Page 81 - Complementarity and Variational Inequalities in Electronics

72 Complementarity and Variational Inequalities in Electronics

4.2.24 Class of (M, ) ∈ Q0 n
n
We define by Q0 n the set of (M, ) ∈ R n×n × (R ;R∪{+∞}) such that there
exists λ 0 > 0 such that

n
(∀ 0 <λ ≤ λ 0 ) : R(λI + M, ) = R ,
n
that is, for any 0 <λ ≤ λ 0 and q ∈ R , problem VI(λI + M,q, ) has at least
one solution.
It is clear that

PD n ⊂ PD0 n , P n ⊂ P0 n , PS n ⊂ PS0 n , DS n ⊂ DS0 n .
We also have the following:
Proposition 13. We have

DS n ⊂ P n
and

DS0 n ⊂ P0 n .
Proof. Let us first remark that if (M, ) ∈ DS n ∪ DS0 n , then condition (4.12)
(or (4.14)) holds. Indeed, here

= K 1 ×K 2 ×···×K n
as in (4.21) (or (4.24)) where K 1 ,K 2 ,...,K n are nonempty closed convex
cones. Thus

D( ∞ ) = D( ) ∞ = K 1 × K 2 × ··· × K n .
If w ∈ K, then for all i ∈{1,...,n}, w i ∈ K i , and since for all j ∈{1,...,n},
j
j
j
j
= i,0 ∈ K i , we obtain w,e e = w j e ∈ K.
If (M, ) ∈ DS n and x
= 0, then
n n

Mx, x = (Mx) i ( x) i = ii (Mx) i x i > 0.
i=1 i=1
Here, for all i ∈{1,...,n}, ii > 0, and thus there necessarily exists α ∈
{1,...,n} such that (Mx) α x α > 0.
If (M, ) ∈ DS0 n and x
= 0, then I ={i ∈{1,...,n}: x i
= 0}
=∅, and

Mx, x = ii (Mx) i x i ≥ 0.
i∈I
So, there necessarily exists α ∈{1,...,n} such that x α
= 0 and (Mx) α x α ≥ 0.