Existence Theory for Nonlinear Ordinary Differential EquationsWe begin our applications of fixed point methods with existence of solutions to certain first order initial initial value problems. This problem is relatively easy to treat, illustrates important methods, and in the end will carry us a good deal further than may first meet the eye. Thus, we seek solutions to Y'. = I(t, y) (1. 1 ) { yeO) = r n where I: I X R n + R and I = 0, b]. We shall seek solutions that are de fined either locally or globally on I,