In the last century many problems which arose in the science, engineer-
ing and technology literature involved nonlinear complex phenomena. In
many situations these natural phenomena give rise to (i). ordinary
differ- ential equations which are singular in the independent and/or
dependent variables together with initial and boundary conditions, and
(ii). Volterra and Fredholm type integral equations. As one might expect
general exis- tence results were difficult to establish for the problems
which arose. Indeed until the early 1990's only very special examples
were examined and these examples were usually tackled using some special
device, which was usually only applicable to the particular problem
under investigation. However in the 1990's new results in inequality and
fixed point theory were used to present a very general existence theory
for singular problems. This mono- graph presents an up to date account
of the literature on singular problems. One of our aims also is to
present recent theory on singular differential and integral equations to
a new and wider audience. The book presents a compact, thorough, and
self-contained account for singular problems. An important feature of
this book is that we illustrate how easily the theory can be applied to
discuss many real world examples of current interest. In Chapter 1 we
study differential equations which are singular in the independent
variable. We begin with some standard notation in Section 1. 2 and
introduce LP-Caratheodory functions. Some fixed point theorems, the
Arzela- Ascoli theorem and Banach's theorem are also stated here.
