What is the condition that the following sum of recursive integrals is bounded?

Question

I’ve come across the following recursive formula $$f_{i}(t)=\int_{t’}G(t,t’)f_{i-1}(t’)dt’$$ for $i>1$ where $f_{1}(t)$ and the kernel $G(t,t’)$ are known. I was wondering under what conditions on the kernel the sum $$S(t)=\sum_{i>1}f_{i}(t)$$ is bounded? I would appreciate any reference material considering properties of such integral summations.