I'm working with a scalar functional differential equations that is T-periodic in time. The precise form is somewhat complicated. I'm interested in T-periodic solutions and already have an existence result, but a want to compute one numerically.
The existence result uses a topological method:
Finding a T-periodic solution is restated as finding a zero of a certain operator $J:C_T \to C_T$, where $C_T$ is the Banach space of continuous T-periodic functions. Then I proved that $J$ has at least one zero in some region of $C_T$ using degree theory.
I was wandering if I could use $J$ to device a numerical method, and I want to know if this as already been done, because I coudn't find it in the literature. My questions is more generally how to combine topological methods to numerical aproximations.
Mi idea is to define $F:C_T\to \mathbb{R}, \ \ F(u)=\left\lVert J(u)\right\rVert_{L^2}^2$. A zero of $J$ is an absolute minimum of $F$, so the problem now is an optimization problem.
I would apreciate references. Thanks!