SELF-ADJOINT REALIZATION OF A CLASS OF THIRD-ORDER DIFFERENTIAL OPERATORS WITH AN EIGENPARAMETER CONTAINED IN THE BOUNDARY CONDITIONS

The present paper deals with a class of third-order differential operators with eigenparameter dependent boundary conditions. Using operator theoretic formulation, the self-adjointness of this operator is proved, the properties of spectrum are investigated, its Green function and the resolvent operator are also obtained.


Introduction
Spectral theory of ordinary differential operators with an eigenparameter contained in the boundary conditions has been discussed for a long time. The feature of such problems is that the eigenparameter appears both in the differential equation and boundary conditions. Due to its wide applications in mechanics and mathematical physics, such as electric circuits, mechanical vibrations, acoustic scattering theory etc, [8,10,11,17], more and more researchers have been paid attention to such problems. Moreover, many excellent results of such problems for second-order or fourth-order differential operators have been obtained, such as self-adjointness, asymptotical formula of the eigenvalues and eigenfunctions, oscillation of the eigenfunction, inverse spectral problems and so on, see, for example [1, 2, 4-6, 12, 18-21, 25, 27, 33, 35, 36]. However, little is known for the case of third-order differential operators.
The characterization of self-adjoint boundary conditions is an important part in the differential operators theory. Such problems have been well established for regular or singular differential operators, see [3,24,31,32] and references cited therein. Wang etc characterized the self-adjoint domain of even order differential operators by using real parameter solutions [31] and gave the classification of boundary condi-tions, that is, separated, coupled and mixed [30]. Similar results have been obtained by Hao etc for the case of odd differential operators [15]. Third-order differential equations appear in many physical phenomenons, for example, the deflection of a curved beam varying cross section, three-layer beam and so on [13]. Hao etc in [23] proved that there is no strict separated boundary conditions for self-adjoint third-order differential operators, that is to say, there only mixed and coupled boundary conditions exist for third-order self-adjoint differential operators. Ugurlu investigated a class of third-order differential operators with mixed boundary conditions, and gave the dependence of eigenvalues on the data, and then generalized these results to differential operators with interface conditions [28,29]. There are also many literatures focusing on other issues for the third-order equation, see, for instance, [16,22,26,34].
In the present paper, we study third-order symmetric differential equation with boundary conditions and L 3 y := (sin β + i)y [1] (a) + (i sin β + 1)y [1] where λ is the spectral parameter, q 0 , q 1 , p 0 , p 1 , w satisfy the following conditions α k , α k , β k , β k (k = 1, 2) are arbitrary real numbers and satisfying Here we consider a third-order differential equation with mixed boundary conditions (1.1)-(1.4), where two boundary conditions are of separated, affinely dependent on the eigenparameter and the rest one is of coupled. By using the classical analysis techniques and spectral theory of linear operator, we define a new linear operator T associated with the problem (1.1)-(1.4) in an appropriate Hilbert space H such that the eigenvalues of the problem (1.1)-(1.4) coincide with those of T. The paper is organized as follows: In Section 2, we investigate some basic notations and preliminaries. In Section 3, we introduce a new Hilbert space and construct an operator T associated with the problem (1.1)-(1.4), and discuss the self-adjointness, the properties of eigenvalues of this operator. The Green function and the resolvent operator are discussed in Section 4.

Notations and preliminaries
Let the quasi-derivatives of y be defined as [14] y [0] = y, y [1] = − 1 + i √ 2 q 0 y , y [2] = iq 0 (q 0 y ) + p 0 y − iq 1 y and H w = L 2 w [a, b] be the weighted Hilbert space consisting of functions y which satisfy b a |y| 2 wdx < ∞ under the inner product y, z w = b a yzwdx. Denote by L max the maximal operator with the domain , y [1] , y [2] ∈ AC[a, b], y ∈ L 2 w [a, b]}, and the rule L max y = y, y ∈ D max .
Then for arbitrary y, z ∈ D max , integration by parts yields Lagrange identity By the definition of quasi-derivatives, we can transfer the equation (1.1) to the following first-order system Y + QY = λWY, Then by (1.5),(2.1), we have the following result.

Operator theoretic formulation and self-adjointness
Motivated by Friedman [9], Mukhtarov [20,25] and Fulton [11], we can construct a new Hilbert space H = L 2 w [a, b] C 2 under a suitable inner product by combining the parameters in the boundary conditions. With this goal, the inner product is defined by We shall use the following notations: Define the operator T in the Hilbert space H with domain Then we get that the eigenvalue problem of BVP (1.1)-(1.4) is transferred to the spectra problem of the operator T.
Considering the operator T we have the following properties.
by the inner product in H. Through the arbitrariness of y 1 , then f 1 = 0. Moreover, for all Z = (z(t), z 1 , z 2 ) ∈ D(T), we have By the arbitrariness of z 2 , we have f 2 = 0. Hence F = (0, 0, 0), and the proof is completed.
Proof. For any U, V ∈ D(T), integration by parts yields Therefore, the operator T is symmetric.
Then by (3.9), (iii) holds. Hence, the operator T is selfadjoint. By the self-adjointness of the operator T, we have the following conclusions.
Corollary 3.1. The eigenvalues of T are real-valued.
by (3.11) and (3.12). Since c 1 , c 2 , c 3 are not all zero, det(A λ + B λ Φ(b, λ 0 )) = 0. On the contrary, if det(A λ + B λ Φ(b, λ 0 )) = 0, then the system of the linear equations (3.13) for the constants c i (i = 1, 2, 3) has non-zero solution (c 1 , c 2 , c 3 ).  Proof. The zeros of ∆(λ) are the eigenvalues of operator T by Lemma 3.4, and all the eigenvalues of T are real by the self-adjointness of T, that is to say, for any λ ∈ C with its imaginary part not vanishing, then ∆(λ) = 0. Therefore, by the distribution of zeros of entire functions, the first part holds. The second conclusion follows from the fact that there at most 3 linearly independent solutions exist for the equation (1.1).
The operator (T − λI) −1 is defined in the whole space by Theorem 4.1. It follows from the facts T is symmetric and Closed Graph Theorem that (T − λI) −1 is bounded. Therefore, λ is a regular point of T provided that it is not an eigenvalue of T.
Theorem 4.2. The operator T has only point spectrum, that is to say, σ(T) = σ p (T).