Geodesic
Neumann problem for fourth order degenerate ordinary differential equations
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (2010), pp. 22-26

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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.
Keywords: Neumann problem, weighted Sobolev spaces, generalized solution, spectrum of linear operators. 
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L. P. Tepoyan; Kalvand Daryoush. Neumann problem for fourth order degenerate ordinary differential equations. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (2010), pp. 22-26. http://geodesic.mathdoc.fr/item/UZERU_2010_1_a3/