A linear inverse problem for a mixed type operator-differential equation with a parameter

We study the inverse problem
$$Bu_t+pLu=\varphi (t)+f(t, p),\quad u(0, p)=u(T,p)=0.$$
The operators $B,\,L$ are selfadjoint in the Hilbert space $E$ and the spectrum of the operator $L$ is semibounded. The unique solvability of this problem is proved with using a series expansion in eigen and associated elements of the pencil $L-\lambda B.$