In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Let $f$ be fixed such that $E$ is non-empty, $x \in E$ and $a > 0$. We set $h(t) = \displaystyle\int_0^t e^{-xu} f(u)\,du$ for all $t \geqslant 0$.
VI.B.1) Show that $Lf(x+a) = a\displaystyle\int_0^{+\infty} e^{-at} h(t)\,dt$.
VI.B.2) We assume that for all $n \in \mathbb{N}$, we have $Lf(x + na) = 0$.
Show that, for all $n \in \mathbb{N}$, the integral $\displaystyle\int_0^1 u^n h\!\left(-\frac{\ln u}{a}\right)du$ converges and that it is zero.
VI.B.3) What do we deduce for the function $h$?