Let $a \in \mathbb{R}$ and $ f \colon [0,1] \to \mathbb{R}$ a continuous function. Solve the integral equation:
$u(t) = f(t) + a\int_{0}^{t}u(s) \, ds $
finding a closed form for the solution.
I was trying to solve this exercise but I am having some trouble: the exercise does not say wheter $u$ should be continuous or not. If I assume continuity, the substitution $h(t) = \int_{0}^{t}u(s)\,ds $ turns the equation in a linear ODE and I am done . If $u$ is not continuous there the trouble begins: Since I can not use fundamental theorem of calculus, I followed two paths : I tred to mollify $u$ by convolution and taking the sequence of solutions $\{u_{\epsilon}\}_{\epsilon > 0}$ to the equation
$u_{\epsilon}(t) = f(t) + a\int_{0}^{t}u_{\epsilon}(s) \, ds $
and setting $u= \lim_{\epsilon \to 0 } u_{\epsilon }$ I tried to prove this function solves the original but failed.
I tried also ,since $u$ is integrable, to consider the fourier series of the left and the right side but I can not switch integral and series in the right side. I am out of ideas I would like a full solution more than some hints.