Page 93 - Complementarity and Variational Inequalities in Electronics

84 Complementarity and Variational Inequalities in Electronics

This last relation, together with (4.39) and (4.40), implies that z ∈ B((1−t)M +
tI, ). This is a contradiction to our assumption requiring that (M, ) is AC n
well-posed since z
= 0.
Thus, for R ≥ R 0 , (4.37) holds, and the Brouwer degree with respect to the
set
n
D R ={x ∈ R :||x|| <R}
and 0 of the map (t,u) à u−H(t,u) is well defined for all t ∈[0,1]. Set R 1 =
||P (0)|| and let R> max{R 0 ,R 1 }. Using the homotopy invariance property
and the normalized property of the Brouwer degree, we obtain

deg(id R − P (id R − (M. + q),D R ,0) = deg(id R − H(1,.),D R ,0)
n
n
n
= deg(id R − H(0,.),D R ,0) = deg(id R − P (0),D R ,0) = 1.
n
n
From the solution property of Brouwer degree we get that SOL(M,q, )
= ∅,
and the result follows.
The following result is a direct consequence of Theorem 4 and Proposi-
tion 17.

n
Corollary 3 (Spectral condition). Let ∈ (R ;R∪{+∞}) and M ∈ R n×n .If
σ ∞ (M, ) ⊂]0,+∞[,

n
then for each q ∈ R , problem VI(M,q, ) has at least one solution.
4.5 A BOUNDEDNESS CONDITION
n
Let : R → R ∪{+∞} be a proper convex and lower semicontinuous func-
tion with closed domain, and let M ∈ R n×n .Using Theorem 4 together with
Proposition 18, we get

D( ) bounded =⇒ (M, ) ∈ AC n ⊂ Q n .

n
Corollary 4. Let ∈ (R ;R ∪{+∞}) and M ∈ R n×n .If D( ) is bounded,
then
n
R(M, ) = R .
Example 42. Let
⎛ ⎞
−1 1 0
M = ⎝ 1 2 −1 ⎠
⎟
⎜
0 −2 −1