Singular Integral Equations

Abstract

In this chapter we discuss some recent results for Fredholm and Volterra integral equations, which deal with the existence of positive (and possibly multiple) solutions of certain classes of these equations. In Section 3.2 we provide some existence results for the nonsingular Fredholm integral equations. The existence of a positive solution of the singular Fredholm integral equations is considered in Section 3.3. A certain class of kernels is examined and an application to a certain boundary value problem is then discussed. The results of Section 3.3 place rather restrictive conditions on the kernels, but allow us to consider a rather large class of functions which appear in the integral equation. In Section 3.4 we consider the opposite scenario, the conditions on the kernels are quite general, but the class of functions which appear in the integral equation is restricted. In Section 3.5 we examine a new class of singular integral equation. The conditions placed on the kernels are motivated by a problem in Homann flow. Here for singular Volterra integral equations the existence of positive, continuous solutions is examined. We continue with the singular Volterra integral equation in Section 3.6. The existence theory presented in this section relies on the ability to obtain solutions of certain integral inequalities.