Classical Fredholm Theory

Abstract

In the previous chapter, we derived the solution of the Fredholm integral equation

$$g(s) = f(s) + \lambda \int {K(s,t)g(t)dt}$$

(4.1.1)

as a uniformly convergent power series in the parameter λ for |λ| suitably small. Fredholm gave the solution of Equation (4.1.1) in general form for all values of the parameter λ. His results are contained in three theorems that bear his name. We have already studied them in Chapter 2 for the special case when the kernel is separable. In this chapter, we study Equation (4.1.1) when the function f(s) and the kernel K(s, t) are any integrable functions. Furthermore, the present method enables us to get explicit formulas for the solution in terms of certain determinants.