My title - Departamento de Matemática da Universidade do Minho

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Dependence on initial data and parameters
Consider a family of ODEs
ẋ = v(t, x, λ)
where λ is a real parameter. We want to understand how solutions depend on the parameter λ. A
basic instrument is the 15
Let φ(t) and ψ(t) be two non-negative real valued functions defined in in-
Grönwall’s lemma.
terval [a, b] such that
φ(t) ≤ K +
∫ t
for any a ≤ t ≤ b and some constant K ≥ 0. Then
a
ψ(s)φ(s)ds
φ(t) ≤ Ke R t
a ψ(s)ds .
Proof.
First, assume K > 0. Define
Φ(t) = K +
∫ t
a
ψ(s)φ(s)ds
and observe that Φ(a) = K > 0, hence Φ(t) > 0 for all a ≤ t ≤ b. The logarithmic derivative is
d
ψ(t)φ(t)
log Φ(t) = ≤ ψ(t)
dt Φ(t)
where we used the hypothesis φ(t) ≤ Φ(t). Integrating the inequality we get, for a ≤ t ≤ b,
Exponentiation gives the result, since
log Φ(t) ≤ Φ(a) +
∫ t
a
ψ(s)ds .
φ(t) ≤ Φ(t) ≤ K · e R t
a ψ(s)ds
The case K = 0 follows taking the limit of the above inequalities for a sequence of K n > 0
decreasing to zero. q.e.d.
Continuous dependence on initial conditions.
If x(t) and y(t) are two solutions of
then
ẋ = v(t, x)
∫ t
x(t) − y(t) = x(0) − y(0) + (v(s, x(s)) − v(s, y(s))) ds
t 0
If L(s) denotes the Lipschitz constant of v(s, ·), we get
The Gronwall’s lemma gives the control
∫ t
‖x(t) − y(t)‖ ≤ ‖x(0) − y(0)‖ + L(s)‖x(s) − y(s)‖ds
t 0
R t
t
‖x(t) − y(t)‖ ≤ e
L(s)ds 0 ‖x(0) − y(0)‖
Observe that the above control also gives an alternative proof of uniqueness of solutions given
a Lipschitz condition on the vector field.
15 T. H. Gronwall, Note on the derivative with respect to a parameter of the solutions of a system of differential
equations, Ann. of Math 20 (1919), 292-296.