Neumann problem for fourth order degenerate ordinary differential equations

In the present paper the Neumann problem for the equation $Lu\equiv(t^{\alpha}u'')''+au=f$, where $0\leqslant\alpha\leqslant4$, $t\in[0,b]$, $f\in L_2(0,b)$ is considered. Firstly, the weighted Sobolev space $W^2_{\alpha}$ and generalized solution for the above-mentioned equation are defined. Then, the existence and uniqueness of the generalized solution is studied, as well as the spectrum and the domain of corresponding operator are described.