A’priori Bounds and Existence Theorems

Abstract

Before discussing in detail the various classes of singularly perturbed boundary value problems, let us give an outline of the principal method of proof that we will use throughout. This method employs the theory of differential inequalities which was developed by M. Nagumo [66] and later refined by Jackson [49]. It enables one to prove the existence of a solution, and at the same time, to estimate this solution in terms of the solutions of appropriate inequalities. Such an approach has been found to be very useful in a number of different applications (see, for example, [5] and [83]). It will be seen that for the general classes of problems which we will study in later chapters, this inequality technique leads elegantly (and easily) to some fairly general results about existence of solutions and their asymptotic behavior. Many results which have been obtained over the years by a variety of methods can now be obtained by this method, which we hope will also very clearly reveal the fundamental asymptotic processes at work.