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By G Teschl

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X(t) .. ..... . .. II .... ... Similarly, solutions can only get from III to II but not from II to III. This already has important consequences for the solutions: • For solutions starting in region I there are two cases; either the solution stays in I for all time and hence must converge to +∞ (maybe in finite time) or it enters region II. ). Since it must stay above x = −t this cannot happen in finite time. • A solution starting in III will eventually hit x = −t and enter region II.

We expect that xh (t) converges to a solution as h ↓ 0. But how should we prove this? Well, the key observation is that, since f is continuous, it is bounded by a constant on each compact interval. Hence the derivative of xh (t) is bounded by the same constant. Since this constant is independent of h, the functions xh (t) form an equicontinuous family of functions which converges uniformly after maybe passing to a subsequence by the Arzel`aAscoli theorem. 17 (Arzel` a-Ascoli). Suppose the sequence of functions fn (x), n ∈ N, on a compact interval is (uniformly) equicontinuous, that is, for every ε > 0 there is a δ > 0 (independent of n) such that |fn (x) − fn (y)| ≤ ε if |x − y| < δ.

11) can be defined. , f is Lipschitz). 11) defined on the open intervals I1 , I2 , respectively. Let I = I1 ∩ I2 = (T− , T+ ) and let (t− , t+ ) be the maximal open interval on which both solutions coincide. I claim that (t− , t+ ) = (T− , T+ ). In fact, if t+ < T+ , both solutions would also coincide at t+ by continuity. 2. This contradicts maximality of t+ and hence t+ = T+ . Similarly, t− = T− . 63) defined on I1 ∪ I2 . In fact, this even extends to an arbitrary number of solutions and in this way we get a (unique) solution defined on some maximal interval.

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Differential Equation - Ordinary Differential Equations by G Teschl


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