Dependence of Eigenvalues of Discontinuous Fourth-order Differential Operators with Eigenparameter Dependent Boundary Conditions

In this paper, we investigate a fourth-order differential operator with eigenparameter dependent boundary conditions and transmission conditions. To study the eigenvalues of the problem, we establish a new operator associated with the considered problem. Furthermore, we prove that the eigenvalues are differentiable depending on the parameters of the problem. Finally, the differential expressions of the eigenvalues with respect to all parameters are given.


Introduction
As we all know, differential operator problems with interior discontinuity have important application prospects. Many actual physical and medical problems can be transformed into differential operator problems with interior discontinuity. For example, the diffraction problem of light, the vibration problem of string with node, the differential operator problem of potential functions are generalized functions, and the heat conduction and mass transfer problems [1][2][3][4]. In order to solve Journal of Nonlinear Mathematical Physics (2022) 29:776-793 interior discontinuities, some conditions are imposed on the discontinuous points, and these conditions are usually called interface conditions, point interactions or transmission conditions. In addition, there are also some physical problems need to be transformed into interior discontinuity high-order differential operator problems. For example, the fourth-order differential equation describing the vibration of a grid of beams. It is necessary to consider the continuity at the nodes and the balance at both ends of the nodes, that is, the transmission conditions are added at these nodes to transform the problem into a differential operator with interior discontinuity. We have noticed that many mathematicians are also very interested in differential operator problems with transmission conditions in recent years, and some considerable results have been obtained [5][6][7][8][9]. Among them, the dependence of eigenvalues on the problem with transmission conditions plays an important role in the spectrum theory of differential operators. Recently, Uǧurlu [10] and Zinsou [11] considered the dependence of eigenvalues of third-order and fourth-order differential equations with transmission conditions respectively, and obtained that the eigenvalues of the problems are differentiable functions of all the data.
In recent years, the differential operators with boundary conditions depending on eigenparameter are also widely used in acoustic scattering, quantum mechanics theory and so on. As far as we know, some partial differential equations can be transformed into differential operator problems with eigenparameter dependent boundary conditions through the method of separation of variables. Particularly, the fourthorder differential operators with eigenparameter dependent boundary conditions appear in elastic beam models, free bending vibrations of rod and so on [12,13]. Up to now, the differential operators with boundary conditions depending on eigenparameter have become an important research topic and some excellent results have been obtained [14][15][16][17]. In particular, the dependence of eigenvalues on the problem with eigenparameter dependent boundary conditions is also concerned by researchers. Recently, Zhang and Li in [18] showed that the eigenvalues of Sturm-Liouville problems with eigenparameter dependent boundary conditions are differential functions of all the data.
The dependence of eigenvalues and eigenfunctions on problems has great significance in differential operator theory, which provides theoretical support for numerical calculation of eigenvalues and eigenfunctions [19,20]. While some researches have been carried on the dependence of the eigenvalues for the differential equations [21][22][23][24][25][26][27][28][29], there is no research on a class of fourth-order differential operator with transmission conditions and containing eigenparameter in the boundary conditions at two endpoints. For such a problem, using the classical analysis techniques and spectral theory of linear operator, a new linear operator F associated with the problem in an appropriate Hilbert space H is defined such that the eigenvalues of the problem coincide with those of F . Furthermore, we not only prove that each of the eigenvalues of the problem can be embedded in a continuous eigenvalue branch but also obtain the differential expressions of the eigenvalues with respect to all data in the sense of Fréchet derivative.
The rest of this paper is organized as follows: In Sect. 2, we introduce a discontinuous fourth-order boundary value problem and define a new self-adjoint operator F such that the eigenvalues of such a problem coincide with those of F . In Sect. 3, with the domain

Lemma 2.1 The operator F is a self-adjoint operator in the Hilbert space H if and only if
where Proof The proof is similar to that of [7], the equation we considered is more complicated and the derivatives in boundary conditions are quasi-derivatives, here we omit the details. ◻

Remark 2.1
The eigenvalues of the fourth-order boundary value problem (2.1)-(2.6) are the same as the eigenvalues of the operator F , and the eigenfunctions are the first component of the corresponding vector eigenfunctions of the operator F . Therefore, the study of the boundary value problem (2.1)-(2.6) can be transformed into that of the operator F . Furthermore, the eigenvalues are all real-valued.
f [3] (x,̃) →f [3] (x, ), at c 0 for some c 0 ∈ J since a linear combination of j linearly independent eigenfunctions can be chosen to satisfy arbitrary initial conditions. Similarly, we obtain (3.4) as (i). This completes the proof. ◻

Differential Expression of Eigenvalues
In this section, we show that the eigenvalues of the operator F are differentiable functions with respect to all data and give expressions for their derivatives.

6.
Let all the data of be fixed except q 0 and (q 0 ) ∶= ( ) . Then
and eigenfunctions, and differentiating dependence to the spectral data are obtained by using Fréchet derivatives. Since both endpoints are containing spectral parameter, these kinds of spectral problems are more complicated and have profound theoretical importance.