Page 102 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 93

Remark 24. The famous result on complementarity problems with P-matrices
(see e.g. [37]) can be easily deduced from Theorem 7. Indeed, if = R , then
n
+
(1 ≤ i ≤ n), and if M is a
can be written as in (4.48)–(4.50) with i = R +
n
P-matrix, then Theorem 7 can be applied to ensure that for each q ∈ R , there
exists a unique u ≥ 0 such that Mu + q ≥ 0 and u,Mu + q = 0.
Example 47. Let
⎛ ⎞
1 −2 −8
M = ⎝ 0 1 2 ⎠
⎟
⎜
0 0 2
and
3
(∀x ∈ R ) : (x) =|x 1 |+|x 3 |.
3
The matrix M is a P-matrix, and ∈ D (R ;R ∪{+∞}). Thus, for each
3
q ∈ R , problem V I(M,q, ) has a unique solution.
4.9 SEMICOERCIVITY AND SOLVABILITY CONDITIONS
The results discussed in the previous section require that B(M, ) ={0}.If
B(M, )
= {0}, then in assuming some semicoercivity condition on the ma-
trix M, we may determine conditions on q ensuring the solvability of problem
VI(M,q, ).

n
Theorem 8. Let : R → R∪{+∞} be a proper convex lower semicontinuous
function with closed domain, and let M ∈ R n×n . Suppose in addition that

(∀x ∈ D( ) ∪ D( ) ∞ ) : Mx,x ≥ 0.

If there exists x 0 ∈ D( ) such that
T
(∀v ∈ B(M, ), v
= 0) : q − M x 0 ,v + ∞ (v) > 0, (4.60)

then problem VI(M,q, ) has at least one solution.
n
Proof. Let q ∈ R .Here (M, ) ∈ PD0 n , and using Proposition 20 and Theo-
n
rem 4, we see that for all i ∈ N,i
= 0, there exists u i ∈ R such that
1 n
( I + M)u i + q,v − u i + (v) − (u i ) ≥ 0, ∀v ∈ R . (4.61)
i
We claim that the sequence {u i }≡{u i ;i ∈ N\{0}} is bounded. Suppose on the
contrary that ||u i || → +∞ as i →+∞. Then, for i large enough, ||u i ||
= 0,