A Singular Sturm-Liouville Problem with Limit Circle Endpoints and Eigenparameter Dependent Boundary Conditions

Cai, Jinming; Zheng, Zhaowen

doi:https://doi.org/10.1155/2017/9673846

Discrete Dynamics in Nature and Society

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Volume 2017 | Article ID 9673846 | https://doi.org/10.1155/2017/9673846

A Singular Sturm-Liouville Problem with Limit Circle Endpoints and Eigenparameter Dependent Boundary Conditions

Jinming Cai¹and Zhaowen Zheng¹

Academic Editor: Chris Goodrich

Received05 Dec 2016

Accepted23 Jan 2017

Published21 Mar 2017

Abstract

In this paper, we investigate a class of discontinuous singular Sturm-Liouville problems with limit circle endpoints and eigenparameter dependent boundary conditions. Operator formulation is constructed and asymptotic formulas for eigenvalues and fundamental solutions are given. Moreover, the completeness of eigenfunctions is discussed.

1. Introduction

It is well known that many topics in mathematical physics require the investigation of eigenvalues and eigenfunctions of the Sturm-Liouville problems. The theory of regular Sturm-Liouville problems is well built; since the foundation work of Weyl on limit-point/limit-circle classification [1], the singular Sturm-Liouville problems (see [2–7] for real coefficients and [8] for complex coefficients) and more general Hamiltonian systems (see [9, 10]) are widely researched. Meanwhile, a large number of researchers are interested in the discontinuous Sturm-Liouville problem with inner discontinuous points, since these problems are of wide applications in engineering and mechanics (see [11–25]). Various physics applications of this kind of problems are found, such as oscillation of linear or nonlinear equation (see [26–29]) and heat and mass transfer problems (see [30]).

The regular Sturm-Liouville problems with transmission conditions containing an eigenparameter on one of the boundary conditions have received a lot of attention in research (see [18–22]). Based on these results, some researchers studied the regular Sturm-Liouville problems with eigenparameter on both of the boundary conditions (see [23–25]). In these papers, Yang and Wang in [18] considered a Sturm-Liouville problem with discontinuities at two points and eigenparameter dependent boundary condition at one endpoint; they obtained the fundamental solutions and gave the asymptotic formulas of eigenvalues and fundamental solutions. Further, they studied the discontinuous Sturm-Liouville problem with eigenparameter boundary conditions at two endpoints in [24] and extended the results of [18] to finite discontinuities case. In papers [19, 22, 23, 25], the authors obtained the estimations of eigenvalues and eigenfunctions of the discontinuous Sturm-Liouville problem with one inner point, containing an eigenparameter in the boundary condition. Şen et al. considered the Sturm-Liouville problem with two inner points containing an eigenparameter in the boundary condition and got similar result, respectively (see [20, 21]). Besides, the authors also discussed the completeness of the eigenfunctions of a regular discontinuous Sturm-Liouville problem in papers [18, 25]. All of them researched the regular Sturm-Liouville problem. However, little is known about the singular Sturm-Liouville problems with limit-circle endpoints.

We will consider the following singular discontinuous Sturm-Liouville problem with two limit-circle endpoints and eigenparameter in the boundary conditions:with eigenparameter dependent conditions at the endpoints and :the transmission conditions at are(we assume that not all are equal to zero, since, in this case, the boundary value problem (1)–(3) has no transmission conditions), where both and are limit-circle points, and are linearly independent real-valued solutions of equation on , and and are linearly independent real-valued solutions of equation on and satisfy , , where is the sesquilinear form; for , for , and is a spectral parameter; is a real-valued continuous function on and has finite limits ; , and are nonzero real numbers.

For the convenience, we setMoreover, we assume that

Here, we research a singular Sturm-Liouville problem with two limit-circle endpoints and the parameter is not only in the equation but also in the boundary conditions. Based on the modified inner product, we define a new self-adjoint operator such that the eigenvalues of such a problem are coincided with those of . We rebuild its fundamental solutions, get the asymptotic formulas for eigenvalues and eigenfunctions, and also discuss the completeness of its eigenfunctions.

At first, we introduce the following lemmas.

Lemma 1 (the Lagrange identity, see [5]). Let , where, for any interval , denotes the set of functions on which are square integrable on . For any , , one has where is the adjoint expression of , which is given by .

Lemma 2 (Green’s formula, see [5]). Let ; for arbitrary , and are defined as above; one has Since is of limit-circle type at , and exist . So exists. Similarly and exist.

Lemma 3 (see [5]). For any , while is defined in Lemma 1, one has

2. Operator Formulation

In this section, we introduce the Hilbert space ; the inner product on is defined by for . For the convenience, we use the following notations: , Besides, we introduce the operator in the Hilbert space as follows: which acts by the rulewith Now we can rewrite problem (1)–(4) in the operator form , for . Obviously, we have the following lemmas.

Lemma 4. The eigenvalues and eigenfunctions of problem (1)–(4) are corresponding to the eigenvalues and the first component of the corresponding eigenfunctions of operator , respectively.

Lemma 5. The domain is dense in .

Proof. Let and let be a functional set which has compact support and can be differential infinitely such that . Since , for is orthogonal to , namely, so . For all , , since is arbitrary. Similarly, one gets . Above all, one has which proves the assertion.

Definition 6. Let denote the Wronskian of functions and . One has

Theorem 7. Operator is symmetric.

Proof. For each , by the definition of inner product and operator , we can getthen (4) implies thatusing boundary condition (2), we getand, by boundary condition (3), we haveSubstituting (17)–(19) into (16) yields . This completes the proof.

Moreover, we have the following conclusion.

Theorem 8. Operator is self-adjoint.

Proof. is self-adjoint if and only if, for each , for some implying that and , where . Concretely, we should prove that the following properties hold:(i) are absolutely continuous on .(ii), .(iii).(iv).(v); .For an arbitrary point , we have that is, . According to the classical Sturm-Liouville theory, (i) and (iv) hold. So .Besides, we haveSubstituting (22) into (21), we getBy Naimark’s Patching Lemma [4], there exists such thatSubstituting (24) into (23), we have . Further, there exists such thatAnalogously, we can get . So (ii) holds. Similarly, one proves (v). Next, let such thatThen, by (23), we have . Similarly, we can get . So is a self-adjoint operator.

From the properties of self-adjoint operators, we have the following corollaries.

Corollary 9. All eigenvalues of the singular Sturm-Liouville problem (1)–(4) are real.

Corollary 10. Let and be two different eigenvalues of the singular Sturm-Liouville problem (1)–(4); then the corresponding eigenfunctions and are orthogonal in the sense of

3. Asymptotic Approximation of Fundamental Solutions

In this section, we construct the fundamental solutions of problem (1)–(4) and get the asymptotic approximation for fundamental solutions.

Lemma 11 (see [5]). Let the real-valued function be continuous on , and let be given entire functions. Then, for any , the equation has unique solution satisfying the initial conditions For each fixed is an entire function of .

Here, we define fundamental solutions and of (1) by the following procedure:

Let be the solution of (1) on the interval , which satisfies the initial conditionsby virtue of Lemma 11, we can define the solution of (1) on by the initial conditionsAnalogously, we define the solutions and of (1) by the initial conditions

Now, we consider Wronskian By the dependence of solutions of initial value problems on the parameter, one has that are entire functions of and are independent of .

Lemma 12. For every ,

Proof. By the definition of , we have using the transmission conditions (4), simple computation gives Thus, for each , we have . This completes the proof.

Besides, we set .

Theorem 13. The eigenvalues of problem (1)–(4) coincide with the zeros of the function .

Proof. Let be any eigenfunction corresponding to eigenvalue ; then the function may be represented in the form where at least one of the constants is not zero. We should show that . Suppose to the contrary that there exists such that . Since eigenfunction satisfies both boundary and transmission conditions (2)–(4), we have , while the determinant of coefficient matrix is not zero, so we obtain , which is a contradiction; then . Conversely, let be a zero of function ; then ; therefore, , for some . Since both and satisfy the boundary condition (3),satisfies problem (1)–(4). So function is an eigenfunction of problem (1)–(4) corresponding to eigenvalue . This completes the proof.

Theorem 14. The eigenvalues of problem (1)–(4) are analytically single.

Proof. Let , and we use the following notions for simplicity: , , and . We differentiate the equation with respect to to obtainUsing integration by parts, we getSubstituting (39) and into the left side of (40), we have Moreover, By (31), we observe that then (40) becomesNext, let be an arbitrary zero of . Since , we obtain , . Noting that is real, a short calculation (44) becomes Since , , , and , . Hence, the analytic multiplicity of is one, which completes the proof.

Lemma 15. Let . Then the following integral and differential equations hold for and :

Proof. For the case of , consider as the solution of the following nonhomogeneous problem:Using the method of constant variation, satisfies Then, differentiating it with respect to , we have (46). The proof for (47) is similar, so we omit the details.

Similarly, we have the following theorem.

Lemma 16. Let . Then the following integral and differential equations hold for and :

Lemma 17. Let . Then, for , have the following estimations.
Case 1. If , thenCase 2. If , thenEach of these asymptotic equalities holds uniformly for as .

Proof. The proof of formulas for is identical to those of Titchmarsh’s proof for (see [6]), so we only give the proof of formulas for , namely, equality (51); the other equalities are similar.
For , by the estimations of and , we have then, substituting (53) into (47) and observing (32), we have differentiating (54) with respect to , we have (51).

Lemma 18. Let . Then, for , have the following estimations.
Case 1. If , thenCase 2. If , then Each of these asymptotic equalities holds uniformly for as .

Theorem 19. Let , . Then the function has the following asymptotic representations.
Case 1. If and , thenCase 2. If and , thenCase 3. If and , thenCase 4. If and , then

Proof. By the definition of , we have According to the equalities of and in Lemma 17, we can obtain the formulas of in this theorem.

Corollary 20. The eigenvalues of the boundary value problem (1)–(4) are bounded below.

Proof. Putting in Theorem 19, we can obtain (). Hence, for sufficiently negative and sufficiently large . This completes the proof.

4. Asymptotic Formulas for Eigenvalues and Eigenfunctions

In this section, we can get the asymptotic formulas for the eigenvalues and eigenfunctions of the singular Sturm-Liouville problem (1)–(4). Since the eigenvalues coincide with the zeros of the entire function , it follows that they have no finite limits.

Theorem 21. The eigenvalues () of problem (1)–(4) have the following asymptotic representations as .
Case 1. If and , thenCase 2. If and , thenCase 3. If and , thenCase 4. If and , then

Proof. By applying the well-known Rouche theorem, we can obtain these conclusions (see [3] Theorem ).

According to Theorem 21 and Lemmas 17 and 18, we can obtain the following asymptotic representations of the eigenfunctions and .

Theorem 22. The eigenfunctions and of problem (1)–(4) have the following asymptotic representations as .
Case 1. If and , thenCase 2. If and , thenCase 3. If and , thenCase 4. If and , then

5. Completeness of Eigenfunctions

In this section, we get the property of spectrum for the operator and discuss the completeness of the eigenfunctions of problem (1)–(4).

Theorem 23. The operator has only point spectrum; that is, .

Proof. We only need to prove that if is not an eigenvalue of , then . Here we investigate the equation , where , , and . Consider the initial value problemLet be the solution of the equation satisfying the transmission conditions (4). Letbe the solution of the equation satisfying Then (70) has the general solutionwhere .
As is not an eigenvalue of problem (1)–(4), we haveThe second and third components of mean thatSubstituting (74) into (76), we can get that is uniquely solvable. So is uniquely determined. Observing that is defined on all of , we get that is bounded by Theorem 8 and the closed graph theorem. Thus, . Hence, .

Lemma 24. The eigenvalues of problem (1)–(4) are countably infinite and can cluster only at .

Lemma 25. The operator has compact resolvent; that is, for each , is compact on (see [14] Theorem ).

By the above lemmas and the spectral theorem for compact operator, we obtain the following theorem.

Theorem 26. The eigenfunctions of problem (1)–(4), expanded to become eigenfunctions of , are complete in ; that is, let be a maximum set of orthogonal eigenfunctions of , where are eigenfunctions of problem (1)–(4). Then, for all , .

Remark 27. In this paper, the spectral properties of singular Sturm-Liouville problems with one inner discontinuous point are considered; if there are two or even multi-inner discontinuous points in the interval , we can obtain similar results by defining a more complicated Hilbert space.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

Jinming Cai prepared the manuscript and corrected the main results and Zhaowen Zheng gave the main thought and revised the manuscript.

Acknowledgments

This research was partially supported by the NSF of China (Grants 11271225 and 11671227).

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Copyright © 2017 Jinming Cai and Zhaowen Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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