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

In this work we consider equations of the form: y(t)=g(t)+∞tok(t−s)ϕ(y(s))ds, tεR+, and their discretised versions (obtained through application of quadrature to (†)) of the form: yn=gn+hσj=0n wn−jϕj, ϕj=ϕ(yj), n,j ε N, subject to certain conditions on k, ϕ, 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 Corduneanuʹs (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.