Fredholm formulas for linear integral equations (Q2703907)

The authors suggest necessary and sufficient conditions on the kernel \(k(t,s)\) of the second-kind linear Fredholm integral equation \(x(t)=\lambda \int_{\Omega} k(t,s)x(s)ds +f(t)\), considered on the space \(L_2(\Omega)\), under which the classical Fredholm formulas \(x(t)= f(t) + \lambda \int_{\Omega} d(\lambda)^{-1} {\mathcal D}(t,s,\lambda) f(s) ds\) for solutions remain valid. Here \(d(\lambda)\) and \({\mathcal D}(t,s,\lambda)\) are the so-called Fredholm determinant and Fredholm function, respectively.