**MSC:**- 65R20 Integral equations
- 45M10 Stability theory

\[

y(t) = g(t) + \int_0^tK(t,s,y(s))ds, \hspace{5mm}{ t \in { \sRR}^+},

\hspace*{5mm} (\dagger)

\]

analytically and when solved numerically.

In some recent work the long-term behaviour of numerical solutions of

the nonlinear convolution equation

\[

y(t) = g(t) + \int_0^tk(t-s)\varphi(y(s))ds, \hspace{5mm}{ t \in {

\sRR}^+}, \hspace*{5mm} (\ddagger)

\]

has been considered. In the present paper, we consider examples of the

form ($\ddagger$) which do not satisfy classical conditions

guaranteeing existence and uniqueness of the exact solutions and

suitable behaviour of approximate solutions. We are able to give a

strengthened version of a theorem given originally by Corduneanu for

certain kernels and nonlinearities. On that basis we consider how

certain numerical codes may fail. We explore and test methods known to

preserve properties of the solutions in the linear case.