Consider the Volterra integral equation of the second kind:
$ z(t) = h(t) + \int_0^t G(t-s)(\phi(z(s)) + f(s))ds$,
where $f:\mathbb{R} \rightarrow Z$ is a continuous T-periodic function, G(t) is a piecewise-differentiable operator function, $h \in W_{-\rho}^{1,2}$ and $\phi$ is a nonlinearity. We could proof, that there exists a unique $h_∗ \in W^{1,2}_{-\rho}$ such that the associated solution $z_∗$ of the volterra integral for $h = h_∗$ is T -periodic. For any other $h_∗ \in W^{1,2}_{-\rho}$ the solution tends to $z_∗$ when $t \to \infty$.
I need to provide a numerical example of an application to the theorem considering a practical example. Unfortunately, I was not able to find sufficient ways for the tools I know (python, JAVA), e.g. inteq for python only works for Volterra integral equations of the first kind.
Do you know suitable ways or environments to implement such a problem in a numerical example experiment (in particular, with checking a frequency-domain condition)?
Thanks in advance!