## 1 Introduction  
We suppose that \(H\) is a real Hilbert space. We use \(\langle\cdot,\cdot\rangle\) to stand for the inner product and \(\|\cdot\|\) the norm. We suppose that \(C\) is a convex closed nonempty set in the Hilbert space \(H\), and \(P_{C}\) is the well-known metric projection from the space \(H\) onto the set \(C\). Here, we also suppose that \(T\) is a nonlinear self mapping defined in \(C\). Let \(\text{Fix}(T)\) be the set of all fixed points of \(T\), that is, \(\text{Fix}(T)=\{x\in C:x=Tx\}\). We use the notations \(\rightarrow\) and \(\rightharpoonup\) to indicate the norm convergence and the weak convergence, respectively. Now, we suppose that \(A:C\to H\) is a nonlinear nonself mapping in \(C\) to \(H\). The well-known classical variational inequality (VI), whose set of all solutions denoted by \(\text{VI}(C,A)\), is to find \(x^{*}\in C\) such that  
\[\langle Ax^{*},x-x^{*}\rangle\geq 0,\quad\forall x\in C. \tag{1}\]  
A mapping \(T:C\to C\) is said to be asymptotically nonexpansive if there exists a sequence \(\{\theta_{n}\}\subset[0,+\infty)\) with \(\lim_{n\rightarrow\infty}\theta_{n}=0\) such that  
\[\|T^{n}x-T^{n}y\|\leq\|x-y\|+\theta_{n}\|x-y\|,\quad\forall n\geq 0,\ x,y\in C. \tag{2}\]  
This mapping is Lipschitz continuous with the Lipschitz constant \(L>1\). Fixed points of Lipschitz continuous mappings are a hot topic and have a lot of applications both in theoretical research, such as in differential equations, control theory, equilibrium problems, and in engineering applications; see References [1; 2; 3; 4; 5; 6] and the references therein. In particular, \(T\) is said to be nonexpansive if \(|Tx-Ty\|\leq\|x-y\|,\quad\forall x,y\in C\), that is, \(\theta\equiv 1\) for all \(n\). Recently, the variational inequality problem (1) has been extensively studied via the iterative methods of Lipschitz continuous mappings, in particular, (asymptotically) nonexpansive mappings; see References [7; 8; 9; 10; 11; 12] and the references therein. We suppose that \(B_{1},B_{2}:C\to H\) are two nonlinear monotone mappings. We also suppose that \(\mu_{1}\) and \(\mu_{2}\) are two positive real constants. We consider the problem of finding \((x^{*},y^{*})\in C\times C\) such that  
\[\left\{ \begin{array}{rl}&\langle\mu_{1}B_{1}y^{*}+x^{*}-y^{*},x-x^{*}\rangle \geq 0,\quad\forall x\in C,\\ &\langle\mu_{2}B_{2}x^{*}+y^{*}-x^{*},x-y^{*}\rangle\geq 0,\quad\forall x\in C.\end{array} \right. \tag{3}\]  
Problem (3) is called a general system of variational inequalities (GSVI). From Reference [8], the GSVI (3) can be translated into a fixed point problem of a Lipschitz continuous nonlinear operator in the following way. **Lemma 1** ([8]). : _We suppose that \(C\) is a convex subset in a Hilbert space \(H\). Fix two elements \(x^{*}\) and \(y^{*}\) in \(C\), \((x^{*},y^{*})\) is a solution of GSVI (3) if and only if \(x^{*}\in\mathrm{GSVI}(C,B_{1},B_{2})\), where \(\mathrm{GSVI}(C,B_{1},B_{2})\) is the fixed point set of the mapping \(G:=P_{C}(I-\mu_{1}B_{1})P_{C}(I-\mu_{2}B_{2})\), and \(y^{*}=P_{C}(I-\mu_{2}B_{2})x^{*}\)._  
The GSVI (3), which includes the variational inequality (1) as a special case, has been investigated via fixed-point algorithms recently in real or complex Hilbert spaces; see References [13; 14; 15; 16; 17; 18] and the references therein. A self mapping \(f:C\to C\) is said to be a strict contraction on \(C\) if there is a number \(\delta\in[0,1)\) such that \(\|f(x)-f(y)\|\leq\delta\|x-y\|\) for all \(x,y\in C\). A nonself mapping \(F:C\to H\) is called monotone if \(\langle Fx-Fy,x-y\rangle\geq 0\)\(\forall x,y\in C\).