We 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, according to the assumptions imposed on I. Notice that (1. 1) is a system of first order equations because I takes its values in Rn. In section 3. 2 we will first establish some basic existence theorems which guarantee that a solution to (1. 1) exists for t > 0 and near zero. Familiar examples show that the interval of existence can be arbi­ trarily short, depending on the initial value r and the nonlinear behaviour of I. As a result we will also examine in section 3. 2 the dependence of the interval of existence on I and r. We mention in passing that, in the results which follow, the interval I can be replaced by any bounded interval and the initial value can be specified at any point in I. The reasoning needed to cover this slightly more general situation requires minor modifications on the arguments given here.

1. Introduction, notation and preliminaries.- 2. Fixed point theory.- 3. Initial value problems.- 4. First order periodic problems.- 5. Existence principles for second order boundary value problems.- 6. Boundary value problems without growth restrictions.- 7. Positone boundary value problems.- 8. Semi-positone boundary value problems.- 9. Differential equations singular in the solution variable.- 10. Existence principle for singular boundary problems.- 11. Nonresonance problems in the limit circle case.- 12. Resonance problems in the limit circle case.- 13. Boundary value problems on the half line.- 14. Existence theory for ordinary differential equations on compact and noncompact intervals.- 15. Impulsive differential equations.- 16. Differential equations in abstract spaces.