Page 162 - Complementarity and Variational Inequalities in Electronics

The Nonregular Dynamical System Chapter | 5 153

that for some ω ≥ 0, F + ωI is monotone, that is,

2
n
(∀x,y ∈ R ) :
F(x) − F(y),x − y ≥−ω||x − y|| .
n
0
Let t 0 ∈ R and x 0 ∈ D(∂ϕ). Then there exists a unique x ∈ C ([t 0 ,+∞);R )
satisfying the system (t 0 ,x 0 ,F,f,ϕ):
dx n
∞
∈ L (t 0 ,+∞;R ), (5.20)
loc
dt
x is right-differentiable on [t 0 ,+∞), (5.21)
x(t 0 ) = x 0 , (5.22)
x(t) ∈ D(∂ϕ), t ≥ t 0 , (5.23)
dx

(t) + F(x(t)) + f(t),v − x(t) +
dt
n
+ ϕ(v) − ϕ(x(t)) ≥ 0, ∀v ∈ R , a.e. t ≥ t 0 . (5.24)
n
n
Remark 31. Note that if F : R → R is Lipschitz continuous with Lipschitz
constant k> 0, then F is continuous, and F + kI is monotone.
n
n
Remark 32. Suppose that F : R → R can be written as
n
(∀x ∈ R ) : F(x) = Ax +∇ (x) + F 1 (x),

1
n
where A ∈ R n×n is a real matrix, ∈ C (R ;R) is convex, and F 1 is Lipschitz
continuous, that is,
n
(∀x ∈ R ) :||F 1 (x) − F 1 (y)|| ≤ k||x − y||

for some constant k> 0. Then F is continuous, and F + ωI is monotone, pro-
vided that ω ≥ 0 is chosen great enough, that is,

ω ≥ sup ||Ax|| + k.
||x||=1

3
For instance, F(x) =−2x + 4x + cos(x) satisfies the requirements with A =
4
−2, (x) = x , and F 1 (·) = cos(·).
Let us denote by x(.;t 0 ,x 0 ) the unique solution of problem (t 0 ,x 0 ,F,f,ϕ).
We prove further that for fixed t ≥ t 0 , the function x 0 → x(t;t 0 ,x 0 ) is uniformly
continuous on D(∂ϕ). Let us first recall some Gronwall inequality that is used
in our next result (see e.g. Lemma 4.1 in [84]).