An important application of the global existence theorem just given is to the initial value problem

for a linear system, where the $m\times m$ matrix-valued function A(t) and the vector-valued function g(t) are continuous on a (bounded or unbounded) open interval I containing the point $t=a.$ In order to apply Theorem 1 to the linear system in (29), we note first that the proof of Theorem 1 requires only that, for each closed and bounded subinterval J of I, there exists a Lipschitz constant k such that

for all t in J (and all ${\mathbf{\text{x}}}_1$ and ${\mathbf{\text{x}}}_2$). Thus we do not need a single Lipschitz constant for the entire open interval I.

In (29) we have $\mathbf{\text{f}}(\mathbf{\text{x}},t)=\mathbf{\text{A}}(t)\mathbf{\text{x}}+\mathbf{\text{g}},$ so

It therefore suffices to show that, if A (t) is continuous on the closed and bounded interval J, then there is a constant k such that

for all t in J. But this follows from the fact (Problem 17) that

where the norm $\Vert \mathbf{\text{A}}\Vert$ of the matrix A is defined to be

Because A(t) is continuous on the closed and bounded interval J, the norm $\Vert \mathbf{\text{A}}\Vert$ is bounded on J, so Eq. (31) follows, as desired. Thus we have the following global existence theorem for the linear initial value problem in (29).

As we saw in Section 4.1, the mth-order initial value problem

is readily transformed into an equivalent $m\times m$ system of the form in (29). It therefore follows from Theorem 2 that if the functions $a_1(t),a_2(t),\dots ,a_m(t)$ and p(t) in (34) are all continuous on the (bounded or unbounded) open interval I containing $t=a,$ then the initial value problem in (34) has a solution on the (entire) interval I.