Page 172 - Complementarity and Variational Inequalities in Electronics

The Nonregular Dynamical System Chapter | 5 163

−1
Setting z n = x n ||x n || , we obtain
1
(∀n ∈ N) :
Mz n ,Gz n < .
n
such that
We have (∀n ∈: N) :||z n || = 1, and thus there exists a subsequence z n k
→ z with ||z|| = 1. Taking the limit as k →∞, we get
Mz,Gz ≤ 0 and a
z n k
contradiction since z = 0.
We end this section by remarking that some of the hypotheses stated in The-
orem 14 can also be used to obtain some additional information on the set of
stationary solutions of (5.23)–(5.24).

Proposition 24. Suppose that the assumptions of Theorem 11 hold together
n
with conditions (5.36), (5.42), and (5.43). Let be a subset of R . Suppose that
n
1
there exists V ∈ C (R ;R) such that
(∀x ∈ D(∂ϕ) ∩ ) :
F(x),∇V(x) + ϕ(x) − ϕ(x −@ V(x)) ≥ 0.

Then
S(F,ϕ) ∩ ⊂ E(F,ϕ,V ).

Proof. Let z ∈ ∩ S(F,ϕ).Wehave z ∈ D(∂ϕ) ∩ and

n

F(z),v − z + ϕ(v) − ϕ(z) ≥ 0,∀v ∈ R . (5.47)
Setting v = z −@ V(z) in (5.47), we get

F(z),∇V(z) + ϕ(z) − ϕ(z −@ V(z)) ≤ 0.

Then, we obtain

F(z),∇V(z) + ϕ(z) − ϕ(z −@ V(z)) = 0.
Proposition 25. Suppose that the assumptions of Theorem 11 hold together
with conditions (5.36), (5.42) and (5.43). Suppose that there exist σ> 0 and
1
n
V ∈ C (R ;R) such that
(∀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 stationary solution of (5.37)–(5.40) is isolated in S(F,ϕ).