1. The Existence and Uniqueness of Solution of a
System of Ordinary Differential Equations
Define the norm of a vector x = (x1, ... , xn) in n-dimensional Euclidean space Rn by
||x|| = ∑i = 1, ..., n|xi|
(1)
Then the function d: Rn × Rn → R defined by
d(x,y) = ||x - y||
is a metric.
Let U be an open connected subset of Rn and f: Rn → Rn be continuous. We consider the differential equation
dx/dt = f(x)
(2)
Here dx/dt = (dx1/dt, ... , dxn/dt) ; equation (2) is in fact a system of n ordinary differential equations in n unknowns. We seek a solution of (2) subject to the initial condition
x = c for t = 0
(3)
which exists in an interval 0 ≤ t ≤ b for some b > 0 and for which x remains in U.
We make use of the following well known result from metric topology:
Contraction mapping principle. Let (X, d) be a complete metric space and let f: X → X satisfy the condition
d(f(x1), f(x2)) ≤ k d(x1, x2)
for all x1, x2 ∈ X, where 0 < k < 1. (Such an f is called a contraction mapping.)Then there exists a unique a ∈ X such that a = f(a).
[See http://qwerty-2009.angelfire.com/contraction-mapping.html and http://mathreference.com/top-ms,contract.html for further information.]
Reduction to an integral equation. The problem of finding a solution to (2) with the initial condition (3) is equivalent to finding a solution of the integral equation
x(t) = c + ∫[0,t]f(x(s)) ds
(4)
for 0 ≤ t ≤ b. The latter is more amenable to analysis.
A transformation on the space of continuous functions. The function f:U → Rn is said to be a Lipschitz function if there exists a constant m such that
||f(x) - f(y)|| ≤ m ||x - y||
(5)
for all x, y ∈ U.
Define a transformation T on the space S of continuous functions x: [0, a] → Rn such that x(t) ∈ N, where N is a compact neighbourhood of
T(x(t)) = c + ∫[0,t]f(x(s)) ds
(6)
[Note that S is a closed subspace of the complete metric space of all continuous functions from [0, a] to Rn. The space of all continuous functions from a compact Hausdorff space C to a Banach space V is a Banach space with norm ||f|| = supx ∈ C||f(x)||.]
Then
T(x(t)) - T(y(t)) = ∫[0,t][f(x(s)) - y(s)] ds
≤ m ∫[0,t]||x - y|| ds
= mt ||x - y||
≤ ma ||x - y||
(7)
If a is chosen so that ma < 1, then T is a contraction mapping. Thus we get
Theorem 1. Let x(0) be a continuous function from [0, a] to Rn satisfying ||x(0) - c|| ≤ a for 0 ≤ t ≤ b. For k = 0,1, 2, ... define
x(k+1) = T(x(k)) = c + ∫[0,t]f(x(k)(s)) ds
(8)
If f is a Lipschitz function, then the sequence {x(k)} converges to a solution of the initial value of the initial value of the initial value problem (2), (3). This solution is unique.
Nonautonomous systems. Consider the nonautonomous system
dx/dt = f(x, t)
(9)
This can be dealt with in the same way as the autonomous system considered in Theorem 1. The result is
Theorem 2. Let f(x, t) be continuous in x and t for ||x - c|| ≤ a and 0 < t ≤ b. Further, let f(x, t) satisfy the inequalities
||f(x, t|| ≤ K(t)
(10)
and
||f(x, t) - f(y, t)|| ≤ m(t) ||x - y||
(11)
for ||x - c|| ≤ a and 0 < t ≤ b where each of K(t) and m(t) is integrable on 0 ≤ t ≤ b. Let b1 ∈ (0, b] be chosen so that
∫[0,b1]K(t) dt ≤ a
(12)
and
∫[0,b1]m(t) dt = δ < 1
(13)
Then if x(0)(t) is continuous, with ||x(0) - c|| ≤ a for 0 < t ≤ b1 and
x(k+1)(t) = c + ∫[0,t]f(x(k)(s), s) ds
(14)
for k = 0, 1, 2, ... , the sequence{x(k)} converges on [0, b1] to a unique solution x of the integral equation
x(t) = c + ∫[0,t]f(x(s), s) ds
(15)
Indication of proof. If (T(x))(t) = c + ∫[0,t]f(x(s), s) ds then
||(T(x) - T(y)|| = ||∫[0,t][f(x(s), s) - f(y(s), s)] ds||
≤ ∫[0,t]m(s) ||x(s) - y(s)|| ds
≤ δ ||x - y||
or
||T(x) - T(y)|| ≤ δ ||x - y||
so the contraction mapping principle can be used.