In the case of a nonlinear initial value problem
the hypothesis in Theorem 1 that f satisfies a Lipschitz condition on a slab (x, t) (t in I, all x) is unrealistic and rarely satisfied. This is illustrated by the following simple example.
Consider the initial value problem
As we saw in Example 6, the equation does not satisfy a “strip Lipschitz condition.” When we solve (36) by separation of variables, we get
Because the denominator vanishes for Eq. (37) provides a solution of the initial value problem in (36) only for despite the fact that the differential equation “looks nice” on the entire real line—certainly the function appearing on the right-hand side of the equation is continuous everywhere. In particular, if b is large, then we have a solution only on a very small interval to the right of .
Although Theorem 2 assures us that linear equations have global solutions, Example 7 shows that, in general, even a “nice” nonlinear differential equation can be expected to have a solution only on a small interval around the initial point and it also shows that the length of this interval of existence can depend on the initial value as well as on the differential equation itself. The reason is this: If f(x, t) is continuously differentiable in a neighborhood of the point (b, a) in -dimensional space, then—as indicated in the discussion preceding Example 6—we can conclude that f(x, t) satisfies a Lipschitz condition on some rectangular region R centered at (b, a), of the form
In the proof of Theorem 1, we need to apply the Lipschitz condition on the function f in analyzing the iterative formula
The potential difficulty is that, unless the values of t are suitably restricted, the point appearing in the integrand in (39) may not lie in the region R where f is known to satisfy a Lipschitz condition. On the other hand, it can be shown that—on a sufficiently small open interval J containing the point —the graphs of the functions given iteratively by the formula in (39) remain within the region R, so the proof of convergence can then be carried out as in the proof of Theorem 1. A proof of the following local existence theorem can be found in Chapter 6 of G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, 4th ed. (New York: John Wiley, 1989).
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