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

Abstract

In this paper we consider a spectral problem for ordinary differential equations of fourth order with the spectral parameter contained in three of the boundary conditions. We study the oscillatory properties of the eigenfunctions and, using these properties, we obtain sufficient conditions for the system of eigenfunctions of the problem in question to form a basis in the space $L_p(0,1),1<p<\infty$, after removing three functions.

Introduction

We consider the following eigenvalue problem

$$\ell (y)(x)\equiv y^{(4)}(x)-{(q(x)y^{\prime }(x))}^{\prime }=\lambda y(x),0<x<1,$$

$$y^{″}(0)=0,$$

$$Ty(0)-a\lambda y(0)=0,$$

$$y^{″}(1)-b\lambda y^{\prime }(1)=0,$$

$$Ty(1)-c\lambda y(1)=0,$$

where $\lambda \in \mathbb{C}$ is a spectral parameter, $Ty\equiv y^{‴}-q{y}^{\prime }$, $q(x)$ is a positive absolutely continuous function on $[0,1]$, $a,b$ and c are real constants such that $a>0,b>0$ and $c<0$.

The eigenvalue problem (1.1)-(1.5) describes the bending vibrations of a homogeneous rod, on the left end of which a tracing force acts and on the right end an inertial mass is concentrated (see [18], [37]).

Eigenvalue problems for ordinary differential operators with a spectral parameter in the boundary conditions of different statements have been studied by many authors (see, for example, [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [14], [15], [16], [19], [20], [21], [22], [23], [24], [25], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [38], [39], [40], [41]). Sturm-Liouville problems with a spectral parameter in the boundary conditions considered in the works of J. Walter [41], C.T. Fulton [20], D.B. Hinton [27], P.A. Binding and P.J. Browne [15] were reduced to eigenvalue problems for operators acting in a Hilbert space $L_2\oplus \mathbb{C}$. In the works of E.M. Russakovskii [38], P.A. Binding, P.J. Browne and B.A. Watson [17] the case of nonlinear (i.e. polynomial) occurrence of the spectral parameter in the boundary conditions and the corresponding operators were constructed in the space $L_2\oplus {\mathbb{C}}^{\mathbb{N}}$, where $N\ge 1$. In these papers, Riesz basis property of the system of root vectors of the corresponding operators in $L_2\oplus \mathbb{C}$ and in $L_2\oplus {\mathbb{C}}^{\mathbb{N}}$ was also established. In [39], a general theory of spectral problems was constructed for ordinary differential equations with a parameter in boundary conditions, where various classes of problems (regular, almost regular and normal) were distinguished and spaces were constructed in which these problems admit a natural linearization. He proved theorems on the completeness, expansions, and basis property of the root vectors of linearizing operators in these spaces.

Thus, in the works mentioned above, spaces are constructed in which these problems admit natural linearization, and theorems on the basis property of root vectors in these spaces are proved. Naturally, the question arises of establishing the Riesz basis property in $L_2$ and the basis property in $L_p,1<p<\infty$, the systems of root functions of the initial problems.

For the first time, E.I. Moiseev and N.Yu. Kapustin [30], [31] for the Sturm-Liouville problem (without potential) with a spectral parameter in the boundary condition established the Riesz basis property in $L_2$, later (see [32]) in $L_p,1<p<\infty$, of eigenfunction systems with one arbitrary remote function. Given the potential, this result was also established by N.Yu. Kapustin [28].

In the future, N.B. Kerimov and R.G. Poladov [35], N.Yu. Kapustin [29], Z.S. Aliev [2], [5], Z.S. Aliev and A.A. Dunyamaliyeva [6], Z.S. Aliyev, A.A. Dunyamaliyeva and Ya.T. Mehraliyev [7] investigated the Sturm-Liouville problem with a spectral parameter in both boundary conditions. They established the conditions under which the system of eigenfunctions after removing two functions forms a basis in the space $L_p,1<p<\infty$. N.B. Kerimov and Z.S. Aliyev [9], [34], Z.S. Aliyev [3], [4], Z.S. Aliyev and S.B. Guliyeva [8], Z.S. Aliyev and F.M. Namazov [9], [10] established a necessary and sufficient, and also sufficient conditions for the systems of root functions of spectral problems for ordinary differential equations of fourth order with a spectral parameter in the boundary conditions without one or two chosen functions to form a basis in the space $L_p,1<p<\infty$.

The above question in a more general form was considered in the works of T.B. Gasymov [26] and Z.S. Aliyev [1]. In these papers, certain conditions ensuring the basis property of the system of root functions of the initial problems were found. It should be noted that these results were repeated in the recent paper [40].

The purpose of the present paper is to study the oscillation properties of eigenfunctions, and using these properties established the conditions under which the subsystems of eigenfunctions of problem (1.1)-(1.5) to forms a basis in the space $L_p,1<p<\infty$.

The structure of this paper is as follows. In Section 2, the existence and uniqueness of a solution $y(x,\lambda )$ to problem (1.1)-(1.3), (1.5) is proved for $\lambda \in \mathbb{C}\backslash \{0\}$, then it is determined everywhere in $[0,1]\times \mathbb{C}$, and its oscillatory properties are studied depending on $\lambda ,\lambda \in (0,+\infty )$. In Section 3, we show that the eigenvalues of the boundary value problem (1.1)-(1.5) are real (nonnegative), simple and form an infinitely increasing sequence. Here by using results of previous section the oscillatory properties of the corresponding eigenfunctions are completely studied. In Section 4 we give operator interpretation of problem (1.1)-(1.5), where we reduce this problem to the spectral problem for a self-adjoint operator in Hilbert space $H=L_2(0,1)\oplus {\mathbb{C}}^3$ and we study some properties of this operator. Next, with the use of oscillatory properties of eigenfunctions we obtain sufficient conditions for the system of eigenfunctions of problem (1.1)-(1.5) to form a basis in the space $L_p(0,1),1<p<\infty$ after removing three functions.