Basis Properties of Root Functions of a Vibrational Boundary Value Problem with Boundary Conditions Depending on the Spe
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NARY DIFFERENTIAL EQUATIONS
Basis Properties of Root Functions of a Vibrational Boundary Value Problem with Boundary Conditions Depending on the Spectral Parameter Z. S. Aliyev1,2∗ and F. M. Namazov1∗∗ 1
2
Baku State University, Baku, AZ1148 Azerbaijan Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, AZ1141 Azerbaijan ∗ e-mail: [email protected], ∗∗ [email protected] Received January 24, 2020; revised January 24, 2020; accepted May 14, 2020
Abstract—We study the basis properties of root functions of a spectral problem describing the bending vibrations of a homogeneous rod with a longitudinal force acting in its cross sections. Both rod ends are elastically fixed and either there is a lumped mass or a follower force acts on each of the ends. We establish a sufficient condition for the basis property of the system of root functions of this problem in the space Lp (0, 1), 1 < p < ∞, after removing two functions. DOI: 10.1134/S0012266120080017
1. INTRODUCTION Consider the spectral problem 0
`(y)(x) ≡ y (4) (x) − (q(x)y 0 (x)) = λy(x), 00
0 < x < 1,
(1.1)
00
U1 (λ, y) ≡ y (0), U2 (λ, y) ≡ y (1) = 0, (1.2) U3 (λ, y) ≡ T y(0) − aλy(0) = 0, (1.3) U4 (λ, y) ≡ T y(1) − cλy(1) = 0, (1.4) 000 0 where λ ∈ C is the spectral parameter, T y ≡ y − qy , q(x) is a positive absolutely continuous function on [0, 1], and a and c are nonzero real constants. The basis properties of the system of root functions of eigenvalue problems for ordinary differential equations with spectral parameter in the boundary conditions have been the subject of study in various function spaces in many papers (see, e.g., the papers [1–14] and the bibliography therein). Problems of such kind are encountered in mechanics and physics (see, e.g., [15, 16]). Problem (1.1)–(1.4) arises when separating variables in a dynamic boundary value problem (for a fourth-order partial differential equation) describing the bending vibrations of a homogeneous rod with a longitudinal force acting in its cross sections and with its both ends elastically fixed, each of them either containing a lumped mass or subjected to a follower force [15]. When studying the corresponding boundary value problem for the partial differential equation, it is necessary to study the basis properties of the system of root functions of problem (1.1)–(1.4) in the space Lp (0, 1), 1 < p < ∞. Problem (1.1)–(1.4) was considered in the papers [10–12], where, in particular, it was proved that the eigenvalues are real and simple, except for at most one double eigenvalue (only in the cases of a > 0, c > 0 and a < 0, c > 0 with c = a + 1, the algebraic multiplicity of the eigenvalue λ = 0 is 2), and form an unboundedly increasing sequence {λk }∞ k=1 such that λk > 0 for k ≥ 4. The eigenfunction yk (x) corresponding to the eigenvalue λk has, within the interval (0, 1), precisely k − 1 simple zeros in the case of a > 0 and c < 0 with k ≥ 1, precisely k − 2 simple zeros in the case of a > 0 and c > 0 with k ≥ 3, and precisely k − 3 simple zeros in the case of a < 0
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