An embedding technique for the solution of linear integral equations

Abstract

Starting from the classical Fredholm theory, it is shown that the solution of a linear integral equation at an eigenvalue λ = λ 0 (say) can be expressed as the limit function (when this exists) of the unique solution of a standard non-homogeneous linear integral equation with λ ≠ λ 0, as λ → λ 0. Developing this idea, most of the standard properties are readily derived, and also, some which are believed to be new. Although, for simplicity, the present theory is restricted to Riemann integrals and kernels of only limited discontinuity, the fundamental principle here explained is obviously capable of a far wider generality in application, and it is hoped to so extend it in later papers.