Page 86 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 77

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c) If D( ∞ ) = R , then
B(M, ) = ker(M).
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d) If D( ∞ ) = R and M is invertible, then
B(M, ) ={0}.
e) If D( ) is bounded, then

B(M, ) ={0}.

f) If (M, ) ∈ PD0 n ∪ P0 n , then
B(M, ) = D( ) ∞ ∩ N 0 (M) ∩ K(M, ).

g) If (M, ) ∈ PS0 n , then

B(M, ) = ker(M).
Proof. a) Part (a) is a direct consequence of the definition of the set B(M, ).
∗
b) Part (b) is a direct consequence of Proposition 14.c)Here (D( ∞ )) ={0}
and K(M, ) = ker(M) ⊂ N − (M).Wealsohave D( ∞ ) ⊂ D( ) ∞ so that
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D( ) ∞ = R . Thus B(M, ) = ker(M). d) Part (d) is a direct consequence of
part (c). e) If D( ) is bounded, then D( ) ∞ ={0}.f)If (M, ) ∈ PD0 n , then

(∀x ∈ D( ) ∞ ) : Mx,x ≥ 0,
and thus

D( ) ∞ ∩ N − (M) = D( ) ∞ ∩ N 0 (M).
Let us now suppose that (M, ) ∈ P0 n .Let w ∈ B(M, ). It suffices to check
that Mw,w = 0. We know that
Mw,h ≥ 0,∀h ∈ D( ∞ ).

j
j
Let j ∈{1,...,n}.Wemayset h = w,e e to get
(Mw) j w j ≥ 0.
This last relation holds for all j ∈{1,...,n}, and since

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0 ≥ Mw,w = (Mw) j w j ,
j=1
we finally obtain that Mw,w = 0. g) Part (g) is a direct consequence of
part (c).