Introductory Differential Equations using Sage - William Stein

1.3. EXISTENCE OF SOLUTIONS TO ODES 17
• the function f(x) = x 2 defined on [−3,7] is Lipschitz continuous, with Lipschitz
constant K = 14;
• the function f defined by f(x) = x 3/2 sin(1/x) (x ≠ 0) and f(0) = 0 restricted to
[0,1], gives an example of a function that is differentiable on a compact set while not
being Lipschitz.
Theorem 1.3.2. (“Picard’s existence and uniqueness theorem” [PL-intro]) Suppose f is
bounded, Lipschitz continuous in x, and continuous in t. Then, for some value ǫ > 0,
there exists a unique solution x = x(t) to the initial value problem (1.1) within the range
[a − ǫ,a + ǫ].
Charles Émile Picard (1856-1941) was a leading French mathematician. Picard made his
most important contributions in the fields of analysis, function theory, differential equations,
and analytic geometry. In 1885 Picard was appointed to the mathematics faculty at the
Sorbonne in Paris. Picard was awarded the Poncelet Prize in 1886, the Grand Prix des
Sciences Mathmatiques in 1888, the Grande Croix de la Légion d’Honneur in 1932, the
Mittag-Leffler Gold Medal in 1937, and was made President of the International Congress
of Mathematicians in 1920. He is the author of many books and his collected papers run to
four volumes.
The proofs of Peano’s theorem or Picard’s theorem go well beyond the scope of this course.
However, for the curious, a very brief indication of the main ideas will be given in the sketch
below. For details, see an advanced text on differential equations.
sketch or the idea of the proof: A simple proof of existence of the solution is obtained by
successive approximations. In this context, the method is known as Picard iteration.
Set x 0 (t) = c and
x i (t) = c +
∫ t
a
f(s,x i−1 (s))ds.
It turns out that Lipschitz continuity implies that the mapping T defined by
T(y)(t) = c +
∫ t
a
f(s,y(s))ds,
is a contraction mapping on a certain Banach space. It can then be shown, by using the
Banach fixed point theorem, that the sequence of “Picard iterates” x i is convergent and
that the limit is a solution to the problem. The proof of uniqueness uses a result called
Grönwall’s Lemma. □
Example 1.3.2. Consider the IVP
x ′ = 1 − x, x(0) = 1,
with the constant solution x(t) = 1. Computing the Picard iterates by hand is easy:
x 0 (t) = 1, x 1 (t) = 1+ ∫ t
0 1 −x 0(s))ds = 1, x 2 (t) = 1+ ∫ t
0 1 −x 1(s))ds = 1, and so on. Since
each x i (t) = 1, we find the solution