On the Spectral Problem Arising in the Mathematical Model of Bending Vibrations of a Homogeneous Rod

Oscillation and basis properties for the equation of vibrating rod at one end of which an inertial mass is concentrated

“…[2][3][4][5][6][7][8][9][10][11][12][13][16][17][18][19][20][21][23][24][25][26][27][28][29][30][31] The study of the basis properties of systems of root functions of ordinary differential operators in the space L p , 1 < p < ∞, plays a fundamental role in the study of the solvability of boundary value problems for partial differential equations. In the works 3,4,7,8,[19][20][21]26,27,29 were studied the basis properties in the space L p , 1 < p < ∞, of subsystems of root functions of Sturm-Liouville problems with spectral parameter in the boundary conditions; in the works 5,6,[9][10][11][12]24,25 were studied the basis properties in the space L p (0, 1), 1 < p < ∞, of subsystems of root functions of the eigenvalue problems for ordinary differential equations of fourth order with spectral parameter in one of the boundary conditions.…”

“…The spectral problems for ordinary differential equations with spectral parameter in the boundary conditions have been studied by many authors 2‐13, 16‐21, 23‐31 . The study of the basis properties of systems of root functions of ordinary differential operators in the space L p , 1 < p < ∞ , plays a fundamental role in the study of the solvability of boundary value problems for partial differential equations.…”

“…Passing to the limit as → 0 in(12) with the use of conditions T 3 (x, 0) ≡ 0 and T 4 (x, 0) ≡ 1, we obtain(x, 0) ≡ C(0) 3 (x, 0).By conditions (s−1) 0, ) = 3s , s = 1, 2, 3 and T 3 (0, ) = 0, it follows from the first part of Lemma 2.1 that 3 (x, ) > 0, ′ 3 (x, ) > 0 for x ∈ (0, 1] and > 0. Then by Lemmas 3.2 and 3.8, we have 3 (x, 0) ≠ 0 and ′ 3 (x, 0) ≠ 0 for x ∈ (0, 1].…”

Spectral properties for the equation of vibrating beam with a spectral parameter in the boundary conditions

“…The spectral properties of the boundary value problems for ordinary differential equations of fourth order with a spectral parameter in the boundary conditions were studied in [1, 2, 7-11, 14, 21-24]. The basis properties of systems of eigenfunctions in L p (0, 1), 1 < p < ∞ of considered problems were studied in [1,2,[8][9][10][11]23]. In these works sufficient conditions were found for subsystems of eigenfunctions to form a basis in L p (0, 1), 1 < p < ∞.…”

“…In [7,[9][10][11], boundary value problems for ordinary differential equations of fourth order with a spectral parameter contained in two of boundary conditions were considered. In [7,10,11] the establishment of sufficient conditions for the systems of eigenfunctions of the problems under consideration after removing two functions to form a basis in the space L p (0, 1), 1 < p < ∞ was based on the rough asymptotic formulas for eigenvalues and eigenfunctions, and the oscillation properties of eigenfunctions and their derivatives. In [9], this was based on finer asymptotic formulas for eigenvalues and eigenfunctions.…”