Page 185 - Complementarity and Variational Inequalities in Electronics

176 Complementarity and Variational Inequalities in Electronics

Corollary 14. Suppose that the assumptions of Theorem 11 hold together with
conditions (5.36), (5.42), and (5.43). Assume also that D(∂ϕ) is closed. Suppose
1 n
that there exist σ> 0 and V ∈ C (R ;R) such that the function ϕ(·) − ϕ(·−
∇V(·)) is lower semicontinuous on D(∂ϕ) ∩ B σ , V(0) = 0,

(∀x ∈ D(∂ϕ) ∩ B σ ) : V(x) ≥ a(||x||)

with a :[0,σ]→ R satisfying a(t) > 0, ∀t ∈ (0,σ),
(∀x ∈ D(∂ϕ) ∩ B σ ) :
F(x),∇V(x) + ϕ(x) − ϕ(x −@ V(x)) ≥ 0,

and
E(F,ϕ,V ) ∩ B σ ={0}.

Then the trivial solution of (5.37)–(5.40) is (a) isolated in S(F,ϕ) and
(b) asymptotically stable.

Proof. Assertion (a) is a direct consequence of Proposition 25. The stability
follows from Theorem 14. The stability ensures the existence of δ> 0 such that
if x 0 ∈ D(∂ϕ) ∩ B δ , then

γ(x 0 ) ⊂ B σ .

Applying Theorem 18 with = B σ , we obtain for x 0 ∈ D(∂ϕ) ∩ B δ that

lim d(x(t;t 0 ,x 0 ),M) = 0,
t→+∞
where M is the largest invariant subset of E (F,ϕ,V ). It is clear that M =
{0} since by assumption, E (F,ϕ,V ) ={0}. The attractivity and assertion (b)
follow.

Corollary 15. Suppose that the assumptions of Theorem 11 hold together with
conditions (5.36), (5.42), and (5.43). Assume that D(∂ϕ) is closed and suppose
that there exists σ> 0 such that

(∀x ∈ D(∂ϕ) ∩ B σ ,x = 0) :
F(x),x + ϕ(x) − ϕ(0)> 0.

Then the trivial stationary solution of (5.37)–(5.40) is (a) isolated in S(F,ϕ)
and (b) asymptotically stable.
1
n
Proof. This follows from Corollary 14 applied with V ∈ C (R ;R) defined by
1
n
2
(∀x ∈ R ) : V(x) = ||x|| .
2