Dependence of Eigenvalues of ( 2 n + 1 ) th Order Boundary Value Problems with Transmission Conditions

This paper deals with some boundary value problems generated by ( 2 n + 1 ) th order differential equation with transmission conditions. After showing that these problems generate self-adjoint operators and the eigenvalues of the problems are real, we introduce the continuous dependence and differentiable dependence of eigenvalues on parameters: coefficient functions and weight function, boundary conditions, transmission conditions, as well as the endpoints and transmission points. In addition, we obtain the differential expressions of all given parameters respectively.


Introduction
The dependence of eigenvalues play an important role in the theory of differential operators, it provides theoretical support for the numerical calculation of eigenvalues [1][2][3].As early as 1987, Poeschel and Trubowitz in [4] considered the = n (q) as a function of potential function q , and showed that is Frechet differentiable of q by using the asymptotic form of the solutions for | | → ∞ .Then Dauge and Helffer in [5] investigated the Neumann eigenvalues = n (q) with respect to the domain, and proved that the eigenvalues of the regular Sturm-Liouville problems with the Neumann boundary conditions are differentiable functions of the right endpoint.On this basis, Kong and Zettl in [6] studied these problems by using more simple methods, and proved that the eigenvalues of Sturm-Liouville problems not only continuously but also differentiably depend on the end points.In the same year, they investigated these problems more systematically in [7], and showed that the 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:1190-1209 eigenvalues of Sturm-Liouville problems not only differentiably depend on the end points, but also differentiably depend on all the given parameters: boundary conditions, coefficient and weight functions.They also obtained the differential expressions of the given parameters by using the similar methods in [6].For a special class of regular boundary value problems studied by Naimark [8] and Weidmann [9], Kong and Zettl in [10] showed that the eigenvalues differentiably depend on the problem data and obtained the differential expressions.Their results extended theorems in [7] from the second-order case to the general even order case.
It is well known that boundary value transmission problems have important applications in physics and engineering, such as heat conduction and mass transfer, string vibration problems with nodes located internally [11], etc., and their physical applications connected with these problems are also found in literature [12][13][14].In recent years, more and more researchers are interested in the study of problems with interior discontinuities, in particular, the dependence of eigenvalues of even order boundary value transmission problems, and some meaningful results have been obtained [15][16][17][18][19][20][21][22][23][24][25][26][27].To deal with interior discontinuities, some conditions are imposed on the discontinuous points, which are often called transmission conditions (see [15,17,22,23,[25][26][27]31]) or interface conditions (see [21,24]).Among them, Zhang and Wang [21] investigated the dependence of eigenvalues of Strum-Liouville problems with interface conditions for second order case and gave the differential expressions of eigenvalues with respect to the given parameters.Li et al. in [22] generalized the results of [21] in fourth order case in 2017.In the same year, Li et al. further generalized these results to the general even order differential operators in [23].Although the general theory and methods for such even order boundary value problems have been highly developed, little is known about the odd order case, especially in the general odd order case.
In 1975, Walker in [28] investigated a vector-matrix formulation for formally symmetric ordinary differential equations.By defining the quasi-derivative y [k] and corresponding matrices, they showed that the differential expression has a first-order vector-matrix formulation as show below: Here M[⋅] is a formally symmetric ordinary differential expression of order m(m can be even or odd),I is an interval of the real line, is a complex number and w is a weight function.A, B and J are m × m matrices.
However, according to the different symmetric expressions in which m is even or odd, the definitions of the quasi-derivatives are different, and the corresponding matrices are also different.For m = 2n + 1 , the formally symmetric ordinary differ- ential expressions can be expressed as show below: and corresponding quasi-derivatives are defined as follows: On this basis, Hinton in [29] studied the deficiency indices of odd order differential operators.By using the form of symmetric differential Eq. ( 1), they got the Lagrange identity for odd order as follows: where . These results provided an important foundation for us to fur- ther study the self-adjointness and boundary value problems of odd order differential operators.
In recent years, Uğurlu in [30] considered a class of formally symmetric boundary value problems generated by the third-order differential equations studied by Walker in [28].After showing that these problems generate self-adjoint operators, the dependence of eigenvalues on the data for these problems was studied and the derivatives of the eigenvalues with respect to some elements of data were introduced.Then they generalized these results to differential operators with transmission conditions in [31].In spired by [30], Li et al. in [32] had studied the selfadjoint of a class of third-order differential operators with an eigenparameter contained in the boundary conditions.At the same year, Bai et al. investigated the dependence of eigenvalues for these problems in [33].Other studies on third-order differential operators were found in literature [34][35][36].
However, up to now, we have not found any study on the dependence of eigenvalues of general odd order boundary value problems with transmission conditions.To further develop the odd order differential operators theory, in this paper, we study the symmetric operators generated by a class of (2n + 1) th order dif- ferential equations with transmission conditions.Combining the quasi-derivatives and the matrices defined in [28] and using the methods in [7], we prove the selfadjointness of the operators, on this basis, we further introduce the continuous dependence of eigenvalues on the problems.In addition, we show the differential properties of the eigenvalues on the given parameters, not only including the boundary conditions and transmission conditions, coefficient functions and weight function, two endpoints,but also including the interior discontinuities points.In particular, we also gave the details of the proof of the differentiability of eigenvalues with respect to the weight function. (2) 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:1190-1209 The rest of this investigation is arranged as follows.In Sect.2, some notations and preliminaries are gave.In Sect.3, we construct an operator T associated with the problems (3)(4)(5), and prove that T is a self-adjoint operator.Then we introduce the continuity results of eigenvalues and eigenfunctions in Sect. 4. In Sect.5, we obtain differential expressions of the eigenvalues with respect to the given parameters.
For y, z ∈ D max , integration by parts yields the Lagrange identity as show below: where T and Z * (x) denotes the complex conjugate transpose of Z(x).

Operator Theoretic Formulation and Self-Adjointness
According to the above analysis, we shall construct the operators related with boundary value problems (3)(4)(5).Consider the operator T defined by Ty = Ly, y ∈ D(T),with the domain Then we have the following Lemma.
Proof Let C ∞ 0 (J) be all the following functions: , where w [a, c) , hence for any  > 0 , there exists g and there exists
Therefore, (iii) hold.This complets the proof.By the self-adjointness of the operator T , we have the following Corollary.

Continuity of Eigenvalues and Eigenfunctions
In this section, we introduce the continuity of eigenvalues and eigenfunctions.
Clearly, the above solutions are linearly independent.Let 1 (x, ), ⋯ , 2n+1 (x, ) be the solutions of Eq. ( 3) on the interval (c, b] and sat- isfy the initial conditions According to the properties of dependence of the solutions on the parameters, the Wronskians W 1 ( ) = W( 1 (x, ), … , 2n+1 (x, )) and W 2 ( ) = W( 1 (x, ), … , 2n+1 (x, )) are independent of the variable x and are entire functions of parameter , short calcula- tion yields that W 2 ( ) = 1 assume that u(x) satisfy the transmission conditions (5), then we have Proof The proof can be given similarly as in [17] Lemma 5.3.1, thus is omitted here.Let and where Φ 1 (c, ) and Φ 2 (c, ) are defined by left and right limits.

Lemma 4.2 A complex number is an eigenvalue of the operator T if and only if
Δ ( ) = det (A + BΦ(b, )) = 0.
In the following, we want to show that a small change of the problem results in only a small change in the eigenvalues and eigenfunctions.To this we introduce the Banach space.Define and where p0 = p 0 , x ∈ J 0, x ∈ J � �J , and p1 , ⋯ , pn , w have similar definitions.
It is clear that Ω is not a subset of X , but Ω 1 is.And with Ω 1 as a subset of X to inherit the norm from X and convergence in Ω that is determined by the norm (29).Then based on the space X , the set Ω and Lemma 4.2, we introduce the following theorems.
(ii) Assume the multiplicity of eigenvalue ( ) is l (l = 2, … , 2n + 1) for all ∈ N , and N ∈ Ω is a neighborhood of 0 .Let u k (⋅, 0 ) be any normalized eigen- functions of ( 0 ).Then there exist l linearly independent normalized eigenfunctions Proof The proof can be given similarly as in [23], with the aid of Theorem 4.1 and Lemma 4.3.

Differential Expressions of Eigenvalues on the Problems
In this section we introduce the derivatives of eigenvalues with respect to the given parameters.Definition 5.1.(see [10]) Let X , Y be Banach space.
) be an eigenvalue of operator T connected with , and let u = u(⋅, ) be the corresponding normalized eigenfunction of ( ).Assume that ( ) has constant geometric multiplicity in some neighborhood of in Ω.Then is continuously differentiable with respect to all the parameters in .More precisely, we have the following.
(ii) Fix all the parameters of except w.Let = (w) and u = u(⋅, w), then is Frechet differentiable at w and (iii) Fix all the parameters of except A. For small K ∈ M 2n+1 ( ) satisfying (viii) Fix all the parameters of except c 1 , here c 1 = c−.Let = (c 1 ) and u = u(⋅, c 1 ), then is Fréchet differentiable at c 1 and (ix) Fix all the parameters of except c 2 , here c 2 = c+.Let = (c 2 ) and u = u(⋅, c 2 ), t h e n i s Fré ch et d i f fe re n t i a b l e a t c 2 a n d (a, b).
Proof First at all, we should emphasize that by ( ) we mean a continuous eigenvalue branch, further be a normalized eigenfunction u(⋅, ) we mean a uniformly convergent normalized eigenfunction branch.
then Eq. ( 3) can be expressed as Let u = u(⋅, w) , v = u(⋅, w + h) , from (30) we get that hence, it follows that then we can get the result as follows by using similar discussion to that of (i) as K → 0 , then by ( 7)-( 9) and ( 12) we have from ( 6), we get that Journal of Nonlinear Mathematical Physics (2023) 30:1190-1209 Combining (32) and ( 33) and letting K → 0 , we obtain that thus the result follows from the (34).
The proof for (iv) is similar to this proof, thus is omitted here.

Conclusion
This paper investigate the eigenvalues dependence of a class of (2n + 1) th order differential equations with transmission conditions.We obtain that the eigenvalues of the problems not only continuously but also differentiably depend on the given parameters of the problems and obtain some new differential expressions of the eigenvalues with respect to the given parameters.This extend the theorems in [31] from the third-order case to general order case with general boundary conditions and transmissions conditions.This further develops the theory of boundary value problems of odd order differential operators.
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