Relative oscillation theory for matrix Sturm-Liouville difference equations extended
© Elyseeva; licensee Springer. 2013
Received: 28 June 2013
Accepted: 18 October 2013
Published: 19 November 2013
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.
is obviously equivalent to (1.3).
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 .
In the recent paper , 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 ) 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  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 .
and the conjoined bases , of (2.1), (2.2) with the initial conditions , at are said to be the principal solutions at M.
for conjoined bases of (2.1), (2.2).
The comparative index is defined by , where and . Introduce the dual index , where .
where is the Wronskian given by (2.5).
where , are the numbers of focal points in , , respectively.
because of the definition in (2.11).
for the numbers of focal points of in and .
for the case .
In [, 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 [, Theorem 2.1].
where , are the numbers of focal points in .
holds for the relative oscillation numbers in (2.16) (see the proof of Theorem 2.1 in ), 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 [, Corollary 2.4]).
where the relative oscillation numbers are defined by (2.17) for , .
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  (see [, 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).
By (2.14), , then summing (2.25) from to , we derive (2.24). The proof is completed. □
- (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
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).
where the Wronskian is given by (2.5) and is defined by (2.11) with . So we have proved the following lemma.
3 Relative oscillation theory for matrix Sturm-Liouville difference equations
3.1 Relative oscillation numbers for matrix Sturm-Liouville difference equations
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 main result of this section is based on the consideration of the following particular cases.
Lemma 3.1 (Case I)
where is given by (2.11) with and defined by (3.2).
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)
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 .
Evaluating according to (2.22), where should be replaced by , we derive (3.9) with given by (3.10). The proof is completed. □
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)
are evaluated according to (3.9), (3.10) and (3.6), respectively, with for case II and for case I.
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).
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 case , , , we additionally have and in (3.16) . Then the proof of (ii) is completed.
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
with an arbitrary symmetric matrix , then, according to [, Theorem 3.1], solves the symplectic difference system with coefficients depending on . Here we assume (see ) that , are defined so that conditions (1.3) hold.
where is given by (3.2).
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 , , , .
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 .
where and is the multiplicity of .
The global oscillation theorem in  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 [, 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).
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)
and , , .
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)
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. □
- (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 , .
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).
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.
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 .
and the finite eigenvalues of (4.1) are the zeros of : , , .
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).
and say that the Wronskian has a weighted node at if .