both the existence of solutions of the original problem and the convergence of iterative sequences generated by the algorithm are proved. Our results improve and extend some known results.

We first recall some concepts and definitions.

Definition 1.1

The mapping \(\eta:H\times H\rightarrow H\) is said to be ε-Lipschitz continuous, if there exists a constant \(\varepsilon>0\) such that
$$\bigl\Vert \eta(x,y)\bigr\Vert \leq\varepsilon \Vert x-y\Vert ,\quad \forall x,y\in H. $$

Definition 1.2

Let \(N:H\times H\rightarrow H\) be a single-valued mapping, and \(A, B:H\rightarrow CB(H)\) be set-valued mappings.
  1. (1)
    N is said to be \((\beta,\xi)\)-Lipschitz continuous if there exists a pair of constants \(\beta,\xi>0\) such that
    $$\bigl\Vert N(u_{1},v_{1})-N(u_{2},v_{2}) \bigr\Vert \leq\beta \Vert u_{1}-u_{2}\Vert +\xi \Vert v_{1}-v_{2}\Vert ,\quad \forall u_{1},v_{1},u_{2},v_{2} \in H; $$
     
  2. (2)
    N is said to be α-strongly mixed monotone with set-valued mappings A and B if there exists a constant \(\alpha>0\) such that
    $$\begin{aligned}& \bigl\langle N(u_{1},v_{1})-N(u_{2},v_{2}),x-y \bigr\rangle \geq\alpha \Vert x-y\Vert ^{2}, \\& \quad\forall x, y\in H, u_{1}\in A(x), v_{1}\in B(x), u_{2}\in A(y), v_{2}\in B(y). \end{aligned}$$
     
  3. (3)
    A is said to be λ-H-Lipschitz continuous if there exists a constant \(\lambda>0\) such that
    $$H(Au,Av)\leq\lambda \Vert u-v\Vert , \quad\forall u,v\in H, $$
    where \(H(\cdot,\cdot)\) is the Hausdorff metric on \(CB(H)\);
     

Definition 1.3

The mapping \(m:H\rightarrow H\) is said to be δ-Lipschitz continuous if there exists a constant \(\delta>0\) such that
$$\Vert mx-my\Vert \leq\delta \Vert x-y\Vert ,\quad \forall x,y\in H. $$

Definition 1.4

The set-valued mapping \(F(\cdot,\cdot):H\times H\rightarrow CB(H)\) is said to be \((l,k)\)-H-Lipschitz continuous if there exist constants \(l, k>0\) such that
$$H\bigl(F(x_{1},y_{1}),F(x_{2},y_{2}) \bigr)\leq l\Vert x_{1}-x_{2}\Vert +k\Vert y_{1}-y_{2}\Vert ,\quad \forall x_{1},x_{2},y_{1},y_{2} \in H, $$
where \(H(\cdot,\cdot)\) is the Hausdorff metric on \(CB(H)\).

To obtain our results, we need the following assumption.

Assumption 1.1

The mapping \(\eta_{i}:H_{i}\times H_{i}\rightarrow H_{i}\) satisfies the following conditions:
  1. (1)

    \(\eta_{i}(u,v)=\eta_{i}(u,z)+\eta_{i}(z,v)\), \(\forall u,v\in H_{i}\);

     
  2. (2)

    \(\eta_{i}(u+v,w)=-\eta_{i}(w-u,v)\), \(\forall u,v,w\in H_{i}\);

     
  3. (3)

    the functionals \(h_{1}\mapsto\langle N_{1}(u_{1},v_{1})+w_{1},\eta_{1}(h_{1},x)\rangle_{1}\) and \(h_{2}\mapsto\langle N_{2}(u_{2},v_{2})+w_{2},\eta_{2}(h_{2}, y)\rangle _{2}\) are both continuous and linear for all \(h_{1}\in H_{1}\) and \(h_{2}\in H_{2}\).

     

Remark 1.1

If \(\eta_{i}(u,v)=u-v\), obviously, Assumption 1.1(2) holds. In a word, Assumption 1.1 is justified.

We also need the following lemma.

Lemma 1.1

[15]

Let E be a normed vector space, \(CB(E)\) be the family of all closed bounded subsets of E, and \(T:E\rightarrow CB(E)\) be a set-valued mapping. Then for any given \(\varepsilon>0\), \(x,y\in E\), and \(u\in Tx\), there exists \(v\in Ty\) such that
$$d(u,v)\leq(1+\varepsilon)H(Tx,Ty), $$
where \(H(\cdot,\cdot)\) is the Hausdorff metric on \(CB(E)\).

2 Auxiliary problem and iterative algorithm

In this section, we extend the auxiliary principle technique to study problem (1.1) and prove the existence of solutions of the auxiliary problem for (1.1). Then, by using the existence theorem we construct an iterative algorithm for problem (1.1).

For each \(i\in I\), given \((x,y)\in K_{1}(x)\times K_{2}(y)\), \(u_{1}\in A_{1}x\), \(v_{1}\in B_{1}x\), \(u_{2}\in A_{2}y\), \(v_{2}\in A_{2}y\), and \(w_{i}\in F_{i}(x,y)\), we consider the following problem: Find \((p_{1},p_{2})\in K_{1}(x)\times K_{2}(y)\) such that
$$ \textstyle\begin{cases} \langle p_{1},h_{1}-p_{1}\rangle_{1}\geq \langle x,h_{1}-p_{1}\rangle_{1}-\rho_{1}\langle N_{1}(u_{1},v_{1})+w_{1},\eta_{1}(h_{1},p_{1})\rangle_{1}\\ \hphantom{\langle p_{1},h_{1}-p_{1}\rangle_{1}\geq}{}+ \rho_{1}[b_{1}(x,p_{1})-b_{1}(x,h_{1})], \quad \forall h_{1}\in K_{1}(x),\\ \langle p_{2},h_{2}-p_{2}\rangle_{2}\geq \langle y,h_{2}-p_{2}\rangle_{2}-\rho_{2}\langle N_{2}(u_{2},v_{2})+w_{2},\eta_{2}(h_{2},p_{2})\rangle_{2}\\ \hphantom{\langle p_{2},h_{2}-p_{2}\rangle_{2}\geq}{} +\rho_{2}[b_{2}(y,p_{2})-b_{2}(y,h_{2})], \quad \forall h_{2}\in K_{2}(y), \end{cases} $$
(2.1)
where \(\rho_{1},\rho_{2}>0\) are constants. Problem (2.1) is called the system of auxiliary variational inequalities for problem (1.1).

Theorem 2.1

For each \(i \in I\), let \(K_{1}:H_{1}\rightarrow 2^{H_{1}}\) and \(K_{2}:H_{2}\rightarrow2^{H_{2}}\) be two set-valued mappings such that for each \(x\in H_{1}\), \(K_{1}(x)\) is a nonempty closed convex subset of \(H_{1}\) and for each \(y\in H_{2}\), \(K_{2}(y)\) is also a nonempty closed convex subset of \(H_{2}\). Let \(N_{i}, \eta _{i}:H_{i}\times H_{i}\rightarrow H_{i}\) be nonlinear single-valued mappings, \(A_{i}, B_{i}:H_{i}\rightarrow CB(H_{i})\), \(F_{i}:H_{1}\times H_{2}\rightarrow CB(H_{i})\) be set-valued mappings, and \(b_{i}:H_{i}\times H_{i}\rightarrow R\) be a mapping such that for any given \((x,y)\in H_{1}\times H_{2}\), the functionals \(h_{1}\mapsto b_{1}(x,h_{1})\) and \(h_{2}\mapsto b_{2}(y,h_{2})\) are proper convex and lower semicontinuous. If Assumption  1.1 holds, then for any given \((x,y)\in H_{1}\times H_{2}\), \(u_{1}\in A_{1}x\), \(v_{1}\in B_{1}x\), \(u_{2}\in A_{2}y\), \(v_{2}\in B_{2}y\), and \(w_{i}\in F_{i}(x,y)\), define the functionals \(J_{1}:K_{1}(x)\rightarrow R\) and \(J_{2}:K_{2}(y)\rightarrow R\) as follows:
$$J_{1}(h_{1})=\frac{1}{2}\langle h_{1},h_{1} \rangle_{1}+j_{1}(h_{1}), $$
where
$$\begin{aligned}& j_{1}(h_{1})=\rho_{1}\bigl\langle N_{1}(u_{1},v_{1})+w_{1},\eta _{1}(h_{1},x)\bigr\rangle _{1}+\rho_{1}b_{1}(x,h_{1})- \langle x,h_{1}\rangle_{1}, \\& J_{2}(h_{2})=\frac{1}{2}\langle h_{2},h_{2} \rangle_{2}+j_{2}(h_{2}), \end{aligned}$$
where
$$j_{2}(h_{2})=\rho_{2}\bigl\langle N_{2}(u_{2},v_{2})+w_{2},\eta _{2}(h_{2},y)\bigr\rangle _{2}+\rho_{2}b_{2}(y,h_{2})- \langle y,h_{2}\rangle_{2}. $$
Then we have:
  1. (i)

    \(J_{1}\) has a unique minimum point \(p_{1}\) in \(K_{1}(x)\), and \(J_{2}\) has a unique minimum point \(p_{2}\) in \(K_{2}(y)\).

     
  2. (ii)

    \(J_{1}\) and \(J_{2}\) have unique minimum points \(p_{1}\) in \(K_{1}(x)\) and \(p_{2}\) in \(K_{2}(y)\), respectively, if and only if \((p_{1},p_{2})\) is a unique solution of the system of auxiliary variational inequalities (2.1).

     

Proof

By Assumption 1.1(3) the functional \(h_{1}\mapsto\langle N_{1}(u_{1},v_{1})+w_{1},\eta_{1}(h_{1},x)\rangle_{1}\) is continuous and linear. Since \(h_{1}\mapsto b_{1}(x,h_{1})\) is proper convex lower semicontinuous on \(K_{1}(x)\), it is easy to see that \(j_{1}(h_{1})\) is proper convex lower semicontinuous on \(K_{1}(x)\), and so \(J_{1}(h_{1})\) is a strictly convex and lower semicontinuous functional on \(K_{1}(x)\). Thus, \(j_{1}(h_{1})\) is bounded from below by a hyperplane \(\langle q_{1},h_{1}\rangle_{1}+a_{1}\) (see [14]), where \(q_{1}\in H_{1}\) and \(a_{1}\in R\). Hence, we have
$$ J_{1}(h_{1}) = \frac{1}{2}\langle h_{1},h_{1} \rangle_{1}+j_{1}(h_{1})\geq \frac{1}{2} \Vert h_{1}\Vert _{1}^{2}+\langle q_{1},h_{1}\rangle_{1}+a_{1} = \frac{1}{2}\Vert h_{1}+q_{1}\Vert _{1}^{2}-\frac{1}{2}\Vert q_{1}\Vert _{1}^{2}+a_{1}. $$
It follows that
$$ J_{1}(h_{1})\rightarrow\infty \quad\bigl( \mbox{as } \Vert h_{1}\Vert _{1}\rightarrow\infty\bigr). $$
(2.2)
Now let \(\{h_{n}^{1}\}\) be a minimizing sequence of \(J_{1}\) on \(K_{1}(x)\), that is,
$$\underset{n\rightarrow\infty}{\lim}J_{1}\bigl(h_{n}^{1} \bigr)=\underset{h_{1}\in K_{1}(x)}{\inf}J_{1}(h_{1}). $$
We claim that \(\{h_{n}^{1}\}\) is bounded. If it is not true, then there exists its subsequence \(\{h_{n_{k}}^{1}\}\) such that \(\Vert h_{n_{k}}^{1}\Vert _{1}\geq k\), \(k=1,2,\ldots\) . By (2.2) we have \(J_{1}(h_{n_{k}}^{1})\rightarrow\infty\), which contradicts the fact that \(\{h_{n}^{1}\}\) is a minimizing sequence of \(J_{1}\) on \(K_{1}(x)\). By the Weierstrass theorem (see [14]) there exists \(p_{1}\in K_{1}(x)\) such that
$$J_{1}(p_{1})=\underset{h_{1}\in K_{1}(x)}{\min}J_{1}(h_{1}). $$
Again from the strict convexity of \(J_{1}\) we have that \(J_{1}\) has a unique minimum point \(p_{1}\) in \(K_{1}(x)\). Using a similar argument, we get that \(J_{2}\) has a unique minimum point \(p_{2}\) in \(K_{2}(y)\).
Now suppose that \(J_{1}\) has a unique minimum point \(p_{1}\) in \(K_{1}(x)\) and \(J_{2}\) has a unique minimum point \(p_{2}\) in \(K_{2}(y)\). Let us show that \((p_{1},p_{2})\) is a unique solution of the system of auxiliary variational inequalities (2.1). For any \(h_{1}\in K_{1}(x)\) and \(t\in[0,1]\), since \(j_{1}\) is convex and \(J_{1}\) is lower semicontinuous on \(K_{1}(x)\), we have
$$\begin{aligned} J_{1}(p_{1}) =&\frac{1}{2}\langle p_{1},p_{1} \rangle _{1}+j_{1}(p_{1})\leq J_{1} \bigl(p_{1}+t(h_{1}-p_{1})\bigr) \\ =&{1} {2}\bigl\langle p_{1}+t(h_{1}-p_{1}),p_{1}+t(h_{1}-p_{1}) \bigr\rangle _{1}+j_{1}\bigl(p_{1}+t(h_{1}-p_{1}) \bigr) \\ \leq& \frac{1}{2}\langle p_{1},p_{1} \rangle_{1}+\frac{t^{2}}{2}\langle h_{1}-p_{1},h_{1}-p_{1} \rangle_{1}+t\langle p_{1},h_{1}-p_{1} \rangle _{1}+j_{1}(p_{1}) +t\bigl(j_{1}(h_{1})-j_{1}(p_{1})\bigr) \\ =&J_{1}(p_{1})+\frac{t^{2}}{2}\langle h_{1}-p_{1},h_{1}-p_{1} \rangle _{1}+t\langle p_{1},h_{1}-p_{1} \rangle_{1}+t\bigl(j_{1}(h_{1})-j_{1}(p_{1}) \bigr). \end{aligned}$$
It follows that
$$\frac{t}{2}\langle h_{1}-p_{1},h_{1}-p_{1} \rangle_{1}+\langle p_{1},h_{1}-p_{1} \rangle_{1}+j_{1}(h_{1})-j_{1}(p_{1}) \geq0. $$
Letting \(t\rightarrow0\) in this inequality, we obtain
$$\begin{aligned}& \langle p_{1},h_{1}-p_{1}\rangle_{1}+j_{1}(h_{1})-j_{1}(p_{1}) \\& \quad= \langle p_{1},h_{1}-p_{1}\rangle_{1}+ \rho_{1}\bigl\langle N_{1}(u_{1},v_{1})+w_{1}, \eta_{1}(h_{1},x)\bigr\rangle _{1}+ \rho_{1}b_{1}(x,h_{1})-\langle x,h_{1} \rangle_{1} \\& \quad\quad{}-\rho_{1}\bigl\langle N_{1}(u_{1},v_{1})+w_{1}, \eta_{1}(p_{1},x)\bigr\rangle _{1}- \rho_{1}b_{1}(x,p_{1})+\langle x,p_{1} \rangle_{1} \\& \quad\geq 0. \end{aligned}$$
By Assumption 1.1(1) we observe that \(\eta_{1}(p_{1},x)-\eta _{1}(h_{1},x)=\eta_{1}(h_{1},p_{1})\), so
$$\begin{aligned} \langle p_{1},h_{1}-p_{1}\rangle_{1} \geq&\langle x,h_{1}-p_{1}\rangle_{1}- \rho_{1}\bigl\langle N_{1}(u_{1},v_{1})+w_{1}, \eta_{1}(h_{1},p_{1})\bigr\rangle _{1} \\ &{}+\rho_{1}\bigl[b_{1}(x,p_{1})-b_{1}(x,h_{1}) \bigr], \quad \forall h_{1}\in K_{1}(x). \end{aligned}$$
In a similar way, we have
$$\begin{aligned} \langle p_{2},h_{2}-p_{2}\rangle_{2} \geq&\langle y,h_{2}-p_{2}\rangle_{2}- \rho_{2}\bigl\langle N_{2}(u_{2},v_{2})+w_{2}, \eta_{2}(h_{2},p_{2})\bigr\rangle _{2} \\ &{}+\rho_{2}\bigl[b_{2}(y,p_{2})-b_{2}(y,h_{2}) \bigr], \quad \forall h_{2}\in K_{2}(y). \end{aligned}$$
So, \((p_{1},p_{2})\) is a unique solution of the system of auxiliary variational inequalities (2.1).
Conversely, let \((p_{1},p_{2})\) be a unique solution of the system of auxiliary variational inequalities (2.1). By the first inequality of (2.1) we have
$$\begin{aligned}& \frac{1}{2}\bigl(\langle h_{1},h_{1} \rangle_{1}-\langle p_{1},p_{1}\rangle _{1}\bigr) \\& \quad= \frac{1}{2}\bigl(\bigl\langle p_{1}+(h_{1}-p_{1}),p_{1}+(h_{1}-p_{1}) \bigr\rangle _{1}-\langle p_{1},p_{1} \rangle_{1}\bigr) \\& \quad= \langle p_{1},h_{1}-p_{1}\rangle_{1}+ \frac{1}{2}\Vert h_{1}-p_{1}\Vert _{1}^{2} \\& \quad\geq \langle p_{1},h_{1}-p_{1} \rangle_{1} \\& \quad\geq \langle x,h_{1}-p_{1}\rangle_{1}- \rho_{1}\bigl\langle N_{1}(u_{1},v_{1})+w_{1}, \eta_{1}(h_{1},p_{1})\bigr\rangle _{1} \\& \quad\quad{}+\rho_{1}\bigl[b_{1}(x,p_{1})-b_{1}(x,h_{1}) \bigr]. \end{aligned}$$
By Assumption 1.1(1) this implies that
$$\begin{aligned}& \frac{1}{2}\langle h_{1},h_{1}\rangle_{1}- \langle x,h_{1}\rangle_{1}+\rho_{1}\bigl\langle N_{1}(u_{1},v_{1})+w_{1}, \eta_{1}(h_{1},x)\bigr\rangle _{1} + \rho_{1}b_{1}(x,h_{1}) \\& \quad\geq \frac{1}{2}\langle p_{1},p_{1} \rangle_{1}-\langle x,p_{1}\rangle_{1}+ \rho_{1}\bigl\langle N_{1}(u_{1},v_{1})+w_{1}, \eta_{1}(p_{1},x)\bigr\rangle _{1} + \rho_{1}b_{1}(x,p_{1}), \end{aligned}$$
that is,
$$J_{1}(h_{1})\geq J_{1}(p_{1}), \quad \forall h_{1}\in K_{1}(x). $$
From the strict convexity of \(J_{1}\) we conclude that \(J_{1}\) has a unique minimum point \(p_{1}\) in \(K_{1}(x)\). Using a similar argument, we obtain that \(J_{2}\) has a unique minimum point \(p_{2}\) in \(K_{2}(y)\). This completes the proof. □

By virtue of Lemma 1.1 and Theorem 2.1 we now construct an iterative algorithm for solving problem (1.1).

Algorithm 2.1

For given \((x_{0},y_{0})\in H_{1}\times H_{2}\), \(u_{0}^{1}\in A_{1}x_{0}\), \(v_{0}^{1}\in B_{1}x_{0}\), \(u_{0}^{2}\in A_{2}y_{0}\), \(v_{0}^{2}\in B_{2}y_{0}\), and \(w_{0}^{i}\in F_{i}(x_{0},y_{0})\), let the sequences \(\{(x_{n},y_{n})\}\subset K_{1}(x_{n-1})\times K_{2}(y_{n-1})\), \(\{u_{n}^{1}\}\), \(\{v_{n}^{1}\}\), \(\{u_{n}^{2}\}\), \(\{ v_{n}^{2}\}\), \(\{w_{n}^{1}\}\), and \(\{w_{n}^{2}\}\) satisfy the following conditions:
$$\begin{aligned}& \langle x_{n+1},h_{1}-x_{n+1}\rangle_{1} \geq \langle x_{n},h_{1}-x_{n+1}\rangle_{1}- \rho_{1}\bigl\langle N_{1}\bigl(u_{n}^{1},v_{n}^{1} \bigr)+w_{n}^{1},\eta_{1}(h_{1},x_{n+1}) \bigr\rangle _{1} \\& \hphantom{\langle x_{n+1},h_{1}-x_{n+1}\rangle_{1} \geq }{} +\rho_{1}\bigl[b_{1}(x_{n},x_{n+1})-b_{1}(x_{n},h_{1}) \bigr],\quad \forall h_{1}\in K_{1}(x_{n+1}); \end{aligned}$$
(2.3)
$$\begin{aligned}& \langle y_{n+1},h_{2}-y_{n+1}\rangle_{2} \geq \langle y_{n},h_{2}-y_{n+1}\rangle_{2}- \rho_{2}\bigl\langle N_{2}\bigl(u_{n}^{2},v_{n}^{2} \bigr)+w_{n}^{2},\eta_{2}(h_{2},y_{n+1}) \bigr\rangle _{2} \\& \hphantom{\langle y_{n+1},h_{2}-y_{n+1}\rangle_{2} \geq}{} +\rho_{2}\bigl[b_{2}(y_{n},y_{n+1})-b_{2}(y_{n},h_{2}) \bigr],\quad \forall h_{2}\in K_{2}(y_{n+1}); \end{aligned}$$
(2.4)
$$\begin{aligned}& u_{n}^{1}\in A_{1}x_{n}, \quad \bigl\Vert u_{n+1}^{1}-u_{n}^{1}\bigr\Vert _{1}\leq\biggl(1+\frac {1}{n+1}\biggr)H(A_{1}x_{n+1},A_{1}x_{n}); \\& v_{n}^{1}\in B_{1}x_{n},\quad \bigl\Vert v_{n+1}^{1}-v_{n}^{1}\bigr\Vert _{1}\leq\biggl(1+\frac {1}{n+1}\biggr)H(B_{1}x_{n+1},B_{1}x_{n}); \\& u_{n}^{2}\in A_{2}y_{n}, \quad \bigl\Vert u_{n+1}^{2}-u_{n}^{2}\bigr\Vert _{2}\leq\biggl(1+\frac {1}{n+1}\biggr)H(A_{2}y_{n+1},A_{2}y_{n}); \\& v_{n}^{2}\in B_{2}y_{n}, \quad \bigl\Vert v_{n+1}^{2}-v_{n}^{2}\bigr\Vert _{2}\leq\biggl(1+\frac {1}{n+1}\biggr)H(B_{2}y_{n+1},B_{2}y_{n}); \\& w_{n}^{i}\in F_{i}(x_{n},y_{n}),\quad \bigl\Vert w_{n+1}^{i}-w_{n}^{i}\bigr\Vert _{1}\leq \biggl(1+\frac{1}{n+1}\biggr)H \bigl(F_{i}(x_{n+1},y_{n+1}),F_{i}(x_{n},y_{n}) \bigr), \end{aligned}$$
(2.5)
for every \(n=0,1,2,\ldots\) , where \(\rho_{1},\rho_{2}>0\) are constants.

3 Existence and convergence theorem

In this section, we prove the existence of the solution of problem (1.1) and the convergence of the iterative sequences generated by Algorithm 2.1.

Theorem 3.1

For each \(i \in I\), let \(H_{i}\) be a Hilbert space, and let \(K_{1}:H_{1}\rightarrow2^{H_{1}}\) and \(K_{2}:H_{2}\rightarrow2^{H_{2}}\) be two set-valued mappings such that for each \(x\in H_{1}\), \(K_{1}(x)\) is a nonempty closed convex subset of \(H_{1}\) and for each \(y\in H_{2}\), \(K_{2}(y)\) is also a nonempty closed convex subset of \(H_{2}\). Let \(N_{i}, \eta_{i}:H_{i}\times H_{i}\rightarrow H_{i}\), \(F_{i}:H_{1}\times H_{2}\rightarrow CB(H_{i})\), \(A_{i} , B_{i}:H_{i}\rightarrow CB(H_{i})\) be mappings, let mappings \(m_{i}:H_{i}\rightarrow H_{i}\) satisfy (1.2), and let \(b_{i}:H_{i}\times H_{i}\rightarrow R\) be real-valued functionals satisfying the properties in Theorem  2.1 and properties (i)-(iv). Assume that the following conditions are satisfied:
  1. (1)

    \(N_{i}\) is \((\alpha_{i},\beta_{i})\)-Lipschitz continuous and \(\xi_{i}\)-strongly mixed monotone with respect to \(A_{i}\) and \(B_{i}\);

     
  2. (2)

    \(F_{i}\) is \((l_{i},k_{i})\)-H-Lipschitz continuous;

     
  3. (3)

    \(A_{i}\) is \(\tau_{i}\)-H-Lipschitz continuous;

     
  4. (4)

    \(B_{i}\) is \(\omega_{i}\)-H-Lipschitz continuous;

     
  5. (5)

    \(m_{i}\) is \(\delta_{i}\)-Lipschitz continuous;

     
  6. (6)

    \(\eta_{i}\) is \(\varepsilon_{i}\)-Lipschitz continuous.

     
If Assumption  1.1 holds and there exist constants \(\rho_{1},\rho_{2}>0\) such that
$$ \textstyle\begin{cases} \frac{1}{1-2\delta_{1}}[1+\rho_{1}\gamma_{1} +\varepsilon_{1}(1+\sqrt{1-2\rho_{1} \xi_{1}+\rho_{1}^{2}(\alpha_{1}\tau_{1}+\beta_{1}\omega_{1})^{2}} +\rho_{1}l_{1})]\\ \quad{} +\frac{1}{1-2\delta_{2}}\varepsilon_{2}\rho_{2}l_{2}< 1,\\ \frac{1}{1-2\delta_{2}}[1+\rho_{2}\gamma_{2} +\varepsilon_{2}(1+\sqrt{1-2\rho_{2} \xi_{2}+\rho_{2}^{2}(\alpha_{2}\tau_{2}+\beta_{2}\omega_{2})^{2}} +\rho_{2}k_{2})]\\ \quad{}+\frac{1}{1-2\delta_{1}}\varepsilon_{1}\rho_{1}k_{1}< 1, \end{cases} $$
(3.1)
then there exist \((x,y)\in K_{1}(x)\times K_{2}(y)\), \(u_{1}\in A_{1}x\), \(v_{1}\in B_{1}x\), \(u_{2}\in A_{2}y\), \(v_{2}\in B_{2}y\), and \(w_{i}\in F_{i}(x,y)\) satisfying problem (1.1), and
$$x_{n}\rightarrow x,\quad\quad y_{n}\rightarrow y, \quad\quad u_{n}^{i} \rightarrow u_{i},\quad\quad v_{n}^{i}\rightarrow v_{i},\quad\quad w_{n}^{i}\rightarrow w_{i} \quad\textit{as } n\rightarrow \infty, $$
where the sequences \(\{x_{n}\}\), \(\{y_{n}\}\), \(\{u_{n}^{i}\}\), \(\{v_{n}^{i}\}\), and \(\{w_{n}^{i}\}\) are generated by Algorithm 2.1.

Proof

First, it follows from (2.3) in Algorithm 2.1 that, for any \(h_{1}\in K_{1}(x_{n})\),
$$\begin{aligned} \langle x_{n},h_{1}-x_{n}\rangle_{1} \geq&\langle x_{n-1},h_{1}-x_{n}\rangle_{1}- \rho_{1}\bigl\langle N_{1}\bigl(u_{n-1}^{1},v_{n-1}^{1} \bigr)+w_{n-1}^{1},\eta_{1}(h_{1},x_{n}) \bigr\rangle _{1} \\ &{}+\rho_{1}\bigl[b_{1}(x_{n-1},x_{n})-b_{1}(x_{n-1},h_{1}) \bigr] \end{aligned}$$
(3.2)
and, for any \(h_{1}\in K_{1}(x_{n+1})\),
$$\begin{aligned} \langle x_{n+1},h_{1}-x_{n+1}\rangle_{1} \geq&\langle x_{n},h_{1}-x_{n+1}\rangle_{1}- \rho_{1}\bigl\langle N_{1}\bigl(u_{n}^{1},v_{n}^{1} \bigr)+w_{n}^{1},\eta_{1}(h_{1},x_{n+1}) \bigr\rangle _{1} \\ &{}+\rho_{1}\bigl[b_{1}(x_{n},x_{n+1})-b_{1}(x_{n},h_{1}) \bigr]. \end{aligned}$$
(3.3)
Adding \(\langle-m_{1}(x_{n}),h_{1}-x_{n}\rangle_{1}\) to the two sides of inequality (3.2) and then taking \(h_{1}=m_{1}(x_{n})+x_{n+1}-m_{1}(x_{n+1})\in K_{1}(x_{n})\), we get
$$\begin{aligned}& \bigl\langle x_{n} -m_{1}(x_{n}),m_{1}(x_{n})+x_{n+1}-m_{1}(x_{n+1})-x_{n} \bigr\rangle _{1} \\& \quad\geq \bigl\langle x_{n-1}-m_{1}(x_{n}),m_{1}(x_{n})+x_{n+1}-m_{1}(x_{n+1})-x_{n} \bigr\rangle _{1} \\& \quad\quad{}-\rho_{1}\bigl\langle N_{1}\bigl(u_{n-1}^{1},v_{n-1}^{1} \bigr)+w_{n-1}^{1},\eta _{1}\bigl(m_{1}(x_{n})+x_{n+1}-m_{1}(x_{n+1}),x_{n} \bigr)\bigr\rangle _{1} \\& \quad\quad{}+\rho _{1}\bigl[b_{1}(x_{n-1},x_{n})-b_{1} \bigl(x_{n-1},m_{1}(x_{n})+x_{n+1}-m_{1}(x_{n+1}) \bigr)\bigr]. \end{aligned}$$
(3.4)
Adding \(\langle-m_{1}(x_{n+1}),h_{1}-x_{n+1}\rangle_{1}\) to the two sides of inequality (3.3) and then taking \(h_{1}=m_{1}(x_{n+1})+x_{n}-m_{1}(x_{n})\in K_{1}(x_{n+1})\), we get
$$\begin{aligned}& \bigl\langle x_{n+1} -m_{1}(x_{n+1}),m_{1}(x_{n+1})+x_{n}-m_{1}(x_{n})-x_{n+1} \bigr\rangle _{1} \\& \quad\geq \bigl\langle x_{n}-m_{1}(x_{n+1}),m_{1}(x_{n+1})+x_{n}-m_{1}(x_{n})-x_{n+1}) \bigr\rangle _{1} \\& \quad\quad{}-\rho_{1}\bigl\langle N_{1}\bigl(u_{n}^{1},v_{n}^{1} \bigr)+w_{n}^{1},\eta _{1}\bigl(m_{1}(x_{n+1})+x_{n}-m_{1}(x_{n}),x_{n+1} \bigr)\bigr\rangle _{1} \\& \quad\quad{}+\rho_{1}\bigl[b_{1}(x_{n},x_{n+1})-b_{1} \bigl(x_{n},m_{1}(x_{n+1})+x_{n}- m_{1}(x_{n})\bigr)\bigr]. \end{aligned}$$
(3.5)
Adding (3.4) and (3.5), by properties (i) and (iii) of \(b(\cdot ,\cdot)\) and Assumption 1.1(2), we obtain
$$\begin{aligned}& \bigl\langle x_{n} -x_{n+1}-m_{1}(x_{n})+m_{1}(x_{n+1}),x_{n}-x_{n+1}-m_{1}(x_{n})+m_{1}(x_{n+1}) \bigr\rangle _{1} \\& \quad\leq \bigl\langle x_{n-1}-m_{1}(x_{n}),x_{n}-x_{n+1}-m_{1}(x_{n})+m_{1}(x_{n+1}) \bigr\rangle _{1} \\& \quad\quad{}+\bigl\langle -x_{n}+m_{1}(x_{n+1}),m_{1}(x_{n+1})+x_{n}-m_{1}(x_{n})-x_{n+1}) \bigr\rangle _{1} \\& \quad\quad{}+\rho_{1}\bigl\langle N_{1}\bigl(u_{n-1}^{1},v_{n-1}^{1} \bigr)+w_{n-1}^{1},\eta _{1}\bigl(m_{1}(x_{n})+x_{n+1}-m_{1}(x_{n+1}),x_{n} \bigr)\bigr\rangle _{1} \\& \quad\quad{}+\rho_{1}\bigl\langle N_{1}\bigl(u_{n}^{1},v_{n}^{1} \bigr)+w_{n}^{1},\eta _{1}\bigl(m_{1}(x_{n+1})+x_{n}-m_{1}(x_{n}),x_{n+1} \bigr)\bigr\rangle _{1} \\& \quad\quad{}-\rho _{1}\bigl[b_{1}(x_{n-1},x_{n})-b_{1} \bigl(x_{n-1},m_{1}(x_{n})+x_{n+1}-m_{1}(x_{n+1}) \bigr)\bigr] \\& \quad\quad{}-\rho _{1}\bigl[b_{1}(x_{n},x_{n+1})-b_{1} \bigl(x_{n},m_{1}(x_{n+1})+x_{n}-m_{1}(x_{n}) \bigr)\bigr] \\& \quad\leq \bigl\langle x_{n-1}-x_{n}-m_{1}(x_{n})+m_{1}(x_{n+1}),x_{n}-x_{n+1}-m_{1}(x_{n})+m_{1}(x_{n+1}) \bigr\rangle _{1} \\& \quad\quad{}+\rho_{1}\bigl\langle N_{1}\bigl(u_{n-1}^{1},v_{n-1}^{1} \bigr)+w_{n-1}^{1}-\bigl(N_{1}\bigl(u_{n}^{1},v_{n}^{1} \bigr)+w_{n}^{1}\bigr),\eta _{1} \bigl(m_{1}(x_{n})+x_{n+1} \\& \quad\quad{}-m_{1}(x_{n+1}),x_{n}\bigr)\bigr\rangle _{1} +\rho_{1}\bigl[b_{1} \bigl(-x_{n-1},x_{n}-m_{1}(x_{n})-x_{n+1}+m_{1}(x_{n+1}) \bigr) \\& \quad\quad{}+b_{1}\bigl(x_{n},m_{1}(x_{n+1})+x_{n}-m_{1}(x_{n})-x_{n+1} \bigr)\bigr] \\& \quad\leq \bigl\langle x_{n-1}-x_{n}-m_{1}(x_{n})+m_{1}(x_{n+1}),x_{n}-x_{n+1}-m_{1}(x_{n})+m_{1}(x_{n+1}) \bigr\rangle _{1} \\& \quad\quad{}+\rho_{1}\bigl\langle N_{1}\bigl(u_{n-1}^{1},v_{n-1}^{1} \bigr)+w_{n-1}^{1}-\bigl(N_{1}\bigl(u_{n}^{1},v_{n}^{1} \bigr)+w_{n}^{1}\bigr),\eta _{1} \bigl(m_{1}(x_{n})+x_{n+1} \\& \quad\quad{} -m_{1}(x_{n+1}),x_{n}\bigr)\bigr\rangle _{1} +\rho_{1}b_{1}\bigl(x_{n}-x_{n-1},x_{n}-m_{1}(x_{n})-x_{n+1}+m_{1}(x_{n+1}) \bigr). \end{aligned}$$
(3.6)
By properties (i) and (iii) of \(b(\cdot,\cdot)\) this implies
$$\begin{aligned}& \bigl\Vert x_{n}-x_{n+1}-m_{1}(x_{n})+m_{1}(x_{n+1}) \bigr\Vert _{1}^{2} \\& \quad\leq \bigl\Vert x_{n-1}-x_{n}-m_{1}(x_{n})+m_{1}(x_{n+1}) \bigr\Vert _{1}\cdot\bigl\Vert x_{n}-x_{n+1}-m_{1}(x_{n})+m_{1}(x_{n+1}) \bigr\Vert _{1} \\& \quad\quad{}+\bigl[\bigl\Vert x_{n-1}-x_{n}-\rho_{1} \bigl(N_{1}\bigl(u_{n-1}^{1},v_{n-1}^{1} \bigr)-N_{1}\bigl(u_{n}^{1},v_{n}^{1} \bigr)\bigr)\bigr\Vert _{1}+\Vert x_{n-1}-x_{n} \Vert _{1} \\& \quad\quad{}+\rho_{1}\bigl\Vert w_{n-1}^{1}-w_{n}^{1} \bigr\Vert _{1}\bigr]\cdot\bigl\Vert \eta _{1} \bigl(m_{1}(x_{n})+x_{n+1}-m_{1}(x_{n+1}),x_{n} \bigr)\bigr\Vert _{1} \\& \quad\quad{}+\rho_{1}\gamma_{1}\Vert x_{n-1}-x_{n} \Vert _{1}\bigl\Vert x_{n}-x_{n+1} -m_{1}(x_{n})+m_{1}(x_{n+1})\bigr\Vert _{1}. \end{aligned}$$
(3.7)
So, by Algorithm 2.1 and condition (6) we have
$$\begin{aligned}& \Vert x_{n}-x_{n+1}\Vert _{1} \\& \quad\leq \Vert x_{n-1}-x_{n}\Vert _{1}+2\bigl\Vert m_{1}(x_{n})-m_{1}(x_{n+1})\bigr\Vert _{1} \\& \quad\quad{}+\varepsilon_{1}\biggl[\bigl\Vert x_{n-1}-x_{n}- \rho_{1}\bigl( N_{1}\bigl(u_{n-1}^{1},v_{n-1}^{1} \bigr)-N_{1}\bigl(u_{n}^{1},v_{n}^{1} \bigr)\bigr)\bigr\Vert _{1}+\Vert x_{n-1}-x_{n} \Vert _{1} \\& \quad\quad{}+\rho_{1}\biggl(1+\frac{1}{n}\biggr)H\bigl(F_{1}(x_{n-1},y_{n-1}),F_{1}(x_{n},y_{n}) \bigr)\biggr] +\rho_{1}\gamma_{1}\Vert x_{n-1}-x_{n} \Vert _{1}. \end{aligned}$$
(3.8)
By conditions (1), (3), and (4) and Algorithm 2.1 we have
$$\begin{aligned}& \bigl\Vert x_{n-1}-x_{n}-\rho_{1}\bigl( N_{1}\bigl(u_{n-1}^{1},v_{n-1}^{1} \bigr)-N_{1}\bigl(u_{n}^{1},v_{n}^{1} \bigr)\bigr)\bigr\Vert _{1}^{2} \\& \quad=\Vert x_{n-1}-x_{n}\Vert _{1}^{2}-2 \rho_{1}\bigl\langle N_{1}\bigl(u_{n-1}^{1},v_{n-1}^{1} \bigr)-N_{1}\bigl(u_{n}^{1},v_{n}^{1} \bigr),x_{n-1} -x_{n}\bigr\rangle _{1} \\& \quad\quad{}+\rho_{1}^{2}\bigl\Vert N_{1} \bigl(u_{n-1}^{1},v_{n-1}^{1} \bigr)-N_{1}\bigl(u_{n}^{1},v_{n}^{1} \bigr)\bigr\Vert _{1}^{2} \\& \quad\leq(1-2\rho_{1}\xi_{1})\Vert x_{n-1}-x_{n} \Vert _{1}^{2}+\rho_{1}^{2}\bigl[\alpha _{1}\bigl\Vert u_{n-1}^{1}-v_{n}^{1} \bigr\Vert _{1}+\beta_{1}\bigl\Vert v_{n-1}^{1}-v_{n}^{1} \bigr\Vert _{1}\bigr]^{2} \\& \quad\leq\biggl[1-2\rho_{1} \xi_{1}+\rho_{1}^{2}( \alpha_{1}\tau_{1}+\beta_{1}\omega_{1})^{2} \biggl(1+\frac {1}{n}\biggr)^{2}\biggr]\Vert x_{n-1}-x_{n}\Vert _{1}^{2}. \end{aligned}$$
(3.9)
It follows from (3.8) and (3.9) and from conditions (2) and (5) that
$$\begin{aligned} \Vert x_{n}-x_{n+1}\Vert _{1} \leq& \frac{1}{1-2\delta_{1}}\biggl\{ \biggl[1+\rho_{1}\gamma_{1} + \varepsilon_{1}\biggl(1+\sqrt{1-2\rho_{1} \xi_{1}+\rho_{1}^{2}(\alpha_{1} \tau_{1}+\beta_{1}\omega_{1})^{2}\biggl(1+ \frac {1}{n}\biggr)^{2}} \\ &{}+\rho_{1}l_{1}\biggl(1+\frac{1}{n}\biggr)\biggr) \biggr]\Vert x_{n-1}-x_{n}\Vert _{1} + \varepsilon_{1}\rho_{1}k_{1}\biggl(1+ \frac{1}{n}\biggr)\Vert y_{n-1}-y_{n}\Vert _{2}\biggr\} . \end{aligned}$$
(3.10)
Second, it follows from (2.4) in Algorithm 2.1 that, for any \(h_{2}\in K_{2}(y_{n})\),
$$\begin{aligned} \langle y_{n},h_{2}-y_{n}\rangle_{2} \geq&\langle y_{n-1},h_{2}-y_{n}\rangle_{2}- \rho_{2}\bigl\langle N_{2}\bigl(u_{n-1}^{2},v_{n-1}^{2} \bigr)+w_{n-1}^{2},\eta_{2}(h_{2},y_{n}) \bigr\rangle _{2} \\ &{}+\rho_{2}\bigl[b_{2}(y_{n-1},y_{n})-b_{2}(y_{n-1},h_{2}) \bigr] \end{aligned}$$
(3.11)
and, for any \(h_{2}\in K_{2}(y_{n+1})\),
$$\begin{aligned} \langle y_{n+1},h_{2}-y_{n+1}\rangle_{2} \geq&\langle y_{n},h_{2}-y_{n+1}\rangle_{2}- \rho_{2}\bigl\langle N_{2}\bigl(u_{n}^{2},v_{n}^{2} \bigr)+w_{n}^{2},\eta_{2}(h_{2},y_{n+1}) \bigr\rangle _{2} \\ &{}+\rho_{2}\bigl[b_{2}(y_{n},y_{n+1})-b_{2}(y_{n},h_{2}) \bigr]. \end{aligned}$$
(3.12)
Adding \(\langle-m_{2}(y_{n}),h_{2}-y_{n}\rangle_{2}\) to the two sides of inequality (3.11) and then taking \(h_{2}=m_{2}(y_{n})+y_{n+1}-m_{2}(y_{n+1})\in K_{2}(y_{n})\), we get
$$\begin{aligned}& \bigl\langle y_{n} -m_{2}(y_{n}),m_{2}(y_{n})+y_{n+1}-m_{2}(y_{n+1})-y_{n} \bigr\rangle _{2} \\& \quad\geq \bigl\langle y_{n-1}-m_{2}(y_{n}),m_{2}(y_{n})+y_{n+1}-m_{2}(y_{n+1})-y_{n} \bigr\rangle _{2} \\& \quad\quad{}-\rho_{2}\bigl\langle N_{2}\bigl(u_{n-1}^{2},v_{n-1}^{2} \bigr)+w_{n-1}^{2},\eta _{2}\bigl(m_{2}(y_{n})+y_{n+1}-m_{2}(y_{n+1}),y_{n} \bigr)\bigr\rangle _{2} \\& \quad\quad{}+\rho _{2}\bigl[b_{2}(y_{n-1},y_{n})-b_{2} \bigl(y_{n-1},m_{2}(y_{n})+y_{n+1}-m_{2}(y_{n+1}) \bigr)\bigr]. \end{aligned}$$
(3.13)
Adding \(\langle-m_{2}(y_{n+1}),h_{2}-y_{n+1}\rangle_{2}\) to the two sides of inequality (3.12) and then taking \(h_{2}=m_{2}(y_{n+1})+y_{n}-m_{2}(y_{n})\in K_{2}(y_{n+1})\), we get
$$\begin{aligned}& \bigl\langle y_{n+1} -m_{2}(y_{n+1}),m_{2}(y_{n+1})+y_{n}-m_{2}(y_{n})-y_{n+1} \bigr\rangle _{2} \\& \quad\geq \bigl\langle y_{n}-m_{2}(y_{n+1}),m_{2}(y_{n+1})+y_{n}-m_{2}(y_{n})-y_{n+1} \bigr\rangle _{2} \\& \quad\quad{}-\rho_{2}\bigl\langle N_{2}\bigl(u_{n}^{2},v_{n}^{2} \bigr)+w_{n}^{2}, \eta_{2}\bigl(m_{2}(y_{n+1})+y_{n}-m_{2}(y_{n}),y_{n+1} \bigr)\bigr\rangle _{2} \\& \quad\quad{}+\rho_{2}\bigl[b_{2}(y_{n},y_{n+1})-b_{2} \bigl(y_{n},m_{2}(y_{n+1})+ y_{n}-m_{2}(y_{n}) \bigr)\bigr]. \end{aligned}$$
(3.14)
Then repeating the method, we have
$$\begin{aligned} \Vert y_{n}-y_{n+1}\Vert _{2} \leq& \frac{1}{1-2\delta_{2}}\biggl\{ \biggl[1+\rho_{2}\gamma_{2} + \varepsilon_{2}\biggl(1+\sqrt{1-2\rho_{2} \xi_{2}+\rho_{2}^{2}(\alpha_{2} \tau_{2}+\beta_{2}\omega_{2})^{2}\biggl(1+ \frac {1}{n}\biggr)^{2}} \\ &{}+\rho_{2}k_{2}\biggl(1+\frac{1}{n}\biggr)\biggr) \biggr]\Vert y_{n-1}-y_{n}\Vert _{2} + \varepsilon_{2}\rho_{2}l_{2}\biggl(1+ \frac{1}{n}\biggr)\Vert x_{n-1}-x_{n}\Vert _{1}\biggr\} . \end{aligned}$$
(3.15)
From (3.10) and (3.15) we have
$$\begin{aligned}& \Vert x_{n}-x_{n+1}\Vert _{1}+\Vert y_{n}-y_{n+1}\Vert _{2} \\& \quad\leq \biggl\{ \frac{1}{1-2\delta_{1}}\biggl[1+\rho_{1}\gamma_{1} + \varepsilon_{1}\biggl(1+\sqrt{1-2\rho_{1} \xi_{1}+\rho_{1}^{2}(\alpha_{1} \tau_{1}+\beta_{1}\omega_{1})^{2}\biggl(1+ \frac {1}{n}\biggr)^{2}} \\& \quad\quad{}+\rho_{1}l_{1}\biggl(1+\frac{1}{n}\biggr)\biggr) \biggr]+\frac{1}{1-2\delta_{2}}\varepsilon _{2}\rho_{2}l_{2} \biggl(1+\frac{1}{n}\biggr)\biggr\} \Vert x_{n-1}-x_{n} \Vert _{1} \\& \quad\quad{}+\biggl\{ \frac{1}{1-2\delta_{2}}\biggl[1+\rho_{2}\gamma_{2} + \varepsilon_{2}\biggl(1+\sqrt{1-2\rho_{2} \xi_{2}+\rho_{2}^{2}(\alpha_{2} \tau_{2}+\beta_{2}\omega_{2})^{2}\biggl(1+ \frac {1}{n}\biggr)^{2}} \\& \quad\quad{}+\rho_{2}k_{2}\biggl(1+\frac{1}{n}\biggr)\biggr) \biggr]+\frac{1}{1-2\delta_{1}}\varepsilon _{1}\rho_{1}k_{1} \biggl(1+\frac{1}{n}\biggr)\biggr\} \Vert y_{n-1}-y_{n} \Vert _{1} \\& \quad\leq \max \bigl\{ \theta_{n}^{1},\theta_{n}^{2} \bigr\} \bigl(\Vert x_{n-1}-x_{n}\Vert _{1}+ \Vert y_{n-1}-y_{n}\Vert _{2}\bigr), \end{aligned}$$
(3.16)
where
$$\begin{aligned}& \theta_{n}^{1}= \frac{1}{1-2\delta_{1}}\biggl[1+ \rho_{1}\gamma_{1} +\varepsilon_{1}\biggl(1+\sqrt {1-2\rho_{1} \xi_{1}+\rho_{1}^{2}( \alpha_{1}\tau_{1}+\beta_{1}\omega_{1})^{2} \biggl(1+\frac {1}{n}\biggr)^{2}} \\& \hphantom{\theta_{n}^{1}=}{}+\rho_{1}l_{1}\biggl(1+\frac{1}{n}\biggr)\biggr) \biggr]+\frac{1}{1-2\delta_{2}} \varepsilon_{2}\rho_{2}l_{2} \biggl(1+\frac{1}{n}\biggr), \\& \theta_{n}^{2}= \frac{1}{1-2\delta_{2}}\biggl[1+\rho _{2}\gamma_{2} +\varepsilon_{2}\biggl(1+\sqrt {1-2\rho_{2} \xi_{2}+\rho_{2}^{2}( \alpha_{2}\tau_{2}+\beta_{2}\omega_{2})^{2} \biggl(1+\frac {1}{n}\biggr)^{2}} \\& \hphantom{\theta_{n}^{2}=}{}+\rho_{2}k_{2}\biggl(1+\frac{1}{n}\biggr)\biggr) \biggr]+\frac{1}{1-2\delta_{1}} \varepsilon_{1}\rho_{1}k_{1} \biggl(1+\frac{1}{n}\biggr). \end{aligned}$$
Letting
$$\theta_{1}=\frac{1}{1-2\delta_{1}}\Bigl[1+\rho_{1} \gamma_{1} +\varepsilon_{1}\Bigl(1+\sqrt{1-2 \rho_{1} \xi_{1}+\rho_{1}^{2}( \alpha_{1}\tau_{1}+\beta_{1}\omega_{1})^{2}} +\rho_{1}l_{1}\Bigr)\Bigr]+\frac{1}{1-2\delta_{2}} \varepsilon_{2}\rho_{2}l_{2} $$
and
$$\theta_{2}=\frac{1}{1-2\delta_{2}}\Bigl[1+\rho_{2} \gamma_{2} +\varepsilon_{2}\Bigl(1+\sqrt{1-2 \rho_{2} \xi_{2}+\rho_{2}^{2}( \alpha_{2}\tau_{2}+\beta_{2}\omega_{2})^{2}} +\rho_{2}k_{2}\Bigr)\Bigr]+\frac{1}{1-2\delta_{1}} \varepsilon_{1}\rho_{1}k_{1}, $$
we can see that \(\theta_{n}^{1}\rightarrow\theta_{1}\) and \(\theta_{n}^{2}\rightarrow \theta_{2}\) as \(n\rightarrow\infty\). Now, by condition (3.1) we have \(\max\{\theta_{1},\theta_{2}\}<1\). Therefore, it follows from (3.16) that \(\{(x_{n},y_{n})\}\) is a Cauchy sequence in \(H_{1}\times H_{2}\). Let \((x_{n},y_{n})\rightarrow(x,y)\in H_{1}\times H_{2}\) as \(n\rightarrow\infty\). Since \(A_{i}\), \(B_{i}\), and \(F_{i}\) are all H-Lipschitz continuous, by (2.5) and by conditions (2), (3), and (4) we have
$$\begin{aligned}& \bigl\Vert u_{n+1}^{1}-u_{n}^{1}\bigr\Vert _{1} \leq\biggl(1+\frac {1}{n+1}\biggr)H(A_{1}x_{n+1},A_{1}x_{n}) \leq\biggl(1+\frac{1}{n+1}\biggr)\tau_{1}\Vert x_{n+1}-x_{n}\Vert _{1}; \\& \bigl\Vert v_{n+1}^{1}-v_{n}^{1}\bigr\Vert _{1} \leq\biggl(1+\frac {1}{n+1}\biggr)H(B_{1}x_{n+1},B_{1}x_{n}) \leq\biggl(1+\frac{1}{n+1}\biggr)\omega_{1}\Vert x_{n+1}-x_{n}\Vert _{1}; \\& \bigl\Vert u_{n+1}^{2}-u_{n}^{2}\bigr\Vert _{2} \leq\biggl(1+\frac {1}{n+1}\biggr)H(A_{2}y_{n+1},A_{2}y_{n}) \leq\biggl(1+\frac{1}{n+1}\biggr)\tau_{2}\Vert y_{n+1}-y_{n}\Vert _{2}; \\& \bigl\Vert v_{n+1}^{2}-v_{n}^{2}\bigr\Vert _{2} \leq\biggl(1+\frac {1}{n+1}\biggr)H(B_{2}y_{n+1},B_{2}y_{n}) \leq\biggl(1+\frac{1}{n+1}\biggr)\omega_{2}\Vert y_{n+1}-y_{n}\Vert _{2}; \\& \bigl\Vert w_{n+1}^{i}-w_{n}^{i}\bigr\Vert _{i} \leq\biggl(1+\frac {1}{n+1}\biggr)H \bigl(F_{i}(x_{n+1},y_{n+1}),F_{i}(x_{n},y_{n}) \bigr) \\& \hphantom{\bigl\Vert w_{n+1}^{i}-w_{n}^{i}\bigr\Vert _{i}}\leq\biggl(1+\frac{1}{n+1}\biggr) \bigl(l_{i}\Vert x_{n+1}-x_{n}\Vert _{1}+k_{i}\Vert y_{n+1}-y_{n}\Vert _{2}\bigr). \end{aligned}$$
Therefore, \(\{u_{n}^{i}\}\), \(\{v_{n}^{i}\}\), and \(\{w_{n}^{i}\}\) (\(i\in I\)) are also Cauchy sequences. Let \(u_{n}^{i}\rightarrow u_{i}\), \(v_{n}^{i}\rightarrow v_{i}\), and \(w_{n}^{i}\rightarrow w_{i}\) (\(i\in I\)) as \(n\rightarrow\infty\). Since \(u_{n}^{1}\in A_{1}x_{n}\), we have
$$\begin{aligned} d(u_{1},A_{1}x) \leq&\bigl\Vert u_{1}-u_{n}^{1} \bigr\Vert _{1}+d\bigl(u_{n}^{1},A_{1}x \bigr) \\ \leq&\bigl\Vert u_{1}-u_{n}^{1}\bigr\Vert _{1}+H(A_{1}x_{n},A_{1}x) \\ \leq&\bigl\Vert u_{1}-u_{n}^{1}\bigr\Vert _{1}+\tau_{1}\Vert x_{n}-x\Vert _{1} \rightarrow 0 \quad (n\rightarrow \infty). \end{aligned}$$
Hence, we conclude that \(u_{1}\in A_{1}x\). Similarly, we can obtain \(u_{2}\in A_{2}y\), \(v_{1}\in B_{1}x\), \(v_{2}\in B_{2}y\), \(w_{i}\in F_{i}(x,y)\), \(\forall i\in I\).
By Theorem 2.1 we may assume that \((p_{1},p_{2})\in K_{1}(x)\times K_{2}(y)\) is the unique solution of the system of auxiliary variational inequalities (2.1), that is,
$$\begin{aligned} \langle p_{1},h_{1}-p_{1}\rangle_{1} \geq&\langle x,h_{1}-p_{1}\rangle_{1}- \rho_{1}\bigl\langle N_{1}(u_{1},v_{1})+w_{1}, \eta_{1}(h_{1},p_{1})\bigr\rangle _{1} \\ &{}+\rho_{1}\bigl[b_{1}(x,p_{1})-b_{1}(x,h_{1}) \bigr],\quad \forall h_{1}\in K_{1}(x), \end{aligned}$$
(3.17)
and
$$\begin{aligned} \langle p_{2},h_{2}-p_{2}\rangle_{2} \geq&\langle y,h_{2}-p_{2}\rangle_{2}- \rho_{2}\bigl\langle N_{2}(u_{2},v_{2})+w_{2}, \eta_{2}(h_{2},p_{2})\bigr\rangle _{2} \\ &{}+\rho_{2}\bigl[b_{2}(y,p_{2})-b_{2}(y,h_{2}) \bigr],\quad \forall h_{2}\in K_{2}(y). \end{aligned}$$
(3.18)
Now, we prove \(p_{1}=x\) and \(p_{2}=y\). By applying (3.2), (3.17), and a similar argument as in proving (3.8), we can easily get
$$\begin{aligned}& \Vert x_{n}-p_{1}\Vert _{1} \\& \quad\leq \Vert x_{n-1}-x\Vert _{1}+2\bigl\Vert m_{1}(x_{n})-m_{1}(p_{1})\bigr\Vert _{1} \\& \quad\quad{}+\rho_{1}\varepsilon_{1}\bigl[\bigl\Vert N_{1} \bigl(u_{n-1}^{1},v_{n-1}^{1} \bigr)-N_{1}\bigl(u_{n}^{1},v_{n}^{1} \bigr)\bigr\Vert _{1}+\bigl\Vert w_{n-1}^{1}-w_{n}^{1} \bigr\Vert _{1}\bigr] +\rho_{1}\gamma_{1}\Vert x_{n-1}-x\Vert _{1}. \end{aligned}$$
(3.19)
Since \(x_{n}\rightarrow x\), \(w_{n}^{1}\rightarrow w_{1}\), and \(N_{1}(u_{n}^{1},v_{n}^{1})\rightarrow N_{1}(u_{1},v_{1})\), from (3.19) we have \(x_{n}\rightarrow p_{1}\). Therefore, we have \(p_{1}=x\).
Using a similar method, we have \(p_{2}=y\). Finally, taking them into (3.17) and (3.18), we have
$$\bigl\langle N_{1}(u_{1},v_{1})+w_{1}, \eta_{1}(h_{1},x)\bigr\rangle _{1} -b_{1}(x,x)+b_{1}(x,h_{1})\geq0, \quad\forall h_{1}\in K_{1}(x), $$
and
$$\bigl\langle N_{2}(u_{2},v_{2})+w_{2}, \eta_{2}(h_{2},y)\bigr\rangle _{2} -b_{2}(y,y)+b_{2}(y,h_{2})\geq0, \quad\forall h_{2}\in K_{2}(y). $$
This completes the proof. □

Remark 3.1

Theorem 3.1 answers positively the open problem raised by Noor [1, 2] in the setting of a more general system of generalized nonlinear mixed quasi-variational inequalities. We emphasize that A and B may not be compact-valued mappings.

Declarations

Acknowledgements

This work was supported by the Innovation Program of Shanghai Municipal Education Commission (15ZZ068), Ph.D. Program Foundation of Ministry of Education of China (20123127110002) and Program for Shanghai Outstanding Academic Leaders in Shanghai City (15XD1503100).

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, Shanghai, China

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