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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012, Article IDΒ 456517, 28 pages
Research Article

Spectral Properties of the Differential Operators of the Fourth-Order with Eigenvalue Parameter Dependent Boundary Condition

1Department of Mathematics, Baku State University, Baku AZ 1148, Azerbaijan
2Department of Mathematics, Mersin University, 33343 Mersin, Turkey

Received 15 August 2011; Accepted 12 November 2011

Academic Editor: AminΒ Boumenir

Copyright Β© 2012 Ziyatkhan S. Aliyev and Nazim B. Kerimov. 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.


We consider the fourth-order spectral problem 𝑦(4)(π‘₯)βˆ’(π‘ž(π‘₯)𝑦′(π‘₯))ξ…ž=πœ†π‘¦(π‘₯),π‘₯∈(0,𝑙) with spectral parameter in the boundary condition. We associate this problem with a selfadjoint operator in Hilbert or Pontryagin space. Using this operator-theoretic formulation and analytic methods, we investigate locations (in complex plane) and multiplicities of the eigenvalues, the oscillation properties of the eigenfunctions, the basis properties in 𝐿𝑝(0,𝑙), π‘βˆˆ(1,∞), of the system of root functions of this problem.

1. Introduction

The following boundary value problem is considered:𝑦(4)ξ€·(π‘₯)βˆ’π‘ž(π‘₯)π‘¦ξ…žξ€Έ(π‘₯)ξ…žπ‘‘=πœ†π‘¦(π‘₯),π‘₯∈(0,𝑙),β€²βˆΆ=,𝑑π‘₯(1.1)β€‰π‘¦ξ…žπ‘¦(0)=0,(1.2a)𝑦(0)cos𝛽+𝑇𝑦(0)sin𝛽=0,(1.2b)ξ…ž(𝑙)cos𝛾+π‘¦ξ…žξ…ž((𝑙)sin𝛾=0,(1.2c)π‘Žπœ†+𝑏)𝑦(𝑙)βˆ’(π‘πœ†+𝑑)𝑇𝑦(𝑙)=0,(1.2d)where πœ† is a spectral parameter, π‘‡π‘¦β‰‘π‘¦ξ…žξ…žξ…žβˆ’π‘žπ‘¦ξ…ž, π‘ž is absolutely continuous function on [0,𝑙], 𝛽, 𝛾, π‘Ž, 𝑏, 𝑐, and 𝑑 are real constants such that 0≀𝛽, π›Ύβ‰€πœ‹/2 and 𝜎=π‘π‘βˆ’π‘Žπ‘‘β‰ 0. Moreover, we assume that the equationπ‘¦ξ…žξ…žβˆ’π‘žπ‘¦=0,(1.3) is disfocal in [0,𝑙], that is, there is no solution of (1.3) such that 𝑦(π‘Ž)=𝑦′(𝑏)=0 for any π‘Ž,π‘βˆˆ[0,𝑙]. Note that the sign of π‘ž which satisfies the disfocal condition may change in [0,𝑙].

Problems of this type occur in mechanics. If 𝛽=0, 𝛾=πœ‹/2, 𝑏=𝑐=0, and 𝑑=1 in the boundary conditions, then the problem (1.1), (1.2a)–(1.2d) arises when variables are separated in the dynamical boundary value problem describing small oscillations of a homogeneous rod whose left end is fixed rigidly and on whose right end a servocontrol force in acting. In particular, the case when π‘Ž<0 corresponds to the situation where this is a particle of mass π‘Ž at the right end of the rod. For more complete information about the physical meaning of this type of problem see [1–3].

Boundary value problems for ordinary differential operators with spectral parameter in the boundary conditions have been considered in various formulations by many authors (see, e.g., [1, 4–25]). In [14–16, 20, 22] the authors studied the basis property in various function spaces of the eigen- and associated function system of the Sturm-Liouville spectral problem with spectral parameter in the boundary conditions. The existence of eigenvalues, estimates of eigenvalues and eigenfunctions, oscillation properties of eigenfunctions, and expansion theorems were considered in [4, 7, 9, 12, 17, 18, 21, 24] for fourth-order ordinary differential operators with a spectral parameter in a boundary condition. The locations, multiplicities of the eigenvalues, the oscillation properties of eigenfunctions, the basis properties in 𝐿𝑝(0,𝑙),π‘βˆˆ(1,∞), of the system of root functions of the boundary value problem (1.1), (1.2a)–(1.2d) with π‘žβ‰₯0, 𝜎>0, are considered in [18] and, with π‘žβ‰₯0, 𝜎<0, 𝑐=0, are considered in [4, 5].

The subject of the present paper is the study of the general characteristics of eigenvalue locations on a complex plane, the structure of root subspaces, the oscillation properties of eigenfunctions, the asymptotic behaviour of the eigenvalues and eigenfunctions, and the basis properties in 𝐿𝑝(0,𝑙), π‘βˆˆ(1,∞), of the system of root functions of the problem (1.1), (1.2a)–(1.2d).

Note that the sign of 𝜎 plays an essential role. In the case 𝜎>0 we associate with problem (1.1), (1.2a)–(1.2d) a selfadjoint operator in the Hilbert space 𝐻=𝐿2(0,𝑙)βŠ•β„‚ with an appropriate inner product. Using this fact and extending analytic methods to fourth-order problems, we show that all the eigenvalues are real and simple and the system of eigenfunctions, with arbitrary function removed, forms a basis in the space 𝐿𝑝(0,𝑙), π‘βˆˆ(1,∞). For 𝜎<0 problem (1.1), (1.2a)–(1.2d) can be interpreted as a spectral problem for a selfadjoint operator in a Pontryagin space Ξ 1. It is proved below that nonreal and nonsimple (multiple) eigenvalues are possible and the system of root functions, with arbitrary function removed, forms a basis in the space 𝐿𝑝(0,𝑙), π‘βˆˆ(1,∞), except some cases where the system is neither completed nor minimal.

2. The Operator Interpretation of the Problem (1.1), (1.2a)–(1.2d)

Let 𝐻=𝐿2(0,𝑙)βŠ•β„‚ be a Hilbert space equipped with the inner product(̂𝑦,̂𝑒)𝐻=({𝑦,π‘š},{𝑒,𝑠})𝐻=(𝑦,𝑒)𝐿2+||πœŽβˆ’1||π‘šπ‘ ,(2.1) where (𝑦,𝑒)𝐿2=βˆ«π‘™0𝑦𝑒𝑑π‘₯.

We define in the 𝐻 operator𝐿̂𝑦=𝐿{𝑦,π‘š}=(𝑇𝑦(π‘₯))ξ…žξ€Ύ,𝑑𝑇𝑦(𝑙)βˆ’π‘π‘¦(𝑙)(2.2) with domain𝐷(𝐿)=̂𝑦={𝑦,π‘š}∈𝐻/𝑦(π‘₯)βˆˆπ‘Š42(0,𝑙),(𝑇𝑦(π‘₯))ξ…žβˆˆπΏ2ξ€Ύ(0,𝑙),π‘¦βˆˆ(B.C.),π‘š=π‘Žπ‘¦(𝑙)βˆ’π‘π‘‡π‘¦(𝑙),(2.3) that is dense in 𝐻 [23, 25], where (B.C.) denotes the set of separated boundary conditions (1.2a)–(1.2c).

Obviously, the operator 𝐿 is well defined. By immediate verification we conclude that problem (1.1), (1.2a)–(1.2d) is equivalent to the following spectral problem:𝐿̂𝑦=πœ†Μ‚π‘¦,Μ‚π‘¦βˆˆπ·(𝐿),(2.4) that is, the eigenvalue πœ†π‘› of problem (1.1), (1.2a)–(1.2d) and those of problem (2.4) coincide; moreover, there exists a correspondence between the eigenfunctions and the adjoint functions of the two problems:̂𝑦𝑛=𝑦𝑛(π‘₯),π‘šπ‘›ξ€ΎβŸ·π‘¦π‘›(π‘₯),π‘šπ‘›=π‘Žπ‘¦π‘›(𝑙)βˆ’π‘π‘‡π‘¦π‘›(𝑙).(2.5) Problem (1.1), (1.2a)–(1.2d) has regular boundary conditions in the sense of [23, 25]; in particular, it has a discrete spectrum.

If 𝜎>0, then 𝐿 is a selfadjoint discrete lower-semibounded operator in 𝐻 and hence has a system of eigenvectors {{𝑦𝑛(π‘₯),π‘šπ‘›}}βˆžπ‘›=1, that forms an orthogonal basis in 𝐻.

In the case 𝜎<0 the operator 𝐿 is closed and non-selfadjoint and has compact resolvent in 𝐻. In 𝐻 we now introduce the operator 𝐽 by 𝐽{𝑦,π‘š}={𝑦,βˆ’π‘š}. 𝐽 is a unitary, symmetric operator in 𝐻. Its spectrum consists of two eigenvalues: βˆ’1 with multiplicity 1, and +1 with infinite multiplicity. Hence, this operator generates the Pontryagin space Ξ 1=𝐿2(0,𝑙)βŠ•β„‚ by means of the inner products (𝐽-metric) [26]:(̂𝑦,̂𝑒)Ξ 1=({𝑦,π‘š},{𝑒,𝑠})Ξ 1=(𝑦,𝑒)𝐿2+πœŽβˆ’1π‘šπ‘ .(2.6)

Lemma 2.1. 𝐿 is a 𝐽-selfadjoint operator in Π1.

Proof. 𝐽𝐿 is selfadjoint in 𝐻 by virtue of Theorem 2.2 [11]. Then, J-selfadjointness of 𝐿 on Π1 follows from [27, Section 3, Proposition 30].

Lemma 2.2 (see [27, Section 3, Proposition 50]). Let πΏβˆ— be an operator adjoined to the operator 𝐿 in 𝐻. Then, πΏβˆ—=𝐽𝐿𝐽.

Let πœ† be an eigenvalue of operator 𝐿 of algebraic multiplicity 𝜈. Let us suppose that 𝜌(πœ†) is equal to 𝜈 if Imπœ†β‰ 0 and equal to whole part 𝜈/2 if Imπœ†=0.

Theorem 2.3 (see [28]). The eigenvalues of operator 𝐿 arrange symmetrically with regard to the real axis. βˆ‘π‘›π‘˜=1𝜌(πœ†π‘˜)≀1 for any system {πœ†π‘˜}π‘›π‘˜=1(𝑛≀+∞) of eigenvalues with nonnegative parts.

From Theorem 2.3 it follows that either all the eigenvalues of boundary value problem (1.1), (1.2a)–(1.2d) are simple (all the eigenvalues are real or all, except a conjugate pair of nonreal, are real) or all the eigenvalues are real and all, except one double or triple, are simple.

3. Some Auxiliary Results

As in [17, 19, 29, 30] forthe analysis of the oscillation properties of eigenfunctions of the problem (1.1), (1.2a)–(1.2d) we will use a PrΓΌfer-type transformation of the following form:𝑦𝑦(π‘₯)=π‘Ÿ(π‘₯)sinπœ“(π‘₯)cosπœƒ(π‘₯),β€²(𝑦π‘₯)=π‘Ÿ(π‘₯)cosπœ“(π‘₯)sinπœ‘(π‘₯),ξ…žξ…ž(π‘₯)=π‘Ÿ(π‘₯)cosπœ“(π‘₯)cosπœ‘(π‘₯),𝑇𝑦(π‘₯)=π‘Ÿ(π‘₯)sinπœ“(π‘₯)sinπœƒ(π‘₯).(3.1)

Consider the boundary conditions (see [29, 30])𝑦′(0)cosπ›Όβˆ’π‘¦ξ…žξ…ž(0)sin𝛼=0,(1.2aβˆ—)𝑦(𝑙)cosπ›Ώβˆ’π‘‡π‘¦(𝑙)sin𝛿=0,(1.2dβˆ—) where π›Όβˆˆ[0,πœ‹/2], π›Ώβˆˆ[0,πœ‹).

Alongside the spectral problem (1.1), (1.2a)–(1.2d) we will consider the spectral problem (1.1), (1.2a)–(1.2c), and (1.2dβˆ—). In [30], Banks and Kurowski developed an extension of the PrΓΌfer transformation (3.1) to study the oscillation of the eigenfunctions and their derivatives of problem (1.1), (1.2aβˆ—), (1.2b), (1.2c), and (1.2dβˆ—) with π‘žβ‰₯0, π›Ώβˆˆ[0,πœ‹/2] and in some cases when (1.3) is disfocal and 𝛼=𝛾=0, π›Ώβˆˆ[0,πœ‹/2]. In [19], the authors used the PrΓΌfer transformation (3.1) to study the oscillations of the eigenfunctions of the problem (1.1), (1.2aβˆ—), (1.2b), (1.2c), and (1.2dβˆ—) with π‘žβ‰₯0 and π›Ώβˆˆ(πœ‹/2,πœ‹). In this work it is proved that problem (1.1), (1.2aβˆ—), (1.2b), (1.2c), and (1.2dβˆ—) may have at most one negative and simple eigenvalue and sequence of positive and simple eigenvalues tending to infinity, the number of zeros of the eigenfunctions corresponding to positive eigenvalues behaves in that usual way (it is equal to the serial number of an eigenvalue increasing by 1); the function associated with the lowest eigenvalue has no zeros in (0,𝑙) (however in reality, this eigenfunction has no zeros in (0,𝑙) if the least eigenvalue is positive; the number of zeros can by arbitrary if the least eigenvalue is negative). In [31], Ben Amara developed an extension of the classical Sturm theory [32] to study the oscillation properties for the eigenfunctions of the problem (1.1), (1.2a)–(1.2c), and (1.2dβˆ—) with 𝛽=0, in particular, given an asymptotic estimate of the number of zeros in (0,𝑙) of the first eigenfunction in terms of the variation of parameters in the boundary conditions.

Let 𝑒 be a solution of (1.3) which satisfies the initial conditions 𝑒(0)=0, 𝑒′(0)=1. Then the disfocal condition of (1.3) implies that π‘’ξ…ž(π‘₯)>0 in [0,𝑙]. Therefore, if β„Ž denotes the solution of (1.3) satisfying the initial conditions 𝑒(0)=𝑐>0, 𝑒′(0)=1, where 𝑐 is a sufficiently small constant, then we have also β„Žξ…ž(π‘₯)>0 on [0,𝑙]. Thus, β„Ž(π‘₯)>0 in [0,𝑙], and hence the following substitutions [33, Theorem 12.1]:𝑑=𝑑(π‘₯)=π‘™πœ”βˆ’1ξ€œπ‘₯0ξ€œβ„Ž(𝑠)𝑑𝑠,πœ”=𝑙0β„Ž(𝑠)𝑑𝑠,(3.2) transform [0,𝑙] into the interval [0,𝑙] and (1.1) into(π‘Μˆπ‘¦)β‹…β‹…=πœ†π‘Ÿπ‘¦,(3.3) where 𝑝=(π‘™πœ”βˆ’1β„Ž)3, π‘Ÿ=π‘™βˆ’1πœ”β„Žβˆ’1; β„Ž(π‘₯), 𝑦(π‘₯) are taken as functions of 𝑑 and β‹…βˆΆ=𝑑/𝑑𝑑. Furthermore, the following relations are useful in the sequel:̇𝑦=π‘™βˆ’1πœ”β„Žβˆ’1π‘¦ξ…ž,𝑙2πœ”βˆ’2β„Ž3Μˆπ‘¦=β„Žπ‘¦ξ…žξ…žβˆ’β„Žξ…žπ‘¦ξ…ž,ξ‚ξ‚€ξ€·π‘‡π‘¦β‰‘π‘™πœ”βˆ’1β„Žξ€Έ3ξ‚Μˆπ‘¦β‹…=𝑇𝑦.(3.4) It is clear from the second relation (3.4) that the sign of π‘¦ξ…žξ…ž is not necessarily preserved after the transformation (3.2). For this reason this transformation cannot be used in any straightforward way. The following lemma of Leighton and Nehari [33] will be needed throughout our discussion. In [30, Lemma 2.1], Banks and Kurowski gave a new proof of this lemma for π‘žβ‰₯0. However, in the case when (1.3) is disfosal on (0,𝑙], they partially proved it [30, Lemma 7.1], and therefore they were able to study problem (1.1), (1.2a)–(1.2c), and (1.2dβˆ—) with 𝛾=0, π›Ώβˆˆ[0,πœ‹/2]. In [31], Ben Amara shows how Lemma 3.1 together with the transformation (3.2) can be applicable to investigate boundary conditions (1.2a)–(1.2c), and (1.2dβˆ—) with 𝛽=0.

Lemma 3.1 (see [33, Lemma 2.1]). Let πœ†>0, and let 𝑦 be a nontrivial solution of (3.3). If 𝑦, ̇𝑦, Μˆπ‘¦, and 𝑇𝑦 are nonnegative at 𝑑=π‘Ž (but not all zero), they are positive for all 𝑑>π‘Ž. If 𝑦, βˆ’Μ‡π‘¦, Μˆπ‘¦, and βˆ’ξ‚π‘‡π‘¦ are nonnegative at 𝑑=π‘Ž (but not all zero), they are positive for all 𝑑<π‘Ž.

We also need the following results which are basic in the sequel.

Lemma 3.2. All the eigenvalues of problem (1.1), (1.2a)–(1.2c), and (1.2dβˆ—) for π›Ώβˆˆ[0,πœ‹/2) or 𝛿=πœ‹/2, π›½βˆˆ[0,πœ‹/2) are positive.

Proof. In this case, the transformed problem is determined by (3.3) and the boundary conditions̇𝑦(0)=0,(3.5a)𝑦(0)cos𝛽+𝑇𝑦(0)sin𝛽=0,(3.5b)̇𝑦(𝑙)cosπ›Ύβˆ—+𝑝(𝑙)Μˆπ‘¦(𝑙)sinπ›Ύβˆ—ξ‚=0,(3.5c)𝑦(𝑙)cosπ›Ώβˆ’π‘‡π‘¦(𝑙)sin𝛿=0,(3.5d)where π›Ύβˆ—=arctg{π‘™βˆ’2πœ”2β„Žβˆ’1(𝑙)[β„Ž(𝑙)cos𝛾+β„Žβ€²(𝑙)sin𝛾]βˆ’1}∈[0,πœ‹/2).
It is known that the eigenvalues of (3.3), (3.5a)–(3.5d) are given by the max-min principle [13, Page 405] using the Rayleigh quotient 𝑅[𝑦]=ξ‚€βˆ«π‘™0π‘Μˆπ‘¦2[𝑦]𝑑𝑑+π‘ξ‚€βˆ«π‘™0𝑦2𝑑𝑑,(3.6) where 𝑁[𝑦]=𝑦2(0)cot𝛽+̇𝑦2(𝑙)cotπ›Ύβˆ—+𝑦2(𝑙)cot𝛿. It follows by inspection of the numerator 𝑅 in (3.6) that zero is an eigenvalue only in the case 𝛽=𝛿=πœ‹/2. Hence, all the eigenvalues of problem (3.3), (3.5a)–(3.5d) for π›Ώβˆˆ[0,πœ‹/2) or 𝛿=πœ‹/2, π›½βˆˆ[0,πœ‹/2), are positive. Lemma 3.2 is proved.

Lemma 3.3. Let 𝐸 be the space of solution of the problem (1.1), (1.2a)–(1.2c). Then, dim𝐸=1.

The proof is similar to that of [19, Lemma 2] using transformation (3.2), Lemmas 3.1 and 3.2 (see also [31, Lemma 2.2]). However, it is not true if πœ‹/2<𝛾<πœ‹ (see, e.g., [31, Page 9]). Therefore, Lemma 3.1 together with the transformation (3.2) cannot be applicable to investigate more general boundary conditions, for example, (1.2aβˆ—), (1.2b), and (1.2c) for π›Όβˆˆ(0,πœ‹/2].

Lemma 3.4 (see [29, Lemma 2.2]). Let πœ†>0 and u be a solution of (3.3) which satisfies the boundary conditions (3.5a)–(3.5c). If π‘Ž is a zero of 𝑒 and Μˆπ‘’ in the interval (0,l), then ̇𝑒(𝑑)𝑇𝑒(𝑑)<0 in a neighborhood of π‘Ž. If π‘Ž is a zero of ̇𝑒 or 𝑇𝑒 in (0,l), then 𝑒(𝑑)Μˆπ‘’(𝑑)<0 in a neighborhood of π‘Ž.

Theorem 3.5. Let 𝑒 be a nontrivial solution of the problem (1.1), (1.2a) and (1.2c) for πœ†>0. Then the Jacobian 𝐽[𝑒]=π‘Ÿ3cosπœ“sinπœ“ of the transformation (3.1) does not vanish in (0,l).

Proof. Let 𝑒 be a nontrivial solution of (1.1) which satisfies the boundary conditions (1.2a) and (1.2c). Assume first that the corresponding angle πœ“ satisfies πœ“(π‘₯0)=π‘›πœ‹ for some integer 𝑛 and for some π‘₯0∈(0,𝑙). Then, the transformation (3.1) implies that 𝑒(π‘₯0)=𝑇𝑒(π‘₯0)=0. Using the transformation (3.2), the solution 𝑒 of (3.3) also satisfies 𝑒(𝑑0)=𝑇𝑒(𝑑0)=0, where 𝑑0=π‘™βˆ’1πœ”βˆ«π‘₯00β„Ž(𝑠)π‘‘π‘ βˆˆ(0,𝑙). However, it is incompatible with the conclusion of Lemma 3.4.
The proof of the inequality cosπœ“(π‘₯)β‰ 0, π‘₯∈(0,𝑙), proceeds in the same fashion as in the previous case. The proof of Theorem 3.5 is complete.

Let 𝑦(π‘₯,πœ†) be a nontrivial solution of the problem (1.1), (1.2a)–(1.2c) for πœ†>0 and πœƒ(π‘₯,πœ†), πœ‘(π‘₯,πœ†) the corresponding functions in (3.1). Without loss of generality, we can define the initial values of these functions as follows (see [30, Theorem 3.3]):πœ‹πœƒ(0,πœ†)=π›½βˆ’2,πœ‘(0,πœ†)=0.(3.7)

With obvious modifications, the results stated in [30, Sections 3–5] are true for the solution of the problem (1.1), (1.2a)–(1.2c), and (1.2dβˆ—) for π›Ώβˆˆ[0,πœ‹/2]. In particular, we have the following results.

Theorem 3.6. πœƒ(𝑙,πœ†) is a strictly increasing continuous function on πœ†.

Theorem 3.7. Problem (1.1), (1.2a)–(1.2c), and (1.2dβˆ—) for π›Ώβˆˆ[0,πœ‹/2] (except the case 𝛽=𝛿=πœ‹/2) has a sequence of positive and simple eigenvalues πœ†1(𝛿)<πœ†2(𝛿)<β‹―<πœ†n(𝛿)⟢∞.(3.8) Moreover, πœƒ(𝑙,πœ†π‘›(𝛿))=(2π‘›βˆ’1)πœ‹/2βˆ’π›Ώ, π‘›βˆˆβ„•; the corresponding eigenfunctions πœπ‘›(𝛿)(π‘₯) have π‘›βˆ’1 simple zeros in (0,l).

Remark 3.8. In the case 𝛽=𝛿=πœ‹/2 the first eigenvalue of boundary value problem (1.1), (1.2a)–(1.2c), and (1.2dβˆ—) is equal to zero and the corresponding eigenfunction is constant; the statement of Theorem 3.7 is true for 𝑛β‰₯2.

Obviously, the eigenvalues πœ†π‘›(𝛿), π‘›βˆˆβ„•, of the problem (1.1), (1.2a)–(1.2c), and (1.2dβˆ—) are zeros of the entire function 𝑦(𝑙,πœ†)cosπ›Ώβˆ’π‘‡π‘¦(𝑙,πœ†)cos𝛿=0. Note that the function 𝐹(πœ†)=𝑇𝑦(𝑙,πœ†)/𝑦(𝑙,πœ†) is defined for β‹ƒπœ†βˆˆπ΄β‰‘(β„‚/ℝ)βˆͺ(βˆžπ‘›=1(πœ†π‘›βˆ’1(0),πœ†π‘›(0))), where πœ†0(0)=βˆ’βˆž.

Lemma 3.9 (see [19, Lemma 5]). Let πœ†βˆˆπ΄. Then, the following relation holds: π‘‘ξ‚€βˆ«π‘‘πœ†πΉ(πœ†)=𝑙0𝑦2(π‘₯,πœ†)𝑑π‘₯𝑦2.(𝑙,πœ†)(3.9)

In (1.1) we set πœ†=𝜌4. As is known (see [34, Chapter II, Section 4.5, Theorem 1]) in each subdomain 𝑇 of the complex 𝜌-plane equation (1.1) has four linearly independent solutions π‘§π‘˜(π‘₯,𝜌), π‘˜=1,4, regular in 𝜌 (for sufficiently large 𝜌) and satisfying the relationsπ‘§π‘˜(𝑠)ξ€·(π‘₯,𝜌)=πœŒπœ”π‘˜ξ€Έπ‘ π‘’πœŒπœ”π‘˜π‘₯[1],π‘˜=1,4,𝑠=0,3,(3.10) where πœ”π‘˜, π‘˜=1,4, are the distinct fourth roots of unity, [1]=1+𝑂(1/𝜌).

For brevity, we introduce the notation 𝑠(𝛿1,𝛿2)=sgn𝛿1+sgn𝛿2. Using relation (3.10) and taking into account boundary conditions (1.2a)–(1.2c), we obtain⎧βŽͺ⎨βŽͺβŽ©ξ‚€ξ‚€πœ‹π‘¦(π‘₯,πœ†)=sin𝜌π‘₯+2ξ‚ξ‚€πœ‹sinπ›½βˆ’cosπœŒπ‘™+2𝑒𝑠(𝛽,𝛾)𝜌(π‘₯βˆ’π‘™)[1]ξ‚€πœ‹ifπ›½βˆˆ0,2ξ‚„,βˆšξ‚€πœ‹2sin𝜌π‘₯βˆ’4ξ‚βˆ’π‘’βˆ’πœŒπ‘₯+(βˆ’1)1βˆ’sgnπ›Ύβˆšξ‚€2sinπœŒπ‘™+(βˆ’1)sgnπ›Ύπœ‹4ξ‚π‘’πœŒ(π‘₯βˆ’π‘™)[1]⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩if𝛽=0,(3.11)𝑇𝑦(π‘₯,πœ†)=βˆ’πœŒ3ξ‚€ξ‚€πœ‹cos𝜌π‘₯+2ξ‚ξ‚€πœ‹sgn𝛽+cosπœŒπ‘™+2𝑒𝑠(𝛽,𝛾)𝜌(π‘₯βˆ’π‘™)[1]ξ‚€πœ‹ifπ›½βˆˆ0,2ξ‚„,βˆ’πœŒ3ξ‚€βˆšξ‚€πœ‹2sin𝜌π‘₯+4ξ‚βˆ’π‘’βˆ’πœŒπ‘₯βˆ’(βˆ’1)1βˆ’sgnπ›Ύβˆšξ‚€πœ‹2sinπœŒπ‘™+4(βˆ’1)sgnπ›Ύξ‚π‘’πœŒ(π‘₯βˆ’π‘™)[1]if𝛽=0.(3.12)

Remark 3.10. As an immediate consequence of (3.11), we obtain that the number of zeros in the interval (0,𝑙) of function 𝑦(π‘₯,πœ†) tends to +∞ as πœ†β†’Β±βˆž.

Taking into account relations (3.11) and (3.12), we obtain the asymptotic formulas⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ‚€βˆšπΉ(πœ†)=21βˆ’sgnπ›ΎπœŒ3cos(πœŒπ‘™+(πœ‹/2)sgn𝛽+(πœ‹/4)sgn𝛾)[1]ξ‚€πœ‹cos(πœŒπ‘™+(πœ‹/2)sgn𝛽+(πœ‹/4)(1+sgn𝛾))ifπ›½βˆˆ0,2ξ‚„,ξ‚€βˆš21βˆ’sgnπ›ΎπœŒ3cos(πœŒπ‘™+(πœ‹/4)(sgnπ›Ύβˆ’1))[1]cos(πœŒπ‘™+(πœ‹/4)(1+sgn𝛾))if𝛽=0.(3.13) Furthermore, we haveξ‚€βˆšπΉ(πœ†)=βˆ’21βˆ’sgn𝛾4||πœ†||3ξ‚€ξ€·||πœ†||ξ€Έ1+π‘‚βˆ’1/4,asπœ†βŸΆβˆ’βˆž.(3.14)

We define numbers 𝜏, 𝜈, πœ‚, 𝛼𝑛, 𝛽𝑛, πœ‚π‘›, π‘›βˆˆβ„•, and a function 𝑧(π‘₯,𝑑), π‘₯∈[0,𝑙], π‘‘βˆˆβ„, as follows:⎧βŽͺ⎨βŽͺ⎩𝜏=3(1+𝑠(𝛽,𝛿))4ξ‚€πœ‹βˆ’1ifπ›Ύβˆˆ0,2ξ‚„,54βˆ’38ξ€·(βˆ’1)sgn𝛽+(βˆ’1)sgnπ›Ώξ€ΈβŽ§βŽͺ⎨βŽͺβŽ©βˆ’1if𝛾=0,πœ‚=3(2+sgn𝛽)4ξ‚€πœ‹βˆ’1ifπ›Ύβˆˆ0,2ξ‚„,54βˆ’38ξ€·(βˆ’1)sgnπ›½ξ€ΈβŽ§βŽͺ⎨βŽͺβŽ©βˆ’1βˆ’1if𝛾=0,𝜈=3(1+𝑠(𝛽,|𝑐|))4ξ‚€πœ‹ifπ›Ύβˆˆ0,2ξ‚„,54βˆ’38ξ€·(βˆ’1)sgn𝛽+(βˆ’1)sgn|𝑐|𝛼if𝛾=0,𝑛=(π‘›βˆ’πœ)πœ‹π‘™,πœ‚π‘›=(π‘›βˆ’πœ‚)πœ‹π‘™,𝛽𝑛=(π‘›βˆ’πœˆ)πœ‹π‘™,⎧βŽͺ⎨βŽͺβŽ©ξ‚€πœ‹π‘§(π‘₯,𝑑)=sin𝑑π‘₯+2ξ‚ξ‚€πœ‹sgnπ›½βˆ’cos𝑑𝑙+2𝑒𝑠(𝛽,𝛾)βˆ’π‘‘(π‘™βˆ’π‘₯)ξ‚€πœ‹ifπ›½βˆˆ0,2ξ‚„,βˆšξ‚€πœ‹2sin𝑑π‘₯βˆ’4+π‘’βˆ’π‘‘π‘₯+(βˆ’1)sgnπ›Ύβˆšξ‚΅2sin𝑑𝑙+(βˆ’1)sgnπ›Ύπœ‹4ξ‚Άπ‘’βˆ’π‘‘(π‘₯βˆ’π‘™)if𝛽=0.(3.15) By virtue of [18, Theorem 3.1], one has the asymptotic formulas4βˆšπœ†π‘›(𝛿)=𝛼𝑛𝑛+π‘‚βˆ’1ξ€Έ,𝜐(3.16)𝑛(𝛿)(ξ€·π‘₯)=𝑧π‘₯,𝛼𝑛𝑛+π‘‚βˆ’1ξ€Έ,(3.17) where relation (3.17) holds uniformly for π‘₯∈[0,𝑙].

By (3.14), we havelimπœ†β†’βˆ’βˆžπΉ(πœ†)=βˆ’βˆž.(3.18) From Property 1 in [30] and formulas (3.9), one has the relationsπœ†1ξ‚€πœ‹2<πœ†1(0)<πœ†2ξ‚€πœ‹2<πœ†2(0)<β‹―.(3.19)

Remark 3.11. It follows by Theorem 3.7, Lemma 3.9, and relations (3.18) and (3.19) that if πœ†>0 or πœ†=0, π›½βˆˆ[0,πœ‹/2), then 𝐹(πœ†)<0; besides, if πœ†=0 and 𝛽=πœ‹/2, then 𝐹(πœ†)=0.

Let 𝑠(πœ†) be the number of zeros of the function 𝑦(π‘₯,πœ†) in the interval (0,𝑙).

Lemma 3.12. If πœ†>0 and πœ†βˆˆ(πœ†π‘›βˆ’1(0),πœ†π‘›(0)], π‘›βˆˆβ„•, then 𝑠(πœ†)=π‘›βˆ’1.

The proof is similar to that of [19, Lemma 10] using Theorems 3.6 and 3.7 and Remark 3.11.

Theorem 3.13. The problem (1.1), (1.2a)–(1.2c), and (1.2dβˆ—) for π›Ώβˆˆ(πœ‹/2,πœ‹) has a sequence of real and simple eigenvalues πœ†1(𝛿)<πœ†2(𝛿)<β‹―<πœ†π‘›(𝛿)⟢+∞,(3.20) including at most one negative eigenvalue. Moreover, (a) if π›½βˆˆ[0,πœ‹/2), then πœ†1(𝛿)>0 for π›Ώβˆˆ(πœ‹/2,𝛿0); πœ†1(𝛿)=0 for 𝛿=𝛿0;πœ†1(𝛿)<0 for π›Ώβˆˆ(𝛿0,πœ‹), where 𝛿0=arctg𝑇𝑦(𝑙,0)/𝑦(𝑙,0); (b) if 𝛽=πœ‹/2, then πœ†1(𝛿)<0; (c) the eigenfunction πœπ‘›(𝛿)(π‘₯), corresponding to the eigenvalue πœ†π‘›(𝛿)β‰₯0, has exactly π‘›βˆ’1 simple zeros in (0,𝑙).

The proof parallels the proof of [19, Theorem 4] using Theorems 3.5–3.7 and Lemmas 3.9 and 3.12.

Lemma 3.14. The following non-selfadjoint boundary value problem: 𝑦(4)ξ‚€(π‘₯)βˆ’π‘ž(π‘₯)𝑦′(π‘₯)ξ…ž=πœ†π‘¦(π‘₯),π‘₯∈(0,𝑙),𝑦(0)=π‘¦ξ…ž(0)=𝑇𝑦(0)=π‘¦ξ…ž(𝑙)cos𝛾+π‘¦ξ…žξ…ž(𝑙)sin𝛾=0,(3.21) has an infinite set of nonpositive eigenvalues πœŒπ‘› tending to βˆ’βˆž and satisfying the asymptote πœ†π‘›ξ‚€1=βˆ’π‘›βˆ’4(1+sgn𝛾)4πœ‹4𝑙4𝑛+π‘œ4ξ€Έ,π‘›βŸΆβˆž.(3.22)

Setting π‘₯=0 in (3.12), we obtain (3.22).

Remark 3.15. By Remark 3.10 the number of zeros of the eigenfunction 𝑦1(𝛿)(π‘₯) corresponding to an eigenvalue πœ†1(𝛿)<0 can by arbitrary. In views of [31, Corollary 2.5], as πœ†1(𝛿)<0 varies, new zeros of the corresponding eigenfunction 𝑦1(𝛿)(π‘₯) enter the interval (0,𝑙) only through the end point π‘₯=0 (since 𝑦1(𝛿)(𝑙)β‰ 0), and hence the number of its zeros, in the case π›½βˆˆ(0,πœ‹/2], is asymptotically equivalent to the number of eigenvalues of the problem (3.21) which are higher than πœ†1(𝛿). In the case 𝛽=0 see [31, Theorem 5.3].

We consider the following boundary conditions:π‘Žπ‘¦(𝑙)βˆ’π‘π‘‡π‘¦(𝑙)=0,(1.2dβ€²)𝑐𝑦(𝑙)+π‘Žπ‘‡π‘¦(𝑙)=0.(1.2dβ€²ξ…ž)

Note that (π‘Ž,𝑐)β‰ 0 since 𝜎<0. The boundary condition (1.2dβ€²) coincides the boundary condition (1.2dβˆ—) for 𝛿=πœ‹/2 (resp., 𝛿=0) in the case π‘Ž=0 (resp., 𝑐=0), and the boundary condition (1.2dβ€²ξ…ž) coincides the boundary condition (1.2dβˆ—) for 𝛿=0 (resp., 𝛿=πœ‹/2) in the case π‘Ž=0 ( resp., 𝑐=0).

Let π‘Žπ‘β‰ 0. The eigenvalues of the problem (1.1), (1.2a)–(1.2c), and (1.2dβ€²) (resp., (1.1) (1.2a)–(1.2c), and (1.2dβ€²ξ…ž)) are the roots of the equation 𝐹(πœ†)=π‘Ž/𝑐 (resp., 𝐹(πœ†)=βˆ’π‘/π‘Ž). By (3.9), this equation has only simple roots; hence all the eigenvalues of the problems (1.1), (1.2a)–(1.2c), and (1.2dβ€²) and (1.1), (1.2a)–(1.2c), and (1.2dβ€²ξ…ž) are simple. On the base of (3.9), (3.18), and (3.19) in each interval 𝐴𝑛, 𝑛=1,2,…, the equation 𝐹(πœ†)=π‘Ž/𝑐 (resp., 𝐹(πœ†)=βˆ’π‘/π‘Ž) has a unique solution πœ‡π‘› (resp., πœˆπ‘›); moreover,𝜈1<πœ†1ξ‚€πœ‹2<πœ‡1<πœ†1(0)<𝜈2<πœ†2ξ‚€πœ‹2<πœ‡2<πœ†2(0)<β‹―(3.23) if π‘Ž/𝑐>0 andπœ‡1<πœ†1ξ‚€πœ‹2<𝜈1<πœ†1(0)<πœ‡2<πœ†2ξ‚€πœ‹2<𝜈2<πœ†2(0)<β‹―(3.24) if π‘Ž/𝑐<0. Besides, πœ‡1=0 if π‘Ž/𝑐<0 and 𝐹(0)=π‘Ž/𝑐;𝜈1=0 if π‘Ž/𝑐>0 and 𝐹(0)=βˆ’π‘/π‘Ž.

Taking into account (1.2dβ€²), (1.2dβ€²ξ…ž), (3.23), and (3.24) and using the corresponding reasoning [18, Theorem 3.1] we have4βˆšπœ‡π‘›=πœ‚π‘›ξ€·π‘›+π‘‚βˆ’1ξ€Έ,4βˆšπœˆπ‘›=πœ‚π‘›ξ€·π‘›+π‘‚βˆ’1ξ€Έ,πœ‘(3.25)𝑛(π‘₯)=𝑧π‘₯,πœ‚π‘›ξ€Έξ€·π‘›+π‘‚βˆ’1ξ€Έ,πœ“π‘›ξ€·(π‘₯)=𝑧π‘₯,πœ‚π‘›ξ€Έξ€·π‘›+π‘‚βˆ’1ξ€Έ,(3.26) where relation (3.26) holds uniformly for π‘₯∈[0,𝑙] and eigenfunctions πœ‘π‘›(π‘₯) and πœ“π‘›(π‘₯), π‘›βˆˆβ„•, correspond to the eigenvalues πœ‡π‘› and πœˆπ‘›, respectively.

Let us denote π‘š(πœ†)=π‘Žπ‘¦(𝑙,πœ†)βˆ’π‘π‘‡π‘¦(𝑙,πœ†).

Remark 3.16. Note that if πœ† is the eigenvalue of problem (1.1), (1.2a)–(1.2d), then π‘š(πœ†)β‰ 0 since πœŽβ‰ 0.

It is easy to see that the eigenvalues of problem (1.1), (1.2a)–(1.2d) are roots of the equation(π‘Žπœ†+𝑏)𝑦(𝑙)βˆ’(π‘πœ†+𝑑)𝑇𝑦(𝑙)=0.(3.27)

By virtue of Remark 3.16 and formula (3.27), a simple calculation yields that the eigenvalues of the problem (1.1), (1.2a)–(1.2d) can be realized at the solution of the equation𝑐𝑦(𝑙,πœ†)+π‘Žπ‘‡π‘¦(𝑙,πœ†)=π‘Žπ‘Žπ‘¦(𝑙,πœ†)βˆ’π‘π‘‡π‘¦(𝑙,πœ†)2+𝑐2βˆ’πœŽπœ†+π‘Žπ‘+𝑐𝑑.βˆ’πœŽ(3.28)

Denote 𝐡𝑛=(πœ‡π‘›βˆ’1,πœ‡π‘›), π‘›βˆˆβ„•, where πœ‡0=βˆ’βˆž.

We observe that the function 𝐺(πœ†)=(𝑐𝑦(𝑙,πœ†)+π‘Žπ‘‡π‘¦(𝑙,πœ†))/(π‘Žπ‘¦(𝑙,πœ†)βˆ’π‘π‘‡π‘¦(𝑙,πœ†)) is well defined for β‹ƒπœ†βˆˆπ΅=(ℂ⧡ℝ)βˆͺ(βˆžπ‘›=1𝐡𝑛) and is a finite-order meromorphic function and the eigenvalues πœˆπ‘› and πœ‡π‘›, π‘›βˆˆβ„•, of boundary value problems (1.1), (1.2a)–(1.2c), and (1.2dβ€²ξ…ž) and (1.1), (1.2a)–(1.2c), and (1.2dβ€²) are zeros and poles of this function, respectively.

Let πœ†βˆˆπ΅. Using formula (3.9), we getπ‘‘ξ€·π‘Žπ‘‘πœ†πΊ(πœ†)=2+𝑐2ξ€Έπ‘šβˆ’2ξ€œ(πœ†)𝑙0𝑦2(π‘₯,πœ†)𝑑π‘₯.(3.29)

Lemma 3.17. The expansion ⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩𝐺(πœ†)=𝐺(0)+βˆžξ“π‘›=1πœ†π‘π‘›πœ‡π‘›ξ€·πœ†βˆ’πœ‡π‘›ξ€Έifπœ‡1𝑐≠0,0+𝑐1πœ†+βˆžξ“π‘›=2πœ†π‘π‘›πœ‡π‘›ξ€·πœ†βˆ’πœ‡π‘›ξ€Έifπœ‡1=0,(3.30) holds, where 𝑐𝑛, π‘›βˆˆβ„•, are some negative numbers.

Proof. It is known (see [35, Chapter 6, Section 5]) that the meromorphic function 𝐺(πœ†) with simple poles πœ‡π‘› allows the representation 𝐺(πœ†)=𝐺1(πœ†)+βˆžξ“π‘›=1ξ‚΅πœ†πœ‡π‘›ξ‚Άπ‘ π‘π‘›πœ†βˆ’πœ‡π‘›,(3.31) where 𝐺1(πœ†) is an entire function, 𝑐𝑛=resπœ†=πœ‡π‘›ξ€·ξ€·πΊ(πœ†)=𝑐𝑦𝑙,πœ‡π‘›ξ€Έξ€·+π‘Žπ‘‡π‘¦π‘™,πœ‡π‘›ξƒ©π‘Žξ€·ξ€Έξ€Έπœ•π‘¦π‘™,πœ‡π‘›ξ€Έξ€·πœ•πœ†βˆ’π‘π‘‡π‘¦π‘™,πœ‡π‘›ξ€Έξƒͺπœ•πœ†βˆ’1,(3.32) and integers 𝑠𝑛, π‘›βˆˆβ„•, are chosen so that series (3.31) are uniformly convergent in any finite circle (after truncation of terms having poles in this circle).
We consider the case π‘Ž/𝑐>0. By virtue of relation (3.18), we have limπœ†β†’βˆ’βˆžπΊ(πœ†)=βˆ’π‘Ž/𝑐. Hence, 𝐺(πœ†)<0 for πœ†βˆˆ(βˆ’βˆž,𝜈1) and 𝐺(πœ†)>0 for πœ†βˆˆ(𝜈1,πœ‡1). Without loss of generality, we can assume π‘Žπ‘¦(𝑙,πœ†)βˆ’π‘π‘‡π‘¦(𝑙,πœ†)>0 for πœ†βˆˆ(βˆ’βˆž,πœ‡1). Then, 𝑐𝑦(𝑙,πœ†)+π‘Žπ‘‡π‘¦(𝑙,πœ†)<0 for πœ†βˆˆ(βˆ’βˆž,𝜈1). Since the eigenvalues πœ‡π‘› and πœˆπ‘›, π‘›βˆˆβ„•, are simple zeros of functions π‘Žπ‘¦(𝑙,πœ†)βˆ’π‘π‘‡π‘¦(𝑙,πœ†) and 𝑐𝑦(𝑙,πœ†)+π‘Žπ‘‡π‘¦(𝑙,πœ†), respectively, then by (3.29) the relations (βˆ’1)𝑛+1𝑐𝑦𝑙,πœ‡π‘›ξ€Έξ€·+π‘Žπ‘‡π‘¦π‘™,πœ‡π‘›ξ€Έξ€Έ>0,(βˆ’1)𝑛+1ξƒ©π‘Žξ€·πœ•π‘¦π‘™,πœ‡π‘›ξ€Έξ€·πœ•πœ†βˆ’π‘πœ•π‘‡π‘¦π‘™,πœ‡π‘›ξ€Έξƒͺπœ•πœ†<0,π‘›βˆˆβ„•,(3.33) are true.
Taking into account (3.33), in (3.32) we get 𝑐𝑛<0, π‘›βˆˆβ„•. The cases π‘Ž/𝑐<0, π‘Ž=0, and 𝑐=0 can be treated along similar lines.
Denote Ω𝑛(πœ€)={πœ†βˆˆβ„‚βˆ£|4βˆšπœ†βˆ’4βˆšπœ‡π‘›|<πœ€} where πœ€>0 is some small number. From the asymptotic formula (3.25), it follows that for πœ€<πœ‹/4𝑙 the domains Ω𝑛(πœ€) asymptotically do not intersect and contain only one pole πœ‡π‘› of the function 𝐺(πœ†).
By (3.11), (3.12), (3.23), (3.24), and (3.25), we see that outside of domains Ω𝑛(πœ€) the asymptotic formulae are true: ⎧βŽͺ⎨βŽͺβŽ©βˆ’π‘ŽπΊ(πœ†)=π‘ξ€·πœŒ+π‘‚βˆ’1ξ€ΈπœŒifπ‘Žπ‘β‰ 0,3ξ€·ξ€·πœŒπ‘§(𝜌)1+π‘‚βˆ’1ξ€Έξ€Έif𝑐=0,βˆ’πœŒβˆ’3(𝑧(𝜌))βˆ’1ξ€·ξ€·πœŒ1+π‘‚βˆ’1ξ€Έξ€Έifπ‘Ž=0,(3.34) where ⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π‘§(𝜌)=cos((πœ‹/4)sgn𝛾)cos(πœŒπ‘™+(πœ‹/2)sgn𝛽+(πœ‹/4)sgn𝛾)ξ‚€πœ‹sin((πœ‹/4)(1+sgn𝛾))cos(πœŒπ‘™+(πœ‹/2)sgn𝛽+(πœ‹/4)(1+sgn𝛾))ifπ›½βˆˆ0,2ξ‚„,ξ‚€βˆš21βˆ’2sgn𝛾cos(πœŒπ‘™βˆ’(1βˆ’sgn𝛾)(πœ‹/4))cos(πœŒπ‘™+(πœ‹/4)sgn𝛾)if𝛽=0.(3.35)
Following the corresponding reasoning (see [36, Chapter VII, Section  2, formula (27)]), we see that outside of domains Ω𝑛(πœ€) the estimation ||||β‰€βŽ§βŽͺ⎨βŽͺβŽ©ξ‚‹π‘€πΊ(πœ†)1𝑀ifπ‘Žπ‘β‰ 0,24||πœ†||3𝑀if𝑐=0,34||πœ†||βˆ’3ifπ‘Ž=0,(3.36) holds; using it in (3.32) we get ||𝑐𝑛||=||||1ξ€œ2πœ‹π‘–πœ•Ξ©π‘›(πœ€)||||=2𝐺(πœ†)π‘‘πœ†πœ‹||||ξ€œ|πœˆβˆ’4βˆšπœ‡π‘›|=πœ€πœˆ3πΊξ€·πœˆ4ξ€Έ||||β‰€βŽ§βŽͺ⎨βŽͺβŽ©π‘€π‘‘πœˆ1𝑛3𝑀ifπ‘Žπ‘β‰ 0,2𝑛6𝑀if𝑐=0,3ifπ‘Ž=0,(3.37) where 𝑀1, 𝑀2, 𝑀3, 𝑀1, 𝑀2, 𝑀3 are some positive constants. By (3.37) and asymptotic formula (3.25) the series βˆ‘βˆžπ‘›=1𝑐𝑛|πœ‡π‘›|βˆ’2 converges. Then, according to Theorem 2 in [35, Chapter 6, Section 5], in formula (3.31) we can assume 𝑠𝑛=1, π‘›βˆˆβ„•.
Let {Γ𝑛}βˆžπ‘›=1 be a sequence of the expanding circles which are not crossing domains Ω𝑛(πœ€). Then, according to Formula (9) in [37, Chapter V, Section 13], we have 𝐺(πœ†)βˆ’πœ‡π‘˜βˆˆintΞ“π‘›π‘π‘˜πœ†βˆ’πœ‡π‘˜=1ξ€œ2πœ‹π‘–Ξ“π‘›πΊ(πœ‰)ξ“πœ‰βˆ’πœ†π‘‘πœ‰,𝐺(0)+πœ‡k∈intΞ“π‘›π‘π‘˜πœ‡π‘˜=1ξ€œ2πœ‹π‘–Ξ“π‘›πΊ(πœ‰)πœ‰π‘‘πœ‰.(3.38) By (3.38), we get 𝐺(πœ†)βˆ’πΊ(0)=πœ‡π‘˜βˆˆintΞ“π‘›πœ†π‘π‘˜πœ‡π‘˜ξ€·πœ†βˆ’πœ‡π‘˜ξ€Έ=1ξ€œ2πœ‹π‘–Ξ“π‘›πœ†πΊ(πœ‰)πœ‰(πœ‰βˆ’πœ†)π‘‘πœ‰.(3.39)
From (3.36), the right side of (3.39) tends to zero as π‘›β†’βˆž. Then, passing to the limit as π‘›β†’βˆž in (3.39), we obtain 𝐺(πœ†)=𝐺(0)+βˆžξ“π‘›=1πœ†π‘π‘›πœ‡π‘›ξ€·πœ†βˆ’πœ‡π‘›ξ€Έ,(3.40) which implies 𝐺1(πœ†)≑𝐺(0).
Differentiating the right side of the least equality, we have 𝐺(𝑠)(πœ†)=(βˆ’1)𝑠𝑠!βˆžξ“π‘›=1π‘π‘›ξ€·πœ†βˆ’πœ‡π‘›ξ€Έπ‘ +1,𝑠=1,2,3.(3.41) Note that the function 𝐹(πœ†) has the following expansion: 𝐹(πœ†)=𝐹(0)+βˆžξ“π‘›=1πœ†Μƒπ‘π‘›πœ†π‘›ξ€·(0)πœ†βˆ’πœ†π‘›ξ€Έ,(0)(3.42) where ̃𝑐𝑛=resπœ†=πœ†π‘›(0)𝐹(πœ†),𝑛=1,2,….(3.43)
Now let πœ‡1=0, that is, 𝐹(0)=π‘Ž/𝑐. 𝐺(πœ†) has the following expansion: 𝐺(πœ†)=𝐺1(π‘πœ†)+1πœ†+βˆžξ“π‘›=2πœ†π‘π‘›πœ‡π‘›ξ€·πœ†βˆ’πœ‡π‘›ξ€Έ.(3.44) Again, according to Formula (9) in [37, Chapter V, Section 13], we have 𝑐𝐺(πœ†)βˆ’1πœ†βˆ’ξ“πœ‡π‘˜βˆˆintΞ“π‘›π‘˜β‰ 1π‘π‘˜πœ†βˆ’πœ‡π‘˜=1ξ€œ2πœ‹π‘–Ξ“π‘›πΊ(πœ‰)πœ‰βˆ’πœ†π‘‘πœ‰.(3.45) By (2.6) [18] and (3.9), we get 𝑐1=βˆ’π‘βˆ’2ξ€·π‘Ž2+𝑐2𝐹′(0)βˆ’1.(3.46) Using (3.42), (3.41), and (3.46), we obtain 𝑐𝐺(πœ†)βˆ’1πœ†π‘Ž=βˆ’π‘+π‘βˆ’2ξ€·π‘Ž2+𝑐2ξ€Έβˆ‘βˆžπ‘›=1̃𝑐𝑛/πœ†2𝑛(0)πœ†βˆ’πœ†π‘›(0)ξ€Έξ€ΈπΉξ…žβˆ‘(0)βˆžπ‘›=1̃𝑐𝑛/πœ†π‘›ξ€·(0)πœ†βˆ’πœ†π‘›.(0)ξ€Έξ€Έ(3.47) Passing to the limit as πœ†β†’0 in (3.47), we get limπœ†β†’0𝑐𝐺(πœ†)βˆ’1πœ†ξ‚π‘Ž=βˆ’π‘+π‘βˆ’2ξ€·π‘Ž2+𝑐2ξ€Έξƒ©βˆžξ“π‘›=1Μƒπ‘π‘›πœ†2𝑛ξƒͺ(0)βˆ’1ξƒ©βˆžξ“π‘›=1Μƒπ‘π‘›πœ†3𝑛ξƒͺ(0)βˆ’1=π‘Žπ‘+2π‘βˆ’2ξ€·π‘Ž2+𝑐2πΉξ€Έξ€·ξ…žξ€Έ(0)βˆ’2πΉξ…žξ…ž(0)=𝑐0.(3.48) Using (3.48) in (3.45), we have 𝑐0+ξ“πœ‡π‘˜βˆˆintΞ“π‘›π‘˜β‰ 1π‘π‘˜πœ†βˆ’πœ‡π‘˜=1ξ€œ2πœ‹π‘–Ξ“π‘›πΊ(πœ‰)πœ‰π‘‘πœ‰.(3.49)
In view of (3.49) and (3.45), we get 𝐺(πœ†)βˆ’π‘0βˆ’π‘1πœ†βˆ’ξ“πœ‡π‘˜βˆˆintΞ“π‘›π‘˜β‰ 1πœ†π‘π‘˜πœ‡π‘˜ξ€·πœ†βˆ’πœ‡π‘˜ξ€Έ=1ξ€œ2πœ‹π‘–Ξ“π‘›πœ†πΊ(πœ‰)πœ‰(πœ‰βˆ’πœ†)π‘‘πœ‰.(3.50) Passing to the limit as π‘›β†’βˆž in (3.50), we obtain 𝐺(πœ†)=𝑐0+𝑐1πœ†+βˆžξ“π‘›=2πœ†π‘π‘›πœ‡π‘›ξ€·πœ†βˆ’πœ‡π‘›ξ€Έ.(3.51)
Lemma 3.17 is proved.

4. The Structure of Root Subspaces, Location of Eigenvalues on a Complex Plane, and Oscillation Properties of Eigenfunctions of the Problem (1.1), (1.2a)–(1.2d)

For 𝑐≠0, we find a positive integer 𝑁 from the inequality πœ‡π‘βˆ’1<βˆ’π‘‘/π‘β‰€πœ‡π‘.

Theorem 4.1. The problem (1.1), (1.2a)–(1.2d) for 𝜎>0 has a sequence of real and simple eigenvalues πœ†1<πœ†2<β‹―<πœ†π‘›βŸΆ+∞,(4.1) including at most 1+sgn|𝑐| number of negative ones. The corresponding eigenfunctions have the following oscillation properties.(a)If 𝑐=0, then the eigenfunction 𝑦𝑛(π‘₯), 𝑛β‰₯2, has exactly π‘›βˆ’1 zeros in (0,𝑙), the eigenfunction 𝑦1(π‘₯) has no zeros in (0,𝑙) in the case πœ†1β‰₯0, and the number of zeros of 𝑦1(π‘₯) can be arbitrary in the case πœ†1<0.(b)If 𝑐≠0, then the eigenfunction 𝑦𝑛(π‘₯) corresponding to the eigenvalue πœ†π‘›β‰₯0 has exactly π‘›βˆ’1 simple zeros for 𝑛≀𝑁 and exactly π‘›βˆ’2 simple zeros for 𝑛>𝑁 in (0,𝑙) and the eigenfunctions associated with the negative eigenvalues may have an arbitrary number of simple zeros in (0,𝑙).

The proof of this theorem is similar to that of [18, Theorem 2.2] using Remark 3.15.

Throughout the following, we assume that 𝜎<0.

Let πœ†,πœ‡(πœ†β‰ πœ‡) be the eigenvalue of the operator 𝐿. The eigenvectors 𝑦(πœ†)={𝑦(π‘₯,πœ†),π‘š(πœ†)} and 𝑦(πœ‡)={𝑦(π‘₯,πœ‡),π‘š(πœ‡)} corresponding to the eigenvalues πœ† and πœ‡, respectively, are orthogonal in Ξ 1, since the operator 𝐿 is 𝐽-selfadjoint in Ξ 1. Hence, by (2.4), we haveξ€œπ‘™0𝑦(π‘₯,πœ†)𝑦(π‘₯,πœ‡)𝑑π‘₯=βˆ’πœŽβˆ’1π‘š(πœ†)π‘š(πœ‡).(4.2)

Lemma 4.2. Let πœ†βˆ—βˆˆβ„ be an eigenvalue of boundary value problem (1.1), (1.2a)–(1.2d) and πΊξ…ž(πœ†βˆ—)≀𝐴, where 𝐴=βˆ’(π‘Ž2+𝑐2)/𝜎. Then, problem (1.1), (1.2a)–(1.2d) has no nonreal eigenvalues.

Proof. Let πœ‡βˆˆβ„‚β§΅β„ be an eigenvalue of problem (1.1), (1.2a)–(1.2d). Then, from Remark 3.16 and equality (4.2), we obtain ξ€œπ‘™0𝑦π‘₯,πœ†βˆ—ξ€Έπ‘šξ€·πœ†βˆ—ξ€Έξƒ©π‘¦(π‘₯,πœ‡)ξƒͺπ‘š(πœ‡)𝑑π‘₯=βˆ’πœŽβˆ’1,ξ€œπ‘™0𝑦π‘₯,πœ†βˆ—ξ€Έπ‘šξ€·πœ†βˆ—ξ€Έπ‘¦(π‘₯,πœ‡)π‘š(πœ‡)𝑑π‘₯=βˆ’πœŽβˆ’1ξ€œ,(4.3)𝑙0||||𝑦(π‘₯,πœ‡)π‘š||||(πœ‡)2𝑑π‘₯=βˆ’πœŽβˆ’1.(4.4)
In view of formula (3.29), the inequality ξ€œπ‘™0𝑦π‘₯,πœ†βˆ—ξ€Έπ‘šξ€·πœ†βˆ—ξ€Έξƒͺ2𝑑π‘₯β‰€βˆ’πœŽβˆ’1(4.5) is true.
By (4.3), ξ€œπ‘™0𝑦π‘₯,πœ†βˆ—ξ€Έπ‘šξ€·πœ†βˆ—ξ€ΈRe𝑦(π‘₯,πœ‡)π‘š(πœ‡)𝑑π‘₯=βˆ’πœŽβˆ’1.(4.6) From (4.4)–(4.6), we get ξ€œπ‘™0⎧βŽͺ⎨βŽͺβŽ©ξƒ©π‘¦ξ€·π‘₯,πœ†βˆ—ξ€Έπ‘šξ€·πœ†βˆ—ξ€Έβˆ’Re𝑦(π‘₯,πœ‡)π‘šξƒͺ(πœ‡)2+Im2𝑦(π‘₯,πœ‡)π‘šβŽ«βŽͺ⎬βŽͺ⎭(πœ‡)𝑑π‘₯<0ifπΊβ€²ξ€·πœ†βˆ—ξ€Έξ€œ<𝐴,𝑙0⎧βŽͺ⎨βŽͺβŽ©ξƒ©π‘¦ξ€·π‘₯,πœ†βˆ—ξ€Έπ‘šξ€·πœ†βˆ—ξ€Έβˆ’Re𝑦(π‘₯,πœ‡)π‘šξƒͺ(πœ‡)2+Im2𝑦(π‘₯,πœ‡)π‘šβŽ«βŽͺ⎬βŽͺ⎭(πœ‡)𝑑π‘₯=0ifπΊβ€²ξ€·πœ†βˆ—ξ€Έ=𝐴.(4.7)
From the second relation it follows that Im(𝑦(π‘₯,πœ‡)/π‘š(πœ‡))=0, which by (1.1) contradicts the condition πœ‡βˆˆβ„‚β§΅β„. The obtained contradictions prove Lemma 4.2.

Lemma 4.3. Let πœ†βˆ—1,πœ†βˆ—2βˆˆβ„, πœ†βˆ—1β‰ πœ†βˆ—2 be eigenvalues of problem (1.1), (1.2a)–(1.2d) and πΊξ…ž(πœ†βˆ—1)≀𝐴. Then, πΊξ…ž(πœ†βˆ—2)>𝐴.

Proof. Let 𝐺′(πœ†βˆ—2)≀𝐴. By (3.29) and (4.2), we have ξ€œπ‘™0𝑦π‘₯,πœ†βˆ—1ξ€Έπ‘šξ€·πœ†βˆ—1ξ€Έξƒͺ2𝑑π‘₯β‰€βˆ’πœŽβˆ’1,ξ€œπ‘™0𝑦π‘₯,πœ†βˆ—2ξ€Έπ‘šξ€·πœ†βˆ—2ξ€Έξƒͺ2𝑑π‘₯β‰€βˆ’πœŽβˆ’1,ξ€œπ‘™0𝑦π‘₯,πœ†βˆ—1ξ€Έπ‘šξ€·πœ†βˆ—1𝑦π‘₯,πœ†βˆ—2ξ€Έπ‘šξ€·πœ†βˆ—2𝑑π‘₯=βˆ’πœŽβˆ’1.(4.8)
Hence, we get ξ€œπ‘™0𝑦π‘₯,πœ†βˆ—1ξ€Έπ‘šξ€·πœ†βˆ—1𝑦π‘₯,πœ†βˆ—2ξ€Έπ‘šξ€·πœ†βˆ—2ξ€Έξƒͺ𝑑π‘₯<0ifπΊβ€²ξ€·πœ†βˆ—1ξ€Έ<𝐴orπΊβ€²ξ€·πœ†βˆ—2ξ€Έξ€œ<𝐴,𝑙0𝑦π‘₯,πœ†βˆ—1ξ€Έπ‘šξ€·πœ†βˆ—1𝑦π‘₯,πœ†βˆ—2ξ€Έπ‘šξ€·πœ†βˆ—2𝑑π‘₯=0ifπΊβ€²ξ€·πœ†βˆ—1ξ€Έ=πΊξ…žξ€·πœ†βˆ—2ξ€Έ=𝐴.(4.9)
From (4.9), it follows that 𝑦(π‘₯,πœ†βˆ—1)/π‘š(πœ†βˆ—1)=𝑦(π‘₯,πœ†βˆ—2)/π‘š(πœ†βˆ—2) for π‘₯∈[0,𝑙]. Therefore, π‘š(πœ†βˆ—2)𝑦(π‘₯,πœ†βˆ—1)=π‘š(πœ†βˆ—1)𝑦(π‘₯,πœ†βˆ—2).
Since πœ†1β‰ πœ†2, then by (1.1) 𝑦(π‘₯,πœ†1)≑0. The obtained contradictions prove Lemma 4.3.

By Lemmas 4.2 and 4.3 problem (1.1), (1.2a)–(1.2d) can have only one multiple real eigenvalue. From (3.41), we get 𝐺(3)(πœ†)>0, πœ†βˆˆπ΅, whence it follows that the multiplicity of real eigenvalue of problem (1.1), (1.2a)–(1.2d) does not exceed three.

Theorem 4.4. The boundary value problem (1.1), (1.2a)–(1.2d) for 𝜎<0 has only point spectrum, which is countable infinite and accumulates at +∞ and can thus be listed as πœ†π‘›,𝑛β‰₯1 with eigenvalues repeated according to algebraic multiplicity and ordered so as to have increasing real parts. Moreover, one of the following occurs.(1)All eigenvalues are real, at that 𝐡1 contains algebraically two (either two simple or one double) eigenvalues, and 𝐡𝑛, 𝑛=2,3,…, contain precisely one simple eigenvalues.(2)All eigenvalues are real, at that 𝐡1contains no eigenvalues but, for some 𝑠β‰₯2, 𝐡𝑠 contains algebraically three (either three simple, or one simple and one double, or one triple) eigenvalues, and 𝐡𝑛, 𝑛=2,3,…, 𝑛≠𝑠 contain precisely one simple eigenvalue.(3)There are two nonreal eigenvalues appearing as a conjugate pair, at that 𝐡1 contains no eigenvalues, and 𝐡𝑛, 𝑛=2,3,…, contain precisely one simple eigenvalue.

Proof. Remember that the eigenvalues of problem (1.1), (1.2a)–(1.2d) are the roots of the equation 𝐺(πœ†)=π΄πœ†+𝐡, where 𝐴=βˆ’(π‘Ž2+𝑐2)/𝜎, 𝐡=βˆ’(π‘Žπ‘+𝑐𝑑)⧡𝜎 (see (3.28)). From (3.41), it follows that πΊξ…žξ…ž(πœ†)>0 for πœ†βˆˆπ΅1; therefore, the function 𝐺(πœ†) is convex on the interval 𝐡1. By virtue of (3.18) and (3.30), we have limπœ†β†’βˆ’βˆžξƒ―βˆ’π‘ŽπΊ(πœ†)=𝑐if𝑐≠0,βˆ’βˆžif𝑐=0,limπœ†β†’πœ‡π‘›βˆ’0𝐺(πœ†)=+∞.(4.10)
That is why for each fixed number 𝐴 there exists number 𝐡𝐴 such that the lines π΄πœ†+𝐡𝐴, πœ†βˆˆβ„, touch the graph of function 𝐺(πœ†) at some point Μƒπœ†βˆˆπ΅1. Hence, in the interval 𝐡1, (3.28) has two simple roots πœ†1<πœ†2 if 𝐡>𝐡𝐴, one double root πœ†1=Μƒπœ† if 𝐡=𝐡𝐴, and no roots if 𝐡<𝐡𝐴.
By (3.29) and (3.30) we have limπœ†β†’πœ‡π‘›+0𝐺(πœ†)=βˆ’βˆž, limπœ†β†’πœ‡π‘›βˆ’0𝐺(πœ†)=+∞, π‘›βˆˆβ„•. Therefore, (3.28) has at least one solution in the interval 𝐡𝑛, 𝑛=2,3,….
Let 𝐡β‰₯𝐡𝐴. If 𝐡>𝐡𝐴, then 𝐺′(πœ†1)<𝐴, πΊξ…ž(πœ†2)<𝐴; if 𝐡=𝐡𝐴 then πΊξ…ž(πœ†1)=𝐴. By (3.29), (3.28) has only simple root πœ†π‘›+1 for 𝐡>𝐡𝐴,πœ†π‘› for 𝐡=𝐡𝐴 in the interval 𝐡𝑛, 𝑛=2,3,….
Let 𝐡<𝐡𝐴. By Lemma 4.3 either πΊξ…ž(πœ†π‘›)>𝐴 for any πœ†π‘›βˆˆβ„, or there exists π‘˜βˆˆβ„• such that πΉξ…ž(πœ†π‘˜)≀𝐴 and πΉξ…ž(πœ†π‘›)>𝐴, π‘›βˆˆβ„•β§΅{π‘˜}. Assume that πœ†π‘˜βˆˆπ΅π‘ . Obviously, 𝑠β‰₯2. Choose natural number 𝑛0 such that the inequalities 𝐴𝑅𝑛0||ξ€·+𝐡>0,𝐺(πœ†)βˆ’π΄πœ†+𝐡𝐴||>||𝐡𝐴||βˆ’π΅,πœ†βˆˆπ‘†π‘…π‘›0,(4.11) are fulfilled; where 𝑅𝑛=πœπ‘›+𝛿0, πœπ‘›=⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©πœ†π‘›ξ‚€πœ‹2ξ‚πœˆif𝑐=0,π‘›π‘Žif𝑐≠0,π‘πœ†>0,𝑛(𝜈0)βˆ’1if𝑐≠0,π‘Ž=0,π‘›π‘Žβˆ’1if𝑐≠0,𝑐<0,(4.12)𝛿0 is sufficiently small positive number, and 𝑆𝑅𝑛={π‘§βˆˆπΆβˆ£|𝑧|=𝑅𝑛}.
We have Δ𝑆𝑅𝑛0arg(𝐺(πœ†)βˆ’(π΄πœ†+𝐡))=Δ𝑆𝑅𝑛0ξ€·ξ€·arg𝐺(πœ†)βˆ’π΄πœ†+𝐡𝐴+Δ𝑆𝑅𝑛0𝐡arg1+π΄βˆ’π΅ξ€·πΊ(πœ†)βˆ’π΄πœ†+𝐡𝐴ξƒͺ,(4.13) where Δ𝑆𝑅𝑛01arg𝑓(𝑧)=π‘–ξ€œπ‘†π‘…π‘›0𝑓′(𝑧)ξƒͺ𝑓(𝑧)𝑑𝑧(4.14) (see [37, Chapter  IV, Section  10]). By (4.11) ||||ξ€·π΅π΄ξ€Έβˆ’π΅ξ€·ξ€·πΊ(πœ†)βˆ’π΄πœ†+𝐡𝐴||||ξ€Έξ€Έ<1,πœ†βˆˆπ‘†π‘…π‘›0,(4.15) hence, the point ξ€·π΅πœ”=π΄ξ€Έβˆ’π΅ξ€·ξ€·πΊ(πœ†)βˆ’π΄πœ†+𝐡𝐴(4.16) does not go out of circle {|πœ”|<1}. Therefore, vector 𝑀=1+πœ” cannot turn around the point 𝑀=0, and the second summand in (4.13) equals zero. Thus, Δ𝑆𝑅𝑛0arg(𝐺(πœ†)βˆ’(π΄πœ†+𝐡))=Δ𝑆𝑅𝑛0𝐺arg(πœ†)βˆ’π΄πœ†+𝐡𝐴.ξ€Έξ€Έ(4.17) By the argument principle (see [37, Chapter IV, Section 10, Theorem 1]) we have 1Ξ”2πœ‹π‘†π‘…π‘›0ξ€·ξ€·arg𝐺(πœ†)βˆ’π΄πœ†+𝐡𝐴=ξ“ξ€Έξ€Έπœ†π΄)𝑛(𝐡∈int𝑆𝑅𝑛0πœšξ‚€πœ†(𝐡𝐴)π‘›ξ‚βˆ’ξ“πœ‡π‘›βˆˆint𝑆𝑅𝑛0πœšξ€·πœ‡π‘›ξ€Έ,(4.18) where 𝜌(πœ†(𝐡𝐴)𝑛) and 𝜚(πœ‡π‘›) are multiplicity of zero πœ†(𝐡𝐴)𝑛 and pole πœ‡π‘› of the function 𝐺(πœ†)βˆ’(π΄πœ†+B𝐴), respectively (πœ†(𝐡𝐴)1=πœ†(𝐡𝐴)2). Obviously, βˆ‘πœ†π΄)𝑛(𝐡∈int𝑆𝑅𝑛0𝜌(πœ†(𝐡𝐴)𝑛)=𝑛0 and βˆ‘πœ‡π‘›βˆˆint𝑆𝑅𝑛0𝜌(πœ‡π‘›)=𝑛0βˆ’1. Then, by (4.18) we obtain (2πœ‹)βˆ’1Δ𝑆𝑅𝑛0ξ€·ξ€·arg𝐺(πœ†)βˆ’π΄πœ†+𝐡𝐴=1.(4.19) From (4.17) and (4.19) follows the validity of the equality (2πœ‹)βˆ’1Δ𝑆𝑅𝑛0arg(𝐺(πœ†)βˆ’(π΄πœ†+𝐡))=1.(4.20)
Using the argument principle again, by (4.20) we get ξ“πœ†π‘›βˆˆint𝑆𝑅𝑛0πœŒξ€·πœ†π‘›ξ€Έβˆ’ξ“πœ‡π‘›βˆˆint𝑆𝑅𝑛0πœšξ€·πœ‡π‘›ξ€Έ=1,(4.21) whence it follows that ξ“πœ†π‘›βˆˆint𝑆𝑅𝑛0πœŒξ€·πœ†π‘›ξ€Έ=𝑛0,(4.22) where πœ†π‘›, π‘›βˆˆβ„•, are roots of the equation 𝐺(πœ†)=π΄πœ†+𝐡. From the above-mentioned reasoning, by (4.22) we have ξ“πœ†π‘šβˆˆintπ‘†π‘…π‘›πœŒξ€·πœ†π‘šξ€Έ=𝑛,𝑛=𝑛0,𝑛0+1,…,(4.23) and, therefore, problem (1.1), (1.2a)–(1.2d) in the interval 𝐡𝑛 for 𝑛=𝑛0, 𝑛0+1,…, has only one simple eigenvalue.
Consider the following two cases.
Case 1. For all real eigenvalues πœ†π‘› of problem (1.1), (1.2a)–(1.2d) the inequalities 𝐺′(πœ†π‘›)>𝐴, πœ†π‘›βˆˆβ‹ƒβˆžπ‘š=2π΅π‘š, are fulfilled. The problem (1.1), (1.2a)–(1.2d) in every interval π΅π‘š, π‘š=2,3,…,𝑛0βˆ’1, has one simple eigenvalue. Hence, problem (1.1), (1.2a)–(1.2d) in the interval (βˆ’βˆž,𝑆𝑅𝑛), 𝑛β‰₯𝑛0, has π‘›βˆ’2 simple eigenvalues, and hence, by (4.23), this problem in the circle π‘†π‘…π‘›βŠ‚β„‚ has one pair of simple nonreal eigenvalues. In this case, the location of the eigenvalues will be in the following form: πœ†1,πœ†2βˆˆβ„‚β§΅β„, πœ†2=πœ†1, Imπœ†1>0, πœ†π‘›βˆˆπ΅π‘›βˆ’1, 𝑛=3,4,….Case 2. Let πΊξ…ž(πœ†π‘˜)≀𝐴, πΊξ…ž(πœ†π‘›)>𝐴, π‘›βˆˆπ‘βˆ£{π‘˜} and πœ†π‘˜βˆˆπ΅π‘ , 𝑠β‰₯2. By Lemma 4.2 problem (1.1), (1.2a)–(1.2d) has no nonreal eigenvalues. From the above-mentioned reasoning it follows that in each interval 𝐡𝑛, π‘›β‰ π‘˜, 𝑛=2,3,…, problem (1.1), (1.2a)–(1.2d) has one simple eigenvalue.Subcase 1. Let πΊξ…ž(πœ†π‘˜)=𝐴,πΊξ…žξ…ž(πœ†π‘˜)β‰ 0, that is, the eigenvalue πœ†π‘˜ is a double one (by this πœ†π‘˜=πœ†π‘˜+1). Then, from (4.23) it follows that the interval 𝐡𝑠 besides the eigenvalue πœ†π‘˜ contains one more simple eigenvalue: at that it is either πœ†π‘˜βˆ’1 (by this π‘˜=𝑠) or πœ†π‘˜+2 (by this π‘˜=π‘ βˆ’1). Hence, πœ†π‘›βˆˆπ΅π‘›+1, 𝑛=1,2,…,π‘ βˆ’2, πœ†π‘ βˆ’1,πœ†π‘ ,πœ†π‘ +1βˆˆπ΅π‘  (by this either πœ†π‘ βˆ’1<πœ†π‘ =πœ†π‘ +1 or πœ†π‘ βˆ’1=πœ†π‘ <πœ†π‘ +1), πœ†π‘›βˆˆπ΅π‘›βˆ’1, 𝑛=𝑠+2,𝑠+3,….Subcase 2. Let πΊξ…ž(πœ†π‘˜)=𝐴, πΊξ…žξ…ž(πœ†π‘˜)=0. By (3.41), πΊξ…žξ…žξ…ž(πœ†π‘˜)β‰ 0. Hence, πœ†π‘˜ is a triple eigenvalue of the problem (1.1), (1.2a)–(1.2d) (by this πœ†π‘˜=πœ†π‘˜+1=πœ†π‘˜+2). Then, from (4.23) it follows that in the interval 𝐡𝑠 problem (1.1), (1.2a)–(1.2d) has unique triple eigenvalue πœ†π‘˜, and therefore, π‘˜=π‘ βˆ’1. At this πœ†π‘›βˆˆπ΅π‘›+1, 𝑛=1,2,…,π‘ βˆ’2, πœ†π‘ βˆ’1=πœ†π‘ =πœ†π‘ +1βˆˆπ΅π‘ , πœ†π‘›βˆˆπ΅π‘›βˆ’1,𝑛=𝑠+2,𝑠+3,….Subcase 3. Let πΊξ…ž(πœ†π‘˜)<𝐴, that is, the eigenvalue πœ†π‘˜ is simple. Then, by (4.23), in the interval 𝐡𝑠problem (1.1), (1.2) has an eigenvalue πœ†π‘˜ as well as two more simple eigenvalues, which, by Lemma 4.3, are πœ†π‘˜βˆ’1 and πœ†π‘˜+1 (and hence π‘˜=𝑠). In this case, we have πœ†π‘›βˆˆπ΅π‘›+1, 𝑛=1,2,…,π‘ βˆ’2, πœ†π‘ βˆ’1,πœ†π‘ ,πœ†π‘ +1βˆˆπ΅π‘  (πœ†π‘ βˆ’1<πœ†π‘ <πœ†π‘ +1), πœ†π‘›βˆˆπ΅π‘›βˆ’1, 𝑛=𝑠+2,𝑠+3,….
Theorem 4.4 is proved.

By Theorem 4.4 we have 𝜚(πœ†π‘›)=2, that is, πœ†π‘›=πœ†π‘›+1 if 𝑛=π‘ βˆ’1 or 𝑛=𝑠; 𝜚(πœ†π‘›)=3, that is, πœ†π‘›=πœ†π‘›+1=πœ†π‘›+2 if 𝑛=π‘ βˆ’1 (If assertion (2) in Theorem 4.4 holds, then we set 𝑠=1).

Let {𝑦𝑛(π‘₯)}βˆžπ‘›=1 be a system of eigen- and associated functions corresponding to the eigenvalue system {πœ†π‘›}βˆžπ‘›=1 of problem (1.1), (1.2a)–(1.2d), where 𝑦𝑛(π‘₯)=𝑦(π‘₯,πœ†π‘›) if 𝜌(πœ†π‘›)=1; 𝑦𝑛(π‘₯)=𝑦(π‘₯,πœ†π‘›), 𝑦𝑛+1(π‘₯)=π‘¦βˆ—π‘›+1(π‘₯)+𝑐𝑛𝑦𝑛(π‘₯), π‘¦βˆ—π‘›+1(π‘₯)=(πœ•π‘¦(π‘₯,πœ†π‘›))/πœ•πœ†, 𝑐𝑛 is an arbitrary constant, if 𝜚(πœ†π‘›)=2;𝑦𝑛(π‘₯)=𝑦(π‘₯,πœ†π‘›), 𝑦𝑛+1(π‘₯)=π‘¦βˆ—π‘›+1(π‘₯)+𝑑𝑛𝑦𝑛(π‘₯), 𝑦𝑛+2(π‘₯)=π‘¦βˆ—π‘›+2(π‘₯)+π‘‘π‘›π‘¦βˆ—π‘›+1(π‘₯)+β„Žπ‘›π‘¦π‘›(π‘₯), π‘¦βˆ—π‘›+2(π‘₯)=πœ•2𝑦(π‘₯,πœ†π‘›)/2πœ•πœ†2, 𝑑𝑛, β„Žπ‘› are arbitrary constants, if 𝜌(πœ†π‘›)=3. Here, 𝑦𝑛(π‘₯) is an eigenfunction for πœ†π‘› and 𝑦𝑛+1(π‘₯) when 𝜌(πœ†π‘›)=2; 𝑦𝑛+1(π‘₯), 𝑦𝑛+2(π‘₯) when 𝜌(πœ†π‘›)=3 are the associated functions (see [34, Pages 16–20] for more details).

We turn now to the oscillation theorem of the eigenfunctions corresponding to the positive eigenvalues of problem (1.1), (1.2a)–(1.2d) since the eigenfunctions associated with the negative eigenvalues may have an arbitrary number of simple zeros in (0,𝑙).

Theorem 4.5. For each 𝑛<𝑁 (resp., 𝑛>𝑁), 𝑦𝑛 has π‘›βˆ’1 (resp., 𝑛) zeros in the interval (0,l). Similarly 𝑦𝑠, 𝑦𝑠+1 both have π‘ βˆ’1 (resp., 𝑠) zeros if 𝑠<𝑁 (resp., 𝑠>𝑁). Finally, if 𝑐≠0, then each of 𝑦𝑁 (and 𝑦𝑠, 𝑦𝑠+1 if 𝑠=𝑁) has π‘βˆ’1  or  𝑁 zeros according to πœ†π‘, πœ†π‘†, πœ†π‘†+1≀ or >βˆ’π‘‘/𝑐, and if 𝑐=0 and 𝑠=𝑁, then 𝑦𝑠, 𝑦𝑠+1 both have 𝑠 zeros.

The proof of this theorem is similar to that of [11, Theorem 4.4] using Lemma 3.12.

5. Asymptotic Formulae for Eigenvalues and Eigenfunctions of the Boundary Value Problem (1.1), (1.2a)–(1.2d)

For 𝑐≠0, let 𝐾 be an integer such that πœ†π‘˜βˆ’1(πœ‹/2)<𝑏/π‘Žβ‰€πœ†π‘˜(πœ‹/2) (interpreting πœ†0(πœ‹/2)=βˆ’βˆž).

Lemma 5.1. The following relations hold for sufficiently large π‘›βˆˆβ„•, 𝑛>𝑛1=max{𝑠,𝑁,𝐾}+2: πœ†π‘›βˆ’2(0)<πœ†π‘›<πœ†π‘›