On the basis property of the root functions of differential operators with matrix coefficients

Oktay Veliev

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In this paper we consider the problem $\begin{matrix} y^{i v} + p_{2} (x) y^{''} + p_{1} (x) y^{'} + p_{0} (x) y = λ y, 0 < x < 1, \\ y^{(s)} (1) - {(- 1)}^{σ} y^{(s)} (0) + \sum_{l = 0}^{s - 1} α_{s, l} y^{(l)} (0) = 0, s = 1, 2, 3, \\ y (1) - {(- 1)}^{σ} y (0) = 0, \end{matrix}$ 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, $l = \bar{0, s - 1}$ , 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...

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