Page 116 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 107

reduces here to

(∀v 2 > 0) : q 2 v 2 > 0,
which is equivalent to

q 2 > 0.
3
Thus, for all q ∈ R such that q 2 > 0, problem VI(M,q, ) has at least one
solution.

4.11 DIAGONAL STABILITY AND SOLVABILITY CONDITIONS

The following result concerns the case (M, ) ∈ DS0 n .
n
Corollary 10. Let K ⊂ R be defined by

K = K 1 × K 2 × ··· × K n ,
where K i ⊂ R (1 ≤ i ≤ n) is a nonempty closed convex cone.
Let M ∈ R n×n . Suppose that there exists a positive diagonal matrix such
that
(∀x ∈ K) : Mx, x ≥ 0. (4.76)
n
a) If B(M,K) ={0}, then for each q ∈ R , problem CP(M,q,K) has at least
one solution.
b) Suppose that B(M,K)
= {0}. If there exists x 0 ∈ K such that
T
(∀v ∈ B(M,K),v
= 0) : q − M x 0 ,v > 0, (4.77)

then problem CP(M,q,K) has at least one solution.
c) Moreover, if u 1 and u 2 are two solutions of problem CP(M,q,K), then

u 1 − u 2 ∈ N − (M). (4.78)

Proof. From Proposition 13 we know that (M, ) ∈ DS0 n ⊂ P0 n .
Part a) is a direct consequence of Corollary 6.
Part b) Using Proposition 20, we see that, for all i ∈ N,i
= 0, there exists
n
u i ∈ R such that
1
( I + M)u i + q,v − u i ≥ 0, ∀v ∈ K. (4.79)
i
We claim that the sequence {u i }≡{u i ;i ∈ N\{0}} is bounded. Suppose on the
contrary that ||u i || → +∞ as i →+∞. Then, for i large enough, ||u i ||
= 0,