Summary: This paper deals with the following equation \[ {\mathbb{T}}\frac{d}{dx}{\hat \phi}(x)=\lambda (\sigma (x)I-c(x){\mathbb{B}}){\hat \phi}(x),\quad {\hat \phi}(0)={\hat \phi}(a) \] with parameter \(\lambda\), where \({\mathbb{T}}\) and \({\mathbb{B}}\) are self-adjoint operators in a Hilbert space \({\mathcal H}\). We obtain necessary and sufficient conditions concerning the distribution of \(\lambda\), guaranteeing the existence of solutions of the above equation. And the obtained abstract result is used to discuss the distribution of the dominant eigenvalue for a class of transport equations.