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)
\(N_{i}\)
is
\((\alpha_{i},\beta_{i})\)-Lipschitz continuous and
\(\xi_{i}\)-strongly mixed monotone with respect to
\(A_{i}\)
and
\(B_{i}\);
-
(2)
\(F_{i}\)
is
\((l_{i},k_{i})\)-H-Lipschitz continuous;
-
(3)
\(A_{i}\)
is
\(\tau_{i}\)-H-Lipschitz continuous;
-
(4)
\(B_{i}\)
is
\(\omega_{i}\)-H-Lipschitz continuous;
-
(5)
\(m_{i}\)
is
\(\delta_{i}\)-Lipschitz continuous;
-
(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.