Metric and Normed Spaces

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.