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Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter

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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

A kind of fourth-order boundary value problem with eigenparameter-dependent boundary and transmission conditions is investigated. By constructing the characteristic function, we prove that the problems consist of a finite number of eigenvalues. We obtain that the number of eigenvalues of the problems not only depend on the order of the equation but also depend on the partition of the domain interval, the boundary conditions, and the eigenparameter-dependent transmission conditions.

Keywords: fourth-order boundary value problems; boundary and transmission conditions with spectral parameter; eigenvalues; finite spectrum
MSC 2010: 34L99; 34B05; 47A75

1 Introduction

The finite spectrum of various boundary value problems has been widely studied by many authors [1–7]. These studies enrich and extend the finite spectral theory of boundary value problems. Historically, such problems come from Atkinson’s statement in his well-known book [8] in 1964. In 2001, Kong et al. proved that if the coefficients of Sturm-Liouville (S-L) problem satisfy some special conditions, the problem may have a finite number of eigenvalues [6]. Since then, Ao et al. generalized the finite spectrum results to S-L problems with transmission conditions, and S-L problems with transmission conditions and eigenparameter-dependent boundary conditions [9,10], and even higher order boundary value problems such as fourth-order boundary value problems and third-order problems have also been considered [11–14]. The corresponding inverse spectral problems of such problems have also been studied in [15–17].

Boundary value problems with transmission conditions have been an important research topic for their applications in mathematical physics. Hald has studied the discontinuous S-L problem in [18], and he has shown the direct and inverse spectral theory on the S-L problem with internal discontinuous point conditions. Since then the boundary value problems with the internal point conditions have lots of studies in various aspects [9–12,18–22], such as eigenvalue asymptotics, completeness of root functions and so on. In addition, the results of boundary value problems with the spectral parameter in the boundary conditions can be found in many literature and here we only refer to [10,22–24]. In 2019, Guliyev [25] studied direct and inverse spectral problems for the one-dimensional Schrödinger equation with distributional potential and boundary conditions containing the eigenvalue parameter. The construction of differential operators with eigenparameter-dependent boundary conditions is given in 2022 [26]. With the development of such problems and the discontinuity problems mentioned above, the boundary value problems with eigenparameter-dependent transmission conditions have attracted several scholars’ attention. In 2005, Akdoǧan et al. [27,28] studied the discontinuous S-L problems with spectral parameter, in which the spectral parameter can appear not only in the differential equations but also in the boundary conditions and the interior point conditions. In 2007, Akdoǧan et al. [29] generalized some results of the classical regular Sturm-Liouville problems. In particular, they constructed Green’s function and derived the asymptotic approximation formulas and the resolvent of operator for Green’s function. Regarding the boundary value problems with eigenparameter-dependent transmission conditions, some scholars have studied the spectral and inverse spectral problems on such problems [30–33]. The S-L problems with Herglotz-type eigenparameter-dependent transmission conditions have also been investigated in recent works [34–36]. The finite spectrum of S-L problems with transmission conditions dependent on the spectral parameter has been given in [7]. However, the finite spectrum of higher order problems with eigenparameter-dependent transmission conditions has not been concluded. Therefore, we will study the finite spectrum of the fourth-order boundary value problems with eigenparameter-dependent boundary and transmission conditions.

Following [12], we consider the fourth-order problems with eigenparameter-dependent boundary and transmission conditions and try to show that these problems also have finite spectrum though the analysis is more complicated than before.

2 Fourth-order problems

Consider the fourth-order differential equation

(2.1) ( p y ″ ) ″ − ( s y ′ ) ′ + q y = λ w y , x ∈ J = [ a , c ) ∪ ( c , b ] ,

with − ∞ < a < b < + ∞ , c ∈ ( a , b ) , and the coefficients satisfying

(2.2) r = 1 ∕ p , s , q , w ∈ L ( J , C ) ,

where L ( J , C ) denotes the complex valued functions which are Lebesgue integrable on J .

The eigenparameter-dependent boundary conditions are as follows:

(2.3) A λ Y ( a ) + B λ Y ( b ) = 0 , Y = ( y , y ′ , p y ″ , ( p y ″ ) ′ − s y ′ ) T ,

where

A λ = α 1 ′ λ + α 1 α 2 ′ λ + α 2 α 3 ′ λ + α 3 α 4 ′ λ + α 4 β 1 ′ λ + β 1 β 2 ′ λ + β 2 β 3 ′ λ + β 3 β 4 ′ λ + β 4 0 0 0 0 0 0 0 0 , B λ = 0 0 0 0 0 0 0 0 μ 1 ′ λ + μ 1 μ 2 ′ λ + μ 2 μ 3 ′ λ + μ 3 μ 4 ′ λ + μ 4 ν 1 ′ λ + ν 1 ν 2 ′ λ + ν 2 ν 3 ′ λ + ν 3 ν 4 ′ λ + ν 4 ,

with α i ′ , α i , β i ′ , β i , μ i ′ , μ i , ν i ′ , ν i ∈ R , i = 1 , 2 , 3 , 4 satisfying

rank α 1 α 2 α 3 α 4 α 1 ′ α 2 ′ α 3 ′ α 4 ′ = 2 , rank β 1 β 2 β 3 β 4 β 1 ′ β 2 ′ β 3 ′ β 4 ′ = 2 , rank μ 1 μ 2 μ 3 μ 4 μ 1 ′ μ 2 ′ μ 3 ′ μ 4 ′ = 2 , rank ν 1 ν 2 ν 3 ν 4 ν 1 ′ ν 2 ′ ν 3 ′ ν 4 ′ = 2 , rank α 1 α 2 α 3 α 4 β 1 β 2 β 3 β 4 = 2 , rank α 1 ′ α 2 ′ α 3 ′ α 4 ′ β 1 ′ β 2 ′ β 3 ′ β 4 ′ = 2 , rank μ 1 μ 2 μ 3 μ 4 ν 1 ν 2 ν 3 ν 4 = 2 , rank μ 1 ′ μ 2 ′ μ 3 ′ μ 4 ′ ν 1 ′ ν 2 ′ ν 3 ′ ν 4 ′ = 2 .

The eigenparameter-dependent transmission conditions will be given in the following two forms:

(i)

(2.4) C λ Y ( c − ) + D λ Y ( c + ) = 0 , Y = ( y , y ′ , p y ″ , ( p y ″ ) ′ − s y ′ ) T ,

where

C λ = − ξ 0 0 0 0 − ξ − 1 0 0 ζ 1 λ − η 1 0 − 1 0 0 0 0 − 1 , D λ = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 ζ 2 λ − η 2 1 ,

with ξ , ζ k , η k ∈ R , ξ ≠ 0 , ζ k ≠ 0 , k = 1 , 2 ;

(ii)

(2.5) C λ ′ Y ( c − ) + D λ ′ Y ( c + ) = 0 , Y = ( y , y ′ , p y ″ , ( p y ″ ) ′ − s y ′ ) T ,

where

C λ ′ = 1 0 0 θ 1 λ − ϑ 1 0 1 0 0 0 0 1 0 0 0 0 1 , D λ ′ = 0 0 0 1 0 0 1 0 θ 2 λ − ϑ 2 1 0 0 − 1 0 0 θ 3 λ − ϑ 3 ,

with θ k , ϑ k ∈ R , θ k ≠ 0 , k = 1 , 2 , 3 . Here λ is the spectral parameter.

In the present article, we mainly discuss the finite spectrum of the fourth-order problems composed of the differential equation (2.1), the eigenparameter-dependent boundary conditions (2.3), and the eigenparameter-dependent transmission conditions (2.4). For the fourth-order problems composed of the differential equation (2.1), the eigenparameter-dependent boundary conditions (2.3), and the eigenparameter-dependent transmission conditions (2.5), the conclusion of the finite spectrum can be similarly obtained.

Let y 1 = y , y 2 = y ′ , y 3 = p y ″ , and y 4 = ( p y ″ ) ′ − s y ′ , then the system representation of (2.1) is as follows:

(2.6) y 1 ′ = y 2 , y 2 ′ = 1 p y 3 = r y 3 , y 3 ′ = s y 2 + y 4 , y 4 ′ = ( λ w − q ) y 1 .

And equation (2.1) is equivalent to the first-order system as follows:

(2.7) Y ′ = 0 1 0 0 0 0 r ( x ) 0 0 s ( x ) 0 1 λ w ( x ) − q ( x ) 0 0 0 Y , Y = y 1 y 2 y 3 y 4 .

Lemma 1

[14] Let (2.2) hold and Φ ( x , λ ) = [ ϕ i j ( x , λ ) ] denote the fundamental matrix of system (2.7) determined by the initial condition Φ ( a , λ ) = I , then λ ∈ C is an eigenvalue of the fourth-order boundary value problems with eigenparameter-dependent transmission conditions (2.1), (2.3), and (2.4) if and only if the characteristic function Δ ( λ ) = 0 . At this time, the characteristic function can be represented as follows:

(2.8) Δ ( λ ) = det [ A λ + B λ Φ ( b , λ ) ] = ∑ k 1 k 2 k 3 k 4 ; 1 ≤ j 1 , j 2 ≤ 4 , j 1 ≠ j 2 ( − 1 ) τ ( k 1 k 2 k 3 k 4 ) d j 1 j 2 k 1 k 2 k 3 k 4 ( λ ) ϕ j 1 k 3 ϕ j 2 k 4 ,

with d j 1 j 2 k 1 k 2 k 3 k 4 ( λ ) = ( α k 1 ′ λ + α k 1 ) ( β k 2 ′ λ + β k 2 ) ( μ j 1 ′ λ + μ j 1 ) ( ν j 2 ′ λ + ν j 2 ) , where k 1 k 2 k 3 k 4 is a permutation of the natural numbers 1 , 2 , 3 , 4 , τ ( k 1 k 2 k 3 k 4 ) denotes the inversion number of the permutation k 1 k 2 k 3 k 4 , and ∑ k 1 k 2 k 3 k 4 ; 1 ≤ j 1 , j 2 ≤ 4 , j 1 ≠ j 2 denotes the sums of all possible permutations of k 1 k 2 k 3 k 4 and 1 ≤ j 1 , j 2 ≤ 4 , j 1 ≠ j 2 .

Proof

If λ is an eigenvalue of the fourth-order boundary value problems (2.1), (2.3), and (2.4), then there exists a non-trivial solution

(2.9) Y ( x ) = Φ ( x , λ ) C , C = c 1 c 2 c 3 c 4 ,

where c 1 , c 2 , c 3 , c 4 are not all 0. Since Y ( x ) satisfies the boundary condition (2.3), we have

(2.10) A λ Φ ( a , λ ) C + B λ Φ ( b , λ ) C = [ A λ + B λ Φ ( b , λ ) ] C = 0 .

Since c 1 , c 2 , c 3 , c 4 are not all 0, then the characteristic function Δ ( λ ) = det [ A λ + B λ Φ ( b , λ ) ] = 0 .

If the characteristic function Δ ( λ ) = det [ A λ + B λ Φ ( b , λ ) ] = 0 , then equation (2.10) has non-zero solution C . Choose such a solution and define Y ( x ) as in (2.9), then Y ( x ) satisfies the fourth-order problems (2.1), (2.3), and (2.4). Thus λ ∈ C is an eigenvalue of the fourth-order problems (2.1), (2.3), and (2.4).

Finally, according to (2.3) and by direct calculation, we can obtain (2.8). This completes the proof.□

The fourth-order problems with eigenparameter-dependent boundary conditions and transmission conditions (2.1), (2.3), and (2.4), or equivalently (2.6), (2.3), and (2.4), are said to be degenerate if in (2.8), either Δ ( λ ) ≡ 0 for all λ ∈ C or Δ ( λ ) ≠ 0 for any λ ∈ C .

Next we will partition the interval J such that the coefficients of the fourth-order equation (2.1) satisfy certain conditions.

We assume (2.2) hold and there exists a partition of the interval J

(2.11) a = a 0 < a 1 < a 2 < ⋯ < a 2 m < a 2 m + 1 = c , c = b 0 < b 1 < b 2 < ⋯ < b 2 n < b 2 n + 1 = b ,

for some positive integers m and n , such that

(2.12) r = 1 p = 0 on [ a 2 i , a 2 i + 1 ] , ∫ a 2 i a 2 i + 1 w ( x ) d x ≠ 0 , ∫ a 2 i a 2 i + 1 w ( x ) x d x ≠ 0 , ∫ a 2 i a 2 i + 1 w ( x ) x 2 d x ≠ 0 , i = 0 , 1 , … , m , q = w = s = 0 on [ a 2 i + 1 , a 2 i + 2 ] , ∫ a 2 i + 1 a 2 i + 2 r ( x ) d x ≠ 0 , ∫ a 2 i + 1 a 2 i + 2 r ( x ) x d x ≠ 0 , ∫ a 2 i + 1 a 2 i + 2 r ( x ) x 2 d x ≠ 0 , i = 0 , 1 , … , m − 1 ;

and

(2.13) r = 1 p = 0 on [ b 2 j , b 2 j + 1 ] , ∫ b 2 j b 2 j + 1 w ( x ) d x ≠ 0 , ∫ b 2 j b 2 j + 1 w ( x ) x d x ≠ 0 , ∫ b 2 j b 2 j + 1 w ( x ) x 2 d x ≠ 0 , j = 0 , 1 , … , n , q = w = s = 0 on [ b 2 j + 1 , b 2 j + 2 ] , ∫ b 2 j + 1 b 2 j + 2 r ( x ) d x ≠ 0 , ∫ b 2 j + 1 b 2 j + 2 r ( x ) x d x ≠ 0 , ∫ b 2 j + 1 b 2 j + 2 r ( x ) x 2 d x ≠ 0 , j = 0 , 1 , … , n − 1 .

Given (2.11)–(2.13), now let

r i = ∫ a 2 i + 1 a 2 i + 2 r ( x ) d x , i = 0 , 1 , … , m − 1 , q i = ∫ a 2 i a 2 i + 1 q ( x ) d x , w i = ∫ a 2 i a 2 i + 1 w ( x ) d x , s i = ∫ a 2 i a 2 i + 1 s ( x ) d x , i = 0 , 1 , … , m , r ˜ j = ∫ b 2 j + 1 b 2 j + 2 r ( x ) d x , j = 0 , 1 , … , n − 1 , q ˜ j = ∫ b 2 j b 2 j + 1 q ( x ) d x , w ˜ j = ∫ b 2 j b 2 j + 1 w ( x ) d x , s ˜ j = ∫ b 2 j b 2 j + 1 s ( x ) d x , j = 0 , 1 , … , n , r ˆ i = ∫ a 2 i + 1 a 2 i + 2 r ( x ) x d x , i = 0 , 1 , … , m − 1 , q ˆ i = ∫ a 2 i a 2 i + 1 q ( x ) x d x , w ˆ i = ∫ a 2 i a 2 i + 1 w ( x ) x d x , i = 0 , 1 , … , m ,

r ˇ j = ∫ b 2 j + 1 b 2 j + 2 r ( x ) x d x , j = 0 , 1 , … , n − 1 , q ˇ j = ∫ b 2 j b 2 j + 1 q ( x ) x d x , w ˇ j = ∫ b 2 j b 2 j + 1 w ( x ) x d x , j = 0 , 1 , … , n , ŕ i = ∫ a 2 i + 1 a 2 i + 2 r ( x ) x 2 d x , i = 0 , 1 , … , m − 1 , q ́ i = ∫ a 2 i a 2 i + 1 q ( x ) x 2 d x , w ́ i = ∫ a 2 i a 2 i + 1 w ( x ) x 2 d x , i = 0 , 1 , … , m , r ̀ j = ∫ b 2 j + 1 b 2 j + 2 r ( x ) x 2 d x , j = 0 , 1 , … , n − 1 , q ̀ j = ∫ b 2 j b 2 j + 1 q ( x ) x 2 d x , ẁ j = ∫ b 2 j b 2 j + 1 w ( x ) x 2 d x , j = 0 , 1 , … , n .

Then, we can state our main results as follows.

Theorem 1

Let m , n , ∈ N , and let (2.2) and (2.11)–(2.13) hold. Then, the fourth-order boundary value problems with eigenparameter-dependent boundary and transmission conditions (2.1), (2.3), and (2.4) have at most 2 m + 2 n + 8 eigenvalues.

Proof

The proof is given in Section 3.□

Theorem 2

Let m , n , ∈ N , and let (2.2) and (2.11)–(2.13) hold. Then the fourth-order boundary value problems with eigenparameter-dependent boundary and transmission conditions (2.1), (2.3), and (2.5) have at most 2 m + 2 n + 14 eigenvalues.

Proof

The proof is similar to the proof of Theorem 1, hence is omitted here.□

3 Proof of main result

In this section, we will give some lemmas and the proof of Theorem 1.

Lemma 2

[11] Let (2.2) and (2.11)–(2.13) hold, Φ ( x , λ ) = [ ϕ i j ( x , λ ) ] be the fundamental matrix solution of system (2.7) determined by the initial condition Φ ( a , λ ) = I for each λ ∈ C . Let

F i ( x , λ , a i ) = 1 x − a i 0 0 0 1 0 0 ∫ a i x ( λ w − q ) ( x − t ) d t ∫ a i x ( λ w − q ) ( x − t ) ( t − a i ) d t + ∫ a i x s d t 1 x − a i ∫ a i x ( λ w − q ) d t ∫ a i x ( λ w − q ) ( t − a i ) d t 0 1 , i = 0 , 2 , … , 2 m ; F i ( x , λ , a i ) = 1 x − a i ∫ a i x r ( x − t ) d t ∫ a i x r ( x − t ) ( t − a i ) d t 0 1 ∫ a i x r d t ∫ a i x r ( t − a i ) d t 0 0 1 x − a i 0 0 0 1 , i = 1 , 3 , … , 2 m − 1 .

Then, for 1 ≤ i ≤ 2 m + 1 , we have

Φ ( a i , λ ) = F i − 1 ( a i , λ , a i − 1 ) Φ ( a i − 1 , λ ) .

If we let

T 0 = F 0 ( a 1 , λ , a 0 ) , T i = F 2 i ( a 2 i + 1 , λ , a 2 i ) F 2 i − 1 ( a 2 i , λ , a 2 i − 1 ) , i = 1 , 2 , … , m ,

then,

Φ ( a 1 , λ ) = F 0 ( a 1 , λ , a 0 ) = T 0 , Φ ( a 2 i + 1 , λ ) = T i Φ ( a 2 i − 1 , λ ) , i = 1 , 2 , … , m .

Hence, we have the following formula:

Φ ( a 2 i + 1 , λ ) = T i T i − 1 ⋯ T 0 , i = 0 , 1 , 2 , … , m .

Lemma 3

[11] Let (2.2) and (2.11)–(2.13) hold, Ψ ( x , λ ) = [ ψ i j ( x , λ ) ] be the fundamental matrix solution of system (2.7) determined by the initial condition Ψ ( c + , λ ) = I for each λ ∈ C . Let

F ˜ j ( x , λ , b j ) = 1 x − b j 0 0 0 1 0 0 ∫ b j x ( λ w − q ) ( x − t ) d t ∫ b j x ( λ w − q ) ( x − t ) ( t − b j ) d t + ∫ b j x s d t 1 x − b j ∫ b j x ( λ w − q ) d t ∫ b j x ( λ w − q ) ( t − b j ) d t 0 1 , j = 0 , 2 , … , 2 n ; F ˜ j ( x , λ , b j ) = 1 x − b j ∫ b j x r ( x − t ) d t ∫ b j x r ( x − t ) ( t − b j ) d t 0 1 ∫ b j x r d t ∫ b j x r ( t − b j ) d t 0 0 1 x − b j 0 0 0 1 , j = 1 , 3 , … , 2 n − 1 .

Then, for 1 ≤ j ≤ 2 n + 1 , we have

Ψ ( b j , λ ) = F ˜ j − 1 ( b j , λ , b j − 1 ) Ψ ( b j − 1 , λ ) .

Still, let

T ˜ 0 = F ˜ 0 ( b 1 , λ , b 0 ) , T ˜ j = F ˜ 2 j ( b 2 j + 1 , λ , b 2 j ) F ˜ 2 j − 1 ( b 2 j , λ , b 2 j − 1 ) , j = 1 , 2 , … , n ,

then,

Ψ ( b 1 , λ ) = F ˜ 0 ( b 1 , λ , b 0 ) = T ˜ 0 , Ψ ( b 2 j + 1 , λ ) = T ˜ j Ψ ( b 2 j − 1 , λ ) , j = 1 , 2 , … , n .

Hence, we have the following formula:

Ψ ( b 2 j + 1 , λ ) = T ˜ j T ˜ j − 1 ⋯ T ˜ 0 , j = 0 , 1 , 2 , … , n .

Lemma 4

Let (2.2) and (2.11)–(2.13) hold. Φ ( x , λ ) and Ψ ( x , λ ) are defined as in Lemmas 2  and 3, respectively, then,

(3.1) Φ ( b , λ ) = Ψ ( b , λ ) G λ Φ ( c − , λ ) ,

where G λ = [ g λ , i j ] 2 × 2 = − D λ − 1 C λ .

Proof

From (2.4), we have

C λ Y ( c − ) + D λ Y ( c + ) = 0 , det ( C λ ) = 1 > 0 , det ( D λ ) = 1 > 0 ,

therefore,

C λ Φ ( c − , λ ) + D λ Φ ( c + , λ ) = 0 ,

that is,

Φ ( c + , λ ) = G λ Φ ( c − , λ ) , G λ = [ g λ , i j ] 2 × 2 = − D λ − 1 C λ ,

thus,

(3.2) Φ ( c + , λ ) [ G λ Φ ( c − , λ ) ] − 1 = I .

From Lemmas 2 and 3

Φ ( c − , λ ) = Φ ( a 2 m + 1 , λ ) = T m Φ ( a 2 m − 1 , λ ) , Ψ ( b , λ ) = Ψ ( b 2 n + 1 , λ ) = T ˜ n Ψ ( b 2 n − 1 , λ ) ,

by combining with Ψ ( c + , λ ) = I and (3.2), we obtain

Ψ ( b , λ ) = Φ ( b , λ ) [ G λ Φ ( c − , λ ) ] − 1 ,

thus,

□ Φ ( b , λ ) = Ψ ( b , λ ) G λ Φ ( c − , λ ) .

Corollary 1

For the fundamental matrix Φ ( c − , λ ) on interval [ a , c ) in (3.1), we have

(3.3) ϕ i j ( c − , λ ) = K i j λ m − 1 + ϕ ˜ i j ( λ ) , i = 1 , 2 , j = 3 , 4 , ϕ i j ( c − , λ ) = K i j λ m + ϕ ˜ i j ( λ ) , i , j = 1 , 2 , o r i = j = 3 , 4 , ϕ i j ( c − , λ ) = K i j λ m + 1 + ϕ ˜ i j ( λ ) , i = 3 , 4 , j = 1 , 2 ,

where K i j are some constants related to r i , r ˆ i , ŕ i , i = 0 , 1 , ⋯ , m − 1 , w i , w ˆ i , w ́ i , i = 0 , 1 , ⋯ , m , and the endpoints a , c ; ϕ ˜ i j ( λ ) are functions of λ , in which the degrees of λ are smaller than m − 1 , m , or m + 1 , respectively.

Corollary 2

For the fundamental matrix Ψ ( b , λ ) on interval ( c , b ] in (3.1), we have

(3.4) ψ i j ( b , λ ) = K ˜ i j λ n − 1 + ψ ˜ i j ( λ ) , i = 1 , 2 , j = 3 , 4 , ψ i j ( b , λ ) = K ˜ i j λ n + ψ ˜ i j ( λ ) , i , j = 1 , 2 , o r i = j = 3 , 4 , ψ i j ( b , λ ) = K ˜ i j λ n + 1 + ψ ˜ i j ( λ ) , i = 3 , 4 , j = 1 , 2 ,

where K ˜ i j are some constants related to r ˜ j , r ˇ j , r ̀ j , j = 0 , 1 , … , n − 1 , w ˜ j , w ˇ j , ẁ j , j = 0 , 1 , … , n and the endpoints c , b ; ψ ˜ i j ( λ ) are functions of λ , in which the degrees of λ are smaller then n − 1 , n , or n + 1 , respectively.

Now we can prove our main result-Theorem 1.

Proof of Theorem 1

From (2.4) and Lemma 4, we know that

(3.5) G λ = − D λ − 1 C λ = ξ 0 0 0 0 ξ − 1 0 0 − ( ζ 1 λ − η 1 ) 0 1 0 ( ζ 1 λ − η 1 ) ( ζ 2 λ − η 2 ) 0 − ( ζ 2 λ − η 2 ) − 1 .

From Corollary 1, Corollary 2, and (3.1), it is better to understand and represent the degrees of ϕ i j ( b , λ ) in λ as the following matrix:

(3.6) m + n + 1 m + n + 1 m + n m + n m + n + 1 m + n + 1 m + n m + n m + n + 2 m + n + 2 m + n + 1 m + n + 1 m + n + 2 m + n + 2 m + n + 1 m + n + 1 .

While ( α 3 ′ β 4 ′ + α 4 ′ β 3 ′ ) ( μ 3 ′ ν 4 ′ − μ 4 ′ ν 3 ′ ) ≠ 0 , in terms of (2.8) and (3.6), it is easy to see that the maximum of the degree of Δ ( λ ) is 2 m + 2 n + 8 . By the Fundamental Theorem of Algebra, we can conclude that Δ ( λ ) has at most 2 m + 2 n + 8 zeros, hence the fourth-order boundary value problem with boundary and transmission conditions dependent on the spectral parameter (2.1), (2.3), and (2.4) has a finite number of eigenvalues and the maximum of this number is 2 m + 2 n + 8 .□

4 Concluding remarks

This study deals with the finite spectral theory of a class of fourth-order boundary value problems with eigenparameter-dependent boundary and transmission conditions. It is illustrated that under certain conditions, the problems considered here have exactly a finite number of eigenvalues, and the number of eigenvalues depends on the partition of the domain interval, the eigenparameter-dependent boundary conditions, and the eigenparameter-dependent transmission conditions. The results of this work extends the results of [7] and [14] directly.

Compared with the results in [4,7] and [14], it is easy to see that the number of eigenvalues depends on the order of the equation, the partition of the domain interval, the eigenparameter-dependent boundary conditions and the eigenparameter-dependent transmission conditions. Also, from Theorems 1 and 2, it can be concluded that the number of eigenvalues is different due to the different connection matrices in the eigenparameter-dependent transmission conditions.

5 Examples

In order to illustrate the conclusion of this study, we give two mathematical examples with different eigenparameter-dependent transmission conditions.

Example 1

Consider the fourth-order boundary value problem with boundary and transmission conditions dependent on the spectral parameter.

(5.1) ( p y ″ ) ″ − ( s y ′ ) ′ + q y = λ w y , on J = [ − 2 , 1 ) ∪ ( 1 , 4 ] , y ( − 2 ) + λ ( ( p y ″ ) ′ − s y ′ ) ( − 2 ) = 0 , ( y ′ ) ( − 2 ) + ( 1 + 2 λ ) ( p y ″ ) ( − 2 ) = 0 , ( p y ″ ) ( 4 ) + λ ( ( p y ″ ) ′ − s y ′ ) ( 4 ) = 0 , y ( 4 ) + λ ( p y ″ ) ( 4 ) = 0 , − y ( 1 − ) + y ( 1 + ) = 0 , − ( y ′ ) ( 1 − ) + ( y ′ ) ( 1 + ) = 0 , ( 2 λ + 1 ) y ( 1 − ) − ( p y ″ ) ( 1 − ) + ( p y ″ ) ( 1 + ) = 0 , − ( ( p y ″ ) ′ − s y ′ ) ( 1 − ) + λ ( p y ″ ) ( 1 + ) + ( ( p y ″ ) ′ − s y ′ ) ( 1 + ) = 0 .

Let m = 1 , n = 1 , and r = 1 p , q , w , s be piecewise constant functions defined as follows:

(5.2) r ( x ) = 1 p ( x ) = 0 , x ∈ [ − 2 , − 1 ) − 1 , x ∈ [ − 1 , 0 ) 0 , x ∈ [ 0 , 1 ) 0 , x ∈ ( 1 , 2 ) 2 , x ∈ [ 2 , 3 ) 0 , x ∈ [ 3 , 4 ) , q ( x ) = 1 , x ∈ ( − 2 , − 1 ) 0 , x ∈ [ − 1 , 0 ) − 1 , x ∈ [ 0 , 1 ) − 2 , x ∈ ( 1 , 2 ) 0 , x ∈ [ 2 , 3 ) 1 , x ∈ [ 3 , 4 ] ,

(5.3) w ( x ) = 1 , x ∈ [ − 2 , − 1 ) 0 , x ∈ [ − 1 , 0 ) 1 , x ∈ [ 0 , 1 ) 1 , x ∈ ( 1 , 2 ) 0 , x ∈ [ 2 , 3 ) 1 2 , x ∈ [ 3 , 4 ) , s ( x ) = − 1 2 , x ∈ ( − 2 , − 1 ) 0 , x ∈ [ − 1 , 0 ) 1 , x ∈ [ 0 , 1 ) − 1 , x ∈ ( 1 , 2 ) 0 , x ∈ [ 2 , 3 ) 1 , x ∈ [ 3 , 4 ] .

It can be obtained from the given conditions that

A λ = 1 0 0 λ 0 1 1 + 2 λ 0 0 0 0 0 0 0 0 0 , B λ = 0 0 0 0 0 0 0 0 0 0 1 λ 1 0 λ 0 , C λ = − 1 0 0 0 0 − 1 0 0 2 λ + 1 0 − 1 0 0 0 0 − 1 , D λ = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 λ 1 .

Then, the characteristic function can be obtained by derivation.

Δ ( λ ) = det [ A λ + B λ Φ ( b , λ ) ] = 668899 124416 λ 12 + 1485107 248832 λ 11 − 4669891 31104 λ 10 − 12861703 41472 λ 9 − 203077211 248832 λ 8 − 57325795 27648 λ 7 − 70116457 41472 λ 6 + 13798351 20736 λ 5 + 63372817 20736 λ 4 + 53303455 13824 λ 3 + 7700105 10368 λ 2 + 1578923 3456 λ − 4668 24 .

Hence, the fourth-order problem with boundary and transmission conditions dependent on the spectral parameter (5.1), (5.2), and (5.3) has at most 2 m + 2 n + 8 = 12 eigenvalues.

Example 2

Consider the fourth-order boundary value problem with boundary and transmission conditions dependent on the spectral parameter.

(5.4) ( p y ″ ) ″ − ( s y ′ ) ′ + q y = λ w y , on J = [ − 3 , 0 ) ∪ ( 0 , 3 ] , y ( − 3 ) + λ ( p y ″ ) ( − 3 ) = 0 , λ ( y ′ ) ( − 3 ) + ( p y ″ ) ( − 3 ) = 0 , y ( 3 ) + λ ( y ′ ) ( 3 ) = 0 , ( y ′ ) ( 3 ) + λ ( p y ″ ) ( 3 ) = 0 , y ( 0 − ) + ( 3 λ + 1 ) ( ( p y ″ ) ′ − s y ′ ) ( 0 − ) + ( ( p y ″ ) ′ − s y ′ ) ( 0 + ) = 0 , ( y ′ ) ( 0 − ) + ( p y ″ ) ( 0 + ) = 0 , ( p y ″ ) ( 0 − ) + ( 2 λ − 1 ) y ( 0 + ) + ( y ′ ) ( 0 + ) = 0 , ( ( p y ″ ) ′ − s y ′ ) ( 0 − ) − y ( 0 + ) + ( λ − 2 ) ( ( p y ″ ) ′ − s y ′ ) ( 0 + ) = 0 .

Let m = 1 , n = 1 , and r = 1 p , q , s , w be piecewise constant functions defined as follows:

(5.5) r ( x ) = 1 p ( x ) = 0 , x ∈ [ − 3 , − 2 ) 1 , x ∈ [ − 2 , − 1 ) 0 , x ∈ [ − 1 , 0 ) 0 , x ∈ ( 0 , 1 ) − 1 , x ∈ [ 1 , 2 ) 0 , x ∈ [ 2 , 3 ] , q ( x ) = 1 , x ∈ [ − 3 , − 2 ) 0 , x ∈ [ − 2 , − 1 ) 2 , x ∈ [ − 1 , 0 ) − 1 , x ∈ ( 0 , 1 ) 0 , x ∈ [ 1 , 2 ) 1 , x ∈ [ 2 , 3 ] ,

(5.6) w ( x ) = 1 , x ∈ [ − 3 , − 2 ) 0 , x ∈ [ − 2 , − 1 ) − 1 , x ∈ ( − 1 , 0 ) 1 , x ∈ [ 0 , 1 ) 0 , x ∈ [ 1 , 2 ) 2 , x ∈ [ 2 , 3 ] , s ( x ) = 1 , x ∈ [ − 3 , − 2 ) 0 , x ∈ [ − 2 , − 1 ) 1 , x ∈ [ − 1 , 0 ) 1 , x ∈ ( 0 , 1 ) 0 , x ∈ [ 1 , 2 ) − 1 , x ∈ [ 2 , 3 ] .

It can be obtained from the given conditions that

A λ = 1 0 λ 0 0 λ 1 0 0 0 0 0 0 0 0 0 , B λ = 0 0 0 0 0 0 0 0 1 λ 0 0 0 1 λ 0 , C λ = 1 0 0 3 λ + 1 0 1 0 0 0 0 1 0 0 0 0 1 , D λ = 0 0 0 1 0 0 1 0 2 λ − 1 1 0 0 − 1 0 0 λ − 2 .

Then, the characteristic function can be obtained by derivation.

Δ ( λ ) = det [ A λ + B λ Φ ( b , λ ) ] = 275 1728 λ 16 − 3205 2592 λ 15 − 833353 31104 λ 14 + 28606811 62208 λ 13 − 225006739 15552 λ 12 + 14995669601 373248 λ 11 − 164251877 2592 λ 10 − 1087212691 729 λ 9 + 10343830877 18624 λ 8 − 11141856263 5832 λ 7 − 1161962764 11664 λ 6 + 10334476445 373248 λ 5 + 22704194419 373248 λ 4 − 1479991589 46656 λ 3 + 4370750009 373248 λ 2 + 113491567 186624 λ + 164995 15552 .

Hence, the fourth-order problem with boundary and transmission conditions dependent on the spectral parameter (5.4), (5.5), and (5.6) has 2 m + 2 n + 12 = 16 eigenvalues.

Acknowledgments

The authors thank the referees for their comments and detailed suggestions. These have significantly improved the presentation of this paper.

  1. Funding information: This work was supported by National Natural Science Foundation of China (Grant Nos. 12261066 and 11661059) and Natural Science Foundation of Inner Mongolia (Grant Nos. 2021MS01020 and 2023LHMS01015).

  2. Author contributions: The authors equally contributed to this work. All authors read and approved the final manuscript.

  3. Conflict of interest: The authors declare that they have no conflicts of interest.

  4. Data availability statement: No data were used to support this study.

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Received: 2023-05-31
Revised: 2023-08-09
Accepted: 2023-08-10
Published Online: 2023-09-08

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2023-0110
Keywords for this article
fourth-order boundary value problems; boundary and transmission conditions with spectral parameter; eigenvalues; finite spectrum
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  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
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