Differential Equations Singular in the Independent Variable

The plan of this chapter is as follows. We begin with some standard notation in Section 1.2 and introduce L P —Carathéodory functions. Some fixed point theorems, the Arzela—Ascoli theorem and Banach’s theorem are also stated here. These results are used throughout this monograph. In Section 1.3 we discuss general existence theory for the initial value problems. First we state the Picard—Lindelöf, Peano’s and a local existence theorem in the Carathéodory setting, and then we prove some very general existence criteria. In Section 1.4 we provide general principles which can be used to establish existence to second order boundary value problems in the Carathéodory setting. Section 1.5 uses the existence principles of Section 1.4 to present a Bernstein—Nagumo theory for the general Sturm—Liouville problems. In Section 1.6 we discuss the method of upper and lower solutions to singular second order differential equations subject to Sturm—Liouville boundary data. In particular we show that our results are easily applicable to a problem occuring in circular membrane theory. Section 1.7 discusses solutions to singular boundary value problems in a weighted Banach space. In Section 1.8 we develop existence criteria for boundary value problems when the nonlinearity has no growth restriction. The ideas rely on the notion of upper and lower surfaces for the problem, i.e., surfaces having a particular form and on which the nonlinearity has a given sign. In Section 1.9 we discuss in detail nonresonant problems involving second order differential equations together with Sturm—Liouville, Neumann and periodic boundary date. Nonresonant problems of limit circle type and Dirichlet type are considered in Sections 1.10 and 1.11 respectively. In Section 1.12 we examine systematically resonant problems involving second order differential equations together with Dirichlet, mixed, Neumann and periodic boundary data. In Sections 1.13 and 1.14 we establish general existence theory for boundary value problems over infinite intervals. In particular our theory includes a discussion of problems arising (i). in the study of plasma physics, (ii). in determining the electrical potential in an isolated neutral atom, (iii). in modelling phenomena which arises in the theory of shallow membrane caps, (iv). in the theory of colloids, and (v). in the flow and heat transfer over a stretching sheet.