Eigenvalue problem for the derivation in Sobolev spaces

Question

For $\lambda>0$ we have the following ODE problem in sobolev setting:

$\lambda y(t) + y'(t) = f(t),\quad y(0)=0,$

for $f\in L_{loc}^{1}(\mathbb{R}_{+})$. One solution in $H^{1,1}_{loc}(\mathbb{R}_{+})$ is given by the resolvent

$(\lambda+d/dt)^{-1}f=\int^{t}_{0}e^{-\lambda(t-s)}f(s)ds$.

My question now is if the solution of this equation is unique.

Thanks

Answer 1

Suppose $y \in H^{1,1}_{loc}(\mathbb{R}^+)$ is a solution of $$ y' + \lambda y = f,\;\;\; y(0)=0. $$ The function $y$ must be equal a.e. to a continuous function on $[0,\infty)$ that vanishes at $x=0$. For any such $y$, $$ \frac{d}{dx}(e^{\lambda x}y(x))=e^{\lambda x}f \\ e^{\lambda}y(x) = \int_{0}^{x}e^{\lambda t}f(t)dt \\ y(x) = e^{-\lambda x}\int_{0}^{x}e^{\lambda t}f(t)dt. $$