Differential Equations Singular in the Dependent Variable

The plan of this chapter is as follows. In Section 2.2 we present general existence principles for first order singular initial value problems. Here our nonlinearity may change sign. Second order singular initial value problems are studied in Section 2.3. In Section 2.4 we provide existence criteria for second order singular positone boundary value problems. In Section 2.5 we prove a very general existence result for second order semipositone problems. Using the method of upper and lower solutions in Section 2.6 we present as existence result for second order singular problems where the nonlinearity may change sign. A very general existence theory for second order singular positone problems and problems with sign changing nonlinearities is developed in Section 2.7. In Sections 2.8 and 2.9 we use fixed point theory in cones to establish, respectively, an existence criterion for singular semipositone type problems, and multiplicity results for positone problems. In Section 2.10 we discuss in detail second order singular boundary value problems where our nonlinearity involves y′ and may change sign. An existence theory for second order singular problems involving nonlinear boundary data is presented in Section 2.11. In particular we show that our theory is easily applicable to a problem occuring in the membrane response of a spherical cap. In Section 2.12 we study second order singular boundary value problems with sign changing nonlinearities involving mixed boundary data. Section 2.13 provides existence theory for second order singular boundary value problems involving differential equations with a nonlinear left hand side. In Sections 2.14 and 2.15 we provide general existence theory for singular boundary value problems over infinite intervals. In particular our theory includes a discussion of problems arising (i). in the unsteady flow of a gas through a semi—infinite porous medium, and (ii). in the theory of draining flows.