Page 88 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 79

The following result shows that the class AC n is nothing else but the class
of couples (M, ) whose generalized eigenvalues are positive.
Proposition 17. The couple (M, ) is of class AC n if and only if

σ ∞ (M, ) ⊂]0,+∞[. (4.36)

Proof. a) Let us first prove that if (4.36) holds, then the couple (M, ) is of
class AC n .Inviewof Remark 22, it suffices to check that for all t ∈]0,1],
we have B((1 − t)I + tM, ) ={0}. Suppose on the contrary that there exist
(1−t)
t ∈]0,1] and z
= 0 such that z ∈ B((1−t)I +tM, ). Then setting μ =− ,
t
we easily check that z ∈ B(M − μI, ), and this is a contradiction since μ ≤ 0.
b) Let us now check that if the couple (M, ) is of class AC n , then (4.36)
holds. Suppose on the contrary that there exist μ ≤ 0 and z
= 0 such that z ∈
1 1
B(M −μI, ).Weset t = = . It is clear that t ∈]0,1] and z ∈ B(M +
1−μ 1+|μ|
(1−t) I, ), and thus z ∈ B((1 − t)I + tM, ), which is a contradiction.
t
n
Proposition 18. Let : R → R be a proper convex lower semicontinuous
function with closed domain, and let M ∈ R n×n .If D( ) is bounded, then
(M, ) ∈ AC n .

Proof. Here D( ) is assumed to be bounded, and thus D( ) ∞ ={0}.If t ∈
[0,1] and z ∈ B((1−t)I +tM, ), then z ∈ D( ) ∞ , and the result follows.
n
Proposition 19. Let : R → R be a proper convex lower semicontinuous
function with closed domain, and let M ∈ R n×n . Then we have

PD n ∪ P n ∪ PS n ⊂ AC n .

Proof. a) We prove that PD n ⊂ AC n .Let (M, ) ∈ PD n , t ∈[0,1], and z ∈
B((1 − t)I + tM, ). Then z ∈ D( ) ∞ and

(1 − t)z + tMz,z ≤ 0.

If t = 1, then Mz,z ≤ 0 and (4.10) yield z = 0. If 0 ≤ t< 1, then
t
2
||z|| ≤− Mz,z ,
1 − t
and from (4.10) we deduce that necessarily z = 0. The result follows.
b) We prove that P n ⊂ AC n .Let (M, ) ∈ P n , t ∈[0,1], and z ∈ B((1 −
t)I + tM, ). We claim that z = 0. Suppose on the contrary that z
= 0. We
claim that there exists some index k ∈{1,...,n} such that
2
(1 − t)z + t(Mz) k z k > 0.
k