Page 122 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 113

We have

2
3
2
2
2
(∀x ∈ R ) : F(x),x = x + x + 2x ≥||x|| ,
1 2 3
n
and thus, for all q ∈ R , problem VI(F,q, ) has at least one solution.
The recession mapping of Brézis and Nirenberg [19] associated with F is
n
denoted by r : R → R ∪{−∞,+∞} and defined by
F
n
(∀x ∈ R ) : r (x) = liminf F(tv),v .
F
t→+∞
v→x
Using this concept, we prove the following:
n
Theorem 10. Suppose that ∈ (R ;R∪{+∞}) with 0 ∈ D( ). Assume also
that F is continuous and satisfies the property
(∀x ∈ D( )) : F(x),x ≥ 0.

If
(∀v ∈ D( ) ∞ ,v
= 0) : r (v) + q,v + ∞ (v) > 0, (4.83)
F
then problem VI(F,q, ) has at least one solution.

Proof. Using Theorem 9, we see that for all i ∈ N,i
= 0, there exists u i ∈ R n
such that

1 n
u i + F(u i ) + q,v − u i + (v) − (u i ) ≥ 0, ∀v ∈ R . (4.84)
i
We claim that the sequence {u i }≡{u i ;i ∈ N\{0}} is bounded. Indeed, if
we suppose the contrary, then we may find a subsequence, again denoted by
n
{u i } i∈N ⊂ R , such that ||u i || → +∞ and satisfying (4.84). It is clear that
(∀i ∈ N) : u i ∈ D( ). Moreover, for i large enough, ||u i ||
= 0, and we may
set

u i
z i = .
||u i ||
There exists a subsequence, again denoted by {z i }, such that lim z i = z with
i→+∞
||z|| = 1.
Let λ> 0. For i large enough, λ < 1, and thus
||u i ||
λ
u i ∈ D( )
||u i ||