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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.

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