Page 80 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 71

4.2.22 Class of (M, ) ∈ DS0 n
n
We define by DS0 n the set of (M, ) ∈ R n×n × (R ;R ∪{+∞}) such that
n
there exists a positive diagonal matrix ∈ R n×n and a subset K ⊂ R of the
form

K = K 1 × K 2 × ··· × K n , (4.23)
where, for each i ∈{1,...,n}, K i ⊂ R is a nonempty closed convex cone, such
that

(4.24)
= K
and

(∀x ∈ K) : Mx, x ≥ 0. (4.25)
n
Remark 20. If (M, ) ∈ DS0 n then ( M, ) ∈ PD0 n with ∈ D (R ;R ∪
{+∞}).

Example 38. Let K = R + × R + × R + , = K , and

⎛ ⎞
1 −20
⎜ 1 ⎟
M = ⎝ 0 0 ⎠ .
2
0 2 0
Set
⎛ ⎞
100
= ⎝ 020 ⎠ .
⎟
⎜
001
We have

2
3
(∀x ∈ K = R ) : Mx, x = Mx,x = (x 1 − x 2 ) + 2x 2 x 3 ≥ 0.
+
Thus (M, ) ∈ DS0 n .

4.2.23 Class of (M, ) ∈ Q n
n
We define by Q n the set of (M, ) ∈ R n×n × (R ;R ∪{+∞}) such that
n
R(M, ) = R ,
n
that is, for each q ∈ R , problem VI(M,q, ) has at least one solution.