Page 87 - Complementarity and Variational Inequalities in Electronics

78 Complementarity and Variational Inequalities in Electronics

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Definition 2. We define by AC n the set of (M, ) ∈ R n×n × (R ;R ∪{+∞})
such that
∀t ∈[0,1]: B((1 − t)I + tM, ) ={0}.

In other words, we say that the couple (M, ) is of class AC n if for all
t ∈[0,1], 0 is the unique solution of problem SCP ∞ ((1 − t)I + tM, ).
Remark 22. Note that the case t = 0in Definition 2 is immaterial since
B(I, ) ={0}.

This concept that may appear to be technical can in fact be used to recover
various important situations. Let us first introduce a concept of a generalized
eigenvalue for the couple (M, ), which we will use in the next section to state
a general spectral condition ensuring the solvability of problem VI(M, ,q).

Definition 3. We say that μ ∈ R is a generalized eigenvalue of the couple
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(M, ) if there exists a vector z ∈ R , z
= 0, such that z ∈ B(M − μI, ),
that is,
⎧
⎪ z ∈ D( ) ∞
⎪
⎨
Mz − μz ∈ (D( ∞ )) ∗ (4.35)
⎪
⎪
Mz − μz,z ≤ 0.
⎩
Let us now denote by σ ∞ (M, ) the set of generalized eigenvalues of
(M, ), that is,
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σ ∞ (M, ) ={μ ∈ R :∃ z ∈ R \{0} a solution of (4.35)}.

Remark 23. i) Problem (4.35) appears as a generalized eigenvalue problem that
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reduces to the classical one when ≡ 0. Indeed, if ≡ 0, then D( ) ∞ = R ,
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∗
(D( ∞ )) ={0}, and Problem (4.35) consists in finding μ ∈ R and z ∈ R ,
z
= 0, such that Mz − μz = 0. Thus
σ ∞ (M,0) = σ(M) ∩ R.

ii) If D( ) ∞ = D( ∞ ), then Problem (4.35) reduces to the eigenvalue
problem for the complementarity problem

∗
D( ∞ ) z ⊥ Mz − μz ∈ (D( ∞ )) .
iii) If D( ) is bounded, then D( ) ∞ ={0} and σ ∞ (M, ) =∅, whereas if
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D( ∞ ) = R , then
σ ∞ (M, ) = σ(M) ∩ R.