One can use the contraction mapping theorem to simplify the last page of the proof of theorem 7.1.1. That is, supposing that T is chosen as at the top of p. 278, recall that I means the interval (to - T, to +T), and recall that C(IT) means the space of continuous functions In → R. Consider the function 12:C(IT) + C(IT), defined by
> N(u)(t) = tJt0 f(s, u(s))ds
(The notation on the left is weird at first. As an element of C(IT), u is a function not a number, and the same for (u). To describe the function (u), you have to say what values it takes where. The left side means the value of the function (u) at the number 1.)
(a) Mimic the arguments leading to equation (6) in the book, to show: 1 is a contraction
(b) Use the contraction mapping theorem to show there exists a unique function a Y e C(IT) satisfying equation (2) in the book. In particular, the CMT gives existence and uniqueness at the same time, which is cleaner than the argument
