Page 60 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 51

and taking now the limit as λ →+∞, we get

∞ (x) ≤ ∞ (x). (4.4)

The result follows from (4.3) and (4.4).
Remark 14. We have
epi( ∞ ) = (epi( )) ∞ .

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Indeed, let x 0 ∈ R be such that α 0 = (x 0 )< +∞.Wehave
(x,α) ∈ epi( ∞ )
⇔
∞ (x) ≤ α
⇔
(λx + x 0 ) − (x 0 )
(∀λ> 0) : ≤ α
λ
⇔
(∀λ> 0) : (λx + x 0 ) ≤ α 0 + λα
⇔
(∀λ> 0) : λ(x,α) + (x 0 ,α 0 ) ∈ epi( )
⇔
(x,α) ∈ (epi( )) ∞ .

In the following result, we show that ∞ turns out to be proper, convex,
lower semicontinuous, and positively homogeneous of order 1.
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Proposition 8. If ∈ 0 (R ;R ∪{+∞}), then ∞ ∈ 0 (R ;R ∪{+∞}), and
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(∀ α ≥ 0), (∀x ∈ R ) : ∞ (αx) = α ∞ (x).
Proof. Let x 0 ∈ D( ).Wehave

1
∞ (0) = lim (x 0 ) = 0,
λ+∞ λ
and thus ∞ is proper. Let us now prove the convexity of ∞ .Let λ ∈[0,1]
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and x 1 ,x 2 ∈ R .Wehave
∞ (λx 1 + (1 − λ)x 2 )
1
= lim [ (λx 0 + (1 − λ)x 0 + λtx 1 + (1 − λ)tx 2 )
t→+∞ t
− λ (x 0 ) − (1 − λ) (x 0 )]