Convergence of Eigenfunction Expansions for a Boundary Value Problem with Spectral Parameter in the Boundary Conditions. II

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

“…where C k , k = 1, 2, 3, 4, are constants. By virtue of Relations (9) and (10) and boundary conditions (2), we have C 1 = C 2 = 0 and…”

“…[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.…”

“…Problem ()–() was considered in Aliyev and Guliyeva 9 for a 1 > 0, a 2 < 0, and in Aliyev et al 10 for a 1 > 0, a 2 > 0. In Aliyev and Guliyeva, 9 it was shown that the eigenvalues of this problem are positive and simple and form an infinitely increasing sequence.…”

“…Here, the oscillatory properties of the eigenfunctions and their derivatives were studied, asymptotic formulas for eigenvalues and eigenfunctions were derived, and using these results conditions were found under which the system of eigenfunctions with two removed functions forms a basis in the space L p (0, 1), 1 < p < ∞ . In the first part of Aliyev et al, 10 we have proved that the eigenvalues of Problem ()–() in the case a 1 > 0 and a 2 > 0 are real and simple, and all, with the exception of one negative eigenvalue, are positive and form an increasing sequence that goes to infinity. Moreover, refined asymptotic formulas for eigenvalues and eigenfunctions were obtained, and with the help of these asymptotic formulas, sufficient conditions are established for the subsystems of eigenfunctions of this problem to form a basis in the space L p (0, 1), 1 < p < ∞ .…”