Geodesic
On degenerate nonself-adjoint differential equations of fourth order
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (2012), pp. 29-33

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We consider the degenerate nonself-adjoint differential equation of fourth order $Lu\equiv(t^{\alpha}u^{\prime\prime})^{\prime\prime}+au^{\prime\prime\prime}-pu^{\prime}+qu=f$ where $t\in(0, b), \ 0\leq\alpha\leq 2, \alpha\neq 1, a, p, q $ are the constant numbers and $a\neq0, p>0, f\in L_2(0, b)$. We prove that the statement of the Dirichlet problem for the above equation depends on the sign of the number $a$ (Keldysh Teorem).
Keywords: Dirichlet problem, degenerate equations, weighted Sobolev spaces, spectral theory of linear operators. 
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L. P. Tepoyan; H. S. Grigoryan. On degenerate nonself-adjoint differential equations of fourth order. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (2012), pp. 29-33. http://geodesic.mathdoc.fr/item/UZERU_2012_3_a5/