Integral characteristics of the growth of spectral functions for generalized second order boundary problems with boundary conditions at a regular end

For the spectral function $\tau(\lambda)$ of the generalized second order boundary problem
\begin{gather*} -\frac d{dM(x)}\biggl[y'_-(x)-\int_{-0}^{x-0}y(s)\,dQ(s)\biggr]-\lambda y(x)=0\qquad(0\leq x<L),\\ y'_-(0)=m,\qquad y(0)=n, \end{gather*}
and for the function $\eta(\lambda)$, which may belong to an extremely large class of positive functions that are nonincreasing on $[1,+\infty)$, the problem of characterizing the growth of the function $\tau(\lambda)$ as $\lambda\uparrow+\infty$ and of the convergence of the integral $\int^{+\infty}\eta(\lambda)\,d\tau(\lambda)$ is connected with the behavior as $x\downarrow0$ of the function $M(x)$.