BVPs for Nonlinear Second-Order DEs

Abstract

Our discussion of differential equations up to this point has focused on solutions of initial value problems. We have made good use of the fact that these problems have unique solutions under mild conditions. We have also encountered boundary value problems in a couple of chapters. These problems require the solution or its derivative to have prescribed values at two or more points. Such problems arise in a natural way when differential equations are used to model physical problems. An interesting complication arises in that BVPs may have many solutions or no solution at all. Consequently, we will present some results on the existence and uniqueness of solutions of BVPs. A useful device will be the contraction mapping theorem (Theorem 7.5), which is introduced by the definitions below. One of the goals of this chapter is to show how the contraction mapping theorem is commonly used in differential equations.