# Systems of Variational Inequalities with Nonlinear Operators
## 2 Preliminaries  
Throughout this paper we write \(x_{n}\rightharpoonup x\) (respectively, \(x_{n}\to x\)) to indicate that the sequence \(\{x_{n}\}\) converges weakly (respectively, strongly) to \(x\). Without loss of generality, we assume that \(E\) is a real Banach space and the dual will be presented by \(E^{*}\) in this paper. **Definition 1**. : _Let \(\{S_{n}\}_{n=0}^{\infty}\) be a vector sequence of pseudocontractive continuous self-mappings on \(C\), a convex closed convex subset of Banach space \(E\). Recall that \(\{S_{n}\}_{n=0}^{\infty}\) is said to be a countable family of \(\ell\)-uniformly Lipschitzian pseudocontractive self-mappings provided that there exists a constant \(\ell>0\) such that each \(S_{n}\) is a \(\ell\)-Lipschitz continuous mapping._  
In a smooth Banach space \(E\), an operator \(A\) is said to be strongly positive if there exists a constant \(\hat{\gamma}>0\) with the property  
\[\|aI-bA\|=\sup_{\|x\|\leq 1}|\langle(aI-bA)x,j(x)\rangle|,\langle Ax,j(x) \rangle\geq\hat{\gamma}\|x\|^{2}\;a\in[0,1],\;b\in[-1,1],\]  
where \(I\) is the identity mapping and \(j(\cdot)\) is the single-valued normalized duality mapping. Recall that a Banach space \(E\) is said to be strictly convex if for any \(x,y\in\{x\in E:\|x\|=1\}\), \(x\
eq y\;\Rightarrow\;\|\frac{x+y}{2}\|<1\). It is also said to be uniformly convex if for each \(\epsilon\in(0,2]\), there exists \(\delta>0\) such that for any \(x,y\in\{x\in E:\|x\|=1\}\), \(\|x-y\|\geq\epsilon\;\Rightarrow\;\|x+y\|\leq 2-2\delta\). Clearly, if \(E\) is uniformly convex, then it is strictly convex. A Banach space \(E\) is said to have a Gateaux differentiable norm if the limit  
\[\lim_{t\to 0}\frac{\|x+ty\|-\|x\|}{t}\]  
exists for each \(x,y\in\{x\in E:\|x\|=1\}\) and in this case we call \(E\) smooth; \(E\) is said to have a uniformly Gateaux differentiable norm if for each \(y\in\{x\in E:\|x\|=1\}\), the above limit is attained uniformly for \(x\in\{x\in E:\|x\|=1\}\). Moreover, it is said to have a uniformly Frechet differentiable norm if the above limit is attained uniformly for \(x,y\in\{x\in E:\|x\|=1\}\) and in this case we call \(E\) uniformly smooth. The norm of \(E\) is said to be the Frechet differentiable if for each \(x\in\{x\in E:\|x\|=1\}\), the above limit is attained uniformly for \(y\in\{x\in E:\|x\|=1\}\). The modulus of smoothness of \(E\) is defined by  
\[\varrho(\tau)=\sup\{\frac{(\|x+y\|+\|x-y\|)-2}{2}:x,y\in E,\;\|x\|=1,\;\|y\|=\tau\},\]  
where \(\varrho:[0,\infty)\to[0,\infty)\) is a function. It is known that \(E\) is uniformly smooth if and only if \(\lim_{\tau\to 0}\frac{\varrho(\tau)}{\tau}=0\). Let \(q\) be a fixed real number with \(1<q\leq 2\). A Banach space \(E\) is said to be \(q\)-uniformly smooth if there exists a constant \(\kappa>0\) such that \(\frac{\varrho(\tau)}{\kappa}\leq\tau^{q}\) for all \(\tau>0\). From [23], we know the following relation. Let \(q\) be a fixed number with \(1<q\leq 2\) and \(E\) be a Banach space. Then \(E\) is \(q\)-uniformly smooth if and only if there exists a constant \(c>0\) such that  
\[\|x\|^{q}+\|xy\|^{q}\geq\frac{\|x+y\|^{q}+\|x-y\|^{q}}{2},\quad\forall x,y\in E.\]  
The best constant \(\kappa\) in the above inequality is called the \(q\)-uniformly smooth constant of \(E\); see [23] for more details. In addition, no Banach space is \(q\)-uniformly smooth for \(q>2\); see [24] for more details. If \(E\) be a 2-uniformly smooth Banach space. Then  
\[\|x+y\|^{2}-\|x\|^{2}\leq 2\langle y,j(x)\rangle+2\|xy\|^{2},\quad\forall x,y \in E,\]  
where \(\kappa\) is the 2-uniformly smooth constant of \(E\). In particular, if \(E\) is a Hilbert space, then the duality pairing \(\langle\cdot,\cdot\rangle\) reduces to the inner product, \(j=I\) the identity mapping of \(E\), and \(\kappa=\sqrt{2}/2\). For \(q>1\), the generalized duality mapping \(J_{q}:E\to 2^{E^{*}}\) is defined by  
\[J_{q}(x)=\{\varphi\in E^{*}:\|\varphi\|=\|x\|^{q-1},\langle x,\varphi\rangle= \|x\|^{q}\},\quad\forall x\in E.\]  
In particular, \(J=J_{2}\) is called the normalized duality mapping.