Research Article | Open Access
Erdoğan Şen, "A Sturm-Liouville Problem with a Discontinuous Coefficient and Containing an Eigenparameter in the Boundary Condition", Physics Research International, vol. 2013, Article ID 159243, 9 pages, 2013. https://doi.org/10.1155/2013/159243
A Sturm-Liouville Problem with a Discontinuous Coefficient and Containing an Eigenparameter in the Boundary Condition
We study a Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at two interior points. We give an operator-theoretic formulation, construct fundamental solutions, investigate some properties of the eigenvalues and corresponding eigenfunctions of the discontinuous Sturm-Liouville problem and then obtain asymptotic formulas for the eigenvalues and eigenfunctions and find Green function of the discontinuous Sturm-Liouville problem.
In recent years, more and more researchers are interested in the discontinuous Sturm-Liouville problem for its application in physics (see [1–16]). Such problems are connected with discontinuous material properties, such as heat and mass transfer, varied assortment of physical transfer problems, vibrating string problems when the string loaded additionally with point masses, and diffraction problems. Moreover, there has been a growing interest in Sturm-Liouville problems with eigenparameter-dependent boundary conditions; that is, the eigenparameter appears not only in the differential equations but also in the boundary conditions of the problems (see [1–16] and corresponding bibliography).
In this paper, following  we consider the boundary value problem for the differential equation for (i.e., belongs to , but the two inner points are and ), where is a real-valued function, continuous in , , and with the finite limits , ; is a discontinuous weight function such that for , for , and for , together with the standard boundary condition at the spectral parameter-dependent boundary condition at and the four transmission conditions at the points of discontinuity and in the Hilbert space , where is a complex spectral parameter; and all coefficients of the boundary and transmission conditions are real constants. We assume that and . Moreover, we will assume that . Some special cases of this problem arise after application of the method of separation of variables to the diverse assortment of physical problems, heat and mass transfer problems (e.g., see ), vibrating string problems when the string was loaded additionally with point masses (e.g., see ), and a thermal conduction problem for a thin laminated plate.
2. Operator-Theoretic Formulation of the Problem
In this section, we introduce a special inner product in the Hilbert space and define a linear operator in it so that the problem (1)–(5) can be interpreted as the eigenvalue problem for . To this end, we define a new Hilbert space inner product on by for and . For convenience, we will use the notations In this Hilbert space, we construct the operator with domain which acts by the rule Thus we can pose the boundary value transmission problem (1)–(7) in as It is readily verified that the eigenvalues of coincide with those of the problem (1)–(7).
Theorem 1. The operator is symmetric.
Proof. Let and be arbitrary elements of . Twice integrating by parts, we find where, as usual, denotes the Wronskian of and ; that is, Since , the first components of these elements, that is, and , satisfy the boundary condition (2). From this fact, we easily see that Further, as and also satisfy both transmission conditions, we obtain Moreover, the direct calculations give Now, inserting (15)–(17) in (13), we have and so is symmetric.
3. Asymptotic Formulas for Eigenvalues and Fundamental Solutions
Let us define fundamental solutions of (1) by the following procedure. We first consider the next initial value problem: By virtue of [1, Theorem 1.5], the problem (20)–(22) has a unique solution which is an entire function of for each fixed . Similarly, have a unique solution which is an entire function of for each fixed . Continuing in this manner, have a unique solution which is an entire function of for each fixed . Slightly modifying the method of [1, Theorem 1.5], we can prove the initial value problem have a unique solution which is an entire function of spectral parameter for each fixed . Similarly, have a unique solution which is an entire function of for each fixed . Continuing in this manner, have a unique solution which is an entire function of for each fixed .
By virtue of (21) and (22), the solution satisfies the first boundary condition (2). Moreover, by (24), (25), (27), and (28), satisfies also transmission conditions (4)–(7). Similarly, by (30), (31), (33), (34), (36), and (37) the other solution satisfies the second boundary condition (3) and transmission conditions (4)–(7). It is well known from the theory of ordinary differential equations that each of the Wronskians , , and is independent of in , , and , respectively.
Lemma 3. The equality holds for each .
Corollary 4. The zeros of , , and coincide.
In view of Lemma 3, we denote , , and by . Recalling the definitions of and , we infer the next corollary.
Corollary 5. The function is an entire function.
Proof. Let . Then, for all . Consequently, the functions and are linearly dependent; that is, , , for some . By (21) and (22), from this equality, we have and so satisfies the first boundary condition (2). Recalling that the solution also satisfies the other boundary condition (3) and transmission conditions (4)–(7), we conclude that is an eigenfunction of (1)–(7); that is, is an eigenvalue. Thus, each zero of is an eigenvalue. Now, let be an eigenvalue, and let be an eigenfunction with this eigenvalue. Suppose that , whence , , and . From this, by virtue of the well-known properties of Wronskians, it follows that each of the pairs , ; , and , is linearly independent. Therefore, the solution of (1) may be represented as where at least one of the coefficients is not zero. Considering the true equalities as the homogenous system of linear equations in the variables and taking (24), (25), (27), (28), (33), (34), (36), and (37) into account, we see that the determinant of this system is equal to , and so it does not vanish by assumption. Consequently, the system (41) has the only trivial solution . We thus get at a contradiction, which completes the proof.
Theorem 7. Let and . Then, the following asymptotic equalities hold as : for and . Moreover, each of these asymptotic equalities holds uniformly for .
Proof. Asymptotic formulas for and are found in [9, Lemma 1.7] and [8, Theorem 3.2], respectively. But the formulas for need individual considerations, since this solution is defined by the initial condition with some special nonstandard form. The initial-value problem (26)–(28) can be transformed into the equivalent integral equation Inserting (43) in (45), we have Multiplying this by and denoting , we have the next “asymptotic integral equation”: Putting , from the last equation, we derive that for some . Consequently, as , and so = as . Inserting the integral term of (46) yields (44) for . The case of (44) follows at once on differentiating (43) and making the same procedure as in the case .
Proof. Putting in and inserting , , we have the following representation for : Putting in (44) and inserting the result in (50), we derive now that By applying the well-known Rouché theorem which asserts that if and are analytic inside and on a closed contour and on , then and have the same number of zeros inside provided that the zeros are counted with multiplicity on a sufficiently large contour, it follows that has the same number of zeros inside the contour as the leading term in (51). Hence, if are the zeros of and , we have for sufficiently large , where for sufficiently large . By putting in (51), we have , and the proof is completed.
Proof. Inserting (44) in the integral term of (51), we easily see that Inserting in (42) yields We already know that all eigenvalues are real. Furthermore, putting , in (51), we infer that as , and so for sufficiently large . Consequently, the set of eigenvalues is bounded below. Letting in (56), we now obtain since for sufficiently large . After some calculation, we easily see that Consequently, In a similar method, we can deduce that Thus is the proof of the theorem, completed.
4. Asymptotic Formulas for Normalized Eigenfunctions
We will first obtain the norms of the eigenelements It is evident that the two-component vectors are the eigenelements of with eigenvalue . For , since is symmetric. Putting we see easily that the eigenelements are orthonormal. That is, and , where is the Kronecker delta.
Lemma 10. The following asymptotic equality holds:
Theorem 11. Let be defined as in (61). Then, the following asymptotic formula holds for the norms of the eigenelements
Theorem 12. The first components of the normalized eigenelements (65) have the following asymptotic representation as :
5. Green's Function
Let be defined by (10) and (11), and let not be an eigenvalue of . For finding the resolvent operator , consider the operator equation for . This operator equation is equivalent to the inhomogeneous differential equation subject to the inhomogeneous boundary conditions and the homogeneous transmission conditions By applying the same techniques as in , we can prove that the problem (80)–(82) has a unique solution which can be represented asPutting where and , we reduce (73) to On the other hand, by applying to Green's function with respect to the variable and recalling that satisfies the initial conditions (30) and (31), we have Inserting in (85) yields We now introduce which we will call Green’s element of (80)–(82). The last formula (87) then takes the form where by we mean .
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