Volterra integral equations and a new Gronwall inequality (Part II: The nonlinear case)

Abstract

We consider nonlinear singular Volterra integral equations of the second kind. We generalise the transformation method introduced in Part I of this paper [6] to cope with both the nonlinearity and slightly more general singular kernels. We also consider a particular class of nonlinear equation for which the solution behaviour is known. Using this a priori knowledge, we propose a modification of the transformation technique which results in a numerical method with good asymptotic stability properties. Applying the general theory of Part I of this paper, we prove convergence of this scheme.