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Relative oscillation theory for matrix Sturm-Liouville difference equations extended
Advances in Difference Equations volume 2013, Article number: 328 (2013)
Abstract
In this article we establish relative oscillation theorems for two discrete matrix Sturm-Liouville eigenvalue problems with Dirichlet boundary conditions and nonlinear dependence on the spectral parameter λ. This nonlinear dependence on λ is allowed both in the leading coefficients and in the potentials. Relative oscillation theory rather than measuring the spectrum of one single problem measures the difference between the spectra of two different problems. This is done by replacing focal points of conjoined bases of one problem by matrix analogs of weighted zeros of Wronskians of conjoined bases of two different problems.
MSC: 39A21, 39A12.
1 Introduction
We consider the discrete matrix Sturm-Liouville spectral problems
and
where , , , is the spectral parameter, and the real symmetric matrix-valued functions , , , , are differentiable in the variable λ and obey the conditions
Conditions (1.3) imply [1, 2] that problem (1.1) is a special case of the discrete symplectic eigenvalue problem
which depends in general nonlinearly on the spectral parameter (see [2]). Here is a differentiable (hence continuous) symplectic matrix-valued function of the variable λ, such that for , . By [2], the matrix-function is symmetric for all and (1.6) describes the monotonic behavior of with respect to λ. In the case of spectral problem (1.1) for , , , , the symplectic matrix in (1.5) has the form
and the monotonicity assumption in (1.6)
is obviously equivalent to (1.3).
The oscillation and spectral theory for symplectic difference systems with linear dependence on the spectral parameter was successfully developed in [3–5]. For the special case of the matrix Sturm-Liouville difference equations the assumption in [3–5] on the linear dependence on leads to the following restrictions:
on the coefficient matrices , , , in (1.1), (1.2). The so-called global oscillation theorem (see [3, 4]) applied to problem (1.1) with assumptions (1.8) relates the number of finite eigenvalues of (1.1) less than or equal to a given number to the number of focal points (counting multiplicity) of the principal solution of (1.1) with (see [1]). Relative oscillation theory adds new aspects to the classical oscillation results by showing that matrix analogs of weighted zeros [6–10] of the Wronskian for suitable solutions of (1.1), (1.2) can be used to count the difference between the numbers of finite eigenvalues of problems (1.1), (1.2). Recall now some results of relative oscillation theory developed in [6, 7, 10] for the scalar case of (1.1), (1.2), (1.8). Consider the Sturm-Liouville eigenvalue problems
Introduce the Wronskian for two solutions , of the difference equations in (1.9), (1.10): . According to [6], the Wronskian has a node at i (or a generalized zero [[11], p.719] in ) if either or , . Then, by [[6], Theorem 4.3], for problems (1.9), (1.10) with , and , , we have
where denotes the total number of nodes of in , denotes the number of eigenvalues of (1.9) (or (1.10)) between and and the solutions , , of (1.9), (1.10) obey the conditions , at .
The main result in [7] extends (1.11) for the case . According to [[7], Theorem 1.2], the number of weighted nodes of the Wronskian in equals the number of eigenvalues of (1.10) below minus the number of eigenvalues of (1.9) below or equal to :
In the recent paper [10], the previous result is generalized for the case , . Note that relative oscillation theory for scalar spectral problems (1.1), (1.2) with nonlinear dependence on λ (see [12]) has never been developed before.
In [13–16] we derive relative oscillation theory for symplectic difference eigenvalue problems with linear dependence on λ. Note that results in [13, 14] cover the special case of the matrix Sturm-Liouville eigenvalue problem (1.1), (1.8) with the linear dependence on λ only under the additional assumption , . The relative oscillation theory which deals with the case is called extended (see [9, 10]). Results of this paper rely on the concept of finite eigenvalue of (1.5) and the global oscillation theorem which was recently proved in [2, 17] for symplectic eigenvalue problems (1.5) with nonlinear dependence on the spectral parameter λ. We combine these results with Theorem 2.1 in [13] presenting the relation between the numbers of focal points of conjoined bases of two discrete symplectic systems with different coefficient matrices. This opens the door for generalizing relative oscillation theory for the case of spectral problems (1.1), (1.3) and (1.2), (1.4).
The paper is organized as follows. In Section 2 we recall main concepts of oscillation theory of symplectic difference systems and the comparative index theory developed in [13, 18–20]. We introduce the relative oscillation numbers which generalize the concept of a weighted zero of the Wronskian for the matrix case. At the end of Section 2, we prove some properties of the relative oscillation numbers.
Section 3 is devoted to relative oscillation theory for problems (1.1), (1.2). In Section 3.1 we derive the relative oscillation numbers for the pair of symplectic difference systems associated with (1.1), (1.2) (see Theorem 3.4). We also investigate other representations of the relative oscillation numbers connected with different choices of symplectic difference systems associated with (1.1), (1.2) (see Section 3.2). The main theorems (see Theorems 3.8, 3.9) proved in Section 3.3 generalize (1.12), (1.11) for the case of matrix eigenvalue problems (1.1), (1.2).
In Section 4 we provide several examples illustrating the relative oscillation theory for scalar problems (1.1), (1.2) with nonlinear dependence on the spectral parameter.
2 Notation and auxiliary results
We will use the following notation. For a matrix A, we denote by , , , , rankA, indA, , , respectively, its transpose, inverse, transpose and inverse, Moore-Penrose pseudoinverse, rank (i.e., the dimension of its image), index (i.e., the number of its negative eigenvalues), positive semidefiniteness, negative semidefiniteness. By I and 0 we denote the identity and zero matrices of appropriate dimensions. We also use the notation for the product of matrices , where we put .
Introduce the symplectic difference systems
where .
Recall (see [11]) that matrix solutions , of (2.1), (2.2) are said to be conjoined bases if
and the conjoined bases , of (2.1), (2.2) with the initial conditions , at are said to be the principal solutions at M.
Note that for conjoined bases , of (2.1), (2.2) there exist symplectic fundamental matrices , such that
(see [[11], Remark 1(ii)]). Define the Wronskian
for conjoined bases of (2.1), (2.2).
Recall the definition of focal points and their multiplicities for conjoined bases of (2.1). We define the numbers of focal points of conjoined basis in , , respectively (see [[21], Definition 1], [[19], Definition 3.2]): , , where
In particular, according to [[1], Section 5], the number of focal points of a conjoined basis of matrix Sturm-Liouville equation (1.1) in the interval is defined to be the number
The main results of this paper are based on the comparative index theory established in [13, 18–20]. According to [19, 20], we define the comparative index for matrices Y, with conditions (2.3) using the notation
The comparative index is defined by , where and . Introduce the dual index , where .
For the comparative index , we have the estimate (see Property 7 in [[19], p.449])
where is the Wronskian given by (2.5).
If is a symplectic fundamental matrix for (2.1) connected with a conjoined basis (see (2.4)), then, according to [[19], Lemmas 3.1 and 3.2], we have
where , are the numbers of focal points in , , respectively.
In this paper we will use the comparative index which, according to the second formula in (2.10), presents the number of focal points in of the transformed conjoined basis (see [14]) with the upper block associated with the Wronskian (2.5). To emphasize the role of the Wronskian (2.5) in the relative oscillation results, we introduce the notation , where by (2.7), (2.10) we have
Note that according to (2.8), we have the inequality
because of the definition in (2.11).
Introduce the notation
for the numbers of focal points of in and .
For arbitrary matrices Y, with conditions (2.3) and symplectic matrices W, , we define the operator
The main result in [19] (see Theorem 2.2) establishes the equality and then
for the case .
When , are conjoined bases and , are the coefficient matrices of (2.1), (2.2), operator (2.13) takes the form
where, according to (2.9), , are the numbers of focal points in . In [13] we derive the general representations of (2.15) in terms of the comparative index for the coefficient matrices of (2.1), (2.2). For arbitrary symplectic matrix W separated into blocks A, B, C, D introduce the notation
In [[13], Lemma 2.3]) we prove that matrices , associated with symplectic W, obey (2.3) (with n replaced by 2n) and then the comparative index for the pair , is well defined. The results of this paper are based on the following comparison theorem proved in [[13], Theorem 2.1].
Theorem 2.1 Let , be conjoined bases of (2.1), (2.2) associated with symplectic fundamental matrices , such that (2.4) hold. Then
where
and then
where , are the numbers of focal points in .
The numbers in the right-hand side of (2.16) are called the relative oscillation numbers for the symplectic difference systems (2.1), (2.2). Note that the inequality
holds for the relative oscillation numbers in (2.16) (see the proof of Theorem 2.1 in [13]), and then for the case .
For the particular case when , are the principal solutions of (2.1), (2.2), we have the following corollary from Theorem 2.1 (see [[13], Corollary 2.4]).
Corollary 2.2 Let and be the principal solutions of (2.1) and (2.2) associated with symplectic fundamental matrices , such that conditions (2.4) hold. Then
where the relative oscillation numbers are defined by (2.17) for , .
Relative oscillation theory for symplectic eigenvalue problems with linear dependence on λ developed in [13, 14] is based on [[13], Lemma 2.2], [[14], Lemma 1] where we evaluate relative oscillation numbers (2.17) assuming that the following condition
holds for the matrices , in (2.1), (2.2) (here due to symplecticity of , ). Then, by [[14], Lemma 1], we have the following representation
provided (2.21) holds. In particular, for the scalar case of Sturm-Liouville equations (1.9), (1.10) with , relative oscillation numbers (2.22) take the values ±1 if and only if the Wronskian for solutions , of (1.9), (1.10) has a weighted node at i according to the definition in [7] (see [[14], Remark 1]).
Note that the symplectic matrices (1.7) associated with matrix Sturm-Liouville equation (1.1) for two arbitrary values and , obey condition (2.21) only for the case when does not depend on λ. For two spectral problems (1.1), (1.2) condition (2.21) is satisfied under the additional assumption . In the next section, using the special structure of symplectic matrices (1.7), we evaluate the relative oscillation numbers for problems (1.1), (1.2) for the general case , . In the proofs we will use the following ‘multiplicative’ property of operator (2.13).
Lemma 2.3 For arbitrary symplectic matrices and , where , define , , . Then, for any matrices Y, with conditions (2.3), we have
Proof By definition (2.13), in the left-hand side of (2.23) we have
Similarly, for the operator in the right-hand side of (2.23), we derive
and then
By (2.14), , then summing (2.25) from to , we derive (2.24). The proof is completed. □
Remark 2.4
-
(i)
Assume that , are conjoined bases and , are the principal solutions of (2.1), (2.2) at . If we apply Lemma 2.3 for the case , ,
then , , and , , . For the given case, equality (2.23) takes the form
(2.26)In particular, if systems (2.1), (2.2) are disconjugate in , i.e. , then
-
(ii)
Note that (2.26) gives us possibility to replace pointwise evaluation of operators (2.15) associated with the pairs , by computation of only one operator (2.13) associated with the products , . In this paper we apply Lemma 2.3 in the opposite direction. Assume that for any i the following factorizations , hold for the coefficient matrices of (2.1), (2.2) (here p and , can depend on i). Then for , we have , and Lemma 2.3 presents the action of operator (2.15) at the point i as a result of actions of p operators associated with the factors , , (see the proofs of Lemma 3.3 and Theorem 3.4 in the next section).
We will also need the following result which is based on (2.22). It is well known (see [11, 20, 22]) that symplectic transformations with lower block triangular matrices do not change the number of focal points. In particular, if we introduce the symplectic matrices
then , , and for operator (2.15) we have
Note that we can rewrite in the form , where , obey assumption (2.21). Then for we have the representation
where the Wronskian is given by (2.5) and is defined by (2.11) with . So we have proved the following lemma.
Lemma 2.5 The relative oscillation numbers and with , given by (2.27) are connected by the equation
where is presented by (2.29). In particular, for the case and , we have , and then, by Corollary 2.2,
3 Relative oscillation theory for matrix Sturm-Liouville difference equations
3.1 Relative oscillation numbers for matrix Sturm-Liouville difference equations
Consider the evaluation of relative oscillation numbers (2.17) for the special case of the matrix Sturm-Liouville equations. Introduce the conjoined bases of systems (2.1), (2.2) associated with (1.1), (1.2):
Note that the coefficients , are not needed in equations (1.1), (1.2), but for convenience we define them at such that (1.3), (1.4) hold. In Remark 3.2 we will show that the results of this section do not depend on the definition of , .
The Wronskian (2.5) of , for two fixed values and
obeys the equation
The symplectic coefficient matrices (1.7) associated with (1.1), (1.2) can be presented in the special factorized form
The main result of this section is based on the consideration of the following particular cases.
Lemma 3.1 (Case I)
Assume that for spectral problems (1.1), (1.3) and (1.2), (1.4), the matrices , obey the conditions
then relative oscillation numbers (2.17) have the form
where is given by (2.11) with and defined by (3.2).
Proof For case (3.5) we have that condition (2.21) is obviously satisfied for matrices (3.4):
and then applying (2.22) we derive (3.6). □
Remark 3.2 Note that in the definition of symplectic systems (1.7), (3.4) we can put and then . However, for the case when , we have for any choice of , . Indeed, for this case by (3.3), (3.2) and according to (2.11), we have .
Lemma 3.3 (Case II)
Assume that for spectral problems (1.1), (1.3) and (1.2), (1.4) the matrices , obey the conditions
then relative oscillation numbers (2.17) have the form
Here is defined by (2.11) where , and the number does not depend on a, b:
Proof Note that for case (3.8) matrices (3.4) obey the condition
The symplectic upper block triangular factors , in (3.11) can be represented in the form
where , are the symplectic lower block triangular matrices. Assumption (3.8) implies that in (3.4). Consider operator (2.15) for case (3.8). Applying Lemma 2.3 for , , , , , , , , we have
where the addends in the braces correspond to the last sum in (2.23). Taking into account that the right-hand side of operator (2.13) equals zero for (see (2.14)) and evaluating the difference according to the definition of the comparative index, we have
Recall that the symmetric nonsingular matrices , are continuous functions in λ and then their eigenvalues have the constant sign for . So we have , for any .
Note that the symplectic matrices , in the operator are unit lower block triangular and then they obey condition (2.21):
Evaluating according to (2.22), where should be replaced by , we derive (3.9) with given by (3.10). The proof is completed. □
Consider the evaluation of the relative oscillation numbers for the general case. Introduce the following Wronskian:
for , defined as in (3.1). Here we use the intermediate point for the convenient interpretation of subsequent results. Note that
and then summing (3.13), (3.14) we derive (3.3). In particular, if case I takes place (i.e., conditions (3.5) hold), we have by (3.13) and similarly, by (3.14), in case II.
Theorem 3.4 (General case)
For spectral problems (1.1), (1.3) and (1.2), (1.4) associated with symplectic matrices (3.4), relative oscillation numbers (2.17) have the form
where the numbers
are evaluated according to (3.9), (3.10) and (3.6), respectively, with for case II and for case I.
Proof For the proof we use factorizations (3.4) and Lemma 2.3. Putting in Lemma 2.3 , , , , , , , we derive
By (3.11), (3.7) operators and can be evaluated according to cases II and I, respectively. For case II, we have that the conjoined bases , obey the symplectic systems , , , and then we have to use the Wronskian given by (3.12) instead of . Similarly, in case I we use that , obey the symplectic systems , and then we apply (3.6) replacing by . Finally, we point out that such modifications of (3.9) and (3.6) do not touch the matrices , according to their definitions in (3.9) and (3.6). The proof is completed. □
Now we formulate some properties of the relative oscillation numbers given by (3.15), (3.16), (3.17).
Proposition 3.5
-
(i)
If case I takes place (i.e., conditions (3.5) hold), then in (3.15) we have for given by (3.6). Similarly, for case II, with given by (3.9).
-
(ii)
If the conditions
(3.18)hold, then the relative oscillation numbers given by (3.15), (3.16), (3.17) are nonnegative. In particular, for the case , , , the relative oscillation numbers are presented in the form
(3.19)where and are defined by (2.11) with and given by (3.9) and (3.6), respectively.
-
(iii)
For the relative oscillation numbers in (3.15), we have the estimate
(3.20)
Proof For the proof of (i), we use (3.13), (3.14). Case I implies (see (3.13)) that and the matrices , in (3.9), (3.10) equal zero. Then and . In a similar way, for case II, , and we get from (3.14) that . Finally, it follows that , .
For the proof of (ii) we note that under assumptions (3.18) numbers (3.17) are nonnegative because of . For the proof , we use the following index result:
which follows from (2.14) for the case , , . Recall that in the proof of Lemma 3.9 we used , according to monotonicity assumptions (1.3), (1.4). Then, by (3.21), , where we use that . Finally, we have because of
For the case , , , we additionally have and in (3.16) . Then the proof of (ii) is completed.
By (2.19), relative oscillation numbers (3.16), (3.17) obey the inequalities
and then for the relative oscillation numbers in (3.15) we have estimate (3.20). The proof is completed. □
3.2 Other representations of the relative oscillation numbers for matrix Sturm-Liouville difference equations
By expanding of the operators Δ, the matrix Sturm-Liouville equation in (1.1) (or (1.2)) can be rewritten as the three-term recurrence [1, 23]
In [[1], Theorem 3.1] the following result for equation (3.22) is proved. If we put
with an arbitrary symmetric matrix , then, according to [[1], Theorem 3.1], solves the symplectic difference system with coefficients depending on . Here we assume (see [1]) that , are defined so that conditions (1.3) hold.
The choice , leads to the symplectic system with matrix (1.7). The solution of this system is connected with in (3.23) by the following symplectic transformation:
The Wronskian for conjoined bases of the transformed systems also depends on the choice of , . So we have
where is given by (3.2).
In particular, the Wronskian in [10] (for the scalar problems (1.9), (1.10))
corresponds to the choice , in (3.23). Transformation (3.24) in this case leads to the following coefficients matrices:
where is defined in (3.22) and is defined similarly for (1.2). Formula (3.25) for , takes the form
It is easy to see that
where for the case . Slightly modifying the proof of Lemma 1 in [[14], p.1233] (see also [[13], Lemma 2.1]), we have the following representation of the relative oscillation numbers for the transformed systems with matrices (3.27):
Here , and is given by (3.26). The number is defined by (2.11) with , where
In particular, for case I (see Lemma 3.1), relative oscillation numbers (3.30) coincide with given by (3.6). In the general case, by Lemma 2.5, relative oscillation numbers (3.30) are connected with (3.15) by the formula
where is given by (2.29) with . For relative oscillation numbers (3.30), we have the following estimate (compare with (3.20)):
with given by (3.31). For the proof, we apply inequalities (2.8), (2.12) to the addends in the right-hand side of (3.30) such that , . By analogy with Remark 3.2, we can also show that for any choice of , , , .
However, we cannot guarantee that Proposition 3.5(ii) holds for relative oscillation numbers (3.30). In particular, for the scalar case of problems (1.1), (1.2), we show that (3.30) takes the value −1 for the case , , under the monotonicity assumptions in (1.3), (1.4) (see Example 4.2 in Section 4). Moreover, one can verify by direct computations that the monotonicity assumption (1.6) holds for (3.27) only if does not depend on λ. So we have
and the last condition is equivalent to
where we use Lemma 2.7 in [22] to evaluate the index of a symmetric matrix with zero diagonal block (see also index results in [24]).
3.3 Relative oscillation theorems
In this section we prove analogs of (1.12), (1.11) for the case of matrix eigenvalue problems (1.1), (1.2). Recall the notion of the finite eigenvalue introduced for (1.5) in [2].
Definition 3.6 Let be the principal solution of (1.5) at . The number is a finite eigenvalue of (1.5) if
where and is the multiplicity of .
The global oscillation theorem in [17] connects the number of the finite eigenvalues (including their multiplicities) of (1.5) with the number of focal points of the principal solution under the additional assumption , , where is the block of in the upper right corner (see [[17], Theorem 3.2]). The symplectic matrix (1.7) satisfies this condition, and then we can formulate the global oscillation theorem for the special case of problem (1.1).
Theorem 3.7 Assume (1.1), (1.3). Then the finite eigenvalues of (1.1) are isolated, bounded from below, and there exists such that for any
where is the number of finite eigenvalues of (1.1) in , is the number of focal points (2.6) of the principal solution in for , and
Using Corollary 2.2 and the connection (3.34) between the number of focal points of the principal solution and the number of finite eigenvalues we can easily prove the following main theorems.
Theorem 3.8 (Relative oscillation theorem for matrix Sturm-Liouville equations)
Let , be the finite spectra and , be the principal solutions of (1.1), (1.3) and (1.2), (1.4). Then there exists the constant such that for all , the following identity holds:
Here the relative oscillation numbers are defined by (3.15), (3.16), (3.17),
and , , .
Proof According to Theorem 3.7,
where, by (3.35) , . Then
By Corollary 2.2, we derive
for , given by (3.15), (3.16), (3.17). Substituting the last representations into (3.37), we complete the proof of (3.36). □
For the case , , , Theorem 3.8 presents the number of finite eigenvalues of (1.1) in .
Theorem 3.9 (Renormalized oscillation theorem)
For problem (1.1), (1.3) for , the following identity holds:
and , are defined by (2.11) with , and given by (3.9) and (3.6), respectively.
Proof For the case , , , we have in (3.36) that and . Applying Proposition 3.5(ii), we complete the proof of Theorem 3.9. □
Remark 3.10
-
(i)
In the definition of (3.16), we use the number given by (3.10) which does not depend on a, b. Then it makes sense to introduce the new constant
(3.39)and use identity (3.36) in the form
(3.40)For the numbers , we can also improve the estimate (3.20)
for , given by (3.9) and (3.6). Indeed, by analogy with the proof of (3.33), we have
where we use that by (2.12) and . Similarly, we can prove that . Note that (3.36) and (3.40) coincide for the case , for example, under the traditional assumption , .
-
(ii)
According to Lemma 2.5, in the right-hand sides of (3.36), (3.38), we can use other representations of the relative oscillation numbers investigated in Section 3.2. In particular, we can use the relative oscillation numbers given by (3.30).
-
(iii)
Note that Theorem 2.1 presents the connection between the numbers of focal points of conjoined bases of two arbitrary symplectic systems. In particular, we can apply Theorem 2.1 to the general case of two symplectic eigenvalue problems (1.5) with nonlinear dependence on the spectral parameter (see [2, 17]). The main properties of relative oscillation numbers (2.16) for this general case are subject of the present investigation of the author.
4 Examples
This section is devoted to examples which illustrate the applications of Theorems 3.8, 3.9 to the scalar spectral problems (1.1), (1.2). Note that the classical oscillation theory for scalar spectral problems (1.1) with nonlinear dependence on the spectral parameter is developed in [12].
Example 4.1 Consider problem (1.1) for the scalar Sturm-Liouville difference equation
then for the principal solution of (4.1) at 0, defined by the initial conditions , , we have
and the finite eigenvalues of (4.1) are the zeros of : , , .
According to Theorem 3.9, we have , where for the scalar case
Then, according to (4.2) and Theorem 3.9, the number of finite eigenvalues of problem (4.1) in the interval equals the total number of generalized zeros of the Wronskian in all intervals , , , . For example, if , , we have the three sign changes of the Wronskian (see Figure 1), then according to Theorem 3.9, the three eigenvalues of problem (4.1): , , are located in . Note that the relative oscillation number achieves its maximal value 2 at the point .
Example 4.2 In this example we evaluate the number of eigenvalues of problem (4.1) in Example 4.1 using the representation of the relative oscillation numbers in the form of (3.30). Recall that (3.30) are associated with the Wronskian given by (3.26). For the scalar case representation, (3.30) can be simplified as follows:
Then we have
It is possible to show that definition (4.3) is equivalent to the definition of a weighted node of the Wronskian in [10]. According to Remark 3.10(ii), the number of finite eigenvalues of problem (4.1) in the interval equals the total number of weighted nodes of the Wronskian in . For example (see Figure 2), for , , the total number of weighted nodes of the Wronskian in the interval equals 1, and we have only one eigenvalue . Note that the relative oscillation number achieves its minimal value −1 at the point .
Example 4.3 Consider spectral problem (4.1) (problem1) and the following spectral problem (problem2):
The finite eigenvalues of (4.4) are the zeros of the equation
The localization of the eigenvalues of (4.1) and (4.4) is shown in Figure 3 where we present the functions of the number of eigenvalues below or equal to λ, .
According to Theorem 3.8, we can calculate the difference between the numbers of eigenvalues of (4.4) and (4.1) using the relative oscillation numbers . For the scalar case, we have
where the number given by (3.10) is defined as . Then we can say that the Wronskian has a weighted node at i if . According to Remark 3.10(i), we can consider the sum as the parameter of problem (4.4).
Similarly, for the scalar case, we derive
and say that the Wronskian has a weighted node at if .
Denote , and observe that we can calculate the constant given by (3.39) evaluating the sum of (4.5), (4.6) for . By Figure 4 we conclude that , where we take . Then, according to Theorem 3.8, we can evaluate the difference between the numbers of eigenvalues of (4.4) and (4.1) calculating the total number of weighted nodes of the Wronskian for all i and , , .
Figure 5 presents the graphs of the signs of the Wronskian, , and the relative oscillation numbers , . For example, we have
because of , , and , , . Similarly, in the next point , we have , , and , , , and then . Finally, according to Figure 3, we have
and the sum of the relative oscillation numbers by Figure 5 equals 3.
Author’s contributions
The author declares that she has conducted all the research in this paper. She also created the displayed pictures in the software Matlab R210b.
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This research was supported by Federal Programme of Ministry of Education and Science of the Russian Federation under Grant 1.6034.2011. The author also thanks anonymous referees for their suggestions which improved the presentation of the results.
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Elyseeva, J.V. Relative oscillation theory for matrix Sturm-Liouville difference equations extended. Adv Differ Equ 2013, 328 (2013). https://doi.org/10.1186/1687-1847-2013-328
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DOI: https://doi.org/10.1186/1687-1847-2013-328