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Asymptotic expressions of eigenvalues and fundamental solutions of a discontinuous fourth-order boundary value problem

In the present paper, we deal with a fourth-order boundary value problem problem with eigenparameter dependent boundary conditions and transmission conditions at a interior point. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of such a problem coincide with those of A. Following Mukhtarov and his students methods [2,4,6] we obtain asymptotic formulae for its eigenvalues and fundamental solutions. Our applications possess a number of interesting properties for studying in boundary value problems which we state in this paper.


Introduction
It is well-known that many topics in mathematical physics require the investigation of eigenvalues and eigenfunctions of Sturm-Liouville type boundary value problems. In recent years, more and more researchers are interested in the discontinuous Sturm-Liouville problem (see [1][2][3][4][5][6]). Various physics applications of this kind problem are found in many literatures, including some boundary value problem with transmission conditions that arise in the theory of heat and mass transfer (see [8,9]). The literature on such results is voluminous and we refer to [1][2][3][4][5][6][7][8][9][10][11].
Fourth-order discontinuous boundary value problems with eigen-dependent boundary conditions and with two supplementary transmission conditions at the point of discontinuity have been investigated in [12,13]. Note that discontinuous Sturm-Liouville problems with eigen-dependent boundary conditions and with four supplementary transmission conditions at the points of discontinuity have been investigated in [3].

Preliminaries
Firstly we define the inner product in L 2 for every f, g ∈ L 2 (I) as It is easy to see that L 2 (I) , [·, ·] is a Hilbert space. Now we define the inner product in the direct sum of spaces L 2 (I) is the direct sum of modified Krein spaces. A fundamental symmetry on the Krein space is given by Consequently, the considered problem (1.1)-(1.9) can be rewritten in operator form as AF = λF , i.e., the problem (1.1)-(1.9) can be considered as the eigenvalue problem for the operator A. Then, we can write the following conclusions: and for each x ∈ [−1, 1] , u (x, λ) is an entire function of λ.
Proof. Let φ 11 (x, λ) be the solution of Eq. (1.1) on [−1, 0) which satisfies the initial conditions By virtue of Lemma 3.1, after defining this solution, we may define the solution φ 12 (x, λ) of Eq. (1.1) on (0, 1] by means of the solution φ 11 (x, λ) by the initial conditions After defining this solution, we may define the solution φ 21 (x, λ) of Eq. (1.1) on [−1, 0) which satisfies the initial conditions After defining this solution, we may define the solution φ 22 (x, λ) of Eq. (1.1) on (0, 1] by means of the solution φ 21 (x, λ) by the initial conditions Analogically we shall define the solutions χ 11 (x, λ) and χ 12 (x, λ) by the initial conditions Moreover, we shall define the solutions χ 21 (x, λ) and χ 22 (x, λ) by the initial conditions

Asymptotic formulae for eigenvalues and fundamental solutions
We start by proving some lemmas.
Proof. Regard φ 11 (x, λ) as the solution of the following non-homogeneous Cauchy problem: Using the method of constant changing, φ 11 (x, λ) satisfies Then differentiating it with respect to x, we have (4.1) . The proof for (4.2) , Each of these asymptotic formulae holds uniformly for x as |λ| → ∞.

Proof. Let
It is easy to see that F 11 (x, λ) is bounded. Therefore φ 11 (x, λ) = O (e) . Substituting it into (4.1) and differentiating it with respect to x for k = 0, 3, we obtain (4.5). According to transmission conditions (1.4)-(1.7) as |λ| → ∞, we get Substituting these asymptotic formulae into (4.2) for k = 0, we obtain Substituting this back into the integral on the right side of (4.7) yields (4.6) for k = 0. The other cases may be considered analogically.
where k = 0, 3. Each of these asymptotic formulae holds uniformly for x.
Since the eigenvalues coincide with the zeros of the entire function W (λ), it follows that they have no finite limit. Moreover, we know from Corollary 4.5 that all real eigenvalues are bounded below. Hence, we may renumber them as λ 0 ≤ λ 1 ≤ λ 2 ≤ ..., listed according to their multiplicity. Proof. By applying the well-known Rouché's theorem, which asserts that if f (s) and g (s) are analytic inside and on a closed contour C, and |g (s)| < |f (s)| on C, then f (s) and f (s) + g (s) have the same number zeros inside C provided that each zero is counted according to their multiplicity, we can obtain these conclusions.