## 1 Introduction  
We suppose that \(H\) is a real or complex Hilbert space and let \(H\) be with inner product \(\langle\cdot,\cdot\rangle\) and norm \(\|\cdot\|\). We suppose that \(C\) is a convex nonempty closed set of \(H\). We also suppose that \(P_{C}\) is the metric projection from \(H\) onto \(C\). Since \(C\) is a convex nonempty closed set, we conclude that \(P_{C}\) is defined. Let \(T\) be a mapping on convex nonempty closed set \(C\). Denote by \(\mathrm{Fix}(T)\) the set of fixed points of \(T\), i.e., \(\mathrm{Fix}(T)=\{x\in C:(I-T)x=0\}\). \(\rightarrow\) and \(\rightharpoonup\) present strong convergence and weak convergence, respectively. A mapping \(T:C\rightarrow\mathcal{C}\) is named to be asymptotically nonexpansive if there exists a sequence \(\{\theta_{n}\}\subset[0,+\infty)\) with \(\lim_{n\rightarrow\infty}\theta_{n}=0\) such that  
\[\|T^{n}x-T^{n}y\|\leq\|x-y\|+\theta_{n}\|x-y\|,\quad\forall n\geq 0,\ x,y\in C. \tag{1}\]  
If \(\theta\equiv 0\), then \(T\) is named to be nonexpansive, that is,  
\[\|Tx-Ty\|\leq\|x-y\|,\quad\forall x,y\in C. \tag{2}\]  
Suppose that \(A\) is a nonself mapping from convex nonempty closed set \(C\) to entire space \(H\). The classical variational inequality (VI) is to find \(x^{*}\in C\) such that  
\[\langle\mu Ax^{*},x-x^{*}\rangle\geq 0,\quad\forall x\in C, \tag{3}\]  
where \(\mu\) is some positive real number. We denote by \(\mathrm{VI}(C,A)\) the set of solutions of VI (3). Assume that \(B_{1}\) is a nonself mapping from convex nonempty closed set \(C\) to entire space \(H\) and \(B_{2}\) is a nonself mapping from convex nonempty closed set \(C\) to entire space \(H\), respectively. we study the system of approximating \((x^{*},y^{*})\in C\times C\) such that  
\[\left\{ \begin{array}{rl}&\langle\mu_{1}B_{1}y^{*}-y^{*}+x^{*},x-x^{*}\rangle \geq 0,\quad\forall x\in C,\\ &\langle\mu_{2}B_{2}x^{*}-x^{*}+y^{*},x-y^{*}\rangle\geq 0,\quad\forall x\in C.\end{array} \right. \tag{4}\]  
Here, \(\mu_{1}\) and \(\mu_{2}\) are two real numbers. The system (4) is named to be a general system of variational inequalities (GSVI). We note that the system (4) can be transformed into a problem of zero points \((I-T)x=0\), that is, the fixed point of \(T\) as following  
**Lemma 1** ([1]). : _Fix \(x^{*},y^{*}\in C\), where \((x^{*},y^{*})\) satisfies the system (4) if and only if_  
\[x^{*}\in\mathrm{GSVI}(C,B_{1},B_{2}),\]  
_where \(\mathrm{GSVI}(C,B_{1},B_{2})\) is the set of solutions of the mapping \(G:=P_{\mathbb{C}}(I-\mu_{1}B_{1})P_{\mathbb{C}}(I-\mu_{2}B_{2})\), and \(y^{*}=P_{\mathbb{C}}(I-\mu_{2}B_{2})x^{*}\)._  
Recently, the variational inequality (3) and the system (4) have been intensively investigated by many authors via fixed-point methods; see [2; 3; 4; 5; 6; 7; 8; 9; 10; 11] and the references therein. A mapping \(f:C\to C\) is said to be a contraction on \(C\) if there exists a constant \(\delta\in[0,1)\) such that \(\|f(x)-f(y)\|\leq\delta\|x-y\|\) for all \(x,y\in C\). A mapping \(F:C\to H\) is called monotone if \(\langle Fx-Fy,x-y\rangle\geq 0\ \forall x,y\in C\). It is called \(\eta\)-strongly monotone if there exists a constant \(\eta>0\) such that \(\langle Fx-Fy,x-y\rangle\geq\eta\|x-y\|^{2}\ \forall x,y\in C\). Moreover, it is called \(\alpha\)-inverse-strongly monotone (or \(\alpha\)-cocoercive), if there exists a constant \(\alpha>0\) such that  
\[\langle Fx-Fy,x-y\rangle\geq\alpha\|Fx-Fy\|^{2},\quad\forall x,y\in C.\]  
Furthermore, let \(X\) be a real Banach space whose topological dual space is denoted with \(X^{*}\). The normalized duality \(J:X\to 2^{X^{*}}\) is defined through  
\[J(x)=\{\psi\in X^{*}:\langle x,\psi\rangle=\|x\|^{2}=\|\psi\|^{2}\},\quad \forall x\in X,\]  
where \(\langle\cdot,\cdot\rangle\) denotes the generalized duality pairing. We suppose that \(T\) is a mapping. Its domain and range are denoted by \(D(T)\) and range \(R(T)\), respectively.