A Fredholm-type theory for third-kind linear integral equations (Q1060379)

The third-kind linear integral equation \(g(t)\phi (t)=f(t)+\lambda \int^{b}_{a}K(t,t')\phi (t')dt'\) where g(t) vanishes at a finite number of points in (a,b), is considered. In general, the Fredholm alternative theory does not hold good for this type of integral equation. However, imposing certain conditions on g(t) and K(t,t'), the above integral equation was shown by \textit{G. R. Bart} [ibid. 79, 48-57 (1981; Zbl 0452.45001)] to obey a Fredholm-type theory, except for a certain class of kernels for which the question was left open. In this note a theory is presented for the equation under consideration with some additional assumptions on such kernels.