Integral Equations and Functionals

Abstract

Introduction. It would be difficult to think of any two topics in mathematical analysis more central and more widely studied during the last fifty years than the theory of integral equations and functionals. Here we are using the word functional as a noun and not as an adjective. A functional is a generalization of the notion of a function of a finite number of numerical variables while an integral equation is, as its name suggests, an equation in which the unknown, say a function of a numerical variable, occurs under an integral. The early history of integral equations goes back to the special integral equations studied by several mathematicians of the late eighteenth and early nineteenth century Laplace, Fourier, Poisson, Abel and Liouville while the pioneering systematic investigations go back to the late nineteenth and early twentieth century work of Volterra, Fredholm and Hilbert. The precise definition of a functional will be given later. To give an approximate idea of the flavor of the subject, it is instructive to point out a few simple examples of a functional with which the reader is familiar in his elementary calculus courses. The length of a curve depends for its value on the curve and so it is a functional of the curve. The area enclosed by a closed plane curve depends on the closed curve for its value and hence it is a functional of a closed plane curve. The volume enclosed by a closed surface in space is a functional of the closed surface. Finally, the integral of a function depends for its value on the function and so it is a functional of the function. We can continue with many other more complicated examples. However, the above illustrations will suffice to show clearly that some of the special functionals have a history going back to the founders of the differential and integral calculus and even as far back as the ancient Greek mathematicians. The notion of the variation of an integral as first employed by Euler and Lagrange in their treatment of the calculus of variations can be viewed as a first, but partially successful, attempt to define a differential of a functional for the special functionals of the calculus of variations. The notion of a differential of a functional is very fundamental in the modern theory of functionals. In 1887, Volterra published a series of famous papers in which he singled out the notion of a functional and pioneered in the development of a theory of functionals. During the next decade, Volterra laid the foundations for his theory of linear integral equations while at the turn of the century, Fredholm gave the fundamentals of the Fredholm integral equation theory, (first used by him in the solution of the Dirichlet problem) a theory which generalizes to the continuous domain