Page 78 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 69

4.2.17 Class of (M, ) ∈ P n
n
We define by P n the set of (M, ) ∈ R n×n × (R ;R ∪{+∞}) such that
j
j
x ∈ D( ) ∞ =⇒ x,e e ∈ D( ∞ )(1 ≤ j ≤ n) (4.12)
and

(x ∈ D( ) ∞ ,x
= 0) =⇒ ∃ α ∈{1,...,n}: x α (Mx) α > 0. (4.13)

Remark 18. i) The set D( ) is nonempty, closed, and convex since ∈
n
(R ;R ∪{+∞}). Note that both D( ) ∞ and D( ∞ ) areusedin (4.12).
ii) Condition (4.12) means that if w ∈ D( ) ∞ , then its projection
j j j n
w,e e = w j e onto the space X j ={x ∈ R : x k = 0,∀k ∈{1,...,n},k
= j}
belongs to D( ∞ ).
α
n
iii) If = K where K is a subset like R , R n−α × (R + ) (α ∈{1,...,n}),
n
or (R + ) , then condition (4.12) holds.
n
Example 33. If M is a P-matrix and ∈ (R ;R ∪{+∞}) with (4.12), then
(M, ) ∈ P n .

4.2.18 Class of (M, ) ∈ P0 n
n
We define by P0 n the set of (M, ) ∈ R n×n × (R ;R ∪{+∞}) such that
j
j
x ∈ D( ) ∞ =⇒ x,e e ∈ D( ∞ )(1 ≤ j ≤ n) (4.14)
and

(x ∈ D( ) ∞ ,x
= 0) =⇒ ∃ α ∈{1,...,n}: x α
= 0& x α (Mx) α ≥ 0. (4.15)

n
Example 34. If M is a P0-matrix and ∈ (R ;R ∪{+∞}) with (4.14), then
(M, ) ∈ P0 n .

4.2.19 Class of (M, ) ∈ PS n
n
We define by PS n the set of (M, ) ∈ R n×n × (R ;R ∪{+∞}) such that
n
D( ∞ ) = R (4.16)
and
σ(M) ∩ R ⊂]0,+∞[. (4.17)

n
Example 35. If M is a weakly positive definite matrix and ∈ (R ;R ∪
n
{+∞}) with D( ∞ ) = R , then (M, ) ∈ PS n .