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18.950 Handout 3. Existence and Uniqueness of Solutions to O.D.E.’s Let l ∈ (0, ∞], R ∈ (0, ∞), y0 ∈ Rn and t0 ∈ R. Suppose
F : (t0 − l, t0 + l) × BR(y0) → Rn
is a continuous function. Consider the first order O.D.E. initial value prob
lem:
du = F (t, u) and u(t0) = y0. (�) dt
In general, we cannot expect to find a solution to the system (�) that exists for all t. e.g. for n = 1, if we take F (t, u) = 1 + u , t2 0 = 0 and y0 = 0, then the solution (which can be found by separation of variables) is u(t) = tan t which blows up as t → ±π/2. The theorem below gives suffi
cient conditions under which a unique solution to (�) exists locally in some neighborhood of t0. Before stating the theorem, we need the following:
Definition. A map f : Ω ⊆ Rm Rn is said to be Lipschitz if there exists a number L such that �
→
x − y� for all x, y ∈ Ω. The infi
f (x) − f (y)� ≤ L�
mum of the set of such L is called the Lipschitz constant of f , denoted Lip f.
Theorem 1. Suppose F = F (t, y) as above is continuous. Suppose also that F is Lipschitz in the y variable uniformly for all t ∈ (t0 − l, t0 + l).
i.e. F (t, y1) − F (t, y2)� ≤ L�y1 − y2� for some L ∈ (0, ∞) and for all y1, y2 ∈ BR(y0) and all t ∈ (t0 − l, t0 + l). Then there exists c > 0 such that the initial value problem (�) has a unique solution u(t) for t ∈ (t0 − c, t0 + c).
Proof. First note that by the fundametal theorem of calculus, u is a solution to (�) in an interval (t0 − c, t0 + c) if and only if
t
u(t) = y0 + F (s, u(s))ds (1)
t0
for t ∈ (t0−c, t0+c). Let M = maxI×B F (t, y) where I = [t0−l1, t0+l1],
R(y0) | |
R 1
l1 = min {l/2, 1}. Set c = min {l1, M ,2L}. For u ∈ C((t0 − c, t0 + c), BR(y0)) (equipped with the sup metric, see handout 2), define a function φ(u) by
t
φ(u)(t) = y0 + F (s, u(s))ds. (2)
t0
1
�
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Note that φ(u) is continuous (in fact differentiable), and for all t ∈ (t0 − c, t0 + c),
φ(u)(t) − y0 t − t0 ≤ M c ≤ R. (3)
| | ≤ M | |
Thus φ(u) ∈ C ((t0 − c, t0 + c), BR(y0)). Moreover, φ is a contraction because
t
φ(u)(t) − φ(v)(t) F (t, u(t)) − F (t, v(t))dt
| | ≤
t0
t
L u(t) − v(t) dt | |
=
t0
≤ L|t − t0|�u − v�sup ≤ 1
2�u − v�sup (4) for all t ∈ (t0 − c, t0 + c) and hence
�φ(u) − φ(v)�sup ≤ 1
2�u − v�sup.
Thus by the contraction mapping theorem, φ has a unique fixed point, which, by (1), is the desired unique solution to (�) in the interval (t0− c, t0+ c).
Corollary 1. Suppose F (t, y, x1, . . . , xm−1), where t, y, xi ∈ R, is a real val
ued function defined in some neighborhood of the point (t0, y0, y1, . . . , ym−1) ∈ Rm+1 . If F continuous in all variables and Lipschitz in the variables y, x1, . . . , xm−1, then the mthorder O.D.E. initial value problem
m−1u
d u m du d
= F t, u, , . . . , m−1 ,
dtm dt dt
u(t0) = y0, du (t0) = y1,
dt
. . . dm−1u
m−1 (t0) = ym−1
dt
has a unique solution near (t0, y0, y1, . . . , ym−1).
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