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Suppose $K(x,t)$ is known and $$ \int f(x)K(x,t)dx=0 $$ Are there some known sufficient and \ or necessary conditions on $K(x,t)$ such that the only solution is $f(x)=0$ a.s.? ($f$ can be in a space of your choice but to keep things concrete say $f \in \mathcal{L}^p$)

I've been searching around for some answers to this question but the questions are usually unanswered and / or uniqueness is rare (e.g. Uniqueness of the Solution to Fredholm's Integral Equations of the First Kind and https://math.stackexchange.com/questions/873278/existence-of-solution-of-volterra-integral-equation-of-the-first-kind). Are there some known results or is this type of problem very much a case-by-case type of issue?

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user103828
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1 Answers1

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Sufficient condition: given $\phi\in C_0^\infty(a,b)$ there exists $t_0$ such that $K(x,t_0)=\phi(x)$ almost everywhere.

Under the above condition, the unique locally integrable solution of the homogeneous Fredholm equation of the first kind is the zero function.

Proof: We have $$\int_a^bf(x)\phi(x)\;dx=\int_a^bf(x)K(x,t_0)\;dx=0,\quad \forall\ \phi\in C_0^\infty(a,b).$$ It follows from the du Bois-Reymond lemma that $f(x)=0$ almost everywhere.

Pedro
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