Page 94 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 85

and

3
(∀x ∈ R ) : (x) = K (x)
with
K =[0,1]×[0,1]×[0,1].

3
Here D( K ) =[0,1]×[0,1]×[0,1] is bounded, and thus R(M, ) = R .

4.6 POSITIVITY AND SOLVABILITY CONDITIONS
Using Theorem 4 together with Proposition 19, we obtain

PD n ∪ P n ∪ PS n ⊂ AC n ⊂ Q n ,

n
and we get the following result ensuring that for each q ∈ R , problem
VI(M,q, ) has at least one solution.
Corollary 5. If

(M, ) ∈ PD n ∪ P n ∪ PS n ,

then
n
R(M, ) = R .

Note that each matrix discussed in Corollary 5 presents some “positivity
property” and is nonsingular on D( ) ∞ in the sense that

D( ) ∞ ∩ ker(M) ={0}.

Example 43. Let
⎛ ⎞
110
⎜ ⎟
M = ⎝ 110 ⎠
001
and

3
(∀x ∈ R ) : (x) = K (x)
with
3
K ={x ∈ R : x 1 ≥ 1,x 2 ≥ 1,x 3 ≥ 1}.