Page 98 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 89

n
Let v ∈ R be fixed. We have
1
( I + M)u i + q,u i − v − (v) + (u i ) ≤ 0.
i
Taking the limit inferior as i →+∞ and using the lower semicontinuity of ,
we obtain
Mu + q,u − v − (v) + (u) ≤ 0. (4.47)

n
n
Since the vector v has been chosen arbitrarily in R , (4.47) holds for all v ∈ R .
The existence result follows.
From Theorem 4 and Proposition 20 we have

PD0 n ∪ P0 n ⊂ Q0 n ,

and thus if

(M, ) ∈ PD0 n ∪ P0 n
and
B(M, ) ={0},
then

(M, ) ∈ Q n .
This, together with Proposition 16, gives the following:

Corollary 6. If
(M, ) ∈ PD0 n ∪ P0 n
and

D( ) ∞ ∩ N 0 (M) ∩ K(M, ) ={0},
then
n
R(M, ) = R .
Example 46. Let
⎛ ⎞
1 0 0
⎜ ⎟
M = ⎝ −210 ⎠
0 0 1
and
3
(∀x ∈ R ) : (x) = R×R×R + (x).