Page 90 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 81

b) If (M, ) ∈ P0 n , then condition (4.15) on M entails that for each x ∈
D( ) ∞ , x
= 0, there exists α ∈{1,...,n} such that
2
x α (λx + Mx) α = λx + x α (Mx) α > 0;
α
thus λI + M satisfies condition (4.13), and the result follows from Proposi-
tion 19.
c) Let (M, ) ∈ PS0 n .Let t ∈[0,1] and z ∈ B((1−t)I +tM, ). We claim
that z = 0. Suppose on the contrary that z
= 0. From (4.16) we deduce that
∗
(D( ∞ )) ={0}, and the second relation in (4.27) yields
(1 − t)z + tMz + tλz = 0.

If t = 0, then z = 0, a contradiction. If 0 <t ≤ 1, then
(1 − t)
Mz =− z − λz,
t

∗
so that ν =− (1−t) − λ< 0 is a real eigenvalue of M, a contradiction to (4.19).
t
The following theorem is the basic result of this section. It reduces the study
of the general class of variational inequalities VI(M,q, ) to semicomplemen-
tarity problems SCP ∞ (tM + (1 − t)I, )(t ∈[0,1]) involving the convex com-
binations of the matrix M and the identity matrix I. More precisely, we prove
n
that if the couple (M, ) ∈ AC n , then for each q ∈ R , problem VI(M,q, )
has at least one solution. In other words, we prove that
AC n ⊂ Q n .

Let us first recall some basic results from Brouwer degree theory, which we will
n
use to prove the next result. Let D ⊂ R be an open bounded set. If f : D → R n
¯
is continuous and 0 /∈ f(∂D), then the Brouwer topological degree of f with
respect to D and 0 is well defined (see e.g. [63]) and denoted by deg(f,D,0).
Let us recall some properties of the topological degree.

P1. Solution property: If 0 /∈ f(∂D) and deg(f,D,0)
= 0, then there exists
x ∈ D such that f(x) = 0.
n
¯
P2. Homotopy invariance property: Let ϕ :[0,1]× D → R ;(λ,x) à ϕ(λ,x),
be continuous and such that 0 /∈ ϕ(λ,∂D) for all λ ∈[0,1]. Then the map
λ à deg(ϕ(λ,·),D,0)

is constant on [0,1].
P3. Normalized property: If p ∈ D, then deg(id R − p,D,0) = 1.
n