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The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions

  • Jia Li , Xiaoling Hao EMAIL logo , Kun Li and Siqin Yao
Published/Copyright: December 31, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

For any positive integer n and a set of positive integers m i , i = 1 , 2 , … , n + 1 , we construct a class of quadratic eigenparameter-dependent boundary Sturm-Liouville problems with n transmission conditions, which have at most ∑ i = 1 n + 1 m i + n + 5 eigenvalues. The key to this analysis is still the division of intervals and an iterative construction of the characteristic function. Further, some examples are given for a simple explanation.

Keywords: Sturm-Liouville problems; finite spectrum; eigenvalue; transmission conditions; quadratic eigenparameter-dependent
MSC 2010: 34B24; 34L15; 34L05

1 Introduction

The Sturm-Liouville problems with transmission conditions occupy an important position for their widely use in mathematical physics, engineering, etc. Such problems are related to heat and mass transfer and vibrating string problems (see [1,2, 3,4]). It is worth mentioning that the theory of fractional calculus is often used to study these problems, and some results about fractional differential equations can be referred to [5,6,7, 8,9]. In recent years, the research of Sturm-Liouville problems includes not only in one interior point, but also in two or infinite many interior points. The Sturm-Liouville problems with an eigenparameter contained in the boundary conditions were early introduced in [10], and the author mainly studied an expansion theorem of eigenfunctions, which was identical with the regular Sturm-Liouville problem in all respects except for the appearance of the parameter λ in the boundary condition. Further developments of such problems can be found in [2,10, 11,12].

It is well known that the classical Sturm-Liouville theory states that the spectrum of regular or singular, self-adjoint Sturm-Liouville problems is unbounded and therefore infinite, but Atkinson [13] proposed that if the coefficients of Sturm-Liouville problems satisfy some conditions, the problems may have finite eigenvalues. For this reason, in 2001, Kong et al. constructed a class of Sturm-Liouville problems with exactly n eigenvalues for any positive integer n in [14], and in 2009, they gave the matrix representations of these problems with self-adjoint boundary conditions in [15]. They further indicated the fact that Sturm-Liouville problems have a finite spectrum. Since then, Ao et al. extended the problems to the case of Sturm-Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions; moreover, they also gave the corresponding matrix representation (see, for example, [1,16, 17,18,19, 20,21]). In particular, Ao et al. obtained that the Sturm-Liouville problems with transmission conditions have at most m + n + 2 eigenvalues in [16], and the Sturm-Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions have at most m + n + 4 eigenvalues in [18]. In 2018, Xu et al. [22] generalized these results to the case of the Sturm-Liouville problems with n transmission conditions, which have at most ∑ i = 1 n + 1 m i + n + 1 eigenvalues.

However, the following questions remain: For the Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions, will the upper limit of the number of eigenvalues change? What factors will affect the number of eigenvalues? In this article, we conclude the Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions have at most ∑ i = 1 n + 1 m i + n + 5 eigenvalues, where m i are connected with the partition of the considered interval. In other words, the division of intervals, the number of transmission conditions and the boundary conditions of spectral parameters will all have an impact on the number of eigenvalues.

Following this introduction, in Section 2, we give some preliminaries, and main results on the finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions are given in Section 3; moreover, some examples are given to illustrate the problems of this article more concisely.

2 Notation and preliminaries

We study the Sturm-Liouville problem consisting of the following equation:

(2.1) − ( p y ′ ) ′ + q y = λ w y , on J = ( a , c 1 ) ∪ ( c 1 , c 2 ) ∪ ⋯ ∪ ( c n , b ) ,

where

− ∞ < a < b < + ∞ , c i ∈ ( a , b ) , i = 1 , 2 , … , n ,

with the eigenparameter-dependent boundary conditions

(2.2) A λ Y ( a ) + B λ Y ( b ) = 0 , Y = y p y ′ ,

and the transmission conditions

(2.3) C i Y ( c i − ) + D i Y ( c i + ) = 0 , i = 1 , 2 , … , n ,

where

A λ = α 1 λ 2 + β 1 λ + γ 1 α 1 ′ λ 2 + β 1 ′ λ + γ 1 ′ α 2 λ 2 + β 2 λ + γ 2 α 2 ′ λ 2 + β 2 ′ λ + γ 2 ′ , B λ = α 3 λ 2 + β 3 λ + γ 3 α 3 ′ λ 2 + β 3 ′ λ + γ 3 ′ α 4 λ 2 + β 4 λ + γ 4 α 4 ′ λ 2 + β 4 ′ λ + γ 4 ′ ,

with α i , β i , γ i , α i ′ , β i ′ , γ i ′ ∈ R , i = 1 , 2 satisfying ∣ α i ∣ + ∣ α i ′ ∣ ≠ 0 , det ( A λ ) ≠ 0 , det ( B λ ) ≠ 0 ,

(2.4) rank α 1 α 1 ′ α 3 α 3 ′ β 1 β 1 ′ β 3 β 3 ′ γ 1 γ 1 ′ γ 3 γ 3 ′ ≥ 2 , rank α 2 α 2 ′ α 4 α 4 ′ β 2 β 2 ′ β 4 β 4 ′ γ 2 γ 2 ′ γ 4 γ 4 ′ ≥ 2 ,

and C i and D i are real valued 2 × 2 matrices satisfying det ( C i ) = ρ i > 0 , det ( D i ) = θ i > 0 . Here, λ is the so-called spectral parameter, and the coefficients satisfy the conditions:

(2.5) r = 1 p , q , w ∈ L ( J , C ) ,

where L ( J , C ) denotes the complex-valued functions, which are Lebesgue integrable on J . Condition (2.5) ensures the uniqueness of the solutions on [ a , b ] for all initial value problem of equation (2.1) (see [10]).

Let r = 1 p , u = y , v = p y ′ . Then, equation (2.1) is equivalent to the system

(2.6) u ′ = r v , v ′ = ( q − λ w ) u , on J .

Remark 1

As usual, the self-adjoint extension of Sturm-Liouville problems with n transmission conditions needs additional restrictions on C i , D i and a new weighted Hilbert space defined as in [23,24, 25]. With this weighted Hilbert space, the operator associated with Sturm-Liouville problems with n transmission conditions is self-adjoint if and only if the associated new operator is self-adjoint, and they consist of the same eigenvalues and satisfy the condition θ 1 ⋯ θ n A E − 1 A ∗ = ρ 1 ⋯ ρ n B E − 1 B ∗ , with E = 0 − 1 1 0 . For further details of the self-adjointness of Sturm-Liouville problems with n transmission conditions, please see [23]. The construction is based on the characteristic function whose zeros are eigenvalues, which applies to the arbitrary boundary condition, so the results in this article are not restricted to self-adjoint problems, but include non-self-adjoint problems. We defined the partition of the interval, the values of p , q , w , and determined that Δ ( λ ) is the polynomial about λ , which shows that the number of eigenvalues is finite. Moreover, we analyzed the number of eigenvalues.

Remark 2

Condition (2.5) does not restrict the sign of any of the coefficients r , q , w . Also, each of r , q , w is allowed to be identically zero on subintervals of J . If r is identically zero on a subinterval J 1 , then there exists a solution y , which is identically zero on J 1 , but its quasi-derivative v = p y ′ is a nonzero constant on J 1 . Such an interval of zero is counted as a single zero, which is given in [2].

Definition 1

A solution y is called a trivial solution of equation (2.1) on some interval, if y is identically zero and its quasi-derivative v = p y ′ is also zero on this interval.

Lemma 1

Let (2.5) hold and let Φ ( x , λ ) = [ φ s t ( x , λ ) ] denote the fundamental matrix of system (2.6) determined by the initial condition Φ ( a , λ ) = I , I is the identity matrix. Then, a complex number λ is an eigenvalue of the Sturm-Liouville problem with n transmission conditions (2.1)–(2.3) if and only if

(2.7) Δ ( λ ) = det [ A λ + B λ Φ ( b , λ ) ] = 0 .

Moreover, it can be written as follows:

(2.8) Δ ( λ ) = det ( A λ ) + det ( B λ ) + h 11 ( λ ) φ 11 ( b , λ ) + h 12 ( λ ) φ 12 ( b , λ ) + h 21 ( λ ) φ 21 ( b , λ ) + h 22 ( λ ) φ 22 ( b , λ ) ,

where H ( λ ) = h 11 ( λ ) h 12 ( λ ) h 21 ( λ ) h 22 ( λ ) , with

h 11 ( λ ) = ( α 2 ′ λ 2 + β 2 ′ λ + γ 2 ′ ) ( α 3 λ 2 + β 3 λ + γ 3 ) − ( α 1 ′ λ 2 + β 1 ′ λ + γ 1 ′ ) ( α 4 λ 2 + β 4 λ + γ 4 ) , h 12 ( λ ) = ( α 1 λ 2 + β 1 λ + γ 1 ) ( α 4 λ 2 + β 4 λ + γ 4 ) − ( α 2 λ 2 + β 2 λ + γ 2 ) ( α 3 λ 2 + β 3 λ + γ 3 ) , h 21 ( λ ) = ( α 2 ′ λ 2 + β 2 ′ λ + γ 2 ′ ) ( α 3 ′ λ 2 + β 3 ′ λ + γ 3 ′ ) − ( α 1 ′ λ 2 + β 1 ′ λ + γ 1 ′ ) ( α 4 ′ λ 2 + β 4 ′ λ + γ 4 ′ ) , h 22 ( λ ) = ( α 1 λ 2 + β 1 λ + γ 1 ) ( α 4 ′ λ 2 + β 4 ′ λ + γ 4 ′ ) − ( α 2 λ 2 + β 2 λ + γ 2 ) ( α 3 ′ λ 2 + β 3 ′ λ + γ 3 ′ ) .

Proof

The proof of the first part of this lemma is similar to the one in [8], and the second part comes from a straightforward computation.□

Definition 2

The Sturm-Liouville problem with n transmission conditions and boundary conditions depending quadratically on the eigenparameter (2.1)–(2.3), or equivalently (2.2), (2.3), and (2.6), is said to be degenerate if in (2.8) either Δ ( λ ) ≡ 0 for all λ ∈ C or Δ ( λ ) ≠ 0 for any λ ∈ C .

3 The main conclusion

In this section, we assume (2.4) holds and there exists a partition of interval J

(3.1) a = a 0 1 < a 1 1 < a 2 1 < ⋯ < a 2 m 1 1 < a 2 m 1 + 1 1 = c 1 , on [ a , c 1 ) , c 1 = a 0 2 < a 1 2 < a 2 2 < ⋯ < a 2 m 2 2 < a 2 m 2 + 1 2 = c 2 , on ( c 1 , c 2 ) , … … … … … … c n = a 0 n + 1 < a 1 n + 1 < a 2 n + 1 < ⋯ < a 2 m n + 1 n + 1 < a 2 m n + 1 + 1 n + 1 = b , on ( c n , b ] ,

for n and some integers m i , i = 1 , 2 , … , n + 1 , such that

(3.2) r = 0 on ( a 2 k i , a 2 k + 1 i ) , ∫ a 2 k i a 2 k + 1 i w ≠ 0 , k = 0 , 1 , … , m i , i = 1 , 2 , … , n + 1

and

(3.3) q = w = 0 on ( a 2 k + 1 i , a 2 k + 2 i ) , ∫ a 2 k + 1 i a 2 k + 2 i r ≠ 0 , k = 0 , 1 , … , m i − 1 , i = 1 , 2 , … , n + 1 .

Next, we let

(3.4) r k i = ∫ a 2 k + 1 i a 2 k + 2 i r ( x ) d x , k = 0 , 1 , … , m i − 1 , i = 1 , 2 , … , n + 1 ; q k i = ∫ a 2 k i a 2 k + 1 i q ( x ) d x , k = 0 , 1 , … , m i , i = 1 , 2 , … , n + 1 ; w k i = ∫ a 2 k i a 2 k + 1 i w ( x ) d x , k = 0 , 1 , … , m i , i = 1 , 2 , … , n + 1 .

Lemma 2

Let (2.5) and (3.1)–(3.3) hold. Let Φ ( x , λ ) = [ φ i j ( x , λ ) ] be the fundamental matrix of the system (2.6) determined by the initial condition Φ ( a , λ ) = I for each λ ∈ C . Then, we have that

(3.5) Φ ( a 1 1 , λ ) = 1 0 q 0 1 − λ w 0 1 1 Φ ( a , λ ) ,

(3.6) Φ ( a 3 1 , λ ) = 1 + ( q 0 1 − λ w 0 1 ) r 0 1 r 0 1 φ 21 1 ( a 3 1 , λ ) 1 + ( q 1 1 − λ w 1 1 ) r 0 1 ,

where

φ 21 1 ( a 3 1 , λ ) = ( q 0 1 − λ w 0 1 ) + ( q 1 1 − λ w 1 1 ) + ( q 0 1 − λ w 0 1 ) ( q 1 1 − λ w 1 1 ) r 0 1 .

In general, for 1 ≤ k ≤ m 1 ,

(3.7) Φ ( a 2 k + 1 1 , λ ) = 1 r k − 1 1 q k 1 − λ w k 1 1 + ( q k 1 − λ w k 1 ) r k − 1 1 Φ ( a 2 k − 1 1 , λ ) .

Proof

Observe from (2.6) that u is constant on each subinterval, where r is identically zero and v is constant on each subinterval, where both q and w are identically zero. The result follows from repeated applications of (2.6).□

Lemma 3

Let (2.5) and (3.1)–(3.3) hold. Let Φ i ( x , λ ) = [ φ s t i ( x , λ ) ] ( x ∈ ( c i , c i + 1 ) , c n + 1 = b = a 2 m n + 1 + 1 n + 1 ) be the fundamental matrix solution of system (2.6) determined by the initial condition Φ i ( c i + , λ ) = I (here Φ i ( c i + , λ ) = Φ i ( a 0 i + 1 , λ ) = Φ ( c i + , λ ) , i = 1 , 2 , … , n ) denote the right limit at point c i for each λ ∈ C . Then, we have that

(3.8) Φ i ( a 1 i + 1 , λ ) = 1 0 q 0 i + 1 − λ w 0 i + 1 1 Φ i ( a 0 i + 1 , λ ) ,

(3.9) Φ i ( a 3 i + 1 , λ ) = 1 + ( q 0 i + 1 − λ w 0 i + 1 ) r 0 i + 1 r 0 i + 1 φ 21 i + 1 ( a 3 i + 1 , λ ) 1 + ( q 1 i + 1 − λ w 1 i + 1 ) r 0 i + 1 ,

where

φ 21 i + 1 ( a 3 i + 1 , λ ) = ( q 0 i + 1 − λ w 0 i + 1 ) + ( q 1 i + 1 − λ w 1 i + 1 ) + ( q 0 i + 1 − λ w 0 i + 1 ) ( q 1 i + 1 − λ w 1 i + 1 ) r 0 i + 1 .

In general, for 1 ≤ k ≤ m i + 1 ,

(3.10) Φ i ( a 2 k + 1 i + 1 , λ ) = 1 r k − 1 i + 1 q k i + 1 − λ w k i + 1 1 + ( q k i + 1 − λ w k i + 1 ) r k − 1 i + 1 Φ i ( a 2 k − 1 i + 1 , λ ) .

Proof

The proof is similar to the one as in Lemma 2.□

Lemma 4

Let (2.5) and (3.1)–(3.3) hold. Let Φ ( x , λ ) = [ φ s t ( x , λ ) ] be the fundamental matrix solution of the system (2.6) determined by the initial condition Φ ( a , λ ) = I for each λ ∈ C , and Φ i ( x , λ ) = [ φ s t i ( x , λ ) ] be given as in Lemma 3. Then, we have that

(3.11) Φ ( b , λ ) = Φ n ( b , λ ) G n Φ n − 1 ( c n , λ ) G n − 1 Φ n − 2 ( c n − 1 , λ ) ⋯ G 1 Φ ( c 1 , λ ) ,

where

G i = ( g s t ) 2 × 2 = − D i − 1 C i

and

Φ ( c 1 , λ ) = Φ ( c 1 − , λ ) = Φ ( a 2 m 1 + 1 1 , λ ) , Φ i ( c i + 1 , λ ) = Φ i ( c i + 1 − , λ ) = Φ i ( a 2 m i + 1 + 1 i + 1 , λ ) = Φ ( c i + 1 , λ ) , ( i = 1 , 2 , … , n − 1 ) , Φ n ( b , λ ) = Φ n ( a 2 m n + 1 + 1 n + 1 , λ )

denote the left limit at point c i ( i = 1 , 2 , … , n ) .

Proof

From the transmission conditions (2.3), we know that C i Φ ( c i − , λ ) + D i Φ ( c i + , λ ) = 0 ; thus, Φ ( c i + , λ ) = − D i − 1 C i Φ ( c i − , λ ) . By using Lemma 4 in [16], when i = 1 , in ( a , c 1 ) ∪ ( c 1 , c 2 ) , we have that Φ ( c 2 , λ ) = Φ 1 ( c 2 , λ ) G 1 Φ ( c 1 , λ ) , when i = 2 , we have that in ( a , c 1 ) ∪ ( c 1 , c 2 ) ∪ ( c 2 , c 3 ) , Φ ( c 3 , λ ) = Φ 2 ( c 3 , λ ) G 2 Φ 1 ( c 2 , λ ) G 1 Φ ( c 1 , λ ) . By repeated application of Lemmas 2–4 in [16], it can be concluded that (3.11) follows.□

Note that c i = a 2 m i + 1 i = a 0 i + 1 , b = a 2 m n + 1 + 1 n + 1 and (3.11). Then, the structure of the fundamental matrix solution Φ ( b , λ ) is given in Lemmas 2 and 3 and mathematical induction yields the following.

Corollary 1

If g 12 i ≠ 0 , i = 1 , 2 , … , n , then for the fundamental matrix Φ ( b , λ ) , we have that

(3.12) φ 11 ( b , λ ) = G 0 n ⋅ ∏ i = 1 n − 1 g 12 i ⋅ ∏ i = 1 n + 1 R i ⋅ ∏ i = 1 m n + 1 − 1 ⋅ ∏ i = 0 m n − 1 ⋅ ∏ i = 0 m n − 1 ⋯ ∏ i = 0 m 2 ⋅ ∏ i = 0 m 1 ( q 0 1 − λ ω 0 1 ) + φ ˜ 11 ( λ ) ,

(3.13) φ 12 ( b , λ ) = G 0 n ⋅ ∏ i = 1 n − 1 g 12 i ⋅ ∏ i = 1 n + 1 R i ⋅ ∏ i = 1 m n + 1 − 1 ⋅ ∏ i = 0 m n − 1 ⋅ ∏ i = 0 m n − 1 ⋯ ∏ i = 0 m 2 ⋅ ∏ i = 1 m 1 ( q 1 1 − λ ω 1 1 ) + φ ˜ 12 ( λ ) ,

(3.14) φ 21 ( b , λ ) = G 0 n ⋅ ∏ i = 1 n − 1 g 12 i ⋅ ∏ i = 1 n + 1 R i ⋅ ∏ i = 1 m n + 1 ⋅ ∏ i = 0 m n − 1 ⋅ ∏ i = 0 m n − 1 ⋯ ∏ i = 0 m 2 ⋅ ∏ i = 0 m 1 ( q 0 1 − λ ω 0 1 ) + φ ˜ 21 ( λ ) ,

(3.15) φ 22 ( b , λ ) = G 0 n ⋅ ∏ i = 1 n − 1 g 12 i ⋅ ∏ i = 1 n + 1 R i ⋅ ∏ i = 1 m n + 1 ⋅ ∏ i = 0 m n − 1 ⋅ ∏ i = 0 m n − 1 ⋯ ∏ i = 0 m 2 ⋅ ∏ i = 1 m 1 ( q 1 1 − λ ω 1 1 ) + φ ˜ 22 ( λ ) ,

where

G 0 n = [ g 12 n ( q m n n − λ ω m n n ) ( q 0 n + 1 − λ ω 0 n + 1 ) + g 11 n ( q 0 n + 1 − λ ω 0 n + 1 ) + g 22 n ( q m n n − λ ω m n n ) + g 21 n ] , R i = ∏ k = 0 m i − 1 r k i , i = 1 , 2 , … , n + 1 , ∏ i = 1 m n + 1 − 1 = ∏ i = 1 m n + 1 − 1 ( q i n + 1 − λ ω i n + 1 ) , ∏ i = 0 m n − 1 = ∏ i = 0 m n − 1 ( q i n − λ ω i n ) , ∏ i = 0 m k = ∏ i = 0 m k ( q i k − λ ω i k ) , k = 2 , … , n − 1 ,

φ ˜ s t ( λ ) = o ( ∏ i = 1 n + 1 R i ) , s , t = 1 , 2 as min { r k i } → ∞ for fixed q , ω , and λ .

From Corollary 1, we can see that each of the entries in Φ (i.e., φ s t , s , t = 1 , 2 ) is a polynomial of λ .

In the next, we construct regular Sturm-Liouville problems with n transmission conditions (2.3) with general self-adjoint and non-self-adjoint boundary conditions (2.2), which have exactly m eigenvalues for each m ∈ N .

Theorem 1

Let m i ∈ N ( i = 1 , 2 , … , n + 1 ) , g 12 i ≠ 0 , i = 1 , 2 , … , n , and let (2.5) and (3.1)–(3.3) hold. Let H = ( h s t ) 2 × 2 be defined as in Lemma 1. Then:

  1. If α 2 ′ α 3 ′ − α 1 ′ α 4 ′ ≠ 0 in h 21 ( λ ) , then the Sturm-Liouville problem with n transmission conditions (2.1)–(2.3) has at most ∑ i = 1 n + 1 m i + n + 5 eigenvalues.

  2. If α 2 ′ α 3 ′ − α 1 ′ α 4 ′ = 0 and α 2 ′ β 3 ′ + α 3 ′ β 2 ′ − α 1 ′ β 4 ′ − α 4 ′ β 1 ′ ≠ 0 in h 21 ( λ ) , or α 2 ′ α 3 − α 1 ′ α 4 ≠ 0 in h 11 ( λ ) , or α 4 ′ α 1 − α 3 ′ α 2 ≠ 0 in h 22 ( λ ) , then the Sturm-Liouville problem with n transmission conditions (2.1)–(2.3) has at most ∑ i = 1 n + 1 m i + n + 4 eigenvalues.

  3. If α 2 ′ α 3 ′ − α 1 ′ α 4 ′ = 0 , α 2 ′ β 3 ′ + α 3 ′ β 2 ′ − α 1 ′ β 4 ′ − α 4 ′ β 1 ′ = 0 and α 2 ′ γ 3 ′ + β 2 ′ β 3 ′ + α 3 ′ γ 2 ′ − α 1 ′ γ 4 ′ − β 1 ′ β 4 ′ − α 4 ′ γ 1 ′ ≠ 0 in h 21 ( λ ) , or α 2 ′ α 3 − α 1 ′ α 4 = 0 and α 2 ′ β 3 + α 3 β 2 ′ − α 1 ′ β 4 − α 4 β 1 ′ ≠ 0 in h 11 ( λ ) , or α 4 ′ α 1 − α 3 ′ α 2 = 0 and α 1 β 4 ′ + α 4 ′ β 1 − α 2 β 3 ′ − α 3 ′ β 2 ≠ 0 in h 22 ( λ ) , or α 1 α 4 − α 2 α 3 ≠ 0 in h 12 ( λ ) , then the Sturm-Liouville problem with n transmission conditions (2.1)–(2.3) has at most ∑ i = 1 n + 1 m i + n + 3 eigenvalues.

  4. If none of the aforementioned conditions holds, then the Sturm-Liouville problem with n transmission conditions (2.1)–(2.3) either has l eigenvalues for l ∈ 1 , 2 , … , ∑ i = 1 n + 1 m i + n + 2 or is degenerate.

Proof

We prove case 1, and other cases can be proved in the same way. We note from (3.2) that the degrees of φ 11 ( b , λ ) , φ 12 ( b , λ ) , φ 21 ( b , λ ) , φ 22 ( b , λ ) in λ are ∑ i = 1 n + 1 m i + n , ∑ i = 1 n + 1 m i + n − 1 , ∑ i = 1 n + 1 m i + n + 1 , ∑ i = 1 n + 1 m i + n , respectively. Thus, from Lemma 1 and Corollary 1, we get that maximum degree of Δ ( λ ) can be ∑ i = 1 n + 1 m i + n + 5 if α 2 ′ α 3 ′ − α 1 ′ α 4 ′ ≠ 0 in h 21 ( λ ) . Here, from fundamental theorem of algebra, one gets that Δ ( λ ) has at most ∑ i = 1 n + 1 m i + n + 5 roots, i.e., the Sturm-Liouville problem (2.1)–(2.3) has at most ∑ i = 1 n + 1 m i + n + 5 eigenvalues. Then, case 1 of this theorem is proved, and other cases can be proved in the similar way.□

Remark 3

If the rank of the matrix in (2.2) is one, then it turns into the ordinary constants, and it is the result of [22].

Theorem 2

Let m i ∈ N ( i = 1 , 2 , … , n + 1 ) , g 12 n = 0 , but g 11 n ω 0 n + 1 + g 22 n ω m n n ≠ 0 , g 12 i ≠ 0 , i = 1 , 2 , … , n − 1 , and let (2.4) and (3.1)–(3.3) hold. Let H = ( h s t ) 2 × 2 be defined as in Lemma 1. Then:

  1. If α 2 ′ α 3 ′ − α 1 ′ α 4 ′ ≠ 0 in h 21 ( λ ) , then the Sturm-Liouville problem with n transmission conditions (2.1)–(2.3) has at most ∑ i = 1 n + 1 m i + n + 4 eigenvalues.

  2. If α 2 ′ α 3 ′ − α 1 ′ α 4 ′ = 0 and α 2 ′ β 3 ′ + α 3 ′ β 2 ′ − α 1 ′ β 4 ′ − α 4 ′ β 1 ′ ≠ 0 in h 21 ( λ ) , or α 2 ′ α 3 − α 1 ′ α 4 ≠ 0 in h 11 ( λ ) , or α 4 ′ α 1 − α 3 ′ α 2 ≠ 0 in h 22 ( λ ) , then the Sturm-Liouville problem with n transmission conditions (2.1)–(2.3) has at most ∑ i = 1 n + 1 m i + n + 3 eigenvalues.

  3. If α 2 ′ α 3 ′ − α 1 ′ α 4 ′ = 0 , α 2 ′ β 3 ′ + α 3 ′ β 2 ′ − α 1 ′ β 4 ′ − α 4 ′ β 1 ′ = 0 and α 2 ′ γ 3 ′ + β 2 ′ β 3 ′ + α 3 ′ γ 2 ′ − α 1 ′ γ 4 ′ − β 1 ′ β 4 ′ − α 4 ′ γ 1 ′ ≠ 0 in h 21 ( λ ) , or α 2 ′ α 3 − α 1 ′ α 4 = 0 and α 2 ′ β 3 + α 3 β 2 ′ − α 1 ′ β 4 − α 4 β 1 ′ ≠ 0 in h 11 ( λ ) , or α 4 ′ α 1 − α 3 ′ α 2 = 0 and α 1 β 4 ′ + α 4 ′ β 1 − α 2 β 3 ′ − α 3 ′ β 2 ≠ 0 in h 22 ( λ ) , or α 1 α 4 − α 2 α 3 ≠ 0 in h 12 ( λ ) , then the Sturm-Liouville problem with n transmission conditions (2.1)–(2.3) has at most ∑ i = 1 n + 1 m i + n + 2 eigenvalues.

  4. If none of the aforementioned conditions holds, then the Sturm-Liouville problem with n transmission conditions (2.1)–(2.3) either has l eigenvalues for l ∈ 1 , 2 , … , ∑ i = 1 n + 1 m i + n + 1 or is degenerate.

Proof

The proof is similar to Theorem 1 only by noting that g 12 n ≠ 0 , but g 11 n ω 0 n + 1 + g 22 n ω m n n ≠ 0 , and the degree of λ will decrease by one and hence is omitted here.□

Remark 4

If α i = β i = α i ′ = β i ′ = 0 ( i = 1 , … , 4 ) , the Sturm-Liouville problems (2.1)–(2.3) will reduce to the finite spectrum of Sturm-Liouville problems with n transmission conditions (see [22]).

Remark 5

If α i = α i ′ = 0 ( i = 1 , … , 4 ) and β 2 ′ β 3 ′ − β 1 ′ β 4 ′ ≠ 0 , the Sturm-Liouville problems (2.1)–(2.3) will reduce to the Sturm-Liouville problems with n transmission conditions and spectral parameter boundary conditions of first-degree polynomials. Similar to the method of [18], we can prove that the problems have at most ∑ i = 1 n + 1 m i + n + 3 eigenvalues.

Finally, we give examples to illustrate our main results.

Example 1

Let n = 2 and consider the Sturm-Liouville problem with two transmission conditions

(3.16) − ( p y ′ ) ′ + q y = λ w y , on J ∈ ( − 5 , − 2 ) ∪ ( − 2 , 1 ) ∪ ( 1 , 6 ) , A λ Y ( − 5 ) + B λ Y ( 6 ) = 0 , − 2 ( p y ′ ) ( − 2 − ) + y ( − 2 + ) = 0 , y ( − 2 − ) + ( p y ′ ) ( − 2 + ) = 0 , 2 y ( 1 − ) − ( p y ′ ) ( 1 + ) = 0 , ( p y ′ ) ( 1 − ) + y ( 1 + ) = 0 ,

where

A λ = 1 2 λ 2 0 λ 2 + 2 λ , B λ = 1 λ 2 + λ 0 λ 2 .

Choose m 1 = 1 , m 2 = 1 , m 3 = 2 , and p , q , w are piecewise constant functions, which are defined as follows:

(3.17) p ( x ) = ∞ , ( − 5 , − 4 ) , 1 , ( − 4 , − 3 ) , ∞ , ( − 3 , − 2 ) , ∞ , ( − 2 , − 1 ) , 1 2 , ( − 1 , 0 ) , ∞ , ( 0 , 1 ) , ∞ , ( 1 , 2 ) , 1 , ( 2 , 3 ) , ∞ , ( 3 , 4 ) , 1 2 , ( 4 , 5 ) , ∞ , ( 5 , 6 ) , q ( x ) = 0 , ( − 5 , − 4 ) , 0 , ( − 4 , − 3 ) , 1 , ( − 3 , − 2 ) , 2 , ( − 2 , − 1 ) , 0 , ( − 1 , 0 ) , − 1 , ( 0 , 1 ) , 1 , ( 1 , 2 ) , 0 , ( 2 , 3 ) , 3 , ( 3 , 4 ) , 0 , ( 4 , 5 ) , 1 , ( 5 , 6 ) , w ( x ) = 1 , ( − 5 , − 4 ) , 0 , ( − 4 , − 3 ) , 1 , ( − 3 , − 2 ) , 1 , ( − 2 , − 1 ) , 0 , ( − 1 , 0 ) , 1 , ( 0 , 1 ) , 1 , ( 1 , 2 ) , 0 , ( 2 , 3 ) , 1 , ( 3 , 4 ) , 0 , ( 4 , 5 ) , 1 , ( 5 , 6 ) .

From (3.16), we have

C 1 = 0 − 2 1 0 , D 1 = 1 0 0 1 , C 2 = 2 0 0 1 , D 2 = 0 − 1 1 0

and

det ( C 1 ) = 2 > 0 , det ( D 1 ) = 1 > 0 , det ( C 2 ) = 2 > 0 , det ( D 2 ) = 1 > 0 ,

G 1 = − D 1 − 1 C 1 = 0 2 − 1 0 , g 12 1 = 2 ≠ 0 , G 2 = − D 2 − 1 C 2 = 0 1 − 2 0 , g 12 2 = 1 ≠ 0 .

By deduction, it can be obtained that the characteristic function

Δ ( λ ) = λ 2 + 2 λ + λ 2 + ( λ 2 + 2 λ ) φ 11 ( b , λ ) + ( − λ 4 + 3 λ 3 + 2 λ 2 ) φ 21 ( b , λ ) + λ 2 φ 22 ( b , λ ) = 8 λ 11 − 128 λ 10 + 780 λ 9 − 2134 λ 8 + 1792 λ 7 + 3107 λ 6 − 6303 λ 5 + 636 λ 4 + 3171 λ 3 + 109 λ 2 − 446 λ .

Hence, the Sturm-Liouville problem (3.16), (3.17) has exactly m 1 + m 2 + m 3 + n + 5 = 11 eigenvalues:

λ 0 = 0 , λ 1 = 4.9561 , λ 2 = 1.8806 , λ 3 = 0.3886 , λ 4 = 2.072 , λ 5 = − 1.2265 , λ 6 = 2.4901 , λ 7 = − 0.461 − 0.1783 i , λ 8 = 3.2205 − 0.1718 i , λ 9 = − 0.461 + 0.1783 i , λ 10 = 3.2205 + 0.1718 i .

Example 2

Let n = 2 and consider the Sturm-Liouville problem with two transmission conditions:

(3.18) − ( p y ′ ) ′ + q y = λ w y , on J ∈ ( − 5 , − 2 ) ∪ ( − 2 , 1 ) ∪ ( 1 , 6 ) , A λ Y ( − 5 ) + B λ Y ( 6 ) = 0 , − 2 ( p y ′ ) ( − 2 − ) + y ( − 2 + ) = 0 , y ( − 2 − ) + ( p y ′ ) ( − 2 + ) = 0 , 2 y ( 1 − ) − ( p y ′ ) ( 1 + ) = 0 , ( p y ′ ) ( 1 − ) + y ( 1 + ) = 0 ,

where

A λ = 1 2 λ 2 0 λ 2 , B λ = λ 2 λ 2 0 λ 2 .

Choose m 1 = 1 , m 2 = 1 , m 3 = 2 , and p , q , w are defined as in Example 1. Then, we get the characteristic function:

Δ ( λ ) = λ 2 + λ 3 + λ 3 φ 11 ( b , λ ) + λ 2 φ 22 ( b , λ ) = 8 λ 9 − 84 λ 8 + 252 λ 7 − 2 λ 6 − 947 λ 5 + 713 λ 4 + 878 λ 3 − 720 λ 2 .

Hence, the Sturm-Liouville problem (3.17), (3.18) has exactly m 1 + m 2 + m 3 + n + 3 = 9 eigenvalues:

λ 0 = 0 , λ 1 = 0 , λ 2 = 4.907 , λ 3 = 0.7737 , λ 4 = − 1.227 , λ 5 = 2.2893 , λ 6 = 1.9257 , λ 7 = 3.2005 , λ 8 = − 1.3692 .

Figures 1 and 2 show the graph of the Δ ( λ ) in Examples 1 and 2, respectively (the graph of complex eigenvalues is not shown here).

Figure 1 Characteristic function in Example 1.
Figure 1

Characteristic function in Example 1.

Figure 2 Characteristic function in Example 2.
Figure 2

Characteristic function in Example 2.

Acknowledgments

The authors are sincerely grateful to the anonymous referees for their valuable comments and suggestions, which greatly improved the quality of this article.

  1. Funding information: This project was supported by the National Natural Science Foundation of China (No. 11801286, No. 11561050), Natural Science Foundation of Inner Mongolia (No. 2018MS01021), and the Natural Science Foundation of Shandong Province (No. ZR2020QA009).

  2. Author contributions: Jia Li, Xiaoling Hao, Kun Li, and Siqin Yao contributed to the design of the article, the analysis of the results, the drawing of the diagrams, and the writing of the manuscript. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Not applicable.

References

[1] J. Ao and J. Sun , Matrix representations of Sturm-Liouville problems with coupled eigenparameter-dependent boundary conditions, Appl. Math. Comput. 244 (2014), 142–148, https://doi.org/10.1016/j.amc.2014.06.096. Search in Google Scholar

[2] W. J. Code and P. J. Browne , Sturm-Liouville problems with boundary conditions depending quadratically on the eigenparameter, J. Math. Anal. Appl. 309 (2005), no. 2, 729–742, https://doi.org/10.1016/j.jmaa.2004.11.067. Search in Google Scholar

[3] C. T. Fulton , Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 77A (1977), no. 3–4, 293–308, https://doi.org/10.1017/S030821050002521X . 10.1017/S030821050002521XSearch in Google Scholar

[4] C. T. Fulton and S. Pruess , Numerical methods for a singular eigenvalue problem with eigenparameter in the boundary conditions, J. Math. Anal. Appl. 71 (1979), no. 2, 431–462, https://doi.org/10.1016/0022-247X(79)90203-8 . 10.1016/0022-247X(79)90203-8Search in Google Scholar

[5] I. Podlubny , Fractional Differential Equations, Academic Press, London, 1999. Search in Google Scholar

[6] B. J. West and M. Bologna , Physics of Fractal Operators, Springer Verlag, New York, 2003. 10.1007/978-0-387-21746-8Search in Google Scholar

[7] B. J. West , Fractional calculus in bioengineering, J. Stat. Phys. 126 (2007), no. 6, 1285–1286, https://doi.org/10.1007/s10955-007-9294-0 . 10.1007/s10955-007-9294-0Search in Google Scholar

[8] D. Baleanu and E. Uğurlu , Regular fractional dissipative boundary value problems, Adv. Differ. Equ. 2016 (2016), 175, https://doi.org/10.1186/s13662-016-0883-6. Search in Google Scholar

[9] E. Uğurlu , D. Baleanu , and K. Taş , On square integrable solutions of a fractional differential equation, Appl. Math. Comput. 337 (2018), 153–157, https://doi.org/10.1016/j.amc.2018.05.014. Search in Google Scholar

[10] J. Walter , Regular eigenvalue problems with eigenvalue parameter in the boundary condition, Math. Z. 133 (1973), no. 4, 301–312, https://doi.org/10.1007/BF01177870. Search in Google Scholar

[11] P. A. Binding , P. J. Browne , and B. A. Watson , Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter, II, J. Comput. Appl. Math. 148 (2002), no. 1, 147–168, https://doi.org/10.1016/S0377-0427(02)00579-4 . 10.1016/S0377-0427(02)00579-4Search in Google Scholar

[12] G. Freiling and V. A. Yurko , Inverse problems for Sturm-Liouville equations with boundary conditions polynomially dependent on the spectral parameter, Inverse Problems 26 (2010), no. 5, 055003, https://doi.org/10.1088/0266-5611/26/5/055003. Search in Google Scholar

[13] F. V. Atkinson , Discrete and Continuous Boundary Problems, Academic Press, New York, 1964. 10.1063/1.3051875Search in Google Scholar

[14] Q. Kong , H. Wu , and A. Zettl , Sturm-Liouville problems with finite spectrum, J. Math. Anal. Appl. 263 (2001), no. 2, 748–762, https://doi.org/10.1006/jmaa.2001.7661. Search in Google Scholar

[15] Q. Kong , H. Volkmer , and A. Zettl , Matrix representations of Sturm-Liouville problems with finite spectrum, Results Math. 54 (2009), no. 1–2, 103–116, https://doi.org/10.1007/s00025-009-0371-3. Search in Google Scholar

[16] J. Ao , J. Sun , and M. Zhang , The finite spectrum of Sturm-Liouville problems with transmission conditions, Appl. Math. Comput. 218 (2011), no. 4, 1166–1173, https://doi.org/10.1016/j.amc.2011.05.033. Search in Google Scholar

[17] J. Ao , F. Bo , and J. Sun , Fourth order boundary value problems with finite spectrum, Appl. Math. Comput. 244 (2014), 952–958, https://doi.org/10.1016/j.amc.2014.07.054. Search in Google Scholar

[18] J. Ao , J. Sun , and M. Zhang , The finite spectrum of Sturm-Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions, Results Math. 63 (2013), no. 3, 1057–1070, https://doi.org/10.1007/s00025-012-0252-z . 10.1007/s00025-012-0252-zSearch in Google Scholar

[19] J. Ao , J. Sun , and A. Zettl , Finite spectrum of 2nth-order boundary value problems, Appl. Math. Lett. 42 (2015), 1–8, https://doi.org/10.1016/j.aml.2014.10.003. Search in Google Scholar

[20] J. Ao , J. Sun , and M. Zhang , Matrix representations of Sturm-Liouville problems with transmission conditions, Comput. Math. Appl. 63 (2012), no. 8, 1335–1348, https://doi.org/10.1016/j.camwa.2012.01.051. Search in Google Scholar

[21] J. Ao , J. Sun , and A. Zettl , Matrix representations of fourth order boundary value problems with finite spectrum, Linear Algebra Appl. 436 (2012), no. 7, 2359–2365, https://doi.org/10.1016/j.laa.2011.10.001. Search in Google Scholar

[22] M. Xu , W. Wang , and J. Ao , Finite spectrum of Sturm-Liouville problems with n transmission conditions, Iran. J. Sci. Technol. Trans. A Sci. 42 (2018), no. 2, 811–817, https://doi.org/10.1007/s40995-016-0072-1. Search in Google Scholar

[23] J. Sun and A. Wang , Sturm-Liouville operators with interface conditions , The Progress of Research for Math., Mech., Phy. and High New Tech. , vol 12, Science Press, Beijing, 2008. Search in Google Scholar

[24] O. Sh. Mukhtarov , M. Kadakal , and F. S. Muhtarov , Eigenvalues and normolized eigenfunctions of discontinuous Sturm-Liouville problem with transmission conditions, Rep. Math. Phys. 54 (2004), no. 1, 41–56, https://doi.org/10.1016/S0034-4877(04)80004-1 . 10.1016/S0034-4877(04)80004-1Search in Google Scholar

[25] O. Sh. Mukhtarov and M. Kandemir , Asymptotic behavior of eigenvalues for the discontinuous boundary-value problem with functional-transmission conditions, Acta Math. Sci. Ser. B (Engl. Ed.) 22B (2002), no. 3, 335–345, https://doi.org/10.1016/S0252-9602(17)30303-X . 10.1016/S0252-9602(17)30303-XSearch in Google Scholar
Received: 2021-09-01
Revised: 2021-11-09
Accepted: 2021-12-05
Published Online: 2021-12-31

© 2021 Jia Li et al., 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-2021-0107
Keywords for this article
Sturm-Liouville problems; finite spectrum; eigenvalue; transmission conditions; quadratic eigenparameter-dependent
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  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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