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

Abstract

We consider the fourth-order spectral problem 𝑦⁽⁴⁾(𝑥)−(𝑞(𝑥)𝑦′(𝑥))^=𝜆𝑦(𝑥),𝑥∈(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 𝐿_𝑝(0,𝑙), 𝑝∈(1,∞), of the system of root functions of this problem.

1. Introduction

The following boundary value problem is considered:𝑦⁽⁴⁾(𝑥)−𝑞(𝑥)𝑦^(𝑥)^=𝜆𝑦(𝑥),𝑥∈(0,𝑙),′∶=𝑑𝑑𝑥,(1.1) 𝑦𝑦(0)cos𝛽𝑦+^(0𝑇)𝑦=(00),sin𝛽=0,((11..22ba))_(𝑎𝜆+(𝑙𝑏))c𝑦o(s𝑙𝛾)−+(𝑦𝑐𝜆^(+𝑙)𝑑si)n𝑇𝛾𝑦=(𝑙)0=,0,((11..22dc))where 𝜆 is a spectral parameter, 𝑇𝑦≡𝑦^−𝑞𝑦^, 𝑞 is absolutely continuous function on [0,𝑙], 𝛽, 𝛾, 𝑎, 𝑏, 𝑐, and 𝑑 are real constants such that 0≤𝛽, 𝛾≤𝜋/2 and 𝜎=𝑏𝑐−𝑎𝑑≠0. Moreover, we assume that the equation𝑦^−𝑞𝑦=0,(1.3) is disfocal in [0,𝑙], that is, there is no solution of (1.3) such that 𝑦(𝑎)=𝑦′(𝑏)=0 for any 𝑎,𝑏∈[0,𝑙]. Note that the sign of 𝑞 which satisfies the disfocal condition may change in [0,𝑙].

Problems of this type occur in mechanics. If 𝛽=0, 𝛾=𝜋/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 𝑎<0 corresponds to the situation where this is a particle of mass 𝑎 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 𝐿_𝑝(0,𝑙),𝑝∈(1,∞), of the system of root functions of the boundary value problem (1.1), (1.2a)–(1.2d) with 𝑞≥0, 𝜎>0, are considered in [18] and, with 𝑞≥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 𝐿_𝑝(0,𝑙), 𝑝∈(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)⊕ℂ 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 𝐿_𝑝(0,𝑙), 𝑝∈(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 Π₁. 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 𝐿_𝑝(0,𝑙), 𝑝∈(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)⊕ℂ be a Hilbert space equipped with the inner product(𝑦,̂𝑢)̂_𝐻=({𝑦,𝑚},{𝑢,𝑠})_𝐻=(𝑦,𝑢)_𝐿2+||𝜎⁻¹||𝑚𝑠,(2.1) where (𝑦,𝑢)_𝐿2=∫^𝑙0𝑦𝑢𝑑𝑥.

We define in the 𝐻 operator𝐿𝑦̂=𝐿{𝑦,𝑚}=(𝑇𝑦(𝑥))^,𝑑𝑇𝑦(𝑙)−𝑏𝑦(𝑙)(2.2) with domain𝐷(𝐿)=𝑦̂={𝑦,𝑚}∈𝐻/𝑦(𝑥)∈𝑊⁴²(0,𝑙),(𝑇𝑦(𝑥))^∈𝐿₂(0,𝑙),𝑦∈(B.C.),𝑚=𝑎𝑦(𝑙)−𝑐𝑇𝑦(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:𝐿𝑦̂=𝜆𝑦,̂𝑦̂∈𝐷(𝐿),(2.4) that is, the eigenvalue 𝜆_𝑛 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:𝑦_𝑛̂=𝑦_𝑛(𝑥),𝑚_𝑛⟷𝑦_𝑛(𝑥),𝑚_𝑛=𝑎𝑦_𝑛(𝑙)−𝑐𝑇𝑦_𝑛(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 {{𝑦_𝑛(𝑥),𝑚_𝑛}}^∞𝑛=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,𝑚}={𝑦,−𝑚}. 𝐽 is a unitary, symmetric operator in 𝐻. Its spectrum consists of two eigenvalues: −1 with multiplicity 1, and +1 with infinite multiplicity. Hence, this operator generates the Pontryagin space Π₁=𝐿₂(0,𝑙)⊕ℂ by means of the inner products (𝐽-metric) [26]:(𝑦,̂𝑢)̂_Π1=({𝑦,𝑚},{𝑢,𝑠})_Π1=(𝑦,𝑢)_𝐿2+𝜎⁻¹𝑚𝑠.(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 𝐿^∗ be an operator adjoined to the operator 𝐿 in 𝐻. Then, 𝐿^∗=𝐽𝐿𝐽.

Let 𝜆 be an eigenvalue of operator 𝐿 of algebraic multiplicity 𝜈. Let us suppose that 𝜌(𝜆) is equal to 𝜈 if Im𝜆≠0 and equal to whole part 𝜈/2 if Im𝜆=0.

Theorem 2.3 (see [28]). The eigenvalues of operator 𝐿 arrange symmetrically with regard to the real axis. ∑^𝑛𝑘=1𝜌(𝜆_𝑘)≤1 for any system {𝜆_𝑘}^𝑛𝑘=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 Prüfer-type transformation of the following form:𝑦𝑦(𝑥)=𝑟(𝑥)sin𝜓(𝑥)cos𝜃(𝑥),_′𝑦(𝑥)=𝑟(𝑥)cos𝜓(𝑥)sin𝜑(𝑥),_𝑇𝑦((𝑥𝑥))==𝑟𝑟((𝑥𝑥))csoins𝜓𝜓((𝑥𝑥))csoins𝜃𝜑((𝑥𝑥)).,(3.1)

Consider the boundary conditions (see [29, 30])𝑦𝑦^′((0𝑙))ccooss𝛿𝛼−−𝑇𝑦𝑦^((𝑙0))ssiinn𝛿𝛼==00,,((11..22da^∗_∗)) where 𝛼∈[0,𝜋/2], 𝛿∈[0,𝜋).

Alongside the spectral problem (1.1), (1.2a)–(1.2d) we will consider the spectral problem (1.1), (1.2a)–(1.2c), and (1.2d^∗). In [30], Banks and Kurowski developed an extension of the Prüfer transformation (3.1) to study the oscillation of the eigenfunctions and their derivatives of problem (1.1), (1.2a^∗), (1.2b), (1.2c), and (1.2d^∗) with 𝑞≥0, 𝛿∈[0,𝜋/2] and in some cases when (1.3) is disfocal and 𝛼=𝛾=0, 𝛿∈[0,𝜋/2]. In [19], the authors used the Prüfer transformation (3.1) to study the oscillations of the eigenfunctions of the problem (1.1), (1.2a^∗), (1.2b), (1.2c), and (1.2d^∗) with 𝑞≥0 and 𝛿∈(𝜋/2,𝜋). In this work it is proved that problem (1.1), (1.2a^∗), (1.2b), (1.2c), and (1.2d^∗) 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 (1.2d^∗) 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 𝑢 be a solution of (1.3) which satisfies the initial conditions 𝑢(0)=0, 𝑢^′(0)=1. Then the disfocal condition of (1.3) implies that 𝑢^(𝑥)>0 in [0,𝑙]. Therefore, if ℎ denotes the solution of (1.3) satisfying the initial conditions 𝑢(0)=𝑐>0, 𝑢′(0)=1, where 𝑐 is a sufficiently small constant, then we have also ℎ^(𝑥)>0 on [0,𝑙]. Thus, ℎ(𝑥)>0 in [0,𝑙], and hence the following substitutions [33, Theorem 12.1]:𝑡=𝑡(𝑥)=𝑙𝜔⁻¹^𝑥0ℎ(𝑠)𝑑𝑠,𝜔=^𝑙0ℎ(𝑠)𝑑𝑠,(3.2) transform [0,𝑙] into the interval [0,𝑙] and (1.1) into(𝑝𝑦)̈^⋅⋅=𝜆𝑟𝑦,(3.3) where 𝑝=(𝑙𝜔⁻¹ℎ)³, 𝑟=𝑙⁻¹𝜔ℎ⁻¹; ℎ(𝑥), 𝑦(𝑥) are taken as functions of 𝑡 and ⋅∶=𝑑/𝑑𝑡. Furthermore, the following relations are useful in the sequel:𝑦̇=𝑙⁻¹𝜔ℎ⁻¹𝑦^,𝑙²𝜔⁻²ℎ³𝑦̈=ℎ𝑦^−ℎ^𝑦^,𝑇𝑦≡𝑙𝜔⁻¹ℎ³𝑦̈^⋅=𝑇𝑦.(3.4) It is clear from the second relation (3.4) that the sign of 𝑦^ 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 𝑞≥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 (1.2d^∗) with 𝛾=0, 𝛿∈[0,𝜋/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 (1.2d^∗) with 𝛽=0.

Lemma 3.1 (see [33, Lemma 2.1]). Let 𝜆>0, and let 𝑦 be a nontrivial solution of (3.3). If 𝑦, 𝑦̇, 𝑦̈, and 𝑇𝑦 are nonnegative at 𝑡=𝑎 (but not all zero), they are positive for all 𝑡>𝑎. If 𝑦, −𝑦̇, 𝑦̈, and −𝑇𝑦 are nonnegative at 𝑡=𝑎 (but not all zero), they are positive for all 𝑡<𝑎.

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 (1.2d^∗) for 𝛿∈[0,𝜋/2) or 𝛿=𝜋/2, 𝛽∈[0,𝜋/2) are positive.

Proof. In this case, the transformed problem is determined by (3.3) and the boundary conditions𝑦̇(𝑦𝑙)(0c)ocso𝛾s_∗𝛽𝑦+̇(0𝑇)𝑦=(00),sin𝛽=0,((33..55ba))𝑦(𝑙)cos𝛿+−𝑝𝑇(𝑙)𝑦𝑦(𝑙̈()𝑙)sisnin𝛿𝛾=^∗=0,0,((33..55dc))where 𝛾^∗=arctg{𝑙⁻²𝜔²ℎ⁻¹(𝑙)[ℎ(𝑙)cos𝛾+ℎ^′(𝑙)sin𝛾]⁻¹}∈[0,𝜋/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]=∫^𝑙0𝑝𝑦̈²𝑑𝑡+𝑁[𝑦]∫^𝑙0𝑦²𝑑𝑡,(3.6) where 𝑁[𝑦]=𝑦²(0)cot𝛽+𝑦̇²(𝑙)cot𝛾^∗+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/2. Hence, all the eigenvalues of problem (3.3), (3.5a)–(3.5d) for 𝛿∈[0,𝜋/2) or 𝛿=𝜋/2, 𝛽∈[0,𝜋/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/2<𝛾<𝜋 (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, (1.2a^∗), (1.2b), and (1.2c) for 𝛼∈(0,𝜋/2].

Lemma 3.4 (see [29, Lemma 2.2]). Let 𝜆>0 and u be a solution of (3.3) which satisfies the boundary conditions (3.5a)–(3.5c). If 𝑎 is a zero of 𝑢 and 𝑢̈ in the interval (0,l), then 𝑢(̇𝑡)𝑇𝑢(𝑡)<0 in a neighborhood of 𝑎. If 𝑎 is a zero of 𝑢̇ or 𝑇𝑢 in (0,l), then 𝑢(𝑡)𝑢(̈𝑡)<0 in a neighborhood of 𝑎.

Theorem 3.5. Let 𝑢 be a nontrivial solution of the problem (1.1), (1.2a) and (1.2c) for 𝜆>0. Then the Jacobian 𝐽[𝑢]=𝑟³cos𝜓sin𝜓 of the transformation (3.1) does not vanish in (0,l).

Proof. Let 𝑢 be a nontrivial solution of (1.1) which satisfies the boundary conditions (1.2a) and (1.2c). Assume first that the corresponding angle 𝜓 satisfies 𝜓(𝑥₀)=𝑛𝜋 for some integer 𝑛 and for some 𝑥₀∈(0,𝑙). Then, the transformation (3.1) implies that 𝑢(𝑥₀)=𝑇𝑢(𝑥₀)=0. Using the transformation (3.2), the solution 𝑢 of (3.3) also satisfies 𝑢(𝑡₀)=𝑇𝑢(𝑡₀)=0, where 𝑡₀=𝑙⁻¹𝜔∫^𝑥0₀ℎ(𝑠)𝑑𝑠∈(0,𝑙). However, it is incompatible with the conclusion of Lemma 3.4.
The proof of the inequality cos𝜓(𝑥)≠0, 𝑥∈(0,𝑙), proceeds in the same fashion as in the previous case. The proof of Theorem 3.5 is complete.

Let 𝑦(𝑥,𝜆) be a nontrivial solution of the problem (1.1), (1.2a)–(1.2c) for 𝜆>0 and 𝜃(𝑥,𝜆), 𝜑(𝑥,𝜆) 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]):𝜃(0,𝜆)=𝛽−𝜋2,𝜑(0,𝜆)=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 (1.2d^∗) for 𝛿∈[0,𝜋/2]. In particular, we have the following results.

Theorem 3.6. 𝜃(𝑙,𝜆) is a strictly increasing continuous function on 𝜆.

Theorem 3.7. Problem (1.1), (1.2a)–(1.2c), and (1.2d^∗) for 𝛿∈[0,𝜋/2] (except the case 𝛽=𝛿=𝜋/2) has a sequence of positive and simple eigenvalues 𝜆₁(𝛿)<𝜆₂(𝛿)<⋯<𝜆_n(𝛿)⟶∞.(3.8) Moreover, 𝜃(𝑙,𝜆_𝑛(𝛿))=(2𝑛−1)𝜋/2−𝛿, 𝑛∈ℕ; the corresponding eigenfunctions 𝜐^𝑛(𝛿)(𝑥) have 𝑛−1 simple zeros in (0,l).

Remark 3.8. In the case 𝛽=𝛿=𝜋/2 the first eigenvalue of boundary value problem (1.1), (1.2a)–(1.2c), and (1.2d^∗) is equal to zero and the corresponding eigenfunction is constant; the statement of Theorem 3.7 is true for 𝑛≥2.

Obviously, the eigenvalues 𝜆_𝑛(𝛿), 𝑛∈ℕ, of the problem (1.1), (1.2a)–(1.2c), and (1.2d^∗) are zeros of the entire function 𝑦(𝑙,𝜆)cos𝛿−𝑇𝑦(𝑙,𝜆)cos𝛿=0. Note that the function 𝐹(𝜆)=𝑇𝑦(𝑙,𝜆)/𝑦(𝑙,𝜆) is defined for 𝜆∈𝐴≡(ℂ/ℝ)∪(⋃^∞𝑛=1(𝜆_𝑛−1(0),𝜆_𝑛(0))), where 𝜆₀(0)=−∞.

Lemma 3.9 (see [19, Lemma 5]). Let 𝜆∈𝐴. Then, the following relation holds: 𝑑𝑑𝜆𝐹(𝜆)=∫^𝑙0𝑦²(𝑥,𝜆)𝑑𝑥𝑦²(𝑙,𝜆).(3.9)

In (1.1) we set 𝜆=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 𝑧_𝑘(𝑥,𝜌), 𝑘=1,4, regular in 𝜌 (for sufficiently large 𝜌) and satisfying the relations𝑧^𝑘(𝑠)(𝑥,𝜌)=𝜌𝜔_𝑘^𝑠𝑒^{𝜌𝜔𝑘𝑥}[1],𝑘=1,4,𝑠=0,3,(3.10) where 𝜔_𝑘, 𝑘=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 obtain𝑦(𝑥,𝜆)=⎧⎪⎨⎪⎩sin𝜌𝑥+𝜋2sin𝛽−cos𝜌𝑙+𝜋2𝑠(𝛽,𝛾)𝑒^{𝜌(𝑥−𝑙)}√[1]if𝛽∈0,𝜋2,2sin𝜌𝑥−𝜋4−𝑒^−𝜌𝑥+(−1)^1−sgn𝛾√2sin𝜌𝑙+(−1)^sgn𝛾𝜋4𝑒^{𝜌(𝑥−𝑙)}𝑇𝑦(𝑥,𝜆)=⎧⎪⎪⎨⎪⎪⎩−𝜌₃[1]if𝛽=0,(3.11)cos𝜌𝑥+𝜋2sgn𝛽+cos𝜌𝑙i+f𝛽𝜋2𝑠∈(𝛽,0𝛾,)𝜋𝑒^{𝜌(𝑥−𝑙)}[1]−𝜌₃2,√2sin𝜌𝑥+𝜋4−𝑒^−𝜌𝑥−(−1)if𝛽=^1−sgn𝛾√0.2sin𝜌𝑙+𝜋4(−1)^sgn𝛾𝑒^{𝜌(𝑥−𝑙)}[1](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(𝑥,𝜆) tends to +∞ as 𝜆→±∞.

Taking into account relations (3.11) and (3.12), we obtain the asymptotic formulas𝐹(𝜆)=⎧⎪⎪⎨⎪⎪⎩√2^1−sgn𝛾𝜌³cos(𝜌𝑙+(𝜋/2)sgn𝛽+(𝜋/4)sgn𝛾)√cos(𝜌𝑙+(𝜋/2)sgn𝛽+(𝜋/4)(1+sgn𝛾))[1]if𝛽∈0,𝜋2,2^1−sgn𝛾𝜌³cos(𝜌𝑙+(𝜋/4)(sgn𝛾−1))cos(𝜌𝑙+(𝜋/4)(1+sgn𝛾))[1]if𝛽=0.(3.13) Furthermore, we have𝐹(𝜆)=−√2^1−sgn𝛾4||𝜆||³1+𝑂||𝜆||^−1/4,as𝜆⟶−∞.(3.14)

We define numbers 𝜏, 𝜈, 𝜂, 𝛼_𝑛, 𝛽_𝑛, 𝜂_𝑛, 𝑛∈ℕ, and a function 𝑧(𝑥,𝑡), 𝑥∈[0,𝑙], 𝑡∈ℝ, as follows:𝜏=⎧⎪⎨⎪⎩53(1+4𝑠(𝛽,𝛿))−1if𝛾∈0,𝜋2,4−38(−1)^sgn𝛽𝜂=⎧⎪⎨⎪⎩3(2+sgn𝛽)+(−1)^sgn𝛿−1if𝛾=0,54−1if𝛾∈0,𝜋2,4−38(−1)^sgn𝛽𝜈=⎧⎪⎨⎪⎩3(1+𝑠(𝛽,|𝑐|))−1−1if𝛾=0,54if𝛾∈0,𝜋2,4−38(−1)^sgn𝛽𝛼+(−1)^sgn|𝑐|if𝛾=0,_𝑛=(𝑛−𝜏)𝜋𝑙,𝜂^𝑛=(𝑛−𝜂)𝜋𝑙,𝛽^𝑛=(𝑛−𝜈)𝜋𝑧(𝑥,𝑡)=⎧⎪⎨⎪⎩sin𝑡𝑥+𝜋𝑙,2sgn𝛽−cos𝑡𝑙+𝜋2𝑠(𝛽,𝛾)𝑒^{−𝑡(𝑙−𝑥)}if𝛽∈0,𝜋√2,2sin𝑡𝑥−𝜋4+𝑒^−𝑡𝑥+(−1)^sgn𝛾√2sin𝑡𝑙+(−1)^sgn𝛾𝜋4𝑒^{−𝑡(𝑥−𝑙)}if𝛽=0.(3.15) By virtue of [18, Theorem 3.1], one has the asymptotic formulas𝜐√⁴𝜆_𝑛(𝛿)=𝛼_𝑛+𝑂𝑛⁻¹,(3.16)_𝑛(𝛿)(𝑥)=𝑧𝑥,𝛼_𝑛+𝑂𝑛⁻¹,(3.17) where relation (3.17) holds uniformly for 𝑥∈[0,𝑙].

By (3.14), we havelim_{𝜆→−∞}𝐹(𝜆)=−∞.(3.18) From Property 1 in [30] and formulas (3.9), one has the relations𝜆₁𝜋2<𝜆¹(0)<𝜆₂𝜋2<𝜆²(0)<⋯.(3.19)

Remark 3.11. It follows by Theorem 3.7, Lemma 3.9, and relations (3.18) and (3.19) that if 𝜆>0 or 𝜆=0, 𝛽∈[0,𝜋/2), then 𝐹(𝜆)<0; besides, if 𝜆=0 and 𝛽=𝜋/2, then 𝐹(𝜆)=0.

Let 𝑠(𝜆) be the number of zeros of the function 𝑦(𝑥,𝜆) in the interval (0,𝑙).

Lemma 3.12. If 𝜆>0 and 𝜆∈(𝜆_𝑛−1(0),𝜆_𝑛(0)], 𝑛∈ℕ, then 𝑠(𝜆)=𝑛−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 (1.2d^∗) for 𝛿∈(𝜋/2,𝜋) has a sequence of real and simple eigenvalues 𝜆₁(𝛿)<𝜆₂(𝛿)<⋯<𝜆_𝑛(𝛿)⟶+∞,(3.20) including at most one negative eigenvalue. Moreover, (a) if 𝛽∈[0,𝜋/2), then 𝜆₁(𝛿)>0 for 𝛿∈(𝜋/2,𝛿₀); 𝜆₁(𝛿)=0 for 𝛿=𝛿₀;𝜆₁(𝛿)<0 for 𝛿∈(𝛿₀,𝜋), where 𝛿₀=arctg𝑇𝑦(𝑙,0)/𝑦(𝑙,0); (b) if 𝛽=𝜋/2, then 𝜆₁(𝛿)<0; (c) the eigenfunction 𝜐^𝑛(𝛿)(𝑥), corresponding to the eigenvalue 𝜆_𝑛(𝛿)≥0, has exactly 𝑛−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(0𝑦)=⁽⁴⁾(𝑦𝑥_)−𝑞(𝑥)𝑦^′(𝑥)^=𝜆𝑦(𝑥),𝑥∈(0,𝑙),(0)=𝑇𝑦(0)=𝑦^(𝑙)cos𝛾+𝑦^(𝑙)sin𝛾=0,(3.21) has an infinite set of nonpositive eigenvalues 𝜌_𝑛 tending to −∞ and satisfying the asymptote 𝜆_𝑛=−𝑛−14(1+sgn𝛾)⁴𝜋⁴𝑙⁴+𝑜𝑛⁴,𝑛⟶∞.(3.22)

Setting 𝑥=0 in (3.12), we obtain (3.22).

Remark 3.15. By Remark 3.10 the number of zeros of the eigenfunction 𝑦^1(𝛿)(𝑥) corresponding to an eigenvalue 𝜆₁(𝛿)<0 can by arbitrary. In views of [31, Corollary 2.5], as 𝜆₁(𝛿)<0 varies, new zeros of the corresponding eigenfunction 𝑦^1(𝛿)(𝑥) enter the interval (0,𝑙) only through the end point 𝑥=0 (since 𝑦^1(𝛿)(𝑙)≠0), and hence the number of its zeros, in the case 𝛽∈(0,𝜋/2], is asymptotically equivalent to the number of eigenvalues of the problem (3.21) which are higher than 𝜆₁(𝛿). In the case 𝛽=0 see [31, Theorem 5.3].

We consider the following boundary conditions:𝑐𝑎𝑦𝑦((𝑙𝑙))+−𝑎𝑐𝑇𝑇𝑦𝑦((𝑙𝑙))==00,.((11..22dd^′_′))

Note that (𝑎,𝑐)≠0 since 𝜎<0. The boundary condition (1.2d^′) coincides the boundary condition (1.2d^∗) for 𝛿=𝜋/2 (resp., 𝛿=0) in the case 𝑎=0 (resp., 𝑐=0), and the boundary condition (1.2d^′) coincides the boundary condition (1.2d^∗) for 𝛿=0 (resp., 𝛿=𝜋/2) in the case 𝑎=0 ( resp., 𝑐=0).

Let 𝑎𝑐≠0. The eigenvalues of the problem (1.1), (1.2a)–(1.2c), and (1.2d^′) (resp., (1.1) (1.2a)–(1.2c), and (1.2d^′)) are the roots of the equation 𝐹(𝜆)=𝑎/𝑐 (resp., 𝐹(𝜆)=−𝑐/𝑎). By (3.9), this equation has only simple roots; hence all the eigenvalues of the problems (1.1), (1.2a)–(1.2c), and (1.2d^′) and (1.1), (1.2a)–(1.2c), and (1.2d^′) are simple. On the base of (3.9), (3.18), and (3.19) in each interval 𝐴_𝑛, 𝑛=1,2,…, the equation 𝐹(𝜆)=𝑎/𝑐 (resp., 𝐹(𝜆)=−𝑐/𝑎) has a unique solution 𝜇_𝑛 (resp., 𝜈_𝑛); moreover,𝜈₁<𝜆₁𝜋2<𝜇¹<𝜆₁(0)<𝜈₂<𝜆₂𝜋2<𝜇²<𝜆₂(0)<⋯(3.23) if 𝑎/𝑐>0 and𝜇₁<𝜆₁𝜋2<𝜈¹<𝜆₁(0)<𝜇₂<𝜆₂𝜋2<𝜈²<𝜆₂(0)<⋯(3.24) if 𝑎/𝑐<0. Besides, 𝜇₁=0 if 𝑎/𝑐<0 and 𝐹(0)=𝑎/𝑐;𝜈₁=0 if 𝑎/𝑐>0 and 𝐹(0)=−𝑐/𝑎.

Taking into account (1.2d^′), (1.2d^′), (3.23), and (3.24) and using the corresponding reasoning [18, Theorem 3.1] we have𝜑√⁴𝜇_𝑛=𝜂_𝑛+𝑂𝑛⁻¹,√⁴𝜈_𝑛=𝜂_𝑛+𝑂𝑛⁻¹,(3.25)_𝑛(𝑥)=𝑧𝑥,𝜂_𝑛+𝑂𝑛⁻¹,𝜓_𝑛(𝑥)=𝑧𝑥,𝜂_𝑛+𝑂𝑛⁻¹,(3.26) where relation (3.26) holds uniformly for 𝑥∈[0,𝑙] and eigenfunctions 𝜑_𝑛(𝑥) and 𝜓_𝑛(𝑥), 𝑛∈ℕ, correspond to the eigenvalues 𝜇_𝑛 and 𝜈_𝑛, respectively.

Let us denote 𝑚(𝜆)=𝑎𝑦(𝑙,𝜆)−𝑐𝑇𝑦(𝑙,𝜆).

Remark 3.16. Note that if 𝜆 is the eigenvalue of problem (1.1), (1.2a)–(1.2d), then 𝑚(𝜆)≠0 since 𝜎≠0.

It is easy to see that the eigenvalues of problem (1.1), (1.2a)–(1.2d) are roots of the equation(𝑎𝜆+𝑏)𝑦(𝑙)−(𝑐𝜆+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,𝜆)+𝑎𝑇𝑦(𝑙,𝜆)𝑎𝑦(𝑙,𝜆)−𝑐𝑇𝑦(𝑙,𝜆)=𝑎²+𝑐²−𝜎𝜆+𝑎𝑏−+𝜎𝑐𝑑.(3.28)

Denote 𝐵_𝑛=(𝜇_𝑛−1,𝜇_𝑛), 𝑛∈ℕ, where 𝜇₀=−∞.

We observe that the function 𝐺(𝜆)=(𝑐𝑦(𝑙,𝜆)+𝑎𝑇𝑦(𝑙,𝜆))/(𝑎𝑦(𝑙,𝜆)−𝑐𝑇𝑦(𝑙,𝜆)) is well defined for 𝜆∈𝐵=(ℂ⧵ℝ)∪(⋃^∞𝑛=1𝐵_𝑛) and is a finite-order meromorphic function and the eigenvalues 𝜈_𝑛 and 𝜇_𝑛, 𝑛∈ℕ, of boundary value problems (1.1), (1.2a)–(1.2c), and (1.2d^′) and (1.1), (1.2a)–(1.2c), and (1.2d^′) are zeros and poles of this function, respectively.

Let 𝜆∈𝐵. Using formula (3.9), we get𝑑𝑑𝜆𝐺(𝜆)=𝑎²+𝑐²𝑚⁻²(𝜆)^𝑙0𝑦²(𝑥,𝜆)𝑑𝑥.(3.29)

Lemma 3.17. The expansion 𝐺(𝜆)=⎧⎪⎪⎨⎪⎪⎩𝐺(0)+^∞_𝑛=1𝜆𝑐_𝑛𝜇_𝑛𝜆−𝜇_𝑛if𝜇¹𝑐≠0,₀+𝑐¹𝜆+^∞_𝑛=2𝜆𝑐_𝑛𝜇_𝑛𝜆−𝜇_𝑛if𝜇¹=0,(3.30) holds, where 𝑐_𝑛, 𝑛∈ℕ, are some negative numbers.

Proof. It is known (see [35, Chapter 6, Section 5]) that the meromorphic function 𝐺(𝜆) with simple poles 𝜇_𝑛 allows the representation 𝐺(𝜆)=𝐺₁(𝜆)+^∞_𝑛=1𝜇𝜆_𝑛^𝑠𝑐_𝑛𝜆−𝜇_𝑛,(3.31) where 𝐺₁(𝜆) is an entire function, 𝑐_𝑛=res_{𝜆=𝜇𝑛}𝐺(𝜆)=𝑐𝑦𝑙,𝜇_𝑛+𝑎𝑇𝑦𝑙,𝜇_𝑛𝑎𝜕𝑦𝑙,𝜇^𝑛𝜕𝜆−𝑐𝑇𝑦𝑙,𝜇^𝑛𝜕𝜆⁻¹,(3.32) and integers 𝑠_𝑛, 𝑛∈ℕ, 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 𝑎/𝑐>0. By virtue of relation (3.18), we have lim_{𝜆→−∞}𝐺(𝜆)=−𝑎/𝑐. Hence, 𝐺(𝜆)<0 for 𝜆∈(−∞,𝜈₁) and 𝐺(𝜆)>0 for 𝜆∈(𝜈₁,𝜇₁). Without loss of generality, we can assume 𝑎𝑦(𝑙,𝜆)−𝑐𝑇𝑦(𝑙,𝜆)>0 for 𝜆∈(−∞,𝜇₁). Then, 𝑐𝑦(𝑙,𝜆)+𝑎𝑇𝑦(𝑙,𝜆)<0 for 𝜆∈(−∞,𝜈₁). Since the eigenvalues 𝜇_𝑛 and 𝜈_𝑛, 𝑛∈ℕ, are simple zeros of functions 𝑎𝑦(𝑙,𝜆)−𝑐𝑇𝑦(𝑙,𝜆) and 𝑐𝑦(𝑙,𝜆)+𝑎𝑇𝑦(𝑙,𝜆), respectively, then by (3.29) the relations (−1)_𝑛+1(−1)^𝑛+1𝑐𝑦𝑙,𝜇_𝑛+𝑎𝑇𝑦𝑙,𝜇_𝑛>0,𝑎𝜕𝑦𝑙,𝜇^𝑛𝜕𝜆−𝑐𝜕𝑇𝑦𝑙,𝜇^𝑛𝜕𝜆<0,𝑛∈ℕ,(3.33) are true.
Taking into account (3.33), in (3.32) we get 𝑐_𝑛<0, 𝑛∈ℕ. The cases 𝑎/𝑐<0, 𝑎=0, and 𝑐=0 can be treated along similar lines.
Denote Ω_𝑛(𝜀)={𝜆∈ℂ∣|√⁴𝜆−√⁴𝜇_𝑛|<𝜀} where 𝜀>0 is some small number. From the asymptotic formula (3.25), it follows that for 𝜀<𝜋/4𝑙 the domains Ω_𝑛(𝜀) asymptotically do not intersect and contain only one pole 𝜇_𝑛 of the function 𝐺(𝜆).
By (3.11), (3.12), (3.23), (3.24), and (3.25), we see that outside of domains Ω_𝑛(𝜀) the asymptotic formulae are true: 𝐺(𝜆)=⎧⎪⎨⎪⎩−𝑎𝜌𝑐+𝑂𝜌⁻¹if𝑎𝑐≠0,₃−𝜌𝑧(₋₃𝜌)1+𝑂𝜌⁻¹if𝑐=0,(𝑧(𝜌))⁻¹1+𝑂𝜌⁻¹if𝑎=0,(3.34) where 𝑧(𝜌)=⎧⎪⎪⎨⎪⎪⎩cos((𝜋/4)sgn𝛾)cos(𝜌𝑙+(𝜋/2)sgn𝛽+(𝜋/4)sgn𝛾)si√n((𝜋/4)(1+sgn𝛾))cos(𝜌𝑙+(𝜋/2)sgn𝛽+(𝜋/4)(1+sgn𝛾))if𝛽∈0,𝜋2,2^1−2sgn𝛾cos(𝜌𝑙−(1−sgn𝛾)(𝜋/4))cos(𝜌𝑙+(𝜋/4)sgn𝛾)if𝛽=0.(3.35)
Following the corresponding reasoning (see [36, Chapter VII, Section  2, formula (27)]), we see that outside of domains Ω_𝑛(𝜀) the estimation ||𝐺(𝜆)||≤⎧⎪⎨⎪⎩𝑀¹𝑀if𝑎𝑐≠0,₂₄𝑀||𝜆||³if𝑐=0,₃₄||𝜆||⁻³if𝑎=0,(3.36) holds; using it in (3.32) we get ||𝑐_𝑛||=||||12𝜋𝑖_{𝜕Ω𝑛(𝜀)}𝐺(𝜆)𝑑𝜆||||=2𝜋||||_{|𝜈−4√𝜇𝑛|=𝜀}𝜈³𝐺𝜈⁴𝑑𝜈||||≤⎧⎪⎨⎪⎩𝑀¹𝑀𝑛³if𝑎𝑐≠0,₂𝑀𝑛⁶if𝑐=0,₃if𝑎=0,(3.37) where 𝑀₁, 𝑀₂, 𝑀₃, 𝑀₁, 𝑀₂, 𝑀₃ are some positive constants. By (3.37) and asymptotic formula (3.25) the series ∑^∞𝑛=1𝑐_𝑛|𝜇_𝑛|⁻² converges. Then, according to Theorem 2 in [35, Chapter 6, Section 5], in formula (3.31) we can assume 𝑠_𝑛=1, 𝑛∈ℕ.
Let {Γ_𝑛}^∞𝑛=1 be a sequence of the expanding circles which are not crossing domains Ω_𝑛(𝜀). Then, according to Formula (9) in [37, Chapter V, Section 13], we have 𝐺(𝜆)−_{𝜇𝑘∈intΓ𝑛}𝑐_𝑘𝜆−𝜇_𝑘=12𝜋𝑖_Γ𝑛𝐺(𝜉)𝐺(0)+_{𝜇k∈intΓ𝑛}𝜉−𝜆𝑑𝜉,𝑐_𝑘𝜇_𝑘=12𝜋𝑖_Γ𝑛𝐺(𝜉)𝜉𝑑𝜉.(3.38) By (3.38), we get 𝐺(𝜆)−𝐺(0)=_{𝜇𝑘∈intΓ𝑛}𝜆𝑐_𝑘𝜇_𝑘𝜆−𝜇_𝑘=21𝜋𝑖_Γ𝑛𝜆𝐺(𝜉)𝜉(𝜉−𝜆)𝑑𝜉.(3.39)
From (3.36), the right side of (3.39) tends to zero as 𝑛→∞. Then, passing to the limit as 𝑛→∞ in (3.39), we obtain 𝐺(𝜆)=𝐺(0)+^∞_𝑛=1𝜆𝑐_𝑛𝜇_𝑛𝜆−𝜇_𝑛,(3.40) which implies 𝐺₁(𝜆)≡𝐺(0).
Differentiating the right side of the least equality, we have 𝐺^(𝑠)(𝜆)=(−1)^𝑠𝑠!^∞_𝑛=1𝑐_𝑛𝜆−𝜇_𝑛^𝑠+1,𝑠=1,2,3.(3.41) Note that the function 𝐹(𝜆) has the following expansion: 𝐹(𝜆)=𝐹(0)+^∞_𝑛=1𝜆𝑐̃_𝑛𝜆_𝑛(0)𝜆−𝜆_𝑛(0),(3.42) where 𝑐̃_𝑛=res_{𝜆=𝜆𝑛(0)}𝐹(𝜆),𝑛=1,2,….(3.43)
Now let 𝜇₁=0, that is, 𝐹(0)=𝑎/𝑐. 𝐺(𝜆) has the following expansion: 𝐺(𝜆)=𝐺₁(𝜆)+𝑐¹𝜆+^∞_𝑛=2𝜆𝑐_𝑛𝜇_𝑛𝜆−𝜇_𝑛.(3.44) Again, according to Formula (9) in [37, Chapter V, Section 13], we have 𝐺(𝜆)−𝑐¹𝜆−_{𝜇𝑘∈intΓ𝑛𝑘≠1}𝑐_𝑘𝜆−𝜇_𝑘=12𝜋𝑖_Γ𝑛𝐺(𝜉)𝜉−𝜆𝑑𝜉.(3.45) By (2.6) [18] and (3.9), we get 𝑐₁=−𝑐⁻²𝑎²+𝑐²𝐹^′(0)⁻¹.(3.46) Using (3.42), (3.41), and (3.46), we obtain 𝐺(𝜆)−𝑐¹𝜆=−𝑎𝑐+𝑐⁻²𝑎²+𝑐²∑^∞𝑛=1𝑐̃_𝑛/𝜆^2𝑛(0)𝜆−𝜆_𝑛(0)𝐹^(0)∑^∞𝑛=1𝑐̃_𝑛/𝜆_𝑛(0)𝜆−𝜆_𝑛(0).(3.47) Passing to the limit as 𝜆→0 in (3.47), we get lim_𝜆→0𝐺(𝜆)−𝑐¹𝜆=−𝑎𝑐+𝑐⁻²𝑎²+𝑐²^∞_𝑛=1𝑐̃_𝑛𝜆^2𝑛(0)⁻¹^∞_𝑛=1𝑐̃_𝑛𝜆^3𝑛(0)⁻¹=𝑎𝑐+2𝑐⁻²𝑎²+𝑐²𝐹^(0)⁻²𝐹^(0)=𝑐₀.(3.48) Using (3.48) in (3.45), we have 𝑐₀+_{𝜇𝑘∈intΓ𝑛𝑘≠1}𝑐_𝑘𝜆−𝜇_𝑘=12𝜋𝑖_Γ𝑛𝐺(𝜉)𝜉𝑑𝜉.(3.49)
In view of (3.49) and (3.45), we get 𝐺(𝜆)−𝑐₀−𝑐¹𝜆−_{𝜇𝑘∈intΓ𝑛𝑘≠1}𝜆𝑐_𝑘𝜇_𝑘𝜆−𝜇_𝑘=21𝜋𝑖_Γ𝑛𝜆𝐺(𝜉)𝜉(𝜉−𝜆)𝑑𝜉.(3.50) Passing to the limit as 𝑛→∞ in (3.50), we obtain 𝐺(𝜆)=𝑐₀+𝑐¹𝜆+^∞_𝑛=2𝜆𝑐_𝑛𝜇_𝑛𝜆−𝜇_𝑛.(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 𝜇_𝑁−1<−𝑑/𝑐≤𝜇_𝑁.

Theorem 4.1. The problem (1.1), (1.2a)–(1.2d) for 𝜎>0 has a sequence of real and simple eigenvalues 𝜆₁<𝜆₂<⋯<𝜆_𝑛⟶+∞,(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 𝑦_𝑛(𝑥), 𝑛≥2, has exactly 𝑛−1 zeros in (0,𝑙), the eigenfunction 𝑦₁(𝑥) has no zeros in (0,𝑙) in the case 𝜆₁≥0, and the number of zeros of 𝑦₁(𝑥) can be arbitrary in the case 𝜆₁<0.(b)If 𝑐≠0, then the eigenfunction 𝑦_𝑛(𝑥) corresponding to the eigenvalue 𝜆_𝑛≥0 has exactly 𝑛−1 simple zeros for 𝑛≤𝑁 and exactly 𝑛−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 𝜆,𝜇(𝜆≠𝜇) be the eigenvalue of the operator 𝐿. The eigenvectors 𝑦(𝜆)={𝑦(𝑥,𝜆),𝑚(𝜆)} and 𝑦(𝜇)={𝑦(𝑥,𝜇),𝑚(𝜇)} corresponding to the eigenvalues 𝜆 and 𝜇, respectively, are orthogonal in Π₁, since the operator 𝐿 is 𝐽-selfadjoint in Π₁. Hence, by (2.4), we have^𝑙0𝑦(𝑥,𝜆)𝑦(𝑥,𝜇)𝑑𝑥=−𝜎⁻¹𝑚(𝜆)𝑚(𝜇).(4.2)

Lemma 4.2. Let 𝜆^∗∈ℝ be an eigenvalue of boundary value problem (1.1), (1.2a)–(1.2d) and 𝐺^(𝜆^∗)≤𝐴, where 𝐴=−(𝑎²+𝑐²)/𝜎. Then, problem (1.1), (1.2a)–(1.2d) has no nonreal eigenvalues.

Proof. Let 𝜇∈ℂ⧵ℝ be an eigenvalue of problem (1.1), (1.2a)–(1.2d). Then, from Remark 3.16 and equality (4.2), we obtain ^𝑙0𝑦𝑥,𝜆^∗𝑚𝜆^∗𝑦(𝑥,𝜇)𝑚(𝜇)𝑑𝑥=−𝜎⁻¹,^𝑙0𝑦𝑥,𝜆^∗𝑚𝜆^∗𝑦𝑚(𝑥(,𝜇𝜇))𝑑𝑥=−𝜎⁻¹,(4.3)_𝑙0||||𝑦(𝑥,𝜇)𝑚(𝜇)||||²𝑑𝑥=−𝜎⁻¹.(4.4)
In view of formula (3.29), the inequality ^𝑙0𝑦𝑥,𝜆^∗𝑚𝜆^∗²𝑑𝑥≤−𝜎⁻¹(4.5) is true.
By (4.3), ^𝑙0𝑦𝑥,𝜆^∗𝑚𝜆^∗Re𝑦𝑚(𝑥(,𝜇𝜇))𝑑𝑥=−𝜎⁻¹.(4.6) From (4.4)–(4.6), we get ^𝑙0⎧⎪⎨⎪⎩𝑦𝑥,𝜆^∗𝑚𝜆^∗−Re𝑦𝑚(𝑥(,𝜇𝜇))²+Im²𝑦(𝑥,𝜇)𝑚(𝜇)⎫⎪⎬⎪⎭𝑑𝑥<0if𝐺^′𝜆^∗<𝐴,_𝑙0⎧⎪⎨⎪⎩𝑦𝑥,𝜆^∗𝑚𝜆^∗−Re𝑦𝑚(𝑥(,𝜇𝜇))²+Im²𝑦(𝑥,𝜇)𝑚(𝜇)⎫⎪⎬⎪⎭𝑑𝑥=0if𝐺^′𝜆^∗=𝐴.(4.7)
From the second relation it follows that Im(𝑦(𝑥,𝜇)/𝑚(𝜇))=0, which by (1.1) contradicts the condition 𝜇∈ℂ⧵ℝ. The obtained contradictions prove Lemma 4.2.

Lemma 4.3. Let 𝜆^∗1,𝜆^∗2∈ℝ, 𝜆^∗1≠𝜆^∗2 be eigenvalues of problem (1.1), (1.2a)–(1.2d) and 𝐺^(𝜆^∗1)≤𝐴. Then, 𝐺^(𝜆^∗2)>𝐴.

Proof. Let 𝐺′(𝜆^∗2)≤𝐴. By (3.29) and (4.2), we have ^𝑙0𝑦𝑥,𝜆^∗1𝑚𝜆^∗1²𝑑𝑥≤−𝜎⁻¹,^𝑙0𝑦𝑥,𝜆^∗2𝑚𝜆^∗2²𝑑𝑥≤−𝜎⁻¹,^𝑙0𝑦𝑥,𝜆^∗1𝑚𝜆^∗1𝑦𝑥,𝜆^∗2𝑚𝜆^∗2𝑑𝑥=−𝜎⁻¹.(4.8)
Hence, we get ^𝑙0𝑦𝑥,𝜆^∗1𝑚𝜆^∗1𝑦𝑥,𝜆^∗2𝑚𝜆^∗2𝑑𝑥<0if𝐺^′𝜆^∗1<𝐴or𝐺^′𝜆^∗2<𝐴,_𝑙0𝑦𝑥,𝜆^∗1𝑚𝜆^∗1𝑦𝑥,𝜆^∗2𝑚𝜆^∗2𝑑𝑥=0if𝐺^′𝜆^∗1=𝐺^𝜆^∗2=𝐴.(4.9)
From (4.9), it follows that 𝑦(𝑥,𝜆^∗1)/𝑚(𝜆^∗1)=𝑦(𝑥,𝜆^∗2)/𝑚(𝜆^∗2) for 𝑥∈[0,𝑙]. Therefore, 𝑚(𝜆^∗2)𝑦(𝑥,𝜆^∗1)=𝑚(𝜆^∗1)𝑦(𝑥,𝜆^∗2).
Since 𝜆₁≠𝜆₂, then by (1.1) 𝑦(𝑥,𝜆₁)≡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⁾(𝜆)>0, 𝜆∈𝐵, 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 𝜆_𝑛,𝑛≥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 𝐵₁ contains algebraically two (either two simple or one double) eigenvalues, and 𝐵_𝑛, 𝑛=2,3,…, contain precisely one simple eigenvalues.(2)All eigenvalues are real, at that 𝐵₁contains no eigenvalues but, for some 𝑠≥2, 𝐵_𝑠 contains algebraically three (either three simple, or one simple and one double, or one triple) eigenvalues, and 𝐵_𝑛, 𝑛=2,3,…, 𝑛≠𝑠 contain precisely one simple eigenvalue.(3)There are two nonreal eigenvalues appearing as a conjugate pair, at that 𝐵₁ contains no eigenvalues, and 𝐵_𝑛, 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(𝜆)=𝐴𝜆+𝐵, where 𝐴=−(𝑎²+𝑐²)/𝜎, 𝐵=−(𝑎𝑏+𝑐𝑑)⧵𝜎 (see (3.28)). From (3.41), it follows that 𝐺^(𝜆)>0 for 𝜆∈𝐵₁; therefore, the function 𝐺(𝜆) is convex on the interval 𝐵₁. By virtue of (3.18) and (3.30), we have lim_{𝜆→−∞}𝐺l(im𝜆)=_{𝜆→𝜇𝑛−0}−−∞𝑎𝑐iiff𝑐𝑐≠=00,,𝐺(𝜆)=+∞.(4.10)
That is why for each fixed number 𝐴 there exists number 𝐵_𝐴 such that the lines 𝐴𝜆+𝐵_𝐴, 𝜆∈ℝ, touch the graph of function 𝐺(𝜆) at some point 𝜆̃∈𝐵₁. Hence, in the interval 𝐵₁, (3.28) has two simple roots 𝜆₁<𝜆₂ if 𝐵>𝐵_𝐴, one double root 𝜆₁=𝜆̃ if 𝐵=𝐵_𝐴, and no roots if 𝐵<𝐵_𝐴.
By (3.29) and (3.30) we have lim_{𝜆→𝜇𝑛+0}𝐺(𝜆)=−∞, lim_{𝜆→𝜇𝑛−0}𝐺(𝜆)=+∞, 𝑛∈ℕ. Therefore, (3.28) has at least one solution in the interval 𝐵_𝑛, 𝑛=2,3,….
Let 𝐵≥𝐵_𝐴. If 𝐵>𝐵_𝐴, then 𝐺^′(𝜆₁)<𝐴, 𝐺^(𝜆₂)<𝐴; if 𝐵=𝐵_𝐴 then 𝐺^(𝜆₁)=𝐴. By (3.29), (3.28) has only simple root 𝜆_𝑛+1 for 𝐵>𝐵_𝐴,𝜆_𝑛 for 𝐵=𝐵_𝐴 in the interval 𝐵_𝑛, 𝑛=2,3,….
Let 𝐵<𝐵_𝐴. By Lemma 4.3 either 𝐺^(𝜆_𝑛)>𝐴 for any 𝜆_𝑛∈ℝ, or there exists 𝑘∈ℕ such that 𝐹^(𝜆_𝑘)≤𝐴 and 𝐹^(𝜆_𝑛)>𝐴, 𝑛∈ℕ⧵{𝑘}. Assume that 𝜆_𝑘∈𝐵_𝑠. Obviously, 𝑠≥2. Choose natural number 𝑛₀ such that the inequalities ||𝐺(𝜆)−𝐴𝜆+𝐴𝐵𝑅_𝑛0𝐴+𝐵>0,||>||𝐵_𝐴−𝐵||,𝜆∈𝑆_𝑅𝑛0,(4.11) are fulfilled; where 𝑅_𝑛=𝜏_𝑛+𝛿₀, 𝜏_𝑛=⎧⎪⎪⎨⎪⎪⎩𝜆^𝑛𝜈𝜋2if𝑐=0,_𝑛if𝑐≠0,𝑎𝜆𝑐>0,_𝑛𝜈(0)−1if𝑐≠0,𝑎=0,_𝑛−1if𝑐≠0,𝑎𝑐<0,(4.12)𝛿₀ is sufficiently small positive number, and 𝑆_𝑅𝑛={𝑧∈𝐶∣|𝑧|=𝑅_𝑛}.
We have Δ_{𝑆𝑅𝑛0}arg(𝐺(𝜆)−(𝐴𝜆+𝐵))=Δ_{𝑆𝑅𝑛0}arg𝐺(𝜆)−𝐴𝜆+𝐵_𝐴+Δ_{𝑆𝑅𝑛0}arg1+𝐵^𝐴−𝐵𝐺(𝜆)−𝐴𝜆+𝐵_𝐴(4.,13) where Δ_{𝑆𝑅𝑛0}arg𝑓(𝑧)=1𝑖_{𝑆𝑅𝑛0}𝑓^′(𝑧)𝑓(𝑧)𝑑𝑧(4.14) (see [37, Chapter  IV, Section  10]). By (4.11) ||||𝐵_𝐴−𝐵𝐺(𝜆)−𝐴𝜆+𝐵_𝐴||||<1,𝜆∈𝑆^𝑅𝑛0,(4.15) hence, the point 𝜔=𝐵^𝐴−𝐵𝐺(𝜆)−𝐴𝜆+𝐵_𝐴(4.16) does not go out of circle {|𝜔|<1}. Therefore, vector 𝑤=1+𝜔 cannot turn around the point 𝑤=0, and the second summand in (4.13) equals zero. Thus, Δ_{𝑆𝑅𝑛0}arg(𝐺(𝜆)−(𝐴𝜆+𝐵))=Δ_{𝑆𝑅𝑛0}arg𝐺(𝜆)−𝐴𝜆+𝐵_𝐴.(4.17) By the argument principle (see [37, Chapter IV, Section 10, Theorem 1]) we have 12𝜋Δ^{𝑆𝑅𝑛0}arg𝐺(𝜆)−𝐴𝜆+𝐵_𝐴=_{𝜆𝐴)𝑛(𝐵∈int𝑆𝑅𝑛0}𝜚𝜆^{(𝐵𝐴)𝑛}−_{𝜇𝑛∈int𝑆𝑅𝑛0}𝜚𝜇_𝑛,(4.18) where 𝜌(𝜆^{(𝐵𝐴)𝑛}) and 𝜚(𝜇_𝑛) are multiplicity of zero 𝜆^{(𝐵𝐴)𝑛} and pole 𝜇_𝑛 of the function 𝐺(𝜆)−(𝐴𝜆+B_𝐴), respectively (𝜆^(𝐵𝐴)1=𝜆^(𝐵𝐴)2). Obviously, ∑_𝜆^{𝐴)𝑛(𝐵}_{∈int𝑆𝑅𝑛0}𝜌(𝜆^{(𝐵𝐴)𝑛})=𝑛₀ and ∑_{𝜇𝑛∈int𝑆𝑅𝑛0}𝜌(𝜇_𝑛)=𝑛₀−1. Then, by (4.18) we obtain (2𝜋)⁻¹Δ_{𝑆𝑅𝑛0}arg𝐺(𝜆)−𝐴𝜆+𝐵_𝐴=1.(4.19) From (4.17) and (4.19) follows the validity of the equality (2𝜋)⁻¹Δ_{𝑆𝑅𝑛0}arg(𝐺(𝜆)−(𝐴𝜆+𝐵))=1.(4.20)
Using the argument principle again, by (4.20) we get _𝜆𝑛∈_{int𝑆𝑅𝑛0}𝜌𝜆_𝑛−_{𝜇𝑛∈int𝑆𝑅𝑛0}𝜚𝜇_𝑛=1,(4.21) whence it follows that _𝜆𝑛∈_{int𝑆𝑅𝑛0}𝜌𝜆_𝑛=𝑛₀,(4.22) where 𝜆_𝑛, 𝑛∈ℕ, are roots of the equation 𝐺(𝜆)=𝐴𝜆+𝐵. From the above-mentioned reasoning, by (4.22) we have _𝜆𝑚∈_{int𝑆𝑅𝑛}𝜌𝜆_𝑚=𝑛,𝑛=𝑛₀,𝑛₀+1,…,(4.23) and, therefore, problem (1.1), (1.2a)–(1.2d) in the interval 𝐵_𝑛 for 𝑛=𝑛₀, 𝑛₀+1,…, has only one simple eigenvalue.
Consider the following two cases.
Case 1. For all real eigenvalues 𝜆_𝑛 of problem (1.1), (1.2a)–(1.2d) the inequalities 𝐺^′(𝜆_𝑛)>𝐴, 𝜆_𝑛∈⋃^∞𝑚=2𝐵_𝑚, are fulfilled. The problem (1.1), (1.2a)–(1.2d) in every interval 𝐵_𝑚, 𝑚=2,3,…,𝑛₀−1, has one simple eigenvalue. Hence, problem (1.1), (1.2a)–(1.2d) in the interval (−∞,𝑆_𝑅𝑛), 𝑛≥𝑛₀, has 𝑛−2 simple eigenvalues, and hence, by (4.23), this problem in the circle 𝑆_𝑅𝑛⊂ℂ has one pair of simple nonreal eigenvalues. In this case, the location of the eigenvalues will be in the following form: 𝜆₁,𝜆₂∈ℂ⧵ℝ, 𝜆₂=𝜆₁, Im𝜆₁>0, 𝜆_𝑛∈𝐵_𝑛−1, 𝑛=3,4,….Case 2. Let 𝐺^(𝜆_𝑘)≤𝐴, 𝐺^(𝜆_𝑛)>𝐴, 𝑛∈𝑁∣{𝑘} and 𝜆_𝑘∈𝐵_𝑠, 𝑠≥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 𝐵_𝑛, 𝑛≠𝑘, 𝑛=2,3,…, problem (1.1), (1.2a)–(1.2d) has one simple eigenvalue.Subcase 1. Let 𝐺^(𝜆_𝑘)=𝐴,𝐺^(𝜆_𝑘)≠0, that is, the eigenvalue 𝜆_𝑘 is a double one (by this 𝜆_𝑘=𝜆_𝑘+1). Then, from (4.23) it follows that the interval 𝐵_𝑠 besides the eigenvalue 𝜆_𝑘 contains one more simple eigenvalue: at that it is either 𝜆_𝑘−1 (by this 𝑘=𝑠) or 𝜆_𝑘+2 (by this 𝑘=𝑠−1). Hence, 𝜆_𝑛∈𝐵_𝑛+1, 𝑛=1,2,…,𝑠−2, 𝜆_𝑠−1,𝜆_𝑠,𝜆_𝑠+1∈𝐵_𝑠 (by this either 𝜆_𝑠−1<𝜆_𝑠=𝜆_𝑠+1 or 𝜆_𝑠−1=𝜆_𝑠<𝜆_𝑠+1), 𝜆_𝑛∈𝐵_𝑛−1, 𝑛=𝑠+2,𝑠+3,….Subcase 2. Let 𝐺^(𝜆_𝑘)=𝐴, 𝐺^(𝜆_𝑘)=0. By (3.41), 𝐺^(𝜆_𝑘)≠0. Hence, 𝜆_𝑘 is a triple eigenvalue of the problem (1.1), (1.2a)–(1.2d) (by this 𝜆_𝑘=𝜆_𝑘+1=𝜆_𝑘+2). Then, from (4.23) it follows that in the interval 𝐵_𝑠 problem (1.1), (1.2a)–(1.2d) has unique triple eigenvalue 𝜆_𝑘, and therefore, 𝑘=𝑠−1. At this 𝜆_𝑛∈𝐵_𝑛+1, 𝑛=1,2,…,𝑠−2, 𝜆_𝑠−1=𝜆_𝑠=𝜆_𝑠+1∈𝐵_𝑠, 𝜆_𝑛∈𝐵_𝑛−1,𝑛=𝑠+2,𝑠+3,….Subcase 3. Let 𝐺^(𝜆_𝑘)<𝐴, that is, the eigenvalue 𝜆_𝑘 is simple. Then, by (4.23), in the interval 𝐵_𝑠problem (1.1), (1.2) has an eigenvalue 𝜆_𝑘 as well as two more simple eigenvalues, which, by Lemma 4.3, are 𝜆_𝑘−1 and 𝜆_𝑘+1 (and hence 𝑘=𝑠). In this case, we have 𝜆_𝑛∈𝐵_𝑛+1, 𝑛=1,2,…,𝑠−2, 𝜆_𝑠−1,𝜆_𝑠,𝜆_𝑠+1∈𝐵_𝑠 (𝜆_𝑠−1<𝜆_𝑠<𝜆_𝑠+1), 𝜆_𝑛∈𝐵_𝑛−1, 𝑛=𝑠+2,𝑠+3,….
Theorem 4.4 is proved.

By Theorem 4.4 we have 𝜚(𝜆_𝑛)=2, that is, 𝜆_𝑛=𝜆_𝑛+1 if 𝑛=𝑠−1 or 𝑛=𝑠; 𝜚(𝜆_𝑛)=3, that is, 𝜆_𝑛=𝜆_𝑛+1=𝜆_𝑛+2 if 𝑛=𝑠−1 (If assertion (2) in Theorem 4.4 holds, then we set 𝑠=1).

Let {𝑦_𝑛(𝑥)}^∞𝑛=1 be a system of eigen- and associated functions corresponding to the eigenvalue system {𝜆_𝑛}^∞𝑛=1 of problem (1.1), (1.2a)–(1.2d), where 𝑦_𝑛(𝑥)=𝑦(𝑥,𝜆_𝑛) if 𝜌(𝜆_𝑛)=1; 𝑦_𝑛(𝑥)=𝑦(𝑥,𝜆_𝑛), 𝑦_𝑛+1(𝑥)=𝑦^∗𝑛+1(𝑥)+𝑐_𝑛𝑦_𝑛(𝑥), 𝑦^∗𝑛+1(𝑥)=(𝜕𝑦(𝑥,𝜆_𝑛))/𝜕𝜆, 𝑐_𝑛 is an arbitrary constant, if 𝜚(𝜆_𝑛)=2;𝑦_𝑛(𝑥)=𝑦(𝑥,𝜆_𝑛), 𝑦_𝑛+1(𝑥)=𝑦^∗𝑛+1(𝑥)+𝑑_𝑛𝑦_𝑛(𝑥), 𝑦_𝑛+2(𝑥)=𝑦^∗𝑛+2(𝑥)+𝑑_𝑛𝑦^∗𝑛+1(𝑥)+ℎ_𝑛𝑦_𝑛(𝑥), 𝑦^∗𝑛+2(𝑥)=𝜕²𝑦(𝑥,𝜆_𝑛)/2𝜕𝜆², 𝑑_𝑛, ℎ_𝑛 are arbitrary constants, if 𝜌(𝜆_𝑛)=3. Here, 𝑦_𝑛(𝑥) is an eigenfunction for 𝜆_𝑛 and 𝑦_𝑛+1(𝑥) when 𝜌(𝜆_𝑛)=2; 𝑦_𝑛+1(𝑥), 𝑦_𝑛+2(𝑥) when 𝜌(𝜆_𝑛)=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), 𝑦_𝑛 has 𝑛−1 (resp., 𝑛) zeros in the interval (0,l). Similarly 𝑦_𝑠, 𝑦_𝑠+1 both have 𝑠−1 (resp., 𝑠) zeros if 𝑠<𝑁 (resp., 𝑠>𝑁). Finally, if 𝑐≠0, then each of 𝑦_𝑁 (and 𝑦_𝑠, 𝑦_𝑠+1 if 𝑠=𝑁) has 𝑁−1  or  𝑁 zeros according to 𝜆_𝑁, 𝜆_𝑆, 𝜆_𝑆+1≤ or >−𝑑/𝑐, and if 𝑐=0 and 𝑠=𝑁, then 𝑦_𝑠, 𝑦_𝑠+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 𝜆_𝑘−1(𝜋/2)<𝑏/𝑎≤𝜆_𝑘(𝜋/2) (interpreting 𝜆₀(𝜋/2)=−∞).

Lemma 5.1. The following relations hold for sufficiently large 𝑛∈ℕ, 𝑛>𝑛₁=max{𝑠,𝑁,𝐾}+2: 𝜆_𝑛−2(0)<𝜆_𝑛<𝜆_𝑛−1𝜋2<𝜆^𝑛−1(0)if𝑐≠0,𝑎𝜆𝑐≤0,_𝑛−2(0)<𝜆_𝑛−1𝜋2<𝜆^𝑛<𝜆_𝑛−1(0)ifc≠0,𝑎𝑐>0or𝑐=0.(5.1)

Proof. Let𝑎𝑐≠0. Note that the eigenvalues𝜆_𝑛(0) (resp., 𝜆_𝑛(𝜋/2)), 𝑛∈ℕ, of problem (1.1), (1.2a)–(1.2c), and (1.2d^′) for𝛿=0 (resp., for 𝛿=𝜋/2) are roots of the equation 𝐺(𝜆)=−𝑎/𝑐(resp., 𝐺(𝜆)=𝑐/𝑎). The equation 𝐴𝜆+𝐵=−𝑎/𝑐 (resp., 𝐴𝜆+𝐵=𝑐/𝑎) has a unique solution 𝜆=−𝑑/𝑐 (resp., 𝜆=−𝑏/𝑎). Since 𝑛>max{𝑁+2,𝐾+2}, in view of (3.29), 𝐺(𝜆_𝑛)>max{−𝑎/𝑐,𝑐/𝑎}. Hence, by (3.23), (3.24), and (3.29), the following relations hold for 𝑛>𝑛₁: 𝜇_𝑛−2<𝜆_𝑛−2𝜋2<𝜈^𝑛−2<𝜆_𝑛−2(0)<𝜆_𝑛<𝜇_𝑛−1if𝑎𝜇𝑐<0,_𝑛−2<𝜆_𝑛−2(0)<𝜈_𝑛−1<𝜆_𝑛−1𝜋2<𝜆^𝑛<𝜇_𝑛−1if𝑎𝑐>0.(5.2)
Let 𝑎=0. In this case 𝜇_𝑛=𝜆_𝑛(𝜋/2), 𝜈_𝑛=𝜆_𝑛(0), 𝑛∈ℕ. Since 𝑛>𝑁+2, so 𝐺(𝜆_𝑛)>0. Then, by the equality 𝐺(𝜆_𝑛)=𝐴𝜆_𝑛+𝐵, 𝑛∈ℕ, we obtain 𝜆_𝑛−2𝜋2<𝜆^𝑛−2(0)<𝜆_𝑛<𝜆_𝑛−1𝜋2<𝜆^𝑛−1(0),𝑛>𝑛₁.(5.3) Now let 𝑐=0. In this case 𝜇_𝑛=𝜆_𝑛(0), 𝜈_𝑛=𝜆_𝑛(𝜋/2), 𝑛∈ℕ. Since 𝑛>𝐾+2, so 𝐺(𝜆_𝑛)>0. Therefore, using 𝐺(𝜆_𝑛)=𝐴𝜆_𝑛+𝐵, we have 𝜆_𝑛−2(0)<𝜆_𝑛−1𝜋2<𝜆^𝑛<𝜆_𝑛−1(0),𝑛>𝑛₁.(5.4) Relations (5.1) are consequences of relations (5.2)–(5.4).
The proof of Lemma 5.1 is complete.

We define numbers 𝜒, 𝜒_𝑛, 𝑛∈ℕ, as follows:𝜒=⎧⎪⎨⎪⎩53(1+𝑠4(𝛽,|𝑐|))if𝛾∈0,𝜋2,4−38(−1)^sgn𝛽𝜒+(−1)^sgn|𝑐|if𝛾=0,_𝑛=(𝑛−𝜒)𝜋𝑙.(5.5)

Using relations (5.1) and formulas (3.16), (3.17), the following corresponding reasoning [18, Theorem 3.1] can be proved.

Theorem 5.2. The following asymptotic formulae hold: 𝑦_𝑛√⁴𝜆_𝑛=𝜒_𝑛+𝑂𝑛⁻¹,(𝑥)=𝑧𝑥,𝜒_𝑛+𝑂𝑛⁻¹,(5.6) where relation (5.6) holds uniformly for 𝑥∈[0,𝑙].

6. Necessary and Sufficient Conditions of Basicity of Root Function System of Problem (1.1), (1.2a)–(1.2d)

Note that the element 𝑦_𝑛̂={𝑦_𝑛(𝑥),𝑚_𝑛}, 𝑛∈ℕ, of the system {𝑦_𝑛̂}^∞𝑛=1 of the root vectors of operator 𝐿 satisfies the relation𝐿𝑦_𝑛̂=𝜆_𝑛𝑦_𝑛̂+𝜃_𝑛𝑦̂_𝑛−1,(6.1) where 𝜃_𝑛 equals either 0 (at that 𝑦_𝑛̂ is eigenvector) or 1 (at that 𝜆_𝑛=𝜆_𝑛−1 and 𝑦_𝑛̂ is associated vector) (see, e.g., [38]).

Theorem 6.1. The system of eigen- and associated functions of operator 𝐿 is a Riesz basis in the space 𝐻.

Proof. Let 𝜇 be a regular value of operator 𝐿, that is, 𝑅_𝜇=(𝐿−𝜇𝐼)⁻¹ exists and a bounded in 𝐻. Then, problem (2.4) is adequate to the following problem of eigenvalues: 𝑅_𝜇𝑦̂=(𝜆−𝜇)⁻¹𝑦,̂𝑦̂∈𝐷(𝐿).(6.2)
By Lemma 2.1,  𝑅_𝜇 is a completely continuous 𝐽-selfadjoint operator in Π₁. Then, in view of [39] the system of the root vectors of operator 𝑅_𝜇 (hence of operator 𝐿) forms a Riesz basis in 𝐻. Theorem 6.1 is proved.

Let {𝜐̂^∗𝑛}^∞𝑛=1, where 𝜐̂^∗𝑛={𝜐^∗𝑛(𝑥),𝑠^∗𝑛}, be a system of the root vectors of operator 𝐿^∗, that is,𝐿^∗𝜐̂^∗𝑛=𝜆_𝑛𝜐̂^∗𝑛+𝜃_𝑛+1𝜐̂^∗𝑛+1.(6.3)

By Lemma 2.2 and relations (6.1), (6.3) we have the following.

Lemma 6.2. 𝜐̂^∗𝑛=𝐽𝑦_𝑛̂(𝑦_𝑛̂={𝑦_𝑛(𝑥),𝑚_𝑛}) if  𝜚(𝜆_𝑛)=1; 𝜐̂^∗𝑛=𝐽𝑦̂^∗𝑛+1+𝑐̃_𝑛𝐽𝑦_𝑛̂,𝜐̂^∗𝑛+1=𝐽𝑦_𝑛̂ if 𝜚(𝜆_𝑛)=2; 𝜐̂^∗𝑛=𝐽𝑦̂^∗𝑛+2+𝑑_𝑛𝐽𝑦̂^∗𝑛+1ℎ_𝑛𝐽𝑦^∗𝑛̂,𝜐̂^∗𝑛+1=𝐽𝑦̂^∗𝑛+1+𝑑_𝑛𝐽𝑦_𝑛̂,𝜐̂^∗𝑛+2=𝐽𝑦_𝑛̂  if  𝜚(𝜆_𝑛)=3, where 𝑦̂^∗𝑛+1={𝑦^∗𝑛+1(𝑥),𝑚^∗𝑛+1}, 𝑚_𝑛+1=𝑚^′(𝜆_𝑛), 𝑦̂^∗𝑛+2={𝑦^∗𝑛+2(𝑥),𝑚^∗𝑛+2}, 𝑚^∗𝑛+2=(1/2)𝑚^(𝜆_𝑛), 𝑐̃_𝑛, 𝑑_𝑛, ℎ_𝑛 are arbitrary constants.

Lemma 6.3. Let 𝛿=⎧⎪⎪⎨⎪⎪⎩^𝑙0𝑦_𝑛̂𝑦^2𝑛(𝑥)𝑑𝑥+𝜎⁻¹𝑚^2𝑛,if𝜌𝜆_𝑛=1,‖‖𝑦̂,𝑦_∗𝑛+1̂^∗𝑛+1_Π1,if𝜌𝜆_𝑛=2,‖‖^2Π₁if𝜌𝜆_𝑛=3,(6.4) where ‖⋅‖_Π1 is the norm in Π₁. Then, 𝛿_𝑛≠0, 𝑛∈ℕ.

Proof. By Remark 3.16,  𝑚_𝑛≠0 if 𝑦_𝑛̂ is the eigenvector of operator 𝐿. If 𝜚(𝜆_𝑛)=1, then 𝐺^′(𝜆_𝑛)≠𝐴, whence by (3.29), we get 𝛿_𝑛≠0.
Let 𝜚(𝜆_𝑛)=2. Then, 𝐺^(𝜆_𝑛)=𝐴 and 𝐺^(𝜆_𝑛)≠0. Differentiating the right-hand side of equality (3.29) on 𝜆, we obtain 𝐺^(𝜆)=2𝑎²+𝑐²𝑚²(𝜆)^𝑙0𝑦(𝑥,𝜆)𝜕𝑦(𝑥,𝜆)𝜕𝜆𝑑𝑥−𝑚^(𝜆)𝑚(𝜆)^𝑙0𝑦²(𝑥,𝜆)𝑑𝑥.(6.5) Assuming 𝜆=𝜆_𝑛 in (6.5) and taking into account (3.29) and (2.6), we get 𝐺^𝜆_𝑛=2𝑎²+𝑐²𝑚^𝑛−2𝑦_𝑛,𝑦^∗𝑛+1_Π1.(6.6) Since 𝐺^(𝜆_𝑛)≠0, from the last equality it follows that 𝛿_𝑛≠0.
Now let 𝜚(𝜆_𝑛)=3, that is, 𝐺^(𝜆_𝑛)=𝐴 and 𝐺^(𝜆_𝑛)=0, 𝐺^(𝜆_𝑛)≠0. Differentiating the right-hand side of (6.5) on 𝜆, we have 𝐺^(𝜆)=2𝑎²+𝑐²𝑚⁻⁶(𝜆)^𝑙0𝜕𝑦𝜕(𝑥𝜆,𝜆)²𝑑𝑥+^𝑙0𝑦(𝑥,𝜆)𝜕²𝑦(𝑥,𝜆)𝜕𝜆²−𝑚_𝑑𝑥𝑚(𝜆)(𝜆)^𝑙0𝑦(𝑥,𝜆)𝜕𝑦(𝑥,𝜆)𝜕𝜆𝑑𝑥−𝑚^−3𝑚_(𝜆)^𝑙0𝑦²(𝑥,𝜆)𝑑𝑥𝑚(𝜆)(𝜆)𝑚(𝜆)^𝑙0𝑦(𝑥,𝜆)𝜕𝑦(𝑥,𝜆)𝜕𝜆𝑑𝑥−𝑚^(𝜆)^𝑙0𝑦²(𝑥,𝜆)𝑑𝑥,(6.7) whence (supposing in that equality 𝜆=𝜆_𝑛), we obtain 𝐺^𝜆_𝑛=2𝑎²+𝑐²𝑚^𝑛−4‖‖𝑦̂^∗𝑛+1‖‖^2Π₁+2𝑦_𝑛̂,𝑦̂^∗𝑛+2_Π1.(6.8) By (6.1) and (6.3), we have 𝑦_𝑛̂,𝑦̂^∗𝑛+2_Π1=‖‖𝑦̂^∗𝑛+1‖‖^2Π₁;(6.9) then taking into account (6.8), we get 𝛿_𝑛=‖‖𝑦̂^∗𝑛+1‖‖^2Π₁=16𝑎²+𝑐²⁻¹𝑚^4𝑛𝐺^𝜆_𝑛≠0.(6.10)
Lemma 6.3 is proved.

Lemma 6.4. The elements 𝜐̂_𝑛={𝜐_𝑛(𝑥),𝑠_𝑛} of the system {𝜐̂_𝑛}^∞𝑛=1 conjugated to the system {𝑦_𝑛̂}^∞𝑛=1 are defined by the equality 𝜐̂_𝑛=𝛿^𝑛−1𝜐̂^∗𝑛,(6.11) where 𝑐̃_𝑛=−𝑐_𝑛−𝛿^𝑛−1‖𝑦̂^∗𝑛+1‖^2Π₁ if 𝜚(𝜆_𝑛)=2; 𝑑_𝑛=−𝑑_𝑛−𝛿^𝑛−1(𝑦̂^∗𝑛+1,𝑦̂^∗𝑛+2)_Π1, ℎ_𝑛=−ℎ_𝑛−𝛿^𝑛−1‖𝑦̂^∗𝑛+2‖^2Π₁+𝛿^𝑛−2(𝑦̂^∗𝑛+1,𝑦̂^∗𝑛+2)^2Π₁+𝑑_𝑛(𝑑_𝑛+𝛿^𝑛−1(𝑦̂^∗𝑛+1,𝑦̂^∗𝑛+2)_Π1) if 𝜚(𝜆_𝑛)=3.

Proof. On the bases of (6.1), (6.3), (2.1), (2.6), and (6.9), we have 𝑦𝑦_𝑛̂_𝑛̂,𝜐̂^∗𝑛=𝛿_𝑛if𝜚𝜆_𝑛=1;𝑦_𝑛̂,𝜐̂^∗𝑚=0if𝜚𝜆_𝑛=𝜚𝜆_𝑚=1,𝑛≠𝑚;𝑦_𝑘̂,𝜐̂^∗𝑚=0,𝑚=𝑘,𝑘+1,𝑦_𝑘̂,𝜐̂^∗𝑘+1=𝑦̂_𝑘+1,𝜐̂^∗𝑘=‖‖𝑦̂^∗𝑘+1‖‖^2Π₁+𝑐_𝑘+𝑐̃_𝑘𝛿_𝑘,𝑦_𝑛̂,𝜐̂^∗𝑘=𝑦̂_𝑘+1,𝜐̂^∗𝑘+1=𝛿_𝑘if𝜚𝜆_𝑛=1,𝜚𝜆_𝑘=2;𝑦_𝑘̂,𝜐̂^∗𝑚=0,𝑦_𝑚̂,𝜐̂^∗𝑚=𝛿_𝑚,𝑚=𝑘,𝑘+1,𝑘+2,𝑦_𝑘̂,𝜐̂^∗𝑘+1=𝑦̂_𝑘+1,𝜐̂^∗𝑘=𝑦^∗𝑘+1,𝑦^∗𝑘+2_Π1+𝑑_𝑘+𝑑_𝑘𝛿^𝑘,,𝜐̂^∗𝑘+2=𝑦̂_𝑘+2,𝜐̂^∗𝑘=‖‖𝑦̂^∗𝑘+2‖‖^2Π₁+𝑑_𝑘+𝑑_𝑘𝑦̂^∗𝑘+1,𝑦̂^∗𝑘+2_Π1+𝑑_𝑘𝑑_𝑘+ℎ_𝑘+ℎ_𝑘𝛿^𝑘if𝜚𝜆_𝑛=1,𝜚𝜆_𝑘=3.(6.12)
Using relation (6.12) and taking into account (6.11), we get the validity of the equality 𝑦_𝑛̂,𝜐̂_𝑘=𝛿_𝑛𝑘,(6.13) where 𝛿_𝑛𝑘 is the Kronecker delta. The proof of Lemma 6.4 is complete.

Corollary 6.5. (i) If 𝜚(𝜆_𝑛)=1, then 𝑠_𝑛≠0; (ii) if 𝜚(𝜆_𝑛)=2, then 𝑠_𝑛+1≠0, 𝑠_𝑛≠0 at 𝑐_𝑛≠𝑐^𝑛(0), 𝑠_𝑛=0at 𝑐_𝑛=𝑐^𝑛(0), where 𝑐^𝑛(0)=𝑚^𝑛−1𝑚^∗𝑛+1−𝛿^𝑛−1‖𝑦̂^∗𝑛+1‖^2Π₁; (iii) if 𝜚(𝜆_𝑛)=3, then 𝑠_𝑛+2≠0, 𝑠_𝑛+1≠0 at 𝑑_𝑛≠𝑑^𝑛(0), 𝑠_𝑛+1=0 at 𝑑_𝑛=𝑑^𝑛(0); 𝑠_𝑛≠0 at ℎ_𝑛≠ℎ^𝑛(0), 𝑠_𝑛=0 at ℎ_𝑛=ℎ^𝑛(0), where 𝑑^0𝑛=𝑚^𝑛−1𝑚^∗𝑛+1−𝛿^𝑛−1(𝑦̂^∗𝑛+1,𝑦̂^∗𝑛+2)_Π1, ℎ^𝑛(0)=𝑚^𝑛−1𝑚^∗𝑛+2−(𝑑_𝑛+𝛿^𝑛−1(𝑦̂^∗𝑛+1,𝑦̂^∗𝑛+2)_Π1)(𝑑_𝑛−𝑚^𝑛−1𝑚^∗𝑛+1)−𝛿^𝑛−1‖𝑦̂^∗𝑛+2‖^2Π₁+𝛿^𝑛−2(𝑦̂^∗𝑛+1,𝑦̂^∗𝑛+2)^2Π₁.

Theorem 6.6. Let 𝑟 be an arbitrary fixed integer. If 𝑠_𝑟≠0, then the system {𝑦_𝑛(𝑥)}^{∞𝑛=1,𝑛≠𝑟} forms a basis in 𝐿_𝑝(0,𝑙), 𝑝∈(1,∞), and even a Riezs basis in 𝐿₂(0,𝑙); if 𝑠_𝑟=0, the system {𝑦_𝑛(𝑥)}^{∞𝑛=1,𝑛≠𝑟} is neither complete nor minimal in 𝐿_𝑝(0,𝑙), 𝑝∈(1,∞).

Proof. By Theorem 7 in [40, Chapter 1, Section 4] and Theorem 6.1, the system {𝜐̂_𝑛}^∞𝑛=1 is a Riesz basis in 𝐻. Then, for any vector 𝑓={𝑓,𝜉}∈𝐻, the following expansion holds: 𝑓={𝑓,𝜉}=^∞_𝑛=1𝑓,𝑦^𝑛̂_𝐻𝜐̂_𝑛=^∞_𝑛=1𝑓,𝑦^𝑛_𝐿2+𝜎⁻¹𝜉𝑚_𝑛𝜐^𝑛,𝑠_𝑛,(6.14) whence it follows the equalities 𝑓=^∞_𝑛=1𝑓,𝑦^𝑛_𝐿2𝜉=_∞+𝜎⁻¹𝜉𝑚_𝑛𝜐^𝑛,_𝑛=1𝑓,𝑦^𝑛_𝐿2+𝜎⁻¹𝜉𝑚_𝑛𝑠^𝑛.(6.15) If 𝜉=0, then by (6.15), we have 𝑓=^∞_𝑛=1𝑓,𝑦_𝑛_𝐿20=_∞𝜐_𝑛,(6.16)_𝑛=1𝑓,𝑦_𝑛_𝐿2𝑠_𝑛.(6.17)
Let 𝑠_𝑟≠0. Then by (6.17), we obtain 𝑓,𝑦_𝑟_𝐿2=−𝑠^𝑟∞−1_{𝑛=1𝑛≠𝑟}𝑓,𝑦_𝑛_𝐿2𝑠_𝑛,(6.18) considering which in (6.16), we get 𝑓=^∞_𝑛_{=1𝑛≠𝑟}𝑓,𝑦_𝑛_𝐿2𝜐_𝑛−𝑠^𝑟−1𝑠_𝑛𝜐_𝑟.(6.19) By (6.13) and (2.1), we have 𝑦_𝑛,𝜐_𝑘−𝑠^𝑟−1𝑠_𝑘𝜐_𝑟_𝐿2=𝑦_𝑛,𝜐_𝑘_𝐿2−𝑠^𝑟−1𝑠_𝑘𝑦_𝑛,𝜐_𝑟_𝐿2=𝑦_𝑛̂,𝜐̂_𝑘_𝐻−|𝜎|⁻¹𝑚_𝑛𝑠_𝑘−𝑠^𝑟−1𝑠_𝑘𝑦_𝑛,𝜐_𝑟_𝐿2+𝑠^𝑟−1𝑠_𝑘|𝜎|⁻¹𝑚_𝑛𝑠_𝑟=𝛿_𝑛𝑘,𝑛,𝑘≠𝑟,(6.20) that is, the system {𝜐_𝑛(𝑥)−𝑠^𝑟−1𝑠_𝑛𝜐_𝑟(𝑥)}^{∞𝑛=1,𝑛≠𝑟} is conjugated to the system {𝑦_𝑛(𝑥)}^{∞𝑛=1,𝑛≠𝑟}. By virtue of (6.19), the system {𝜐_𝑛(𝑥)−𝑠^𝑟−1𝑠_𝑛𝜐_𝑟(𝑥)}^{∞𝑛=1,𝑛≠𝑟} is a Riesz basis in 𝐿₂(0,𝑙). Then, on the base of Corollary 2 [40, Chapter 1, Section 4] {𝑦_𝑛(𝑥)}^{∞𝑛=1,𝑛≠𝑟} is also in a Riesz basis in 𝐿₂(0,𝑙). The basicity of the system {𝑦_𝑛(𝑥)}^{∞𝑛=1,𝑛≠𝑟} in the space 𝐿_𝑝(0,𝑙), 𝑝∈(1,∞)⧵{2}, can be proved by scheme of the proof of Theorem 5.1 in [18] using Theorem 5.2.
Now let 𝑠_𝑟=0. Then, by (2.1) and (6.13) we have 𝑦_𝑛,𝜐_𝑟_𝐿2=𝑦_𝑛̂,𝜐̂_𝑟_𝐻−|𝜎|⁻¹𝑚_𝑛𝑠_𝑟=0,𝑛∈ℕ,𝑛≠𝑟.(6.21)
So, the function 𝜐_𝜏(𝑥) is orthogonal to all functions of the system {𝑦_𝑛(𝑥)}^{∞𝑛=1,𝑛≠𝑟}, that is, the system {𝑦_𝑛(𝑥)}^{∞𝑛=1,𝑛≠𝑟} is incomplete in 𝐿₂(0,𝑙).
On the basis of Corollary 6.5 there exists 𝑘∈ℕ such that 𝑠_𝑘≠0 (e.g., if 𝜚(𝜆_𝑘)=1). Then, for any 𝑓(𝑥)∈𝐿₂(0,𝑙) the following expansion holds: 𝑓=^∞_𝑛_{=1𝑛≠𝑘}𝑓,𝜐_𝑛−𝑠^𝑘−1𝑠_𝑛𝜐_𝑘_𝐿2𝑦_𝑛.(6.22) By (6.21) and (6.22), we get 𝑦_𝑘=^∞_𝑛_{=1𝑛≠𝑘}𝑦_𝑘,𝜐_𝑛−𝑠^𝑘−1𝑠_𝑛𝜐_𝑘_𝐿2𝑦_𝑛=^∞_𝑛_{=1𝑛≠𝑘,𝑟}𝑦_𝑘,𝜐_𝑛−𝑠^𝑘−1𝑠_𝑛𝜐_𝑘_𝐿2𝑦_𝑛+𝑦_𝑘,𝜐_𝑟−𝑠^𝑘−1𝑠_𝑟𝜐_𝑘_𝐿2𝑦_𝑟=^∞_𝑛_{=1𝑛≠𝑘,𝑟}𝑦^𝑘,𝜐_𝑛_𝐿2−𝑠^𝑘−1𝑠_𝑛𝑦_𝑘,𝜐_𝑘_𝐿2𝑦^𝑛=^∞_𝑛_{=1𝑛≠𝑘,𝑟}𝑦^𝑘̂,𝜐̂_𝑛_𝐻−|𝜎|⁻¹𝑚_𝑘𝑠_𝑛−𝑠^𝑘−1𝑠_𝑛𝑦_𝑘̂,𝜐̂_𝑘+𝑠^𝑘−1𝑠_𝑛|𝜎|⁻¹𝑚_𝑘𝑠_𝑘𝑦^𝑛=−^∞_𝑛_{=1𝑛≠𝑘,𝑟}𝑠^𝑘−1𝑠_𝑛𝑦_𝑛,(6.23) whence it follows the equality ∑^{∞𝑛=1,𝑛≠𝑟}𝑠_𝑛𝑦_𝑛=0, that is, the system {𝑦_𝑛(𝑥)}^{∞𝑛=1,𝑛≠𝑟} is nonminimal in 𝐿₂(0,𝑙).
Obviously, this system is neither complete nor minimal in 𝐿_𝑝(0,𝑙), 𝑝∈(1,∞)⧵{2}. The proof of Theorem 6.6 is complete.

Corollary 6.7. If 𝜎>0, the system {𝑦_𝑛(𝑥)}^{∞𝑛=1,𝑛≠𝑟} (for any 𝑟∈ℕ) is a basis in 𝐿_𝑝(0,𝑙), 𝑝∈(1,∞), and even a Riesz basis for 𝑝=2.