Page 118 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 109

This is a contradiction to condition (4.77) since z
= 0 and we have already
proved that z ∈ B(M,K). The sequence {u i } is thus bounded, and we may con-
clude as in the proof of Theorem 5.
Part c) is obtained as usual from Proposition 15.

Example 57. Let
⎛ ⎞
1 −40
⎟
M = ⎝ 1 1 0 ⎠
⎜
0 0 2
and set
3
K = R .
+
Let
⎛ ⎞
1 0 0
α = ⎝ 0 α 0 ⎠ .
⎜
⎟
0 0 1
We have

2 2 2
Mx, α x = x + (α − 4)x 1 x 2 + αx + 2x .
1 2 3
We may thus choose α> 4to have

3
(∀x ∈ R ) : Mx, α x ≥ 0.
+
We have

x ∈ B(M,K)
if and only if

⎧
x 1 ≥ 0
⎪
⎪
⎪
⎪
x 2 ≥ 0
⎪
⎪
⎪
⎨
x 3 ≥ 0
⎪
⎪
⎪ x 1 − 4x 2 ≥ 0
⎪
⎪
⎪
⎪
x 1 + x 2 ≥ 0
⎩
and
2
x 1 (x 1 − 4x 2 ) + x 2 (x 1 + x 2 ) + 2x = 0,
3