In this paper we consider the problem ıv + 2 () + 1 () + 0 () = λ 0 < < 1 () (1) − (−1) σ () (0) + −1 =0 α () (0) = 0 = 1 2 3 (1) − (−1) σ (0) = 0 where λ is a spectral parameter; () ∈ L 1 (0 1), = 0 1 2, are complex-valued functions; α , = 1 2 3, = 0 − 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 forms a basis in the space L (0 1), 1 < < ∞, when α 3 2 + α 1 0 = α 2 1 , () ∈ W 1 (0 1), = 1 2, and 0 () ∈ L 1 (0 1); moreover, this basis is unconditional for = 2.

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In this paper, we consider the problem alignedrightleftyıv+qMathClass-open(xMathClass-close)y=λy,0

In this paper, we consider the problem y ıv + q (x) y = λy, 0 < x < 1, y (1) − (−1) σ y (0) + αy (0) + γ y (0) = 0, y (1) − (−1) σ y (0) + βy (0) = 0, y (1) − (−1) σ y (0) = 0, y (1) − (−1) σ y (0) = 0 where λ is a spectral parameter; q (x) ∈ L 1 (0, 1) is a complex-valued function; α, β, γ 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 and it is proved that all the eigenvalues, except for a finite number, are simple in the case αβ = 0. It is shown that the system of root functions of this spectral problem forms a basis in the space L p (0, 1), 1 < p < ∞, when αβ = 0; moreover, this basis is unconditional for p = 2.

In this paper, we consider the following boundary value problem y (4) + q (x) y = λy, 0 < x < 1, y ′′′ (1) − (−1) σ y ′′′ (0) + αy (0) = 0, y (s) (1) − (−1) σ y (s) (0) = 0, s = 0, 2, where λ is a spectral parameter, q (x) ∈ L 1 (0, 1) is complex-valued function 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. When α ̸ = 0, we proved that all the eigenvalues, except for finite number, are simple and the system of root functions of this spectral problem forms a Riesz basis in the space L 2 (0, 1). Furthermore, we show that the system of root functions forms a basis in the space L p (0, 1), 1 < p < ∞ (p ̸ = 2), under the conditions α ̸ = 0 and q (x) ∈ W 1 1 (0, 1).