The use of Sherman–Morrison formula in the solution of Fredholm integral equation of second kind

We consider a constructive method for the solution of Fredholm integral equations of second kind. This method is based on a simple generalization of the well-known Sherman–Morrison formula to the infinite dimensional case. In particular, this method constructs a sequence of functions, that converges to the exact solution of the integral equation under consideration. A formal proof of this convergence result is provided for the case of Fredholm integral equations with L2 integral kernel. Finally, a boundary value problem for the Laplace equation is considered as an example of the application of the proposed method.