Linear Algebra, Theory And Applications, 2012a

Applications To Differential
Equations
C.1 Theory Of Ordinary Differential Equations
Here I will present fundamental existence and uniqueness theorems for initial value problems
for the differential equation,
x ′ = f (t, x) .
Suppose that f :[a, b] × R n → R n satisfies the following two conditions.
|f (t, x) − f (t, x 1 )|≤K |x − x 1 | , (3.1)
f is continuous. (3.2)
The first of these conditions is known as a Lipschitz condition.
Lemma C.1.1 Suppose x :[a, b] → R n is a continuous function and c ∈ [a, b]. Thenx is a
solution to the initial value problem,
if and only if x is a solution to the integral equation,
x ′ = f (t, x) , x (c) =x 0 (3.3)
x (t) =x 0 +
∫ t
c
f (s, x (s)) ds. (3.4)
Proof: If x solves (3.4), then since f is continuous, we may apply the fundamental
theorem of calculus to differentiate both sides and obtain x ′ (t) =f (t, x (t)) . Also, letting
t = c on both sides, gives x (c) =x 0 . Conversely, if x is a solution of the initial value
problem, we may integrate both sides from c to t to see that x solves (3.4).
Theorem C.1.2 Let f satisfy (3.1) and (3.2). Then there exists a unique solution to the
initial value problem, (3.3) on the interval [a, b].
Proof: Let ||x|| λ
≡ sup { e λt |x (t)| : t ∈ [a, b] } . Then this norm is equivalent to the usual
norm on BC ([a, b] , F n ) described in Example 14.6.2. This means that for ||·|| the norm given
there, there exist constants δ and Δ such that
||x|| λ
δ ≤||x|| ≤ Δ ||x||
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