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The Fredholm equation of the first kind is $$ f(x)=\int_a^b K(x,t)\phi(t)\, dt \tag 1 $$ Q1:

Does it mean $x$ is a scalar $x\in \mathbb R$?

Or is it a function $x:\mathbb R\rightarrow \mathbb R$, i.e. $x(t)$, so explicit we have $$ f(x(t))=\int_a^b K(x(t),t)\phi(t)\, dt \quad ? \tag 2 $$

Q2:

And the same question for the Volterra equation of the first kind. Is it $$ f(x)=\int_a^x K(x,t)\phi(t)\, dt \tag 3 $$ Or $$ f(x(t))=\int_a^{x(t)} K(x(t),t)\phi(t)\, dt \quad ?\tag 4 $$

Q3:

What about the Fredholm equation of the second kind and the Volterra equation of the second kind?

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    $\begingroup$ Fredholm equation: $a,b$ do not depend on $t$, which means $f(x)$ can't depend on $t$ either. $x$ is another variable, independent on $t$. Volterra equation is the same, despite what it may look like: $x$ is still a separate variable. $t$ is just an integration variable. $\endgroup$ – Yuriy S Jan 23 '18 at 18:28

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