Eigenvalues of Sturm-Liouville Problems with Eigenparameter Dependent Boundary and Interface Conditions

. In this paper, a regular discontinuous Sturm-Liouville problem which contains eigenparameter in both boundary and interface conditions is investigated. Firstly, a new operator associated with the problem is constructed, and the self-adjointness of the operator in an appropriate Hilbert space is proved. Some properties of eigenvalues are discussed. Finally, the continuity of eigenvalues and eigenfunctions is investigated, and the differential expressions in the sense of ordinary or Fr´ e chet of the eigenvalues concerning the data are given.


Introduction
For the classical Sturm-Liouville problems, the eigenparameter λ generally only appears in differential equation. However, for many practical problems in physics, engineering, and other fields, the corresponding mathematical models require that the eigenparameter λ not only appears in differential equation, but also in boundary conditions. For the classical Sturm-Liouville problems with eigenparameter dependent boundary conditions, there have been a sea of research results, (see, for example, [3,7,8,9] and references cited therein).
Recently, there has been increasing interest in Sturm-Liouville problems with discontinuity and eigenparameter dependent boundary conditions, that is, a discontinuous point appears in the interval, namely, problems are considered in two disjoint intervals. To deal with such problems, some conditions are imposed on these points, which are also called transmission conditions, interface conditions etc [20,30]. Such problems arise in many problems of physics and mechanics [14,24]. For example, heat and mass transfer problems. For such problems, many researchers study the asymptotic of eigenvalues and eigenfunctions, inverse problems, the completeness of eigenfunctions and resolvent and so on. Many important results have been obtained for this kind of problems (see [1,2,4,13,15,16,17,18,19,20,22,23,25,27,30]).
The dependence of eigenvalues for regular or singular Sturm-Liouville operators has been well investigated in recent years, see [26,28,29]. This kind of problems consists of a certain second-order differential expression with selfadjoint boundary conditions, and study the continuity and differentiability of eigenvalues with respect to the given parameters appear in the equation and boundary conditions. These results provided a fundamental to the numerical computation of spectrum, for example, the codes SLEUTH and SLEIGN2 constructed by Greenberg, Bailey et al. [6].
Kong and Zettl in [12] showed the continuity of eigenvalue of regular Sturm-Liouville problems and each eigenvalue is differentiable with respect to a given parameter. Such a problem gained various generalizations since then, for example, Sturm-Liouville operators with interface conditions, third-order and fourthorder differential operators, and general even order case, etc. [10,13,21,30]. Particularly, in recent papers, we generalized these results to third-order differential operators with eigen-dependent boundary conditions [5], and Ao et al. considered the case of third-order differential operators with discontinuity [27]. Inspired by the above results, we consider the following differential equation Then we consider a discontinuous Sturm-Liouville problem with eigenparameter contained in both boundary and interface conditions (1.1)-(1.5). Noting that the problem (1.1)-(1.5) can be gotten by using the method of separation of variables to various physical problems in some special cases. For example, some boundary-value problems arising in diffraction problems etc. (see [14,24]). Using operator theory and analysis technique, the problem (1.1)-(1.5) is transferred to a self-adjoint operator in a proper Hilbert space. We introduce some properties of eigenvalues and eigenfunctions of this operator. Moreover, we introduce the dependence of eigenvalues of the problem (1.1)-(1.5) with respect to the parameters given in the problem. The outline of this paper is arranged as follows: in Section 2, a new Hilbert space related to the problem is constructed, and a new operator is defined in this space such that the eigenvalues of the problem are consistent with the eigenvalues of this operator. The fundamental solutions are constructed, and it is proved that the new operator is self-adjoint, and the simplicity of the eigenvalue is proved. The continuity of eigenvalues and eigenfunctions are proved in Section 3, followed by the differential expressions of eigenvalues about each parameter are given in Section 4.

Preliminaries and basic results
In this section, we define a new Hilbert space For convenience, we shall use the following notations: So, the problem (1.1)-(1.5) can be transformed into the following form Then we have the following lemmas.
We can obtain that m(x) is orthogonal to − W (m, n; c+)+ In addition, it is easy to prove that (1) w(x), (pw ′ )(x) ∈ AC(I) and lw ∈ L 2 (I); that is, ⟨lm, w⟩ 1 = ⟨m, t⟩ 1 . According to classical Sturm-Liouville theory, (1) and (4) hold. By (4) By Naimark's Patching Lemma, there is an M ∈ D(T ) satisfying We have w 3 = δw(c−), hence (2) is true. (5) can be proved in the same way. Further, let M ∈ D(T ) and satisfy We have w(c+) − γ 1 w(c−) = 0. Consequently, the operator T is self-adjoint. ⊓ ⊔ Corollary 1. All eigenvalues of the problem (1.1)-(1.5) are real, and for two different eigenvalues, the corresponding eigenfunctions m(x) and n(x) are orthogonal in the following sense In what follows, we define two fundamental solutions of Equation (1.1) We can define the solution θ 2 (x, λ) of Equation (1.1) on the interval (c, b] by the initial conditions Similarly, define the solution η 2 (x, λ) and η 1 (x, λ) by the initial conditions Let us consider the Wronskians where w 1 , w 2 are entire functions of λ on the interval [a, c) and (c, b]. Proof. According to (2.5)-(2.8), by simply calculation we can get Proof. Using similar methods proposed in [1], we can prove the assertion. ⊓ ⊔ Corollary 2. Suppose λ = λ 0 is an eigenvalue, then θ(x, λ 0 ) and η(x, λ 0 ) are linearly independent.
Proof. The proof can be completed by using similar methods in [4], hence we omit it here. ⊓ ⊔

Continuity of eigenvalues and eigenfunctions
In this section, we prove continuity of the eigenvalues and normalized eigenfunctions for the problem (1.1)-(1.5). Denote Consider a Banach space equipped with the norm Let Ω = {Z ∈ B : (1.6)-(1.7) hold}. When considering the variables in the parameter matrix of boundary conditions separately, we use the symbol Then we get the continuous dependence of the eigenvalues on the parameters in the SL problems.
Theorem 4. Let Z = ( 1 p ,q,ω,Ã,B,γ 1 ,γ 2 ,δ,ã,b,c −,c +) and λ(Z) be an eigenvalue of (1.1)-(1.5) with Z. Then, λ is continuous at Z. That is, give any ε > 0 sufficiently small, there exists a σ > 0 such that |λ( Proof. By Theorem 2, λ is an eigenvalue of (1.1)-(1.5) if and only if w(Z, λ(Z)) = 0, for any Z ∈ Ω. It is easy to get that w(Z, λ) is an entire function of λ and is continuous in Z. By Corollary 3, we get that λ(Z) is an isolated eigenvalue, then w(Z, λ) is not a constant. By the well-known theorem on continuity of the roots of an equation, the statements follows. ⊓ ⊔ Next, we give the continuity of the corresponding eigenvector.
Theorem 5. Let λ( Z) be an eigenvalue of problems (1.1)-(1.5) with Z ∈ Ω and (f, f 1 , f 2 , f 3 ) ∈ H be a normalized eigenvector for Z. Then there exists a normalized eigenvector (g, g 1 , g 2 , g 3 ) ∈ H for λ(Z) with Z ∈ Ω, which is specified in Theorem 4, such that when Z → Z ∈ Ω, we have and where Y and V (c 0 , Z) are linearly independent vectors. Let W (x) be the vector solutions of (1.1) with Z = Z, λ = λ( Z) and the initial condition Since Y (·, Z k ) satisfies the conditions Taking the limit k → ∞, we have Therefore, W (x) is a vector eigenfunction for Z = Z, λ = λ( Z), which contradicts that λ( Z) is simple. Thus, (3.1) holds.

Differential expression of eigenvalues
In this section we show that the eigenvalues are differentiable functions of all the parameters of the problem.
Theorem 6. Let Z = (K, M, γ 1 , γ 2 , δ, 1 p , q, ω) ∈ Ω with λ = λ(Z) be an eigenvalue of operator T connected with Z, and let (u, u 1 , u 2 , u 3 ) be a normalized eigenvector for λ(Z). Then λ is differential with respect to all the parameters in Z, and more precisely, the derivative formulas of λ are given as follows: (1) Fix all the parameters of Z except the boundary condition (1.2) parameter matrix and let λ(K) := λ(Z). Then, for all H satisfying det(K + L) = det K = ρ 1 .
(2) Fix all the parameters of Z except the boundary condition (1.3) parameter matrix  Proof. Fix all but one of the parameters in Z and let λ( Z) be the eigenvalue satisfying Theorem 6 when ∥Z − Z∥ ≤ ε for sufficiently small ε > 0. For the above five cases, we replace λ( Z) by It follows from (4.1) and (4.2) that Integrating from a to c and c to b, then we have . Then, it follows from (1.2) that Therefore, It follows from (1.3) that Therefore, It follows from (1.4) that Therefore, By (4.3)-(4.6) we have the following Let L → 0, the desired result can be obtained by Theorem 3. Similarly, we can get that (2) is also true.