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In the context of a physics problem, I am looking at a linear integral equation 2nd kind Volterra equation with deviating (centrosymmetric) argument in the unknown $u(t) \in L^2[a,b]$: \begin{equation} u(x) = f(x) + \int_a^x K_1(x,s) u(s) \mathrm{d}s+\int_a^{b-x+a} K_2(x,s) u(s) \mathrm{d}s \phantom{texttexttexttex}(1) \end{equation} where $x \in [a,b]$, $K_1(x,s)$ and $K_2(x,s)$ are absolutely continuous functions in $[a,b]\times[a,b]$ and $f(x)\in L^2[a,b]$. Does (1) have a unique solution?

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  • $\begingroup$ Sometimes, but, obviously, not always. Take $a=0,b=1, K_1=K_2=1, f=0$. Then any constant function is a solution. $\endgroup$ – fedja Dec 6 '18 at 22:13

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