Dependence of eigenvalues of Sturm Liouville problems on time scales with eigenparameter - dependent boundary conditions

: In this article, we study the eigenvalue dependence of Sturm - Liouville problems on time scales with spectral parameter in the boundary conditions. We obtain that the eigenvalues not only continuously but also smoothly depend on the parameters of the problem. Moreover, the di ﬀ erential expressions of the eigenvalues with respect to the data are given.


Introduction
It is well known, Stefan Hilger, a German mathematician, first proposed the concept of time scales in his doctoral dissertation. Since then there are a considerable number of studies on the problems on time scales, and here we refer to [1][2][3][4][5][6][7][8][9][10][11]. Time scale organically unifies continuous systems and discrete systems. Therefore, the research results on time scales are more general and have a wide application prospect.
In 1999, Agarwal et al. studied the Sturm-Liouville (S-L) problem y qy λy 0 σ σ ΔΔ + + = under separated boundary conditions and proved the existence of the eigenvalues, and the number of generalized zeros of the eigenfunctions [1]. In 2008, Kong considered the S-L problem in the general form and discussed the dependence of eigenvalues of the S-L problem on boundary conditions [3]. In 2011, Zhang and Yang studied the eigenvalues of S-L problem with coupled boundary conditions on time scales [4]. The inverse spectral problems for S-L operators on time scales have been studied by Kuznetsova et al. in [12][13][14] and their references. Besides the aforementioned contents, the self-adjoint even-order differential equations on time scales have also been given, e.g., in [5,6] and their references.
In classical S-L problems, the dependence of the eigenvalues on the problem is widely studied by many authors [15][16][17][18][19]. These studies play an important role in the fundamental theory of differential operators and the numerical computation of spectra. For general theory and methods on these problems, the readers may refer to [18,19].
In most recent years, the research on the dependence of the eigenvalues of a differential operator or boundary value problem on the problem has been extended in various aspects. In 2015, Zhang and Wang showed the eigenvalues of an S-L problem with interface conditions and obtained that the eigenvalues depend not only continuously but also smoothly on the coefficient functions, boundary conditions, and interface conditions [20]. In 2016, Zhu and Shi generalized the problem to the discrete case and considered the dependence of eigenvalues of discrete S-L problems on the problem [21]. They also considered the eigenvalue dependence problems for singular S-L problems in [22]. For higher order boundary value problems, there are several literature on the problems, for example, in [23][24][25][26][27][28][29].
In recent years, there has been a lot of interest in the literature on boundary value problems with eigenparameter-dependent boundary conditions, for example, in some physical problems such as heat conduction problems and vibrating string problems. In particular, the spectral problems having boundary conditions depending on the eigenparameter arising in mechanical engineering can be found in the wellknown textbook [30] of Collatz. For such problems arising in applications, including an extensive bibliography and historical notes, also see [31][32][33]. The recent important achievements on such problems can be found in many literature, and here, we only refer to some of them, for example, in [34][35][36][37].
In 2020, Zhang and Li studied the regular S-L problems with eigenparameter-dependent boundary conditions [38]. They obtained that the eigenvalues not only continuously but also smoothly depend on the parameters of the problem, and further, the differential expressions of the eigenvalues with respect to the data are given.
There are many conclusions about the problems of differential equations on time scales and the dependence of eigenvalues of differential equations; however, to our best knowledge, few people have studied the dependence of eigenvalues of differential equations on time scales. Therefore, it is very meaningful to consider the eigenvalue dependence of S-L problems on time scales with spectral parameter boundary conditions. This article is organized as follows. In Section 2, we introduce the problems studied here and show the continuous dependence of the eigenvalues on the problem. In Section 3, the differential properties of eigenvalues with respect to the data of the problem are given, and in particular, the derivative formulas are listed.

Continuous dependence of eigenvalues and eigenfunctions
Before presenting the main results, we recall the following concepts related to time scales for the convenience of the reader. For further knowledge on time scales, the reader may refer to [1][2][3][4][5][6][7][8][9][10] In this article, we use the notation f t f σ t σ ( ) ( ( )) = for any function f defined on a time scale .
Definition 2. For f : → and t ∈ (if t sup = , assume t is not left-scattered), define the Δ-derivative f t Δ ( ) of f t ( ) to be the number, provided it exists, with the property that, for any ε 0, > there is a neighbor- for all s ∈ .
The following formula involving the graininess function is valid for all points at which f t Definition 3. Let f : → be a function. We say that f C rd ∈ if it is continuous at each right-dense point in and f s lim s t ( ) exists as a finite number for all left-dense points in . exists.
Consider the differential equation with boundary conditions αy ρ a α py ρ a cos sin 0, Where a b −∞ < < < ∞, q is real valued function and λ ∈ is the spectral parameter with coefficients satisfying: η γβ γβ ρ a a μ ρ a p ρ a ρ a a π π 0, 0 0, arctan , andˆ0 . In fact, this is obviously true when ρ a a. ( ) = Now we assume ρ a a. ( ) < Then when α 0 = , (2.2) means that y ρ a p ρ a y a y ρ a μ ρ a y ρ a y a y ρ a y a 0 t a n 0 . Let the weighted space be defined as follows: with the inner product for any x x y y , , , .
Let ϕ t λ ( ) and χ t λ ( ) be the fundamental solutions of the S-L equation (2.1) satisfying the initial conditions ϕ ρ a χ ρ a pϕ ρ a pχ ρ a where c c , 1 2 ∈ . If λ is an eigenvalue of the problem (2.1)-(2.5), then there exists c c 0 , T 1 2
According to the aforementioned initial conditions, we have holds. If the solution y t λ , ( ) described by (2.6) is chosen, then y t λ , ( ) satisfies the S-L problem (2.1)-(2.5); hence, y t λ , ( )is an eigenfunction and λ is the eigenvalue of the S-L problem (2.1)-(2.5). This completes the proof. □ Lemma 4. The spectrum of T consists of isolated eigenvalues, which coincide with those of the S-L problem (2.1)-(2.5). Moreover, all eigenvalues are simple, real, bounded below and can be ordered as follows: Proof. Let y t λ , ( ) be a nontrivial solution of the problem (2.1) and (2.2). The eigenvalues of the problem (2.1)-(2.5) are the roots of the equation: Let λ * be a nonreal eigenvalue of the S-L problem (2.1)-(2.5). Then λ * is also an eigenvalue of this problem, since p t q t w t α β β γ , , , ( ) ( ) ( ) , and γ 2 are real; moreover, y t λ y t λ , , . ( ) ( ) = * * Let ν be another eigenvalue of the S-L problem (2.1)-(2.5), by virtue of (2.1), we have py t λ y t ν py t ν y t λ ν λ w t y t ν y t λ , , , , , , .
By integrating this relation from ρ a ( ) to b, and using the formula for the integration by parts, and taking into account the condition (2.2), we obtain Setting ν λ = * and λ λ = * in (2.10), we obtain Since λ * is a root of equation ( Since λ λ ≠ * * , we have the relation which contradicts the condition η 0 > . Therefore, λ ∈ * . The entire function occurring on the left-hand side in equation (2.9) does not vanish for nonreal λ. Consequently, it does not vanish identically. Therefore, its zeros form an at most countable set without finite limit points [18,39].

Differential expressions of eigenvalues
In this section, we shall show the eigenvalues determined in Theorem 1 are differentiable, and in particular, we give the derivative formulas of the eigenvalues for all parameters.  ( ( ) ) ⋅ be the normalized eigenvector. Then Proof. We fix the data except β 1 on ω , and for arbitrary small ε, let u u , , ) be the corresponding eigenvectors for λ β 1 ( ), λ β ε 1 ( ) + , respectively, and here, u u β Dividing both sides of (3.1) by ε and taking the limit as ε 0 → , by Theorem 2, we obtain λ β λ β λ γ u b ,  Dividing both sides of (3.4) by ε and taking the limit as ε 0 → , then by Theorem 2, we obtain λ α α pu ρ a sec . Let h 0 → , then the desired result can be obtained by Theorem 2. Parts 4 and 5 can be proved in the same way, and here, we omit the details. □