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Date Submitted: 08/08/2013 06:44 PM
We have existence and uniqueness of the solution when f(t,x) is continuous and Lips- chitz with respect to its second variable, i.e. there exists a constant C independent of t, x, y such that
|f(t, x) − f(t, y)| ≤ C|x − y|. (1.2) If the constant is independent of time for t > 0, we can even take T = +∞. Here X
andf(t,X(t))arevectorsinRn forn∈Nand|·|isanynorminRn.
Remark 1.1 Throughout the text, we use the symbol “C” to denote an arbitrary constant. The “value” of C may change at every instance. For example, if u(t) is a function of t bounded by 2, we will write
even if the two constants “C” on the left- and right-hand sides of the inequality may be different. When the value “2” is important, we will replace the above right-hand side by 2C. In our convention however, 2C is just another constant “C”.
Higher-order differential equations can always been put in the form (1.1), which is quite general. For instance the famous harmonic oscillator is the solution to
x′′ + ω2x = 0, ω2 = k , m
(1.4)
(1.5)
Exercise 1.1 Show that the implied function f in (1.5) is Lipschitz. The Lipschitz condition is quite important to obtain uniqueness of the solution. Take
for instance the case n = 1 and the function x(t) = t2. We easily obtain that x′(t) = 2x(t) t ∈ (0, +∞), x(0) = 0.
However, the solution x ̃(t) ≡ 0 satisfies the same equation, which implies non-uniqueness of the solution. This remark is important in practice: when an equation admits several solutions, any sound numerical discretization is likely to pick one of them, but not nec- essarily the solution one is interested in.