Page 124 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 115

Remark 29. Note that if

n
(∀x ∈ R ) : F(x) = Mx,
where M is a positive semidefinite matrix, then
n
T (x).
(∀x ∈ R ) : r (x) = ker(M+M )
F
More generally, if F is a continuous and positively homogeneous mapping such
that
n
(∀x ∈ R ) : F(x),x ≥ 0,
then
n
(∀x ∈ R ) : r (x) = N (F) ,
F
where
n
N(F) ={x ∈ R : F(x),x = 0}.
Remark 30. Suppose that
n
(∀x ∈ R ) : F(x) =∇ (x)
1
n
for some convex function ∈ C (R ;R). Then
n
(∀x ∈ R ) : r (x) = ∞ (x).
F
Indeed, we have
n
(∀x,y ∈ R ,t > 0) : (tx) − (y) ≥ F(y),tx − y .
n
Thus, for all x,y ∈ R and t> 0, we have
(tx) (y) F(y),tx − y
lim ≥ lim + lim ,
t→+∞ t t→+∞ t t→+∞ t
and therefore
∞ (x) ≥ F(y),x .

n
Thus, for all x ∈ R and t> 0, we have
∞ (x) ≥ F(tx),x ,
so that

∞ (x) ≥ liminf F(tx),x ≥ r (x).
F
t→+∞