§ 4. Differential Manifolds
255
y 2 (t) − y 1 (t) ≤ M
y 1 (u)du ≤ MM
t
2 /2! ,
y 3 (t) − y 2 (t) ≤ M
y 2 (u) − y 1 (u)du ≤ MM
2 t
3 /3!
and so on until
y n+1 (t) − y n (t) ≤ MM
n−1 t
n /n! for |t| ≤ a
.
(15.12)
The series
[y n+1 (t) − y n (t)], dominated by an exponential series, thus converges normally in |t| ≤ a
to a solution y(t) = y(t, z) of (7) defined for |t| ≤ a
and |z| ≤ c and with values in |y| ≤ b, proving statement (a). Note that this
result only assumes L to be continuous and to have a continuous derivative
D 2 L(t, x, z).
A very particular and important case is that of a linear differential equation, i.e. for which L is of the form
L(t, y, z) = A(t, z)y + b(t, z)
with functions A(t, z) and b(t, z) defined and continuous for |t| ≤ a, |z| ≤ c. As
L(t, y, z) is defined without any restrictions on y, successive approximations
y n (t, z) are defined and continuous; hence there are no other restrictions
on the domain of existence of solutions apart from those imposed on the
coefficients A(t, z) and b(t, z).
(iii) Uniqueness of the solution. A similar calculation to (9) and (10)
shows that if y
and y
(these are not derivatives) are solutions of (7), then
setting
k(r) = sup
|t|≤r
y
(t) − y
(t) ,
gives
y
(t) − y
(t) ≤
L[u, y
(u), z) − L[u, y
(u), z]du
≤ M
k(r)|t|
for |t| ≤ r as long as y
(u) and y
(u) remain in the ball y ≤ b, which is
the case for sufficiently small r since (7) implies y(0) = 0. Taking the sup
of the left hand side for |t| ≤ r, k(r) ≤ M
k(r)r follows, and so k(r) = 0 if
r ≤ 1/M
. This proves statement (b).
(iv) Dependence on initial conditions. It is now a matter of showing that,
for t and z in the neighbourhood of 0, like L, the solution y(t, z) of (7) is of
class C
k as a function of (t, z). Suppose first that k = 1 ; it all amounts to
showing that the functions (8) are C
1 and that their first derivatives converge
uniformly (Chap. III, n
◦ 22, theorem 23).
255
y 2 (t) − y 1 (t) ≤ M
y 1 (u)du ≤ MM
t
2 /2! ,
y 3 (t) − y 2 (t) ≤ M
y 2 (u) − y 1 (u)du ≤ MM
2 t
3 /3!
and so on until
y n+1 (t) − y n (t) ≤ MM
n−1 t
n /n! for |t| ≤ a
.
(15.12)
The series
[y n+1 (t) − y n (t)], dominated by an exponential series, thus converges normally in |t| ≤ a
to a solution y(t) = y(t, z) of (7) defined for |t| ≤ a
and |z| ≤ c and with values in |y| ≤ b, proving statement (a). Note that this
result only assumes L to be continuous and to have a continuous derivative
D 2 L(t, x, z).
A very particular and important case is that of a linear differential equation, i.e. for which L is of the form
L(t, y, z) = A(t, z)y + b(t, z)
with functions A(t, z) and b(t, z) defined and continuous for |t| ≤ a, |z| ≤ c. As
L(t, y, z) is defined without any restrictions on y, successive approximations
y n (t, z) are defined and continuous; hence there are no other restrictions
on the domain of existence of solutions apart from those imposed on the
coefficients A(t, z) and b(t, z).
(iii) Uniqueness of the solution. A similar calculation to (9) and (10)
shows that if y
and y
(these are not derivatives) are solutions of (7), then
setting
k(r) = sup
|t|≤r
y
(t) − y
(t) ,
gives
y
(t) − y
(t) ≤
L[u, y
(u), z) − L[u, y
(u), z]du
≤ M
k(r)|t|
for |t| ≤ r as long as y
(u) and y
(u) remain in the ball y ≤ b, which is
the case for sufficiently small r since (7) implies y(0) = 0. Taking the sup
of the left hand side for |t| ≤ r, k(r) ≤ M
k(r)r follows, and so k(r) = 0 if
r ≤ 1/M
. This proves statement (b).
(iv) Dependence on initial conditions. It is now a matter of showing that,
for t and z in the neighbourhood of 0, like L, the solution y(t, z) of (7) is of
class C
k as a function of (t, z). Suppose first that k = 1 ; it all amounts to
showing that the functions (8) are C
1 and that their first derivatives converge
uniformly (Chap. III, n
◦ 22, theorem 23).