# Systems of Variational Inequalities with Nonlinear Operators
## 2 Preliminaries
It is known that \(J(x)=\frac{J_{q}(x)}{\|x\|^{q-2}}\) for all \(x\in E\). If \(E\) is a Hilbert space, then \(J=I\) (the identity mapping). Recall that the following statements hold:  
1. if \(E\) is smooth, then \(J\) is norm-to-weak\({}^{*}\) continuous single-valued on \(E\);
2. if \(E\) is uniformly smooth, then \(J\) is norm-to-norm uniformly continuous single-valued on bounded subsets of \(E\);
3. if \(E\) has a uniformly Gateaux differentiable norm, then \(J\) is norm-to-weak\({}^{*}\) uniformly continuous single-valued on bounded subsets of \(E\);  
**Proposition 1**. : _(see [25]). Let \(C\) be a convex nonempty closed set in a Banach space \(E\). Let \(S_{0},S_{1},...\) be a sequence of mappings of \(C\) into itself. Suppose \(\sum_{n=1}^{\infty}\sup\{\|S_{n}x-S_{n-1}x\|:x\in C\}<\infty\). For each \(y\in C\), \(\{S_{n}y\}\) converges in norm to some point of \(C\). Moreover, let \(S\) be a mapping defined by \(Sy=\lim_{n\to\infty}S_{n}y\) for all \(y\in C\). \(\{\|Sx-S_{n}x\|:x\in C\}\to 0\) as \(n\to\infty\)._  
**Proposition 2**. : _(see [26]). Let \(C\) be a convex closed set in a Banach space \(E\) and \(T:C\to C\) be a strong continuous pseudocontraction mapping. Then, \(T\) has a fixed point. Indeed, it is the unique fixed point in \(C\) for \(T\)._  
Let \(D\) be a nonempty set in \(C\) and let \(\Pi\) be a mapping from \(C\) to \(D\). Then \(\Pi\) is said to be a sunny if \(\Pi[(1-t)\Pi(x)+tx]=\Pi(x)\), when \((1-t)\Pi(x)+tx\in C\) for all \(x\in C\) and \(t\geq 0\). A mapping \(\Pi\) of \(C\) into itself is called a retraction if \(\Pi^{2}=\Pi\). If a mapping \(\Pi\) of \(C\) into itself is a retraction, then \(\Pi(z)=z\) for each \(z\in R(\Pi)\), where \(R(\Pi)\) is the range of \(\Pi\). A subset \(D\) of \(C\) is called a sunny nonexpansive retract of \(C\) if there exists a sunny nonexpansive retraction from \(C\) onto \(D\). In a smooth Banach space \(E\), a duality mapping \(J\) is said to be weakly sequentially continuous [27], if for each \(\{x_{n}\}\subset E\) with \(x_{n}\rightharpoonup x\), then \(\{j(x_{n})\}\) converges weakly\({}^{*}\) to \(j(x)\). In [27], [PERSON] and [PERSON] showed that a space with a weakly continuous duality mapping satisfies [PERSON]'s condition. Conversely, we know from [28] that if a space satisfies [PERSON]'s condition and has a uniformly Gateaux differentiable norm, then it has a weakly continuous duality mapping. **Proposition 3**. : _(see [29]). Let \(C\) be a nonempty closed convex subset of a smooth Banach space \(E\), \(D\) be a nonempty subset of \(C\) and \(\Pi\) be a retraction of \(C\) onto \(D\). Then the following are equivalent:_  
1. \(\Pi\) _is sunny and nonexpansive;_
2. \(\langle x-y,j(\Pi(x)-\Pi(y))\rangle\geq\|\Pi(x)-\Pi(y)\|^{2},\forall x,y\in C\)_;_
3. \(\langle x-\Pi(x),j(y-\Pi(x))\rangle\leq 0,\forall x\in C,y\in D\)_._  
If \(E\) is a Hilbert space, then a sunny nonexpansive retraction \(\Pi_{C}\) of \(E\) onto \(C\) coincides with the nearest projection of \(E\) onto \(C\) and it is well known that if \(C\) is a convex closed set in a reflexive Banach space \(E\) with a uniformly Gateaux differentiable norm and \(D\) is a nonexpansive retract of \(C\), then it is a sunny nonexpansive retract of \(C\); see, e.g., [30; 31] and the references therein. To prove our main results, we need to use some lemmas in the sequel. The following Lemma is an immediate consequence of the subdifferential inequality of the function \(\frac{1}{2}\|\cdot\|^{2}\). **Lemma 2**. : _[_32_]_ _Let \(E\) be a real Banach space and \(J\) be the normalized duality mapping on \(E\). Then for any given \(x,y\in E\), the following inequality holds:_  
\[\|x+y\|^{2}-\|x\|^{2}\leq 2\langle y,j(x+y)\rangle,\quad\forall j(x+y)\in J(x+y).\]  
_If \(C\) is a convex closed set in a smooth Banach space \(E\) and \(\Pi_{C}\) a sunny nonexpansive retraction from \(E\) onto \(C\), we have_  
\[\mathrm{VI}(C,B)=\mathrm{Fix}(\Pi_{C}(I-\lambda B)).\]  
_where \(B:C\to E\) be an accretive mapping and \(\lambda>0\),_  
Using Proposition 3, we immediately obtain the following lemmas.