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.

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 $t=a,$ and it also shows that the length of this interval of existence can depend on the initial value $\mathbf{\text{x}}(a)=\mathbf{\text{b}},$ 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 $(m+1)$-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

$(i=1,2,\dots ,m).$ 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 $({\mathbf{\text{x}}}_n(t),t)$ 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 $t=a$—the graphs of the functions $\{{\mathbf{\text{x}}}_n(t)\}$ 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).