# Third kind Fredholm integral equation

Let us consider the following integral equation $$a(x)u(x) + \int\limits_0^1 {K(s,x)u(s)ds} = f(x)$$ Let f in $$L^p(0,1)$$ for some $$p \in [1,\infty] and let$$ $$K \in L^q((0,1) \times (0,1))$$. Assume that $$a$$ doesn't vanishe in any point of (0,1). My question is: What are the optimal assumptions to ensure that this equation has a solution? In my opinion, if $$K$$ and $$f$$ are continuous with $$K$$ is lipschitz kernel then we can apply Picard's iterations to prove the existence, what about $$L^p$$? Thank you.