Abstract
In this paper, we consider the problem y IV + q x y = λ y , 0 < x < 1 , y ″ ′ 1 - - 1 σ y ″ ′ 0 + α y ′ 0 + γ y 0 = 0 , y ″ 1 - - 1 σ y ″ 0 + β y 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 α β ≠ 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 α β ≠ 0 ; moreover, this basis is unconditional for p = 2.
Highlights
Henceforward, by L we denote the differential operator generated by the differential expression l (y) = yıv + q (x) y, x ∈ (0, 1) (1)and the boundary conditionsU3(y) ≡ y (1) − (−1)σ y (0) + αy (0) + γ y (0) = 0, U2(y) ≡ y (1) − (−1)σ y (0) + βy (0) = 0, (2)Us(y) ≡ y(s) (1) − (−1)σ y(s) (0) = 0, s = 0, 1, where q (x) ∈ L1 (0, 1) is complex-valued function; α, β, γ are arbitrary complex constants and σ = 0, 1
Under the condition of α3,2 + α1,0 = α2,1, the asymptotic formulae for eigenvalues and root functions are obtained; all the eigenvalues except for a finite number are simple is proved; 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 p j (x) ∈ W1j (0, 1), j = 1, 2
The existence of a wide class of boundary value problems for second-order ordinary differential operators with regular, but not strongly regular boundary conditions, whose system of root functions does not form a basis in L2, is established in the paper [14]
Summary
The existence of a wide class of boundary value problems for second-order ordinary differential operators with regular, but not strongly regular boundary conditions, whose system of root functions does not form a basis in L2, is established in the paper [14]. Some sharp results on the absence of the basis property are obtained in [3] It is proved in [15] that the system of root functions of the differential operator l (y) = y + q (x) y, y (1) − (−1)σ y (0) + γ y (0) = 0, y (1) − (−1)σ y (0) = 0 forms an unconditional basis of the space L2 (0, 1), where q (x) ∈ L1 (0, 1) is an arbitrary complex-valued function, γ is an arbitrary nonzero complex constant and σ = 0, 1.
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