## 2 Problem Formulation and Lipschitz Constant Estimates  
Let \(F:[0,T]\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) and, for each \(t\in[0,T]\), \(K(t)\) be a nonempty, closed and convex subset of \(\mathbb{R}^{n}\). We denote with \(\langle\cdot,\cdot\rangle\) the scalar product in \(\mathbb{R}^{n}\). The variational inequality on \(\mathbb{R}^{n}\), with parameter \(t\in[0,T]\), is the following problem: for each \(t\in[0,T]\), find \(x(t)\in K(t)\) such that:  
\[\langle F(t,x(t)),y-x(t)\rangle\geq 0,\qquad\forall\;y\in K(t). \tag{1}\]In the case where, for each \(t\in[0,T]\) the solution of (1) is unique, we will prove, assuming the Lipschitz continuity of \(F\) and of the parametric functions describing \(K(t)\), the Lipschitz continuity of the solution map \(t\mapsto x(t)\) and derive an a priori estimate of its Lipschitz constant \(\Lambda\). Specifically, we estimate \(\Lambda\) in the general case where the variables \(t\) and \(x\) cannot be separated, as well as in the separable case where \(F(t,x)=G(x)+H(t)\). Moreover, in the special circumstance where \(K(t)=K\), for any \(t\in[0,T]\), two simplified formulas are derived. We then focus on computing approximations for the mean value of the components \(x_{j}:[0,T]\to\mathbb{R}\), \(j=1,\ldots n\):  
\[\frac{1}{T}\int_{0}^{T}x_{j}(t)dt. \tag{2}\]  
We recall here some useful monotonicity properties. **Definition 1**. : _A map \(F:\mathbb{R}^{n}\to\mathbb{R}^{n}\) is monotone on a set \(A\subset\mathbb{R}^{n}\) if and only if_  
\[\langle F(x)-F(y),x-y\rangle\geq 0,\qquad\forall\ x,y\in A.\]  
_If the equality holds only when \(x=y\), \(F\) is said to be strictly monotone._  
A stronger type of monotonicity is given by the following definition. **Definition 2**. : \(F:\mathbb{R}^{n}\to\mathbb{R}^{n}\) _is \(\alpha\)-strongly monotone on \(A\) if and only if \(\exists\alpha>0\):_  
\[\langle F(x)-F(y),x-y\rangle\geq\alpha\|x-y\|^{2},\qquad\forall\ x,y\in A.\]  
The concepts of strict and strong monotonicity coincide for linear operators and they are equivalent to the positive definiteness of the corresponding matrix. When \(F\) also depends on a parameter, the following property is useful. **Definition 3**. : \(F:[0,T]\times\mathbb{R}^{n}\to\mathbb{R}^{n}\) _is uniformly \(\alpha\)-strongly monotone on \(A\) if and only if \(\exists\alpha>0\):_  
\[\langle F(t,x)-F(t,y),x-y\rangle\geq\alpha\|x-y\|^{2}\qquad\forall\ x,y\in A, \ \forall\ t\in[0,T].\]  
Sufficient conditions for the existence and uniqueness of the solution of a variational inequality problem can be found in [8]. We mention here the following classical theorem which, in the parametric case, has to be applied for each \(t\in[0,T]\). **Theorem 1**. : _Let \(K\subset\mathbb{R}^{n}\) be a closed and convex set and \(F:\mathbb{R}^{n}\to\mathbb{R}^{n}\) continuous on \(K\). If \(K\) is bounded, then the variational inequality problem \(VI(F,K)\) admits at least one solution. If \(K\) is unbounded, then the existence of a solution is guaranteed by the following coercivity condition:_  
\[\lim_{\|x\|\to+\infty}\frac{\langle F(x)-F(x_{0}),x-x_{0}\rangle}{\|x-x_{0}\| }=+\infty,\]  
_for \(x\in K\) and some \(x_{0}\in K\). Furthermore, if \(F\) is \(\alpha\)-strongly monotone on \(K\), then the solution exists and is unique._  
Let us now prove two theorems, along with three corresponding corollaries, which constitute the main contribution of the paper. Such results, at different levels of generality, ensure that the unique solution of (1) is Lipschitz continuous, and also provide an estimate of the Lipschitz constant of the solution. **Theorem 2** (General case).