Necessary and sufficient conditions for discreteness and boundedness below of the spectrum of the singular differential operator $\ell \left(y\right)\equiv \frac{1}{w\left(t\right)}\left(r\right(t\left)y\right)$, $t\in [a,\infty )$ are established. These conditions are based on a recently proved relationship between spectral properties of $\ell$ and oscillation of a certain associated second order differential equation.

In this paper we consider the problem $\begin{array}{c}{y}^{iv}+p_2\left(x\right)y^{\text{'}\text{'}}+p_1\left(x\right)y^{\text{'}}+p_0\left(x\right)y=\lambda y,0<x<1,\\ y^{\left(s\right)}\left(1\right)-{(-1)}^{\sigma }{y}^{\left(s\right)}\left(0\right)+\sum _{l=0}^{s-1}{\alpha }_{s,l}{y}^{\left(l\right)}\left(0\right)=0,s=1,2,3,\\ y\left(1\right)-{(-1)}^{\sigma }y\left(0\right)=0,\end{array}$ where λ is a spectral parameter; p j (x) ∈ L 1(0, 1), j = 0, 1, 2, are complex-valued functions; α s;l, s = 1, 2, 3, $l=\overline{0,s-1}$, are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established in the case α 3,2 + α 1,0 ≠ α 2,1. It is proved that the system of root functions of this spectral problem...

The paper defines and studies the Drazin inverse for a closed linear operator $A$ in a Banach space $X$ in the case that $0$ belongs to a spectral set of the spectrum of $A$. Results are applied to extend a result of Krein on a nonhomogeneous second order differential equation in a Banach space.