C2 FUNDAMENTAL THEORY OF DYNAMICAL SYSTEMS
HANDOUT 3
EXISTENCE OF SOLUTIONS OF DIFFERENTIAL
EQUATIONS
In this handout, we sketch a rigorous proof of the existence of theorem for ordinary differential
equations. It is based on the Contraction Mapping Theorem, a statement of which will be found in
Handout 2.
The material in this handout is not examinable.
Consider the differential equation:
dx(t )
dt
=
F(x(t))
(1)
with initial condition x = x 0 at t = 0. Our aim is to write the solution to (1) as the fixed point of an
appropriate on the collection of all local curves through x 0. We then show that the operator is
contracting when restricted to suitably short curves, and apply the contraction mapping theorem
to obtain the existence of unique local solutions to (1). Recall that we say a function is C r if it is rtimes differentiable and the rth derivative continuous.
STATEMENT OF THE THEOREM
Assume that F is C 1 vector field on R n, and that F(x) ≠ 0. Then there exists an interval (-ε, ε) and a
unique C2 curve ξ*: (-ε, ε) → Rn which solves (1).
In fact the solution also depends smoothly on the initial condition x0. Furthermore if F is Cr then the
solution is also Cr in x0 and Cr+1 in t. To show this requires a refinement of the estimates carried out
below, but adds essentially no new concepts and hence is not carried out here. Details can be found
in most standard textbooks on differential equations.
PREPARATION FOR THE PROOF : CONDITIONS ON F
Since F is C1 there exists a constant K < ∞ and a neighbourhood B(x0,η) = { x : x - x0 ≤ η } on which
DF is bounded by K. This implies that
F(x+h) - F(x)
≤
K h
for all x,y∈ B(x0,η). Let
ε
=
 1
η
min 
,
2
K
2
F
(x 0 )

=
{ ξ : [-ε, ε] → B(x0,η) such that ξ(0) = x0 }



and define
X
Existence of Solutions
2
be the collection of continuous curves passing through x0. We define the distance between two such
curves by
ξ - ξ'
max ξ(s) - ξ'(s)
=
s∈[ − ε ,ε ]
and claim that with this norm X is a complete metric space. A proof of this can be found in any
textbook of real analysis. We are going to search for solutions to (1) in the set X.
THE CONTRACTIVE OPERATOR
Let P : X → X be defined according to the formula
t
P(ξ)(t)
=
x0 +
∫ F(ξ(s)) ds
(2)
0
for all t∈(-ε, ε). Differentiating (2) we get
d
P(ξ)(t)
dt
F(ξ(t))
=
(3)
Hence if ξ is continuous, P(ξ) is C1. Furthermore, ξ∈X is a solution of (1) if and only if P(ξ) = ξ . In
such a case ξ is C 1 and hence using (3) again it must be C 2. Indeed, if F is C r and P(ξ) = ξ then by
induction we see that ξ must in fact be Cr+1.
Since a solution of ( 1) is given by fixed points of P, we concentrate on proving the existence of a such
a fixed point ξ*, using the Contraction Mapping Theorem. We first show that P maps X into X, that is
that if ξ∈X then P(ξ)∈X. To do this, we check that that P(ξ)(t)∈B(x0,η) for all t∈(-ε, ε):
P (ξ)(t) - x0
t
∫ F(ξ(s)) ds
=
0
+ max F(ξ(s)) - F(x0) )
≤
t ( F(x0)
≤
ε ( F(x0) + K ξ(s) - x0 )
≤
η
2
≤
η
s ≤t
+ Kεη
by the choice of ε and the fact that ξ(s)∈B(x0, η ). Now it remains to show that P is a contraction
mapping, this follows from the following estimate:
P (ξ) - P (ξ')
=
=
max
P (ξ)(t) - P (ξ')(t)
max
x0 +
t ∈[ − ε ,ε ]
t ∈[ − ε ,ε ]
∫
t
0
F(ξ(s)) ds - x0 +
t
∫ F(ξ'(s)) ds
0
t
=
≤
1
max
t ∈[ − ε ,ε ]
∫ [F(ξ(s)) - F(ξ'(s))] ds
0
max s F(ξ(s)) - F(ξ'(s))
s∈[ − ε ,ε ]
≤
Kε max ξ(s) - ξ'(s)
≤
Kε ξ - ξ'
s∈[ − ε ,ε ]
1
Because K ε ≤ 2 , this guarantees that P is a contraction mapping, with contraction rate 2. It now
follows from the Contraction Mapping Theorem (Handout 2) that there exists a unique ξ *∈X such
that P(ξ*) = ξ*. This is the required solution of (1).
Existence of Solutions
3
COMMENTS
The underlying simplicity of this proof may be obscured by the fact that it is necessary to check that
the short curves always lie in B(x0, η ). The complexity in attending to these details can be
understood in the light of the example of non-uniqueness which I presented in lectures. For F(x) =
x1/2, the argument does not work with an initial condition x0 = 0 because there is no suitable K, so no
domain in which P is a contraction mapping. You may like to check however that 0 is a fixed point of
P in this case!