Page 128 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 119

Proposition 21 means that if assumptions (H1) and (H2) hold, then problem
(4.87)–(4.89) can be studied via the variational inequality VI(M,q, ) with

M =−PA, q =−PDu, ≡ ◦ C. (4.93)

The following result is then of particular interest to calculate the recession tools
involved in B(M,q, ), which here are D( ) ∞ , ∞ and D( ∞ ) with ≡
◦ C.

Proposition 22. Suppose that assumptions (H1) and (H2) are satisfied and let
be defined as in (4.90). Then

n
D( ) ∞ ={x ∈ R : Cx ∈ D( ) ∞ }, (4.94)
n
(∀x ∈ R ) : ∞ (x) = ∞ (Cx), (4.95)
and
n
D( ∞ ) ={x ∈ R : Cx ∈ D( ∞ )}. (4.96)
Proof. i) Let
n
D ∞ (C, ) ={x ∈ R : Cx ∈ D( ) ∞ }.
It is easy to see that

D( ) ∞ = D ∞ (C, ). (4.97)
Indeed, if e ∈ D( ) ∞ , then (∀λ> 0) : λe+¯x 0 ∈ D( ). Thus (∀λ> 0) : C(λe+
¯ x 0 ) ∈ D( ).Here ¯y 0 = C ¯x 0 ∈ D( ), and thus

1
Ce ∈ (D( ) −¯y 0 ) = D( ) ∞ .
λ
λ>0
Thus e ∈ D ∞ (C, ).
Reciprocally, if e ∈ D ∞ (C, ), then Ce ∈ D( ) ∞ , and thus (∀λ> 0) :
λCe +¯y 0 ∈ D( ). Thus (∀λ> 0) : λe +¯x 0 ∈ D( ), and then

1
e ∈ (D( ) −¯x 0 ) = D( ) ∞ .
λ
λ>0
n
ii) Let x ∈ R be given. We have
1 1
∞ (x) = lim (C(¯x 0 + λx)) = lim ( ¯y 0 + λCx) = ∞ (Cx).
λ→+∞ λ λ→+∞ λ
iii) Relation (4.96) is a direct consequence of (4.95).