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

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 SturmLiouville 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 SturmLiouville 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 problemwith 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: Ly fl − (p (x) y󸀠 (x))󸀠 + q (x) y (x) = λy (x) , x ∈ (a, ξ) ∪ (ξ, b) , −∞ < a < ξ < b < ∞, (1)

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][19][20][21][22]).Based on these results, some researchers studied the regular Sturm-Liouville problems with eigenparameter on both of the boundary conditions (see [23][24][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.S ¸en 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.
For the convenience, we set Moreover, 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 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 .
Moreover, we have the following conclusion.
From the properties of self-adjoint operators, we have the following corollaries.1)-( 4); then the corresponding eigenfunctions () and () are orthogonal in the sense of

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.
Here, we define fundamental solutions (, ) and (, ) of ( 1) by the following procedure: 2 (, ) ,  ∈ (, ) . ( Let  1 (, ) be the solution of (1) on the interval (, ), which satisfies the initial conditions by virtue of Lemma 11, we can define the solution  2 (, ) of ( 1) on (, ) by the initial conditions Analogously, we define the solutions  2 (, ) and  1 (, ) of (1) by the initial conditions ( By the dependence of solutions of initial value problems on the parameter, one has that   () ( = 1, 2) are entire functions of  and are independent of .
Proof.By the definition of   (), we have using the transmission conditions (4), simple computation gives Thus, for each  ∈ C, we have  2 () =  1 ().This completes the proof.
Similarly, we have the following theorem.
Proof.The proof of formulas for  1 (, ) is identical to those of Titchmarsh's proof for (, ) (see [6]), so we only give the proof of formulas for  2 (, ), namely, equality (51); the other equalities are similar.

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

)
Proof.By the definition of  2 (), we have According to the equalities of  2 (, ) and   2 (, ) in Lemma 17, we can obtain the formulas of  2 () in this theorem.