Eigenvalue problem for the derivation in Sobolev spaces 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

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 $y$,
$$
                  \frac{d}{dx}(e^{\lambda x}y(x))=e^{\lambda x}f \\
                e^{\lambda}y(x) = \int_{0}^{x}e^{\lambda t}f(t)dt \\
                 y(x) = e^{-\lambda x}\int_{0}^{x}e^{\lambda t}f(t)dt.
$$
