# Systems of Variational Inequalities with Nonlinear Operators
## 1 Introduction  
Let \(E\) be a real Banach space whose topological dual space is denoted by \(E^{*}\). Recall that the normalized duality mapping \(J:E\to 2^{E^{*}}\) is defined by  
\[J(x)=\{\varphi\in E^{*}:\ \|\varphi\|=\|x\|,\langle x,\varphi\rangle=\|x\|^{2}\}, \quad\forall x\in E,\]  
where \(\langle\cdot,\cdot\rangle\) is the duality pair on \(E\) and \(E^{*}\). \(J\) is single-valued in a smooth Banach space. In the sequel, we shall denote by \(j\) the single-valued duality mapping, that is, \(j(x)\in J(x)\). Let \(C\) be a convex closed set in \(E\). A mapping \(f:C\to C\) is said to be \(\delta\)-Lipschitzian on \(C\) if \(\delta\in(0,+\infty)\) and \(\|f(x)-f(y)\|\leq\delta\|x-y\|\) for all \(x,y\in C\). If \(\delta<1\), then \(f\) is called a \(\delta\)-contraction mapping or a contraction mapping with coefficient \(\delta\). Each contraction \(f:C\to C\) has a unique fixed point from the well known the Banach contractive principal. A mapping \(f:C\to C\) is said to be nonexpansive if it is Lipschitzian with \(\delta=1\). We use \(\operatorname{Fix}(f)\) to denote the set of fixed points of \(f\), i.e., \(\operatorname{Fix}(f)=\{x\in C:f(x)=x\}\). Moreover, a mapping \(T:C\to C\) is said to be asymptotically nonexpansive [1] if there exists a sequence \(\{\theta_{n}\}\subset[0,+\infty)\) with \(\lim_{n\to\infty}\theta_{n}=0\) such that  
\[\|T^{n}x-T^{n}y\|\leq\|x-y\|+\theta_{n}\|x-y\|,\quad\forall x,y\in C,\ \forall n \geq 0.\]  
If  
\[\limsup_{n\to\infty}\big{(}\sup_{x,y\in C}(\|T^{n}x-T^{n}y\|-\|x-y\|)\big{)} \leq 0, \tag{1}\]  
and \(T\) enjoys the continuity, then \(T\) is called an asymptotically nonexpansive mapping in the intermediate sense; see [2]. Throughout this paper, we assume  
\[c_{n}:=\max\{0,\sup_{x,y\in C}(\|T^{n}x-T^{n}y\|-\|x-y\|)\}. \tag{2}\]Hence, \(c_{n}\geq 0\ \forall n\geq 0,\ c_{n}\to 0\ (n\to\infty)\), and the definition is reduced to  
\[\|T^{n}x-T^{n}y\|\leq\|x-y\|+c_{n},\quad\forall x,y\in C,\ \forall n\geq 0.\]  
Recall that a mapping \(T\) with domain \(D(T)\) and range \(R(T)\) in \(E\) is called pseudocontractive if the inequality holds  
\[\|x-y\|\leq\|r((I-T)x-(I-T)y)+(x-y)\|,\quad\forall x,y\in D(T),\forall r>0.\]  
From a result of [PERSON] (1964), we know that the notion of pseudocontraction is equivalent to the one that for each \(x,y\in D(T)\), there exists \(j(x-y)\in J(x-y)\) such that  
\[\langle Tx-Ty,j(x-y)\rangle\leq\|x-y\|^{2}.\]  
It is well known that the class of pseudocontractive mappings is a crucial generation of nonexpansive mappings. Moreover, focus on pseudocontractive mappings are also from their relation with the class of accretive mappings in Banach spaces (monotone in Hilbert spaces). A mapping \(A\) with domain \(D(A)\) and range \(R(A)\) in \(E\) is called accretive if for each \(x,y\in D(T)\), there exists \(j(x-y)\in J(x-y)\) such that  
\[\langle Ax-Ay,j(x-y)\rangle\geq 0.\]  
It will be called a monotone mapping if the space is Hilbert. If for each \(x,y\in D(A)\), there exists \(j(x-y)\in J(x-y)\) such that  
\[\alpha\|Ax-Ay\|^{2}\leq\langle Ax-Ay,j(x-y)\rangle\quad\text{for some $\alpha>0$},\]  
then \(A\) is called \(\alpha\)-inverse-strongly accretive. Recently, fixed/zero points of pseudocontraction/accretive operators were investigated by many authors for solving various convex optimization problems; see Haire and Haire (1996); [PERSON] (1996); Haire and Haire (1997); Haire and Haire (1998); [PERSON] (1999); Haire and Haire (2000); Haire and Haire (2001); [PERSON] (2002) and the references therein. Let \(E\) be a smooth Banach space. Let \(B_{1},B_{2}\) be two non-self-mappings from \(C\) to \(E\). The general system of variational inequalities (GSVI) is to find \((x^{*},y^{*})\in C\times C\) such that  
\[ \begin{cases}&\langle\mu_{1}B_{1}y^{*}-y^{*}+x^{*},j(x-x^{*})\rangle\geq 0, \quad\forall x\in C,\\ &\langle\mu_{2}B_{2}x^{*}-x^{*}+y^{*},j(x-y^{*})\rangle\geq 0,\quad\forall x \in C,\end{cases}  \tag{3}\]  
where \(\mu_{1}\) and \(\mu_{2}\) are two positive coefficients.