Qualitative behaviour and stability of solutions of discretised nonlinear Volterra integral equations of convolution type

摘要

In this work we consider equations of the form: and their discretised versions (obtained through application of quadrature to (?)) of the form: , subject to certain conditions on k, , g and the weights {wj}. One purpose of the paper is to show how the discussion of qualitative behaviour and stability for (?) can be mimicked in a discussion of. We first describe Corduneanu's (1973) discussion of stability for, re-presenting his material in a modified form which lends itself to adaptation for our discussion of (?). We give a stability result for (?). We then demonstrate the stability behaviour of some simple quadrature rules applied to an illustrative example equation and we observe that, for a particular family of quadrature rules, the qualitative behaviour of solutions to the example equation is preserved under the discretisation.