Degenerate first order differential-operator equations

We consider boundary value problem for degenerate first order differential-operator equation $Lu\equiv t^{\alpha}u'-Pu=f, ~u(0)-\mu u(b)=0,$ where $t\in(0,b), \alpha\geq 0$, $P:H\rightarrow H$ is linear operator in separable Hilbert space $H, f\in L_{2,\beta}((0,b),H),~\mu\in\mathbb{C}$. We prove that under some conditions on the operator $P$ and number $\mu$ boundary value problem has unique generalized solution $u\in L_{2,\beta}((0,b),H)$ when $2\alpha+\beta<1$, $\beta\geq 0$ and for any $f\in L_{2,\beta}((0,b),H)$.