# Uniqueness of solution of Volterra Integral Equation with deviating argument

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 $$u(t) \in L^2[a,b]$$: $$$$u(x) = f(x) + \int_a^x K_1(x,s) u(s) \mathrm{d}s+\int_a^{b-x+a} K_2(x,s) u(s) \mathrm{d}s \phantom{texttexttexttex}(1)$$$$ where $$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?

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