K. Frischmuth, N. J. Ford, J. T. Edwards
Volterra Integral Equations with non-Lipschitz Nonlinearity
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 51, 65-82(1997)

MSC:

65R20 Integral equations

45M10 Stability theory

Abstract: In this work we consider equations of the form:
\[
y(t) = g(t) + \int_0^tK(t,s,y(s))ds, \hspace{5mm}{ t \in { \sRR}^+},
\hspace*{5mm} (\dagger)
\]
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.

Keywords: Volterra integral equations, stability, quadrature rules