Page 120 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 111

It is clear that u i ∈ D( ) (i ∈ N). Moreover, for i large enough, ||u i ||
= 0, and
we may set
u i
z i = .
||u i ||
There exist subsequences, again denoted by {t i } and {z i }, such that lim t i =
i→+∞
t ∈[0,1] and lim z i = z with ||z|| = 1.
i→+∞
Setting now v = 0in (4.82), we obtain
2
(1 − t i )||u i || + t i F(u i ),u i + q,u i ≤ (0) − (u i ).
For i great enough, we have u i ≥ σ, and thus

2
2
(1 − t i )||u i || + t i F(u i ),u i ≥ ((1 − t i ) + αt i )||u i || .
If α ≥ 1, then ((1−t i )+αt i ) ≥ ((1−t i )+t i ) = 1, and if α ≤ 1, then ((1−t i )+
αt i ) ≥ (α(1 − t i ) + αt i ) ≥ α. Therefore

2
min{1,α}||u i || + q,u i − (0) + (u i ) ≤ 0.
The function is proper, convex, and lower semicontinuous, and thus there
exist a ≥ 0 and b ∈ R such that
n
(∀x ∈ R ) : (x) ≥−a||x|| + b.
We thus have

2
min{1,α}||u i || + q,u i − (0) − a||u i || + b ≤ 0.
2
Dividing this last relation by ||u i || , we get
q (0) a b
2
min{1,α}||z i || + ,z i − − + ≤ 0.
||z i || ||u i || 2 ||u i || ||u i || 2
Taking the limit as i →+∞, we get

2
min{1,α}||z|| ≤ 0,
a contradiction. Thus, for R ≥ R 0 , (4.81) holds, and the Brouwer degree with
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respect to the set D R ={x ∈ R :||x|| <R} and 0 of the map (t,u) à
u − H(t,u) is well defined for all t ∈[0,1]. Set R 1 = P (−q) and let
R> max{R 0 ,R 1 }. Using the homotopy invariance property and the normalized
property of Brouwer degree, we obtain
deg(id R − P (id R − (F(.) + q),D R ,0) = deg(id R − H(1,.),D R ,0)
n
n
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