Let $F: L^1(\mathbb R, C(\mathbb R\times \mathbb R)) \rightarrow C(\mathbb R\times \mathbb R)$ be defined as: $$F(f)(a,y) = y+ \int_0^a f(\tau-a, \tau,y) d\tau $$
where $f \in L^1(\mathbb R, C(\mathbb R \times \mathbb R))$.
Could I say something about the Fréchet differentiability of $F$ with respect to $f$?
Are there in general some conditions for an intergal operators to be differentiable ?