8 FREDERICK A. HOWES

This theorem is proved in [15] using a lemma on systems of first order equations.

This lemma is itself a useful tool in the study of singularly perturbed systems of

equations.

The final theorem we need to state is a uniqueness theorem for a general two-point

boundary value problem. A more general version, together with a proof, can be found in

the book of Protter and Weinberger [22;Chapter1].

Theorem 1.3. Let x = x(t) be a solution of the boundary value problem

x" = F(t,x,x') , 0 t 1 ,

x(0) = A , x(l) = B .

Assume that F,F and F , are continuous and that F 0. Then the solution x is unique.