Oscillation theorems and Rayleigh principle for linear Hamiltonian and symplectic systems with general boundary conditions

Applied Mathematics and Computation 218 (2012) 8309–8328

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Oscillation theorems and Rayleigh principle for linear Hamiltonian and symplectic systems with general boundary conditions Roman Šimon Hilscher a,⇑, Vera Zeidan b a b

Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlárˇská 2, CZ-61137 Brno, Czech Republic Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA

a r t i c l e

i n f o

a b s t r a c t The aim of this paper is to establish the oscillation theorems, Rayleigh principle, and coercivity results for linear Hamiltonian and symplectic systems with general boundary conditions, i.e., for the case of separated and jointly varying endpoints, and with no controllability (normality) and strong observability assumptions. Our method is to consider the time interval as a time scale and apply suitable time scales techniques to reduce the problem with separated endpoints into a problem with Dirichlet boundary conditions, and the problem with jointly varying endpoints into a problem with separated endpoints. These more general results on time scales then provide new results for the continuous time linear Hamiltonian systems as well as for the discrete symplectic systems. This paper also solves an open problem of deriving the oscillation theorem for problems with periodic boundary conditions. Furthermore, the present work demonstrates the utility and power of the analysis on time scales in obtaining new results especially in the classical continuous and discrete time theories. Ó 2012 Elsevier Inc. All rights reserved.

Keywords: Oscillation theorem Rayleigh principle Linear Hamiltonian system Time scale symplectic system Discrete symplectic system Finite eigenvalue Finite eigenfunction Selfadjoint eigenvalue problem

1. Introduction In this paper, we develop the oscillation and spectral theory for the linear Hamiltonian system

x0 ¼ AðtÞx þ BðtÞu; u0 ¼ CðtÞx  AT ðtÞu  kWðtÞx;

t 2 ½a; b;

ðHk Þ

with general self-adjoint boundary conditions, i.e., with the separated boundary conditions

Ra xðaÞ þ Ra uðaÞ ¼ 0; Rb xðbÞ þ Rb uðbÞ ¼ 0;

ð1:1Þ

or with the joint boundary conditions

R



xðaÞ xðbÞ



 þR

uðaÞ



uðbÞ

¼ 0:

ð1:2Þ

Here we assume that A; B; C; W : ½a; b ! Rnn are given piecewise continuous ðCp Þ matrix-valued functions such that BðtÞ; CðtÞ, and WðtÞ are symmetric on ½a; b, the Legendre condition

BðtÞ P 0 for all t 2 ½a; b holds, and the monotonicity condition ⇑ Corresponding author. E-mail addresses: [email protected] (R. Šimon Hilscher), [email protected] (V. Zeidan). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2012.01.056

ð1:3Þ

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R. Šimon Hilscher, V. Zeidan / Applied Mathematics and Computation 218 (2012) 8309–8328

WðtÞ P 0 for all t 2 ½a; b

ð1:4Þ

is satisfied. The separated boundary conditions (1.1) are given in terms of the real n  n-matrices

    rank Ra ; Ra ¼ rank Rb ; Rb ¼ n;

Ra ; Ra

and

Rb ; Rb

Ra RTa and Rb RTb are symmetric;

satisfying

ð1:5Þ 

while the joint boundary conditions (1.2) are given in terms of the real 2n  2n-matrices R and R such that

rankðR ; RÞ ¼ 2n;

R RT is symmetric:

ð1:6Þ

The oscillation and eigenvalue problems for linear Hamiltonian systems have been of great interest of researchers for many years. In the classical setting, see the results in [27,32], the oscillation and eigenvalue theory for system ðHk Þ contains the controllability (called also normality) and strong observability assumptions, see [27, Remark 7.7.1] and Remark 4.2. The main results we are now concerned with are the following:  the oscillation theorem, which relates the number of focal points of a special conjoined basis of system ðHk Þ in the interval ða; b with the number of eigenvalues which are less than or equal to the given value k,  a geometric characterization of the eigenvalues and its multiplicities in terms of the eigenfunctions,  the Rayleigh principle, which provides a variational characterization of the eigenvalues in terms of the associated quadratic functional, and  the relationship between the coercivity and positivity of the quadratic functional. All these topics are considered for the system ðHk Þ with the boundary conditions (1.1) or (1.2). The key ingredient of this paper is the absence of the controllability and strong observability assumptions. In this new theory, which started with the paper [29], the traditional concepts of focal points and eigenvalues are replaced by their suitable generalizations, i.e., by the notions of proper focal points and finite eigenvalues. In the case of separated boundary conditions (1.1), such an oscillation theorem was derived in [36, Theorem 1.5]. Its proof was then simplified and shortened in [31, Theorem 6.2] and [30, Theorem B.5] for the Dirichlet boundary conditions

xðaÞ ¼ 0; xðbÞ ¼ 0:

ð1:7Þ

On the other hand, the Rayleigh principle for system ðHk Þ without normality is known only for the Dirichlet boundary conditions (1.7), see [30, Theorem 1.1]. In this paper, we first provide an alternative proof of the oscillation theorem in [36, Theorem 1.5], i.e., for the system ðHk Þ with separated boundary conditions (1.1). Our method uses a transformation technique of the separated endpoints to the zero endpoints, which is known from the discrete time and/or time scales setting, see [14,19]. At the same time we improve the result from [36, Theorem 1.5], since we do not exclude a certain discrete set of k’s from our result. This method then also applies for deriving the corresponding Rayleigh principle, which generalizes the result from the zero endpoints in [30, Theorem 1.1] to the separated ones, including a geometric characterization of the finite eigenvalues in terms of the finite eigenfunctions for the separated endpoints case. As an application of the Rayleigh principle we establish the equivalence between the coercivity and positivity of the associated quadratic functional. All four results, i.e., the oscillation theorem, geometric characterization of finite eigenvalues, Rayleigh principle, and coercivity of the quadratic functional for the separated endpoints, are in this paper further generalized to the jointly varying endpoints (1.2) by another transformation, which utilizes the previously obtained results for the separated endpoints. Note that the latter transformation requires working with possibly ‘‘abnormal’’ systems ðHk Þ, so that the traditional methods such as in [27,33,37] are not applicable in this case. As examples of jointly varying endpoints we may consider the periodic boundary conditions xðbÞ ¼ xðaÞ and uðbÞ ¼ uðaÞ or the antiperiodic boundary conditions xðbÞ ¼ xðaÞ and uðbÞ ¼ uðaÞ, for which the above results were missing in the current literature. The above described methods are natural in the discrete time setting, i.e., for the discrete symplectic systems, see [2,6,7,13,14],

xkþ1 ¼ Ak xx þ Bk uk ;

ukþ1 ¼ Ck xk þ Dk uk  kW k xkþ1 ; k 2 ½0; NZ ;

ðDk Þ

as well as for the time scale symplectic systems, see [1,12,21,23,24,31],

xD ¼ AðtÞx þ BðtÞ u;

uD ¼ CðtÞx þ DðtÞu  kWðtÞxr ;

t 2 ½a; qðbÞT :

ðSk Þ

For this reason, our results for the separated and jointly varying endpoints are in this paper derived for the time scale symplectic system ðSk Þ, and then the time scale ½a; bT is reduced to being purely continuous or purely discrete to get the corresponding results in the continuous and discrete time cases. Therefore, the obtained results are new not only for time scales, but also for the continuous and discrete time cases, as well as for special linear Hamiltonian systems on time scales, see e.g. [8,15]. Hence, this paper demonstrates the utility and power of the analysis on time scales in obtaining new results notably in the classical continuous and discrete time theories. The setup of this paper is the following. In Section 2, we present our main results regarding the eigenvalue problems with the linear Hamiltonian system ðHk Þ. In Section 3 we present the corresponding new results for the discrete symplectic system

R. Šimon Hilscher, V. Zeidan / Applied Mathematics and Computation 218 (2012) 8309–8328

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ðDk Þ with jointly varying endpoints. The proofs of the statements in Sections 2 and 3 follow from the more general theory over time scales derived in the subsequent sections. In Section 4, we introduce the time scale symplectic systems and recall the known spectral theory for the Dirichlet boundary conditions. The main results for the time scale symplectic systems with separated and jointly varying endpoints, i.e., the oscillation theorems, geometric characterization of finite eigenvalues, Rayleigh principle, and coercivity results are then established in Sections 5 and 6. In the final section we discuss in detailed references how the results of this paper generalize the known theory of linear Hamiltonian and symplectic systems. 2. Linear Hamiltonian systems In this section we present an overview of the main results of this paper for the linear Hamiltonian system ðHk Þ. Let us first recall the basic terminology. A matrix-valued solution ðXð; kÞ; Uð; kÞÞ of ðHk Þ is a conjoined basis if X T U is symmetric and b ð; kÞ; Uð; b kÞÞ of ðHk Þ is given by the initial conditions rankX T ; U T ¼ n. The principal solution ð X

b ða; kÞ  0; X

b Uða; kÞ  I for all k 2 R:

ð2:1Þ

From [29, Theorem 3] it is known that, under the assumption (1.3), for each k 2 R and every conjoined basis ðXð; kÞ; Uð; kÞÞ of ðHk Þ the kernel of Xð; kÞ is piecewise constant on ½a; b. This allows to define the proper (or generalized) focal points. According to [36, Definition 1.1], a point t 0 2 ða; b is a proper focal point of a conjoined basis ðXð; kÞ; Uð; kÞÞ of ðHk Þ if

mðt 0 Þ :¼ def Xðt0 ; kÞ  def Xðt 0 ; kÞ P 1;

ð2:2Þ

and in this case the number mðt0 Þ is the multiplicity of the proper focal point t 0 . Here defX is the defect of X, i.e., the dimension of its kernel. Since under the assumption (1.3) the kernel of Xð; kÞ is constant in some left neighborhood of t0 for each t 0 2 ða; b, the left-hand limit def Xðt 0 ; kÞ in (2.2) is well defined. As we have mentioned above, the results presented below follow from the more general theory of time scale symplectic systems, which is derived in Sections 5 and 6, by taking the time scale to be the real connected interval ½a; b. In particular, Remark 4.3(ii) states that assumption (4.6) appearing in all the time scale results in Sections 5 and 6 is always satisfied under the Legendre condition (1.3). 2.1. Separated endpoints We first consider the eigenvalue problem with separated boundary conditions (1.1), that is,

ðHk Þ with Ra xðaÞ þ Ra uðaÞ ¼ 0;

Rb xðbÞ þ Rb uðbÞ ¼ 0:

ð2:3Þ

The results will be formulated in terms of a special conjoined basis of ðHk Þ associated with the boundary conditions (1.1). Hence, we define the natural conjoined basis ðXð; kÞ; Uð; kÞÞ as the solution of ðHk Þ starting with the initial conditions

Xða; kÞ  RTa ; Uða; kÞ  ðRa ÞT for all k 2 R:

ð2:4Þ Ra

Rb

For the Dirichlet boundary conditions, i.e., when Ra ¼ Rb ¼ 0 and ¼ ¼ I, the natural conjoined basis reduces to the prinb kÞ; Uð; b kÞÞ. The nonexistence of proper focal points in ða; b for the natural conjoined basis of ðHk Þ is one of cipal solution ð Xð; the conditions required for the positivity of the quadratic functional

F c ðz; kÞ :¼

Z

b

fxT ðC  kWÞx þ uT BugðtÞ dt þ xT ðbÞSb xðbÞ  xT ðaÞSa xðaÞ;

ð2:5Þ

a

see Proposition 2.1 below. The symmetric matrices Sa and Sb in (2.5) are related to the matrices Ra ; Ra ; Rb ; Rb by the equations

Sa ¼ Rya Ra Rya Ra ; Sb ¼ Ryb Rb Ryb Rb ;

ð2:6Þ

see [27, Corollary 3.1.3]. The dagger stands for the Moore–Penrose generalized inverse (the pseudoinverse), see e.g. [3,4]. We say that the functional F c ð; kÞ is positive definite if F c ðz; kÞ > 0 for all admissible z ¼ ðx; uÞ with x – 0 and satisfying

xðaÞ 2 Im RTa ;

xðbÞ 2 Im RTb :

ð2:7Þ C1p ;

We also recall that z ¼ ðx; uÞ is called admissible if x 2 Bu 2 Cp , and the first equation of system ðHk Þ is satisfied on ½a; b. Note that since this equation does not depend on k, the set all admissible pairs is constant with respect to k 2 R. The next auxiliary result is proven in [29, Theorem 1]. Proposition 2.1 (Positivity). Let k0 2 R be fixed. The functional F c ð; k0 Þ is positive definite if and only if the Legendre condition (1.3) holds, the natural conjoined basis ðXð; k0 Þ; Uð; k0 ÞÞ has no proper focal points in ða; b, and

Sb þ Uðb; k0 ÞX y ðb; k0 Þ > 0 on Im RTb \ Im Xðb; k0 Þ:

ð2:8Þ

The following definition is an extension of the finite eigenvalue notion in [31, Definition 2.4], see also [30, Definition 2.6], to the case of separated endpoints.

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R. Šimon Hilscher, V. Zeidan / Applied Mathematics and Computation 218 (2012) 8309–8328

Definition 2.2 (Finite eigenvalue for separated endpoints). Let ðXð; kÞ; Uð; kÞÞ be the natural conjoined basis of ðHk Þ and define the matrix

Kc ðkÞ :¼ Rb Xðb; kÞ þ Rb Uðb; kÞ:

ð2:9Þ

A number k0 2 R is a finite eigenvalue of the eigenvalue problem (2.3) if

hðk0 Þ :¼ r  rank Kc ðk0 Þ P 1; where r :¼ maxk2R rank Kc ðkÞ:

ð2:10Þ

In this case, the number hðk0 Þ is called the algebraic multiplicity of the finite eigenvalue k0 . If we now set for each k 2 R, counting the multiplicities,

  n1 ðkÞ :¼ the number of proper focal points of Xð; kÞ; Uð; kÞ in ða; b;

ð2:11Þ

n2 ðkÞ :¼ the number of finite eigenvalues of ð2:3Þ which are less than or equal to k;

ð2:12Þ

pðkÞ :¼ rank M c ðkÞ; where Mc ðkÞ :¼ ½I  Kc ðkÞKyc ðkÞRb ;

ð2:13Þ

then we have the following generalization of [30, Theorem 2.9] and [36, Theorem 1.5]. Theorem 2.3 (Oscillation theorem for separated endpoints). Assume (1.4) and that the functions n1 ; n2 ; p are given by (2.11), (2.12), (2.13). Then the Legendre condition (1.3) holds and

n1 ðkÞ þ pðkÞ ¼ n2 ðkÞ for all ðor for someÞ k 2 R

ð2:14Þ

if and only if there exists k0 < 0 such that the functional F c ð; k0 Þ is positive definite. Proof. This result follows from Theorem 5.12 and Remark 4.3(ii).

h

The condition on the positivity of F c ð; k0 Þ can be checked by Proposition 2.1. Note that in comparison with the known oscillation theorem for the separated endpoints in [36, Theorem 1.5], our result is an equivalence and the equality in (2.14) is satisfied for all k 2 R, while in [36, Theorem 1.5] it holds for all k 2 R except for k in a certain discrete set. A more precise analysis, such as in [31, Theorem 9.2] and [30, Theorem B.5], also reveals that assumptions (1.3) and (1.4) imply the boundedness of the finite eigenvalues of (2.3) from below and that these finite eigenvalues are isolated (provided there is a finite eigenvalue at all). Following [31, Definition 5.3] and [30, Theorem A.2], we say that a solution ðxð; k0 Þ; uð; k0 ÞÞ of (2.3) with k ¼ k0 is a finite eigenfunction corresponding to the finite eigenvalue k0 , if

WðtÞxðt; k0 ÞX0 on ½a; b:

ð2:15Þ

In this case the dimension xðk0 Þ of the associated eigenspace, i.e., the dimension of the functions Wx where ðx; uÞ is a finite eigenfunction for k0 , is called the geometric multiplicity of k0 as a finite eigenvalue of (2.3). The next statement follows from Theorem 5.14. Theorem 2.4 (Geometric characterization of finite eigenvalues). Assume (1.4). A number k0 2 R is a finite eigenvalue of the eigenvalue problem (2.3) with the algebraic multiplicity hðk0 Þ P 1 if and only if there exists a finite eigenfunction corresponding to k0 . In this case, the algebraic and geometric multiplicities of k0 coincide, i.e., hðk0 Þ ¼ xðk0 Þ. Upon introducing the semi-inner product and the seminorm

hz; ~ziW z :¼

Z

b

xT ðtÞWðtÞ ~xðtÞdt; a

kzkW :¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hz; ziW

~ Þ, the orthogonality relation between such pairs will be denoted by z ? ~z, meanover admissible pairs z ¼ ðx; uÞ and ~z ¼ ð~ x; u ing that hz; ~ziW ¼ 0. As in the case of the Dirichlet boundary conditions, one can easily prove that the eigenvalue problem (2.3) has only real finite eigenvalues, and that the finite eigenfunctions corresponding to different finite eigenvalues are orthogonal with respect to the semi-inner product h; iW , compare with [31, Propositions 5.7–5.8] and [30, Remark 2.7]. This implies that for any set of finite eigenvalues fk1 ; . . . ; km g, the corresponding finite eigenfunctions can be orthonormalized by the standard procedure. The following result is a generalization of [30, Theorem 1.1] to the separated endpoints. Its proof is based on Theorem 5.16 and Remark 4.3(ii). We put

A :¼ fz ¼ ðx; uÞ; z is admissibleg:

ð2:16Þ

Theorem 2.5 (Rayleigh principle for separated endpoints). Assume (1.4) and that the functional F c ð; k0 Þ is positive definite for some k0 < 0. Let k1 6 . . . 6 km 6 . . . be the finite eigenvalues of the eigenvalue problem (2.3) with the corresponding orthonormal finite eigenfunctions z1 ¼ ðx1 ; u1 Þ; . . . ; zm ¼ ðxm ; um Þ; . . .. Then for each m 2 N [ f0g we have

R. Šimon Hilscher, V. Zeidan / Applied Mathematics and Computation 218 (2012) 8309–8328

kmþ1 ¼ min

8313

  F c ðz; 0Þ ; z ¼ ðx; uÞ 2 A with ð2:7Þ; ðWxÞðÞ – 0; and z ? z1 ; . . . ; zm : hz; zW i

Note that by [30, Corollary 4.1] the actual number of finite eigenvalues of (2.3) depends on the dimension of the set of functions Wx, where z ¼ ðx; uÞ 2 A with (2.7). We say that the functional F c ð; 0Þ is coercive if there exists a > 0 such that

F c ðz; 0Þ P a

Z

b

jxðtÞj2 dt for every admissible z ¼ ðx; uÞ with ð2:7Þ:

a

Clearly, every coercive functional F c ð; 0Þ is also positive definite. The result below shows that these two notions are equivalent. This equivalence is known in [38, Theorem 4.2] and [37, Theorem 5.5] under a normality assumption. Here we remove this normality assumption. Note that the result below is also a generalization of [30, Theorem 4.5] from the Dirichlet endpoints to the separated ones. This follows from Theorem 5.17. Theorem 2.6 (Coercivity for separated endpoints). The functional F c ð; 0Þ in (2.5) is coercive if and only if it is positive definite. 2.2. Jointly varying endpoints The eigenvalue problem for a possibly abnormal linear Hamiltonian system ðHk Þ with jointly varying endpoints (1.2), that is,

ðHk Þ with R



xðaÞ xðbÞ



  uðaÞ þR ¼ 0; uðbÞ

ð2:17Þ

has not been studied in the literature so far. Therefore, all the results presented below for the eigenvalue problem (2.17) are b ð; kÞ; Uð; b kÞÞ of ðHk Þ together with another conjoined basis completely new. We shall now consider the principal solution ð X e kÞ; Uð; e kÞÞ of ðHk Þ, which is given by the initial conditions ð Xð;

e ða; kÞ  I; Uða; e X kÞ  0 for all k 2 R:

ð2:18Þ

The quadratic functional associated with this problem has the form

Gc ðz; kÞ :¼

Z

b

fxT ðC  kWÞx þ uT BugðtÞdt þ



T   xðaÞ S ; xðbÞ xðbÞ

xðaÞ

a

ð2:19Þ

where z ¼ ðx; uÞ is admissible and the symmetric matrix S is given by

S ¼ Ry R Ry R:

ð2:20Þ

The functional Gc ð; kÞ is positive definite if Gc ðz; kÞ > 0 for all admissible z ¼ ðx; uÞ with x – 0 and satisfying



xðaÞ



xðbÞ

2 Im RT :

ð2:21Þ

The following auxiliary result is proven in [29, Corollary 2]. Proposition 2.7 (Positivity). Let k0 2 R be fixed. The functional Gc ð; k0 Þ is positive definite if and only if the Legendre condition b ð; k0 Þ; Uð; b k0 Þ has no proper focal points in ða; b, and (1.3) holds, the principal solution ð X

Sþ



I

0

b Uðb; k0 Þ

e Uðb; k0 Þ



0

I

b ðb; k0 Þ X

e Xðb; k0 Þ

y

 0 > 0 on Im RT \ Im b X ðb; k0 Þ

I e ðb; k0 Þ X

 :

Next we define the finite eigenvalues of the eigenvalue problem (2.17), thus extending this notion to the jointly varying endpoints. b kÞ; Uð; b kÞÞ and ð Xð; e kÞ; Uð; e kÞÞ of ðHk Þ be given Definition 2.8 (Finite eigenvalue for joint endpoints). Let the solutions ð Xð; by (2.1) and (2.18). Define

    0 I I 0 þ R : Kc ðkÞ :¼ R b e b e X ðb; kÞ Xðb; kÞ Uðb; kÞ Uðb; kÞ

ð2:22Þ

A number k0 2 R is a finite eigenvalue of the eigenvalue problem (2.17) if

hðk0 Þ :¼ s  rank Kc ðk0 Þ P 1;

where s :¼ maxk2R rank Kc ðkÞ:

ð2:23Þ

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In this case, the number hðk0 Þ is called the algebraic multiplicity of the finite eigenvalue k0 . For each k 2 R we set, counting the multiplicities,

b ð; kÞ; Uð; b kÞÞ in ða; b; n1 ðkÞ :¼ the number of proper focal points of ð X

ð2:24Þ

n2 ðkÞ :¼ the number of finite eigenvalues of ð2:17Þ which are less than or equal to ðkÞ;

ð2:25Þ

qðkÞ :¼ rank Mc ðkÞ;

where M c ðkÞ :¼ ½I  Kc ðkÞKyc ðkÞR:

ð2:26Þ

The following result generalizes [27, Theorem 7.2.2] to a possibly abnormal system ðHk Þ. It is proven via Theorem 6.11 and Remark 4.3(ii). Theorem 2.9 (Oscillation theorem for joint endpoints). Assume (1.4) and that the functions n1 ; n2 ; q are given by (2.24), (2.25), (2.26). Then the Legendre condition (1.3) holds and

n1 ðkÞ þ qðkÞ ¼ n2 ðkÞ for all ðor for someÞ k 2 R if and only if there exists k0 < 0 such that the functional Gc ð; k0 Þ is positive definite. The finite eigenfunctions for the eigenvalue problem (2.17) are defined analogously to the corresponding finite eigenfunctions for the separated endpoints case. More precisely, a solution ðxð; k0 Þ; uð; k0 ÞÞ of (2.17) with k ¼ k0 is a finite eigenfunction corresponding to the finite eigenvalue k0 (according to Definition 2.8) if condition (2.15) holds. Remark 2.10. (i) The geometric characterization of finite eigenvalues of (2.17) is the same as for the separated endpoints. Namely, the statement of Theorem 2.4 remains valid when the finite eigenvalue notion from Definition 2.8 is used in this statement. (ii) As in the separable endpoints case, under (1.4) the finite eigenvalues of (2.17) are real and the finite eigenfunctions corresponding to different finite eigenvalues are orthogonal with respect to the semi-inner product h; iW . (iii) Assumptions (1.3) and (1.4) imply that the finite eigenvalues of (2.17) are bounded from below and isolated. (iv) Both for the separated and jointly varying endpoints the expansion theorem for admissible z ¼ ðx; uÞ in [30, Theorem 4.3] remains valid under the assumption that the corresponding functional F c ð; k0 Þ and Gc ð; k0 Þ, respectively, is positive definite. In our next result we provide a variational characterization of the finite eigenvalues of (2.17). This result generalizes [27, Theorem 7.7.1] to a possibly abnormal system ðHk Þ. Its proof is based on Theorem 6.14 and Remark 4.3(ii). Theorem 2.11 (Rayleigh principle for joint endpoints). Assume (1.4) and that the functional Gc ð; k0 Þ is positive definite for some k0 < 0. Let k1 6 . . . 6 km 6 . . . be the finite eigenvalues of the eigenvalue problem (2.17) with the corresponding orthonormal finite eigenfunctions z1 ¼ ðx1 ; u1 Þ; . . . ; zm ¼ ðxm ; um Þ; . . .. Then for each m 2 N [ f0g we have

kmþ1 ¼ min

  Gc ðz; 0Þ ; z ¼ ðx; uÞ 2 A with ð2:21Þ; ðWxÞðÞ – 0; and z ? z1 ; . . . ; zm : hz; ziW

We say that the functional Gc ð; 0Þ is coercive if there exists a > 0 such that

Gc ðz; 0Þ P a

Z

b

jxðtÞj2 dt

for every admissible z ¼ ðx; uÞ with ð2:21Þ:

a

The result below follows from Theorem 6.15. Theorem 2.12 (Coercivity for joint endpoints). The functional Gc ð; 0Þ in (2.19) is coercive if and only if it is positive definite. 3. Discrete symplectic systems In this section we present a new oscillation theorem and the Rayleigh principle for discrete symplectic systems ðDk Þ introduced in Section 1 with jointly varying endpoints. Note that the oscillation theorems for discrete symplectic systems with Dirichlet and separated endpoints have been derived in [13,14], respectively. Therefore, we consider in this section the eigenvalue problem

ðDk Þ with R



x0 xNþ1



þR



u0 uNþ1



¼ 0;

ð3:1Þ

R. Šimon Hilscher, V. Zeidan / Applied Mathematics and Computation 218 (2012) 8309–8328

8315

with the associated discrete quadratic functional

Gd ðz; kÞ :¼

( T N X xk uk

k¼0

!

CTk Ak

CTk Bk

BTk Ck

DTk Bk

xk uk



)  k xTkþ1 W k xkþ1

 þ

T   x0 S xNþ1 xNþ1 x0

ð3:2Þ

over admissible z ¼ ðx; uÞ satisfying



x0 xNþ1



2 Im RT :

ð3:3Þ

We shall not repeat all the terminology from the previous section and just give the reference to the most important notions b ðkÞ; UðkÞÞ b e ðkÞ; UðkÞÞ e regarding the oscillation theorems. Let ð X and ð X be the principal and associated solutions of system ðDk Þ, b e b e i.e., X 0 ðkÞ  0  U 0 ðkÞ and U 0 ðkÞ  I   X 0 ðkÞ for all k 2 R. Finite eigenvalues of (3.1) are defined similarly to Definition 2.8 via the matrix

    0 I I 0 Kd ðkÞ :¼ R b þR b : e e X Nþ1 ðkÞ X Nþ1 ðkÞ U Nþ1 ðkÞ U Nþ1 ðkÞ

ð3:4Þ

b ðkÞ; UðkÞÞ b b k ðkÞ þ ind P b k ðkÞ, see (4.4) The proper focal points of ð X in the interval ðk; k þ 1Z are counted by the number rank M and also [13,14,26,28], where

b k ðkÞ X b y ðkÞBk Tb k ðkÞ; b k ðkÞ :¼ Tb k ðkÞ X P kþ1

b y ðkÞ M b k ðkÞ; M b k ðkÞ :¼ ½I  X b kþ1 ðkÞ X b y ðkÞBk : Tb k ðkÞ :¼ I  M k kþ1

Following (2.24)–(2.26) we define

b ðkÞ; UðkÞÞ b n1 ðkÞ :¼ the number of proper focal points of ð X inð0; N þ 1Z ;

ð3:5Þ

n2 ðkÞ :¼ the number of finite eigenvalues of ð3:1Þ which are less than or equal to k;

ð3:6Þ

qðkÞ :¼ rank Md ðkÞ þ ind Pd ðkÞ;

ð3:7Þ

where, according to (3.4), (2.26), and (6.9),

M d ðkÞ :¼ ½I  Kd ðkÞ Kyd ðkÞ R; T d ðkÞ :¼ I  Myd ðkÞ M d ðkÞ;   0 I y Pd ðkÞ :¼ T d ðkÞ Md ðkÞ b e Nþ1 ðkÞ Kd ðkÞRT d ðkÞ: X Nþ1 ðkÞ X From Theorem 6.11 we then obtain the following generalization of [14, Theorem 1] to jointly varying endpoints. At the same time we get a generalization of [6, Theorem 3] in which we remove the assumption (A5). Theorem 3.1 (Oscillation theorem for joint endpoints). Assume W k P 0 for all k 2 ½0; NZ . Then with n1 ; n2 ; q defined by (3.5), (3.6), (3.7) we have

n1 ðkÞ þ qðkÞ ¼ n2 ðkÞ for all ðor for someÞ k 2 R if and only if there exists k0 < 0 such that the functional Gd ð; k0 Þ in (3.2) is positive definite. The following result is a generalization of [14, Theorem 2] from separated to joint endpoints, see also [7, Theorem 4.6]. Its proof is based on Theorem 6.14. Theorem 3.2 (Rayleigh principle for joint endpoints). Assume W k P 0 for all k 2 ½0; NZ and that the functional Gd ð; k0 Þ is positive definite for some k0 < 0. Let k1 6 . . . 6 kr be the finite eigenvalues of the eigenvalue problem (3.1) with the corresponding orthonormal finite eigenfunctions z1 ¼ ðx1 ; u1 Þ; . . . ; zr ¼ ðxr ; ur Þ, where r 6 ðN þ 1Þn is the total number of finite eigenvalues of (3.1), including their multiplicities. Then for each m 2 f1; . . . ; r  1g we have

kmþ1 ¼ min

  Gd ðz; 0Þ ; z ¼ ðx; uÞ 2 A with ð3:3Þ; W k xkþ1 X0; and z ? z1 ; . . . ; zm : hz; zW i

We say that the functional Gd ð; 0Þ is coercive if there exists a > 0 such that

Gd ðz; 0Þ P a

N X

jxkþ1 j2

for every admissible z ¼ ðx; uÞ satisfying ð3:3Þ:

k¼0

The next statement follows from Theorem 6.15. Theorem 3.3 (Coercivity for joint endpoints). The functional Gd ð; 0Þ in (3.2) is coercive if and only if it is positive definite.

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4. Time scale symplectic systems In this section we present basic results about time scale symplectic systems known in the literature. The elements of the time scales theory can be found in [9,10] and the theory of time scale symplectic systems in [1,12,11,21,23,24,31,34]. Let T be a bounded time scale with the forward and backward jump operators r and q and with the graininess lðtÞ ¼ rðtÞ  t. If we set a :¼ min T and b :¼ max T, then we may identify T with the time scale interval ½a; bT :¼ ½a; b \ T. Consider the time scale symplectic system ðSk Þ introduced in Section 1, in which the coefficients A; B; C; D; W : ½a; qðbÞT ! Rnn are piecewise rd-continuous (Cprd ) matrix-valued functions on such that WðtÞ is symmetric and

WðtÞ P 0 for all t 2 ½a; qðbÞT :

ð4:1Þ

The matrix functions A; B; C; D satisfy the identity

S T ðtÞJ þ JSðtÞ þ lðtÞS T ðtÞJSðtÞ ¼ 0;

t 2 ½a; qðbÞT ;

ð4:2Þ

where the 2n  2n matrices SðtÞ and J are given by

 SðtÞ :¼

AðtÞ

BðtÞ

CðtÞ

DðtÞ

 ;

 J :¼

0

I

I

0

 ð4:3Þ

:

Indentity (4.2) implies that the matrix I þ lðtÞSðtÞ is symplectic, i.e.,

½I þ lðtÞS T ðtÞJ ½I þ lðtÞSðtÞ ¼ J ; which motivates, together with the fact that the fundamental matrix of ðSk Þ is symplectic, the name of this system. With this notation and with z ¼ ðxT ; uT ÞT , abbreviated as z ¼ ðx; uÞ, we can write the system (Sk ) as

zD ¼ Sðt; kÞ z; where

t 2 ½a; qðbÞT ;

 Sðt; kÞ :¼ SðtÞ  k

0

ðSk Þ 

0

lðtÞWðtÞBðtÞ WðtÞ½I þ lðtÞAðtÞ

;

see [31, Remark 2.1]. The solutions of the system ðSk Þ are piecewise rd-continuously D-differentiable (C1prd ) functions on ½a; bT . The composition of a function f with the forward and backward jump operators will be abbreviated by f r :¼ f r and f q :¼ f q, respectively. Remark 4.1. (i) In the continuous time case, we have lðtÞ  0 and rðtÞ ¼ t ¼ qðtÞ. Therefore, in this case identity (4.2) implies that DðtÞ ¼ AT ðtÞ, and BðtÞ and CðtÞ are symmetric. Thus, the time scale symplectic system ðSk Þ reduces to the linear Hamiltonian system ðHk Þ, in which AðtÞ :¼ AðtÞ; BðtÞ :¼ BðtÞ, and CðtÞ :¼ CðtÞ. (ii) In the discrete case, we have lðtÞ  1; rðtÞ ¼ t þ 1, and qðtÞ ¼ t  1. Therefore, with the notation Ak :¼ I þ AðkÞ; Bk :¼ BðkÞ, Ck :¼ CðkÞ; Dk :¼ I þ DðkÞ, and W k :¼ WðkÞ we can see that the time scale symplectic system ðSk Þ reduces to the discrete symplectic system ðDk Þ. Remark 4.2. The normality notion mentioned in Section 1 has for the time scale symplectic systems the following form. If ðxð; kÞ; uð; kÞÞ is a solution of ðSk Þ with xðt; kÞ  0 on ½a; bT (or on a subinterval of ½a; bT ), then also uðt; kÞ  0 on ½a; bT . Such a normality assumption is not required in this paper. Conjoined bases of system ðSk Þ are defined in usual way, i.e., as the solutions ðXð; kÞ; Uð; kÞÞ of ðSk Þ such that b kÞ; Uð; b kÞÞ of ðSk Þ is given by the initial condirankðX T ; U T Þ ¼ n and X T U is symmetric. As before, the principal solution ð Xð; tions (2.1). The concept of proper focal points for conjoined bases of ðSk Þ is in details discussed in [31, Section 3]. In particular, following [31, Definition 3.1] we say that a conjoined basis ðXð; kÞ; Uð; kÞÞ has a proper focal point in ðqðt 0 Þ; t 0 T if

  mðt 0 Þ :¼ def Xðt0 ; kÞ  dim Ker X q ðt 0 ; kÞ \ Ker Xðt 0 ; kÞ þ ind Pq ðt 0 ; kÞ;

ð4:4Þ

where (suppressing the arguments t and k in the notation)

P :¼ TXðX r Þy BT;

T :¼ I  My M;

M :¼ ½I  X r ðX r Þy B;

q

q  and where the notation Ker X ðt  0 ; kÞ and ind P ðt 0 ; kÞ mean

Ker X

q

ðt 0 ;

kÞ ¼

8 < Ker X q ðt0 ; kÞ;

if t 0 is left-scattered;

 lim Ker Xðt; kÞ; if t 0 is left-dense; : Ker Xðt0 ; kÞ ¼ t!t 0 8 < ind Pq ðt 0 ; kÞ; if t0 is left-scattered; ind Pq ðt 0 ; kÞ ¼ lim ind Pðt; kÞ; if t is left-dense: 0 :  t!t 0

ð4:5Þ

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The following hypothesis regarding the conjoined bases ðXð; kÞ; Uð; kÞÞ of ðSk Þ was used in [31] and will also be used in this paper.

For any k 2 R; ðiÞ the kernel of Xð; kÞ is piecewise constant on ½a; bT ;

9 > > > =

e > 0 such that Pðt; kÞ P 0 for all t 2 ðt0 ; t0 þ eT ; > > > ; ðiiiÞ for every left-dense point t 0 2 ða; bT there exists e > 0 such that Pðt; kÞ P 0 for all t 2 ½t0  e; t0 ÞT :

ðiiÞ for every right-dense point t0 2 ½a; bÞT there exists

ð4:6Þ Condition (4.6) implies that the conjoined basis ðXð; kÞ; Uð; kÞÞ has finitely many proper focal points in ða; bT . This is a standing requirement in order to be able to count the proper focal points and their multiplicities at all. Remark 4.3. (i) In the theory of symplectic systems it is known, see [21, Lemma 3.1] or [31, Formula (3.8)], that the matrix Pðt; kÞ can be written as (suppressing the arguments t and k)

P ¼ T½ðI þ lDT ÞB  kl2 BT WB  lBT U r ðX r Þy B T: (ii) From the above formula it follows that in the continuous time setting (in which lðtÞ  0) as in Section 2 we have P ¼ TBT, so that under the Legendre condition (1.3) the matrix Pðt; kÞ is nonnegative definite on ½a; b. This means that in the continuous time case there are no proper focal points coming from the index of the matrix Pðt; kÞ. The definition of proper focal points in (4.4) then reduces to the definition in (2.2). In addition, the result of [29, Theorem 3] shows that assumption (4.6)(i) is satisfied by every conjoined basis of ðHk Þ. Note that in this case Mðt; kÞ ¼ 0 and so Tðt; kÞ ¼ I if and only if Ker X T ðt; kÞ # Ker BðtÞ. (iii) In the discrete time setting there are no dense points in ½a; bT at all, so that assumption (4.6) holds trivially. We say that z ¼ ðx; uÞ is admissible if x 2 C1prd on ½a; bT ; Bu 2 Cprd on ½a; qðbÞT , and the first equation of system ðSk Þ is satisfied on ½a; qðbÞT . If, as in (2.16), we again denote by A the set of all admissible pairs z ¼ ðx; uÞ, then the set A is independent of k 2 R. Let us summarize the definitions and results known in [31] for the Dirichlet boundary conditions (1.7), i.e., for the eigenvalue problem

ðSk Þ with xðaÞ ¼ 0 ¼ xðbÞ:

ð4:7Þ

By [31, Definition 2.4], a number k0 2 R is called a finite eigenvalue of (4.7) if

b ðb; k0 Þ P 1; hðk0 Þ :¼ r 0  rank X

b ðb; kÞ: where r 0 :¼ max rank X k2R

ð4:8Þ

In this case, the number hðk0 Þ is the algebraic multiplicity of the finite eigenvalue k0 . Note that the quantity r0 is well defined, since by [16, Corollary 4.5] the solutions of ðSk Þ are entire functions in the argument k (as a consequence of the linear depenb kÞ; Uð; b kÞÞ of ðSk Þ. Following [31, Definition dence of the coefficients on k). The above notion uses the principal solution ð Xð; 5.3], a solution ðxð; k0 Þ; uð; k0 ÞÞ of (4.7) satisfying

WðtÞxr ðt; k0 ÞX0 on½a; qðbÞT

ð4:9Þ

is called a finite eigenfunction corresponding to the finite eigenvalue k0 . The dimension xðk0 Þ of the associated eigenspace, i.e., the dimension of the functions Wxr where ðx; uÞ is a finite eigenfunction for k0 , is called the geometric multiplicity of k0 . The following result is from [31, Theorem 5.2]. Proposition 4.4 (Geometric characterization of finite eigenvalues). Assume (4.1). A number k0 2 R is a finite eigenvalue of the eigenvalue problem (4.7) according to (4.8) with the algebraic multiplicity hðk0 Þ P 1 if and only if there exists a corresponding finite eigenfunction for k0 . In this case, the algebraic and geometric multiplicities of k0 coincide, i.e., hðk0 Þ ¼ xðk0 Þ. Remark 4.5. (i) In [31, Proposition 4.5] it is proven under (4.1) that for each t 2 ½a; bT the kernel of Xðt; kÞ is piecewise constant with respect to k. This fact implies that the finite eigenvalues of (4.7) are isolated. (ii) The result of [31, Propositions 5.7] shows that under (1.4), the finite eigenvalues of the eigenvalue problem (4.7) are real. (iii) In [31, Propositions 5.8] it is shown that under (1.4), the finite eigenfunctions corresponding to different finite eigenvalues are orthogonal with respect to the semi-inner product

hz; ~ziW :¼

Z

b

½xr ðtÞT WðtÞ ~xr ðtÞ Dt;

a

~ Þ. where z ¼ ðx; uÞ and ~z ¼ ð~ x; u

kzkW :¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi hz; ziW :

ð4:10Þ

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For each k 2 R we set, counting the multiplicities,

b ð; kÞ; Uð; b kÞÞ in ða; b ; n1 ðkÞ :¼ the number of proper focal points of ð X T

ð4:11Þ

n2 ðkÞ :¼ the number of finite eigenvalues of ð2:3Þ which are less than or equal to k:

ð4:12Þ

The following two global oscillation theorems are proven in [31, Theorem 6.2 and Corollary 6.4]. b kÞ; Uð; b kÞÞ of Proposition 4.6 (Oscillation theorem for Dirichlet endpoints). Assume (4.1) and that the principal solution ð Xð; ðSk Þ satisfies condition (4.6). Then with n1 and n2 defined by (4.11), (4.12) we have for all k 2 R

n2 ðkþ Þ ¼ n2 ðkÞ < 1;

ð4:13Þ

n2 ðkþ Þ  n2 ðk Þ ¼ n1 ðkþ Þ  n1 ðk Þ P 0;

ð4:14Þ

and there exists m 2 N [ f0g such that

n1 ðkÞ ¼ n2 ðkÞ þ m for all k 2 R: Moreover, for a suitable k0 < 0 we have

n2 ðkÞ  0 and n1 ðkÞ  m for all k 6 k0 : Remark 4.7. Condition n2 ðkÞ  0 for all k 6 k0 means that there are no finite eigenvalues of (4.7) before the number k0 . Hence, under the assumptions of Proposition 4.6, the finite eigenvalues of (4.7) are bounded from below (provided there exists a finite eigenvalue at all). The quadratic functional associated with the eigenvalue problem (4.7) has the form

F 0 ðz; kÞ :¼

Z

b

Wðt; zðtÞÞDt  khz; ziW ;

a

ð4:15Þ

where for z ¼ ðx; uÞ the value hz; ziW is given in (4.10) and where

Wðt; zÞ :¼ xT CT ðtÞ½I þ lðtÞAðtÞx þ 2lðtÞxT CT ðtÞBðtÞu þ uT ½I þ lðtÞDT ðtÞBðtÞu:

ð4:16Þ

The functional F 0 ð; kÞ is positive definite if F 0 ðz; kÞ > 0 for all admissible z ¼ ðx; uÞ satisfying (1.7) and x – 0. The oscillation theorem below is a combination of Proposition 4.6 and [35, Proposition 2.2]. b kÞ; Uð; b kÞÞ of Proposition 4.8 (Oscillation theorem for Dirichlet endpoints). Assume (4.1) and that the principal solution ð Xð; ðSk Þ satisfies condition (4.6). Then with n1 and n2 defined by (4.11) we have

n1 ðkÞ ¼ n2 ðkÞ for all ðor for someÞ k 2 R if and only if there exists k0 < 0 such that the quadratic functional F 0 ð; k0 Þ is positive definite. The positivity of the functional F 0 ð; k0 Þ in Proposition 4.8 can be tested by the following condition on proper focal points from [21, Theorem 4.1]. Proposition 4.9 (Positivity). Let k0 2 R be fixed. The functional F 0 ð; k0 Þ is positive definite if and only if the principal solution b k0 Þ; Uð; b k0 Þ ( ) has no proper focal points in ða; b . Xð; T A variational characterization of the finite eigenvalues of (4.7) in terms of the Rayleigh quotient is established in [35, Theorem 4.1] as follows. b kÞ; Uð; b kÞÞ of ðSk Þ Proposition 4.10 (Rayleigh principle for Dirichlet endpoints). Assume that the principal solution ð Xð; satisfies condition (4.6), the functional F 0 ð; k0 Þ is positive definite for some k0 < 0, and (4.1) holds. Let k1 6 . . . 6 km 6 . . . be the finite eigenvalues of the eigenvalue problem (4.7) with the corresponding orthonormal finite eigenfunctions z1 ¼ ðx1 ; u1 Þ; . . . ; zm ¼ ðxm ; um Þ; . . .. Then for each m 2 N [ f0g

kmþ1 ¼ min

  F 0 ðz; 0Þ ; z ¼ ðx; uÞ 2 A with ð3:3Þ; ðWxr ÞðÞ – 0; and z ? z1 ; . . . ; zm : hz; ziW

The total number of finite eigenvalues of (4.7) depends on the dimension of the functions Wxr , where z ¼ ðx; uÞ 2 A with (1.7), see [35, Corollary 4.6]. We say that the functional F 0 ð; 0Þ is coercive if there exists a > 0 such that

F 0 ðz; 0Þ P a

Z a

b

jxr ðtÞj2 Dt for every admissible z ¼ ðx; uÞ with ð1:7Þ:

R. Šimon Hilscher, V. Zeidan / Applied Mathematics and Computation 218 (2012) 8309–8328

8319

Theorem 4.11 (Coercivity for Dirichlet endpoints). The functional F 0 ð; 0Þ in (4.15) is coercive if and only if it is positive definite. Proof. Of course, the coercivity of F 0 ð; 0Þ implies its positive definiteness. Hence, we assume that the functional F 0 ð; 0Þ is positive definite. Then F 0 ðz; 0Þ > 0 for every admissible z ¼ ðx; uÞ with xðaÞ ¼ 0 ¼ xðbÞ and x – 0. Consider the eigenvalue problem (4.7) with WðtÞ  I on ½a; qðbÞT . Then Proposition 4.10 implies that the smallest finite eigenvalue k1 of (4.7) is nonnegative. Suppose that k1 ¼ 0. Then, by Proposition 4.10, F 0 ðz; 0Þ P k1 hz; ziW for every admissible z ¼ ðx; uÞ with xðaÞ ¼ 0 ¼ xðbÞ and x – 0, and the finite eigenfunction z1 ¼ ðx1 ; u1 Þ of (4.7) corresponding to the finite eigenvalue k1 ¼ 0 satisfies F 0 ðz1 ; 0Þ ¼ k1 hz1 ; z1 iW ¼ 0. Note that the finite eigenfunction z1 exists due to Proposition 4.4. The equality F 0 ðz1 ; 0Þ ¼ 0 however contradicts the positive definiteness of F 0 ð; 0Þ. Thus, we must have k1 > 0. From Proposition 4.10 we then have F 0 ðz; 0Þ P k1 hz; ziW for every admissible z ¼ ðx; uÞ with xðaÞ ¼ 0 ¼ xðbÞ, which shows the coercivity of the functional F 0 ð; 0Þ with a :¼ k1 > 0. h In the subsequent sections we will discuss how the problems with separated and jointly varying endpoints can be transformed to the Dirichlet and separated endpoints, respectively. 5. Spectral theory for problems with separated endpoints In this section we study the spectral properties of the eigenvalue problem

ðSk Þ with Ra xðaÞ þ Ra uðaÞ ¼ 0; Rb xðbÞ þ Rb uðbÞ ¼ 0; Ra ;

ð5:1Þ

Rb

where the matrices Ra ; Rb ; satisfy (1.5). The natural conjoined basis ðXð; kÞ; Uð; kÞÞ of ðSk Þ is determined by the initial conditions (2.4). The associated quadratic functional is

F ðz; kÞ :¼ F 0 ðz; kÞ þ xT ðbÞ Sb xðbÞ  xT ðaÞ Sa xðaÞ;

ð5:2Þ

where z ¼ ðx; uÞ, the functional F 0 ð; kÞ is defined by (4.15), and where the symmetric matrices Sa and Sb are given by (2.6). We say that F ð; kÞ is positive definite if F ðz; kÞ > 0 for all admissible z ¼ ðx; uÞ satisfying (2.7) and x – 0. The dependence of the functional F ð; kÞ on the parameter k is discussed in the next result. Its proof is essentially the same as for the zero endpoints case in [35, Corollary 5.2], when the appropriate separated endpoint conditions are incorporated into the functional, compare also with [23, Theorem 3.2]. Proposition 5.1 (Comparison theorem). Let k0 2 R and (4.1) hold. The functional F ð; k0 Þ is positive definite (nonnegative) if and only if the functional F ð; kÞ is positive definite (nonnegative) for every k 6 k0 . The following statement is proven in [21, Theorem 4.1]. Proposition 5.2 (Positivity). Let k0 2 R be fixed. The functional F ð; k0 Þ is positive definite if and only if the natural conjoined basis ðXð; k0 Þ; Uð; k0 ÞÞ has no proper focal points in ða; bT , and inequality (2.8) holds. Following the results in [14,19], we shall now develop a transformation of the eigenvalue problem (5.1) which has separated endpoints into an eigenvalue problem of the form (4.7) which has Dirichlet endpoints. In particular, the transformation from [14] shows to be very useful. Given the time scale ½a; bT and the matrices Ra ; Ra ; Rb ; Rb satisfying condition (1.5), we consider the extended time scale

½a  1; b þ 1T :¼ ½a; bT [ fa  1; b þ 1g; in which qðaÞ ¼ a  1; rðbÞ ¼ b þ 1, so that lða  1Þ ¼ lðbÞ ¼ 1. Let Rb ¼ QD be the polar decomposition of Rb with Q orthogonal and D ¼ ðRTb Rb Þ1=2 symmetric. From (1.5) it follows that the matrices

K a :¼ ½Ra ðRa ÞT þ Ra RTa 1 ; K b :¼ Q T ½Rb ðRb ÞT þ Rb RTb 1 Q exist and are symmetric and positive definite. For t ¼ 9a  1 and t ¼ b we set

Aða  1Þ :¼ ðRa ÞT K a  I; AðbÞ :¼ Q T Rb  I; T

Bða  1Þ :¼ RTa ; Cða  1Þ :¼ RTa K a ; Dða  1Þ :¼

ðRa ÞT

Wða  1Þ :¼ 0;

BðbÞ :¼ Q Rb ; CðbÞ :¼ K b Q T Rb ;  I;

DðbÞ :¼ K b Q WðbÞ :¼ 0:

T

Rb

> > > > > > > =

> > > >  I; > > > ;

ð5:3Þ

With the matrices Sða  1Þ and SðbÞ defined according to (4.3), it follows by direct calculations from (1.5) that the matrices I þ lða  1Þ Sða  1Þ and I þ lðbÞ SðbÞ are symplectic. Remark 5.3. The coefficients AðtÞ; BðtÞ; CðtÞ; DðtÞ; WðtÞ are in the original time scale defined on ½a; qðbÞT . Thus, if the point b is left-scattered, then their values at t ¼ b do not exists. On the other hand, if b is left-dense, then although their values at t ¼ b do exist, they are never used, since all the computations depend in this case on the values on ½a; bÞT . Therefore,

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by the above formula (5.3) we define the coefficients AðtÞ; BðtÞ; CðtÞ; DðtÞ; WðtÞ at t ¼ b if b is left-scattered, and redefine them at t ¼ b if b is left-dense. In both cases, the new coefficients are uniquely determined on the time scale interval ½a  1; qðb þ 1ÞT ¼ ½a  1; bT . In addition, the new coefficient matrix SðÞ belongs to Cprd on ½a  1; bT , so that it defines on this interval a time scale symplectic system. On the extended time scale we now consider the time scale symplectic system

xD ¼ AðtÞ x þ BðtÞ u;

uD ¼ CðtÞ x þ DðtÞ u  k WðtÞ xr ;

ðSext k Þ

t 2 ½a  1; bT :

Since Wða  1Þ ¼ 0, the system ðSext k Þ at t ¼ a  1 does not depend on k. The relationship between the values of a solution ðXð; kÞ; Uð; kÞÞ of ðSext k Þ at t ¼ a  1 and t ¼ a is given by



Xða; kÞ





¼ ½I þ Sða  1Þ

Uða; kÞ

Xða  1; kÞ Uða  1; kÞ

 :

b kÞ; Uð; b kÞÞ of ðSext Þ satisfies In particular, the principal solution ð Xð; k

b ða; kÞ X b Uða; kÞ

!

¼ ½I þ Sða  1Þ

b ða  1; kÞ X b  1; kÞ Uða

!

¼

ðRa ÞT K a

RTa

RTa K a

ðRa ÞT

!  0 I

¼

RTa ðRa ÞT

! :

ð5:4Þ

This means, in view of the uniqueness of solutions of ðSk Þ, that the following holds. b kÞ; Uð; b kÞÞ of ðSext Þ coincides on ½a; b with the natural conjoined basis ðXð; kÞ; Uð; kÞÞ of Lemma 5.4. The principal solution ð Xð; k T ðSk Þ. In addition, we have

9 b ðb þ 1; kÞ ¼ Q T KðkÞ; X > > > > T  b > Uðb þ 1; kÞ ¼ K b Q ½Rb Uðb; kÞ  Rb Xðb; kÞ; > > = T y T b Mðb; kÞ ¼ Q ½I  KðkÞ K ðkÞ Rb ¼ Q MðkÞ; > > > > > Tb ðb; kÞ ¼ TðkÞ :¼ I  M y ðkÞ MðkÞ; > > ; y b Pðb; kÞ ¼ PðkÞ :¼ TðkÞ Xðb; kÞ K ðkÞ Rb TðkÞ;

ð5:5Þ

where, similarly to (2.9) and (2.13),

KðkÞ :¼ Rb Xðb; kÞ þ Rb Uðb; kÞ;

MðkÞ :¼ ½I  KðkÞ Ky ðkÞ Rb :

ð5:6Þ

Proof. The first conclusion of this lemma follows from the identity (5.4) by the uniqueness of solutions of systems ðSk Þ and ext ðSext k Þ. The first two formulas in (5.5) follow from the system ðSk Þ at t ¼ b. For the third and fourth ones we have ð4:5Þ b b r ðb; kÞ ½ X b r ðb; kÞy BðbÞ ¼ Q T Rb  Q T KðkÞ Ky ðkÞ Q Q T Rb ¼ Q T MðkÞ; Mðb; kÞ ¼ BðbÞ  X ð4:5Þ b y ðb; kÞ Mðb; b kÞ ¼ I  M y ðkÞ QQ T MðkÞ ¼ TðkÞ: Tb ðb; kÞ ¼ I  M

Here we used the formula ðQ T KÞy ¼ Ky Q , which holds when Q is an orthogonal matrix. Finally, the last equation in (5.5) is proven by the calculation ð4:5Þ b b b r ðb; kÞy BðbÞ Tb ðb; kÞ ¼ TðkÞ Xðb; kÞ Ky ðkÞ Rb TðkÞ: Pðb; kÞ ¼ Tb ðb; kÞ Xðb; kÞ ½ X

This completes the proof.

h

Similar formulas is in (5.5) hold also for vector-valued solutions ðxð; kÞ; uð; kÞÞ of system ðSext k Þ. In particular, we have

xða  1Þ ¼ Ra xðaÞ þ Ra uðaÞ;

xðb þ 1Þ ¼ Q T ½Rb xðbÞ þ Rb uðbÞ;

uða  1Þ ¼ K a ½Ra uðaÞ  Ra xðaÞ; uðb þ 1Þ ¼ K b Q T ½Rb uðbÞ  Rb xðbÞ:

)

ð5:7Þ

Lemma 5.5. A function ðxð; kÞ; uð; kÞÞ 2 C1prd defined on ½a  1; b þ 1T is a solution of the system ðSext k Þ if and only if the function ðxð; kÞ; uð; kÞÞ restricted to ½a; bT is a solution of the system ðSk Þ and (5.7) holds. In this case ðxð; kÞ; uð; kÞÞ satisfies the Dirichlet boundary conditions

xða  1Þ ¼ 0; xðb þ 1Þ ¼ 0

ð5:8Þ

if and only if it satisfies the separated boundary conditions (1.1). Moreover, for a given value k0 2 R the condition WðtÞ xr ðt; k0 Þ X0 on ½a  1; bT holds if and only if the corresponding condition (4.9) is satisfied. Proof. This result follows immediately from the identities in (5.7) and from the definition of the extended coefficients in (5.3). h

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Let us consider the quadratic functional associated with system ðSext k Þ, i.e.,

F ext ðz; kÞ :¼

Z

bþ1

a1

Wðt; zðtÞÞ Dt  k hz; ziext ;

hz; ~ziext :¼

Z

bþ1

½xr ðtÞT WðtÞ ~xr ðtÞ Dt;

a1

~ Þ are admissible over ½a  1; b þ 1T and Wðt; zÞ is defined in (4.16). Since Wða  1Þ ¼ 0 ¼ WðbÞ, where z ¼ ðx; uÞ and ~z ¼ ð~ x; u we have hz; ~ziext ¼ hz; ~ziW for every admissible z and ~z on ½a  1; b þ 1T . Lemma 5.6. The pair z ¼ ðx; uÞ is admissible on ½a  1; b þ 1T with (5.8) if and only if it is admissible on ½a; bT with (2.7). In this case, for every k 2 R we have F ext ðz; kÞ ¼ F ðz; kÞ, where F ð; kÞ is given in (5.2). Proof. This result follows by direct calculations, in which we prove that

F ext ðz; kÞ ¼ F 0 ðz; kÞ þ lða  1Þ Wða  1; zða  1ÞÞ þ lðbÞ Wðb; zðbÞÞ ¼ F ðz; kÞ; because

lða  1Þ Wða  1; zða  1ÞÞ ¼ xT ðaÞ Sa xðaÞ and lðbÞ Wðb; zðbÞÞ ¼ xT ðbÞ Sb xðbÞ: In the last formula we use the admissibility of z at t ¼ b, identity DT þ A þ lDT A ¼ lBT C from (4.2), and the boundary conditions (2.7). h The above lemma yields the following, compare with [19, Lemmas 4.7–4.8]. Corollary 5.7. Let k 2 R be fixed. The functional F ext ð; kÞ is positive definite (nonnegative) over the endpoints (5.8) if and only if the functional F ð; kÞ in (5.2) is positive definite (nonnegative) over the endpoints (2.7). In view of Lemma 5.5, it follows that the eigenvalue problem (5.1) is completely equivalent to the extended eigenvalue problem

ðSext with xða  1Þ ¼ 0 ¼ xðb þ 1Þ: k Þ

ð5:9Þ

Therefore, we define the finite eigenvalues and finite eigenfunctions for problem (5.1) according to the corresponding notions for the extended problem (5.9). Definition 5.8 (Finite eigenvalue for separated endpoints). Let ðXð; kÞ; Uð; kÞÞ be the natural conjoined basis of ðSk Þ. A number k0 2 R is a finite eigenvalue of the eigenvalue problem (5.1) if condition (2.10) holds with the matrix KðkÞ defined in (5.6). In this case, the number hðk0 Þ is called the algebraic multiplicity of the finite eigenvalue k0 . Remark 5.9. b b b (i) Since for the extended system ðSext k Þ we have Kðk0 Þ ¼ Q Xðb þ 1; k0 Þ where ð Xð; kÞ; Uð; kÞÞ is the principal solution of ðSext Þ, it follows that k is a finite eigenvalue of (5.1) according to Definition 5.8 with the algebraic multiplicity 0 k hðk0 Þ P 1 if and only if it is a finite eigenvalue of the same algebraic multiplicity for the extended eigenvalue problem (5.9), compare with (4.8). (ii) From Remark 4.5(ii) we have that the finite eigenvalues of (5.1) are real. (iii) Note that the eigenvalue problem (5.1) may be singular, i.e., we allow det KðkÞ ¼ 0 for all k 2 R, so that r < n is also included in our theory. For each k 2 R we define, counting the multiplicities,

n1 ðkÞ :¼ the number of proper focal points of ðXð; kÞ; Uð; kÞÞ in ða; bT ;

ð5:10Þ

n2 ðkÞ :¼ the number of finite eigenvalues of ð5:1Þ which are less than or equal to k;

ð5:11Þ

pðkÞ :¼ rank MðkÞ þ ind PðkÞ;

ð5:12Þ

where the matrices MðkÞ and PðkÞ are defined in (5.6) and (5.5), respectively. The following result is a generalization of Proposition 4.6 to the separated endpoints case. Theorem 5.10 (Oscillation theorem for separated endpoints). Assume (4.1) and that the natural conjoined basis ðXð; kÞ; Uð; kÞÞ of ðSk Þ satisfies condition (4.6). Then with n1 ; n2 ; p defined by (5.10), (5.11), (5.12), conditions (4.13) and

n2 ðkþ Þ  n2 ðk Þ ¼ n1 ðkþ Þ  n1 ðk Þ þ pðkþ Þ  pðk Þ P 0; hold for all k 2 R, and there exists m 2 N [ f0g such that

n1 ðkÞ þ pðkÞ ¼ n2 ðkÞ þ m for all k 2 R:

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Moreover, for a suitable k0 < 0 we have

n2 ðkÞ  0 and n1 ðkÞ þ pðkÞ  m for all k 6 k0 : Proof. Assumption (4.6) on ½a; bT for the natural conjoined basis ðXð; kÞ; Uð; kÞÞ of ðSk Þ is equivalent to the same condition for b kÞ; Uð; b kÞÞ of the extended system ðSext Þ on ½a  1; b þ 1 . By Remark 5.9(i), the number of finite the principal solution ð Xð; T k eigenvalues of (5.1) which are less than or equal to k is equal to the number of finite eigenvalues of the extended problem (5.9) which are less than or equal to k. Therefore, the result will follow from Proposition 4.6, once we count the number of proper b ð; kÞ; Uð; b kÞÞ in ða  1; b þ 1 . From X b ða  1; kÞ  0 for all k 2 R it follows that Xða; b focal points of ð X kÞ ¼ Bða  1Þ, so that T

b  1; kÞ  0; Mða

Tb ða  1; kÞ  I;

b  1; kÞ  0 for all k 2 R: Pða

b kÞ; Uð; b kÞÞ does not have any proper focal point in ða  1; a . On the other hand, from Lemma 5.4 we This means that ð Xð; T b b get that ð X ð; kÞ; Uð; kÞÞ has exactly ð5:12Þ b b rank Mðb; kÞ þ ind Pðb; kÞ ¼ rank MðkÞ þ ind PðkÞ ¼ pðkÞ

proper focal points in ðb; b þ 1T , while its number of proper focal points in ða; bT is equal to the number of proper focal points of the natural conjoined basis ðXð; kÞ; Uð; kÞÞ of ðSk Þ in ða; bT . Consequently, we need to replace the quantity n1 ðkÞ in Proposition 4.6 by the quantity n1 ðkÞ þ pðkÞ as defined in (5.10) and (5.12). This completes the proof. h Remark 5.11. As in Remark 4.7, condition n2 ðkÞ  0 for all k 6 k0 means that there are no finite eigenvalues of (5.1) before the number k0 . Therefore, under the assumptions of Theorem 5.10, the finite eigenvalues of (5.1) are bounded from below (provided there exists a finite eigenvalue at all). The following is a generalization of Proposition 4.8. Theorem 5.12 (Oscillation theorem for separated endpoints). Assume (4.1) and that the natural conjoined basis ðXð; kÞ; Uð; kÞÞ of ðSk Þ satisfies condition (4.6). Then with n1 ; n2 ; p defined by (5.10), (5.11), (5.12) we have

n1 ðkÞ þ pðkÞ ¼ n2 ðkÞ for all ðor for someÞ k 2 R if and only if there exists k0 < 0 such that the quadratic functional F ð; k0 Þ is positive definite. Proof. This result follows from Proposition 4.8 applied to (5.9), or alternatively directly from Theorem 5.10 by the aid of Proposition 5.2. h Remark 5.13. In [14, Proposition 2] for the discrete case with separated endpoints, as well as in the special time scale case with zero endpoints in [31, Section 9], one can prove that the positive definiteness of the quadratic functional F ð; k0 Þ used in Theorem 5.12 can be actually deduced from the positive definiteness of WðtÞ on ½a; qðbÞT . In this case and under the stronger assumption WðtÞ > 0 on ½a; qðbÞT , the oscillation theorem in Theorem 5.12 can be improved. However, for arbitrary time scales (for the zero endpoints case) or even for the continuous time case (with separated or jointly varying endpoints), this result remains open. Note that the method presented in this section is not suitable to derive this result, since the extension of WðtÞ at t ¼ a  1 and t ¼ b is zero (i.e., not positive definite). Following [31, Definition 5.3], we introduce the notion of finite eigenfunctions for the eigenvalue problem (5.1). A solution ðxð; k0 Þ; uð; k0 ÞÞ of (5.1) is called a finite eigenfunction corresponding to the finite eigenvalue k0 , if (4.9) is satisfied. The dimension xðk0 Þ of the associated eigenspace, i.e., the dimension of the functions Wxr where ðx; uÞ is a finite eigenfunction for k0 , is called the geometric multiplicity of k0 . The result below is a generalization of [31, Theorem 5.2] to the separated endpoints case. Theorem 5.14 (Geometric characterization of finite eigenvalues). Assume (4.1). A number k0 2 R is a finite eigenvalue of the eigenvalue problem (5.1) according to Definition 5.8 with algebraic multiplicity hðk0 Þ P 1 if and only if there exists a corresponding finite eigenfunction for k0 . In this case, the algebraic and geometric multiplicities of k0 coincide, i.e., hðk0 Þ ¼ xðk0 Þ. Proof. The statement is a direct consequence of Proposition 4.4 and Lemma 5.5, since the finite eigenfunctions for the extended eigenvalue problem (5.9) generate uniquely the finite eigenfunctions of (5.1). h Remark 5.15. By Lemma 5.5 and Remarks 4.5 and 5.9(i), we have that under (4.1) the finite eigenfunctions of (5.1) corresponding to different finite eigenvalues of (5.1) are orthogonal with respect to the semi-inner product (4.10). We conclude this section by deriving the Rayleigh principle for the eigenvalue problem (5.1). This is a generalization of [35, Theorem 4.1] to the separated endpoints case.

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Theorem 5.16 (Rayleigh principle for separated endpoints). Assume that the natural conjoined basis ðXð; kÞ; Uð; kÞÞ of ðSk Þ satisfies condition (4.6), the functional F ð; k0 Þ is positive definite for some k0 < 0, and (4.1) holds. Let k1 6 . . . 6 km 6 . . . be the finite eigenvalues of the eigenvalue problem (5.1) with the corresponding orthonormal finite eigenfunctions z1 ; . . . ; zm ; . . .. Then for each m 2 N [ f0g

kmþ1 ¼ min

  F ðz; 0Þ ; z ¼ ðx; uÞ 2 A with ð1:7Þ; ðWxr ÞðÞX0; and z ? z1 ; . . . ; zm : hz; ziW

Proof. This statement follows directly from Proposition 4.10 combined with Lemma 5.6 and Corollary 5.7, which are applied to the extended functional F ext . h We say that the functional F ð; 0Þ is coercive if there exists a > 0 such that

F ðz; 0Þ P a

Z

b

jxr ðtÞj2 Dt

for every admissiblez ¼ ðx; uÞwithð2:7Þ:

a

Theorem 5.17 (Coercivity for separated endpoints). The functional F ð; 0Þ in (5.2) is coercive if and only if it is positive definite. Proof. This result is a consequence of Theorem 4.11 combined with Lemma 5.6. h Remark 5.18. In [19, Section 4], another transformation of the separated endpoints problem (in the form of projections) to a problem with the Dirichlet boundary conditions was proposed. This alternative transformation extends the time scale interval ½a; bT by fa  1g at the left endpoint, and by fb þ 1; b þ 2g at the right endpoint. Thus, this new extended problem has ½a  1; b þ 2T as the base time scale interval. However, the oscillation theorem for the separated endpoints resulting from the application of Proposition 4.6 to this extended problem yields exactly the same result as displayed in Theorem 5.10. b kÞ; Uð; b kÞÞ of the extended system again coincides on ½a; b with the natural The reason is that the principal solution ð Xð; T conjoined basis ðXð; kÞ; Uð; kÞÞ of ðSk Þ, and it has no proper focal points in ða  1; aT and in ðb; b þ 1T , while it has exactly pðkÞ proper focal points in ðb þ 1; b þ 2T . The method in the present paper is however shorter and more transparent. 6. Spectral theory for problems with jointly varying endpoints In this section we further extend the results in Section 5 to the most general self-adjoint boundary conditions, i.e., to the jointly varying endpoints (1.2), where the 2n  2n matrices R and R satisfy (1.6). This will allow to include, for example, the problems with the periodic and antiperiodic boundary conditions, which could not be treated by the results in the previous section, see Remark 6.17 below. Consider the eigenvalue problem

ðSk Þ with R



xðaÞ xðbÞ



þR



uðaÞ uðbÞ



¼ 0;

ð6:1Þ

and the associated quadratic functional

 Gðz; kÞ :¼ F 0 ðz; kÞ þ

T   xðaÞ S ; xðbÞ xðbÞ

xðaÞ

ð6:2Þ

where z ¼ ðx; uÞ, the functional F 0 ð; kÞ is given in (4.15), and where the symmetric matrix S is defined by (2.20). We say that Gð; kÞ is positive definite if Gðz; kÞ > 0 for all admissible z ¼ ðx; uÞ satisfying (2.21) and x – 0. The following two results correspond to Propositions 5.1 and 5.2, see [23, Theorem 4.1]. In the second statement we utilize the augmented 2n  2n mab ð; kÞ; Uð; b kÞÞ and trix-valued functions X and U, which are defined through the principal and associated solutions ð X e kÞ; Uð; e kÞÞ of ðSk Þ, respectively, by ð Xð;

 Xðt; kÞ :¼

0

I

b ðt; kÞ X

e ðt; kÞ X

 ;

 Uðt; kÞ :¼

I

0

b kÞ Uðt;

e kÞ Uðt;

 :

ð6:3Þ

b kÞ; Uð; b kÞÞ and ð X e ð; kÞ; Uð; e kÞÞ of ðSk Þ are determined by the initial Recall that the principal and associated solutions ð Xð; conditions (2.1) and (2.18). Proposition 6.1 (Comparison theorem). Let k0 2 R and (4.1) hold. The functional Gð; k0 Þ is positive definite (nonnegative) if and only if the functional Gð; kÞ is positive definite (nonnegative) for every k 6 k0 . Proposition 6.2 (Positivity). Let k0 2 R be fixed. The functional Gð; k0 Þ is positive definite if and only if the principal solution b ð; k0 Þ; Uð; b k0 Þ ( ) has no proper focal points in ða; b , and the matrices Xðb; k0 Þ and Uðb; k0 Þ defined in (6.3) satisfy the inequality X T

S þ Uðb; k0 Þ X y ðb; k0 Þ > 0 on Im RT \ Im Xðb; k0 Þ:

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The technique for handling problems jointly varying endpoints is known to be (both in the discrete and time scale environment) the augmenting of the coefficients of the system ðSk Þ into dimension 2n to obtain a problem with separated endpoints, see [19,23,24] in the time scales case and [5,17,18,20,22] in the discrete case. And to this augmented problem the results from Section 5 can be applied. Define on ½a; bT the 2n  2n matrices



AðtÞ :¼

0

0

0 AðtÞ

 ;



BðtÞ :¼

0

0

0 BðtÞ

 ;

 CðtÞ :¼

0

0

0 CðtÞ

 ;

 DðtÞ :¼

0

0

0 DðtÞ

 ;

WðtÞ :¼



0

0

0 WðtÞ

 ; ð6:4Þ

and the augmented time scale symplectic system

xD ¼ AðtÞ x þ BðtÞ u;

uD ¼ CðtÞ x þ DðtÞ u  k WðtÞ xr ;

t 2 ½a; qðbÞT

with 2n-vector-valued solutions ðxð; kÞ; uð; kÞÞ. Since ðSaug k Þ is a time scale symplectic system in the traditional sense of (4.2), all the theory of time scale symplectic systems from Section 4 can be used to study the system ðSaug k Þ. In particular, we have for system ðSaug Þ the notions of conjoined bases, proper focal points, admissible pairs, and since WðtÞ P 0 on ½a; bT under (4.1), also k the notions of finite eigenvalues and finite eigenfunctions when the appropriate boundary conditions are introduced. Note that since the augmented coefficients in (6.4) have zeros in the first row, the solutions of system ðSaug k Þ are of the form



Xðt; kÞ ¼

X2

X1

X 3 ðt; kÞ X 4 ðt; kÞ

 ;



Uðt; kÞ ¼

U2

U1

U 3 ðt; kÞ U 4 ðt; kÞ

 ;

t 2 ½a; bT ;

where X 1 ; X 2 ; U 1 ; U 2 are constant n  n matrices and ðX 1 ð; kÞ; U 1 ð; kÞÞ and ðX 2 ð; kÞ; U 2 ð; kÞÞ are solutions of the system ðSk Þ. In particular, it is easily seen that the solution ðXð; kÞ; Uð; kÞÞ defined by (6.3) is a conjoined basis of ðSaug k Þ. 4.2. Namely, the normality Remark 6.3. The augmented system ðSaug k Þ does not satisfy the normality condition in Remark   c on ½a; bT , where c 2 Rn is a condition is violated by the constant solutions ðxð; kÞ; uð; kÞÞ with xðt; kÞ  0 and uðt; kÞ  0 nonzero constant. Therefore, it is essential for the theory with jointly varying endpoints to have the previous results on separated endpoints ready without this normality assumption. Remark 6.4. Continuing the discussion in Remark 5.13, one cannot expect to derive the oscillation theorems for jointly varying endpoints under WðtÞ > 0 on ½a; qðbÞT by the methods of this section, since the augmented matrix WðtÞ in (6.4) is only nonnegative definite on ½a; qðbÞT . Following (4.4), the definition of proper focal points for the conjoined basis ðXð; kÞ; Uð; kÞÞ of ðSaug k Þ in (6.3) contains the matrices

P :¼ TXðX r Þy BT;

T :¼ I  My M;

M :¼ ½I  X r ðX r Þy  B;

from which the first two reduce as in [17, Lemma 2], see also [23, pg. 458], to the form

 Pðt; kÞ ¼

0

0

0

b kÞ Pðt;

 ;

 Tðt; kÞ ¼

I

0

0

Tb ðt; kÞ

 ;

t 2 ½a; qðbÞT ;

b kÞ and T b ðt; kÞ are defined in (4.5) through the principal solution ð X b ð; kÞ; Uð; b kÞÞ of ðSk Þ. And since where Pðt;   I b kÞ holds, it follows that the conjoined basis ðXð; kÞ; Uð; kÞÞ defined in (6.3) has the same number Ker Xðt; Ker Xðt; kÞ ¼ 0 b ð; kÞ; Uð; b kÞÞ. of proper focal points in ða; b as the principal solution ð X T

Define now the 2n  2n matrices

Ra :¼



0

0

I

I

 ;

Ra :¼



I

I

0 0

 ;

Rb :¼ R;

Rb :¼ R ;

Sa :¼ 0;

Sb :¼ S:

It is easy to check that the required properties in (1.5) and (2.6) are satisfied with the above data, since Rya ¼ 12 RTa . Consider the augmented eigenvalue problem with separated endpoints

ðSaug with Ra xðaÞ þ Ra uðaÞ ¼ 0; k Þ

Rb xðbÞ þ Rb uðbÞ ¼ 0;

ð6:5Þ

and the augmented quadratic functional, compare with (5.2),

F ðz; kÞ :¼ F 0 ðz; kÞ þ xT ðbÞ Sb xðbÞ  xT ðaÞ Sa xðaÞ with the separated endpoints, compare with (2.7),

xðaÞ 2 Im RTa ;

xðbÞ 2 Im RTb ;

ð6:6Þ

R. Šimon Hilscher, V. Zeidan / Applied Mathematics and Computation 218 (2012) 8309–8328

8325

where z ¼ ðx; uÞ and where the functional F 0 ð; kÞ is defined according to (4.15) through the augmented coefficients in (6.4). Since Xða; kÞ ¼ RTa and Uða; kÞ ¼ ðRa ÞT , compare with (2.4), it follows that ðXð; kÞ; Uð; kÞÞ is the natural conjoined basis of system ðSaug k Þ. Lemma 6.5. (i) A function ðxð; kÞ; uð; kÞÞ is a solution of the eigenvalue problem (6.5) if and only if it has the form

xðt; kÞ ¼



xðaÞ xðt; kÞ

 ;

uðt; kÞ ¼





uðaÞ uðt; kÞ

;

t 2 ½a; bT ;

ð6:7Þ

where ðxð; kÞ; uð; kÞÞ is a solution of the eigenvalue problem (6.1). (ii) A function ðx; uÞ is admissible for the functional F ð; kÞ and satisfies (6.6) if and only if it has the form

xðtÞ ¼



xðaÞ xðtÞ

 ;

t 2 ½a; bT ;

uðtÞ ¼



bðtÞ uðtÞ

 ;

t 2 ½a; qðbÞT ;

where ðx; uÞ is admissible for Gð; kÞ and satisfies (2.21) and bðtÞ 2 Rn for all t 2 ½a; qðbÞT . In addition, in this case we have F ðz; kÞ ¼ Gðz; kÞ. Proof. The statements follow by direct calculations.

h

The above lemma yields the following result, compare with [19, Lemma 5.5]. Corollary 6.6. Let k 2 R be fixed. The functional F ð; kÞ is positive definite (nonnegative) over the endpoints (6.6) if and only if the functional Gð; kÞ in (6.2) is positive definite (nonnegative) over the endpoints (2.21). In view of Lemma 6.5, the eigenvalue problem (6.1), which has jointly varying endpoints, is completely equivalent to the eigenvalue problem ðSaug k Þ with separated endpoints. Therefore, we define the finite eigenvalues and finite eigenfunctions for problem (6.1) in terms of the corresponding notions for the augmented problem (6.5). Definition 6.7 (Finite eigenvalue for joint endpoints). Let ðXð; kÞ; Uð; kÞÞ be defined by (6.3) through the principal and b kÞ; Uð; b kÞÞ and ð Xð; e kÞ; Uð; e kÞÞ of ðSk Þ. A number k0 2 R is a finite eigenvalue of the eigenvalue associated solutions ð Xð; problem (6.1) if condition (2.23) holds with the matrix KðkÞ defined by

KðkÞ :¼ R Xðb; kÞ þ R Uðb; kÞ with Xðb; kÞ and Uðb; kÞ from ð6:3Þ:

ð6:8Þ

In this case, the number hðk0 Þ is called the algebraic multiplicity of the finite eigenvalue k0 . Remark 6.8. (i) Since Kðk0 Þ ¼ Rb Xðb; k0 Þ þ Rb Uðb; k0 Þ, the number k0 is a finite eigenvalue of (6.1) according to Definition 6.7 with the algebraic multiplicity hðk0 Þ P 1 if and only if it is a finite eigenvalue of the same algebraic multiplicity for the augmented eigenvalue problem (6.5), see Definition 5.8. (ii) By part (i) and Remark 5.9(ii), the finite eigenvalues of (6.1) are real. (iii) Note that the eigenvalue problem (6.1) may be singular, i.e., we allow det KðkÞ ¼ 0 for all k 2 R, so that s < n is also included in our theory. Define the 2n  2n matrices MðkÞ; TðkÞ, and PðkÞ by

MðkÞ :¼ ½I  KðkÞ Ky ðkÞ R; TðkÞ :¼ I  My ðkÞ MðkÞ; PðkÞ :¼ TðkÞ Xðb; kÞ Ky ðkÞ R TðkÞ;

9 > = > ;

ð6:9Þ

compare with (5.5) and (2.26). For each k 2 R we define, counting the multiplicities,

b ð; kÞ; Uð; b kÞ in ða; b ; n1 ðkÞ :¼ the number of proper focal points of ð X T

ð6:10Þ

n2 ðkÞ :¼ the number of finite eigenvalues of ð6:1Þ which are less than or equal to k;

ð6:11Þ

qðkÞ :¼ rank MðkÞ þ ind PðkÞ:

ð6:12Þ

The following result is a generalization of Proposition 4.6 and Theorem 5.10 to joint endpoints. b ð; kÞ; Uð; b kÞÞ of ðSk Þ Theorem 6.9 (Oscillation theorem for joint endpoints). Assume (4.1) and that the principal solution ð X satisfies condition (4.6). Then with n1 ; n2 ; q defined by (6.10), (6.11), (6.12), conditions (4.13) and

n2 ðkþ Þ  n2 ðk Þ ¼ n1 ðkþ Þ  n1 ðk Þ þ qðkþ Þ  qðk Þ P 0

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hold for all k 2 R, and there exists m 2 N [ f0g such that

n1 ðkÞ þ qðkÞ ¼ n2 ðkÞ þ m for all k 2 R: Moreover, for a suitable k0 < 0 we have

n2 ðkÞ  0 and n1 ðkÞ þ qðkÞ  m for all k 6 k0 : b ð; kÞ; Uð; b kÞÞ of ðSk Þ is equivalent to the same condition for the natural Proof. Assumption (4.6) for the principal solution ð X conjoined basis ðXð; kÞ; Uð; kÞÞ of the augmented system ðSaug k Þ. And since by Remark 6.8(i) the numbers of finite eigenvalues of (6.1) and (6.5) which are less than or equal to k are the same, the statement then follows from Theorem 5.10 applied to the problem (6.5). h Remark 6.10. Similarly to Remark 5.11, the assumptions of Theorem 6.9 imply that the finite eigenvalues of (6.1) are bounded from below (provided there exists a finite eigenvalue at all). The following is a generalization of Proposition 4.8 and Theorem 5.12 to jointly varying endpoints. b ð; kÞ; Uð; b kÞÞ of ðSk Þ Corollary 6.11 (Oscillation theorem for joint endpoints). Assume (4.1) and that the principal solution ð X satisfies condition (4.6). Then with n1 ; n2 ; q defined by (6.10), (6.11), (6.12) we have

n1 ðkÞ þ qðkÞ ¼ n2 ðkÞ for all ðor for someÞ k 2 R if and only if there exists k0 < 0 such that the quadratic functional Gð; k0 Þ is positive definite. Proof. This result follows from Theorem 5.12 by the aid of Corollary 6.6, or alternatively directly from Theorem 6.9. h A solution ðxð; k0 Þ; uð; k0 ÞÞ of (6.1) is called a finite eigenfunction corresponding to the finite eigenvalue k0 , if (4.9) is satisfied. The dimension xðk0 Þ of the associated eigenspace is called the geometric multiplicity of k0 . The next result is a generalization of Theorem 5.14 to the jointly varying endpoints case. Theorem 6.12 (Geometric characterization of finite eigenvalues). Assume (4.1). A number k0 2 R is a finite eigenvalue of the eigenvalue problem (6.1) according to Definition 6.7 with algebraic multiplicity hðk0 Þ P 1 if and only if there exists a corresponding finite eigenfunction for k0 . In this case, the algebraic and geometric multiplicities of k0 coincide, i.e., hðk0 Þ ¼ xðk0 Þ. Proof. This result is a consequence of Theorem 5.14 and Lemma 6.5(i), since the finite eigenfunctions for the augmented eigenvalue problem (6.5) and the finite eigenfunctions of (6.1) are in one-to-one correspondence via (6.7). h Remark 6.13. From Lemma 6.5(i) and Remark 5.15 we have that under (1.4) the finite eigenfunctions of (6.1) corresponding to different finite eigenvalues of (6.1) are orthogonal with respect to the semi-inner product (4.10). Our next result in this section represents a generalization of Proposition 4.10 and Theorem 5.16 to the jointly varying endpoints case. b kÞ; Uð; b kÞÞ of ðSk Þ Theorem 6.14 (Rayleigh principle for jointly varying endpoints). Assume that the principal solution ð Xð; satisfies condition (4.6), the functional Gð; k0 Þ is positive definite for some k0 < 0, and (4.1) holds. Let k1 6 . . . 6 km 6 . . . be the finite eigenvalues of the eigenvalue problem (6.1) with the corresponding orthonormal finite eigenfunctions z1 ; . . . ; zm ; . . .. Then for each m 2 N [ f0g

kmþ1 ¼ min



 Gðz; 0Þ ; z ¼ ðx; uÞ 2 A with ð2:21Þ; ðWxr ÞðÞX0; and z ? z1 ; . . . ; zm : hz; ziW

Proof. This result follows directly from Theorem 5.16, Lemma 6.5, and Corollary 6.6 applied to the augmented functional F. h We say that the functional Gð; 0Þ is coercive if there exists a > 0 such that

Gðz; 0Þ P a

Z

b

jxr ðtÞj2 Dt

for every admissible z ¼ ðx; uÞ with ð2:21Þ:

a

Theorem 6.15 (Coercivity for joint endpoints). The functional Gð; 0Þ in (6.2) is coercive if and only if it is positive definite.

R. Šimon Hilscher, V. Zeidan / Applied Mathematics and Computation 218 (2012) 8309–8328

8327

Proof. This result is a consequence of Theorem 5.17 combined with Lemma 6.5(ii). h Remark 6.16. The expansion of the admissible pairs z ¼ ðx; uÞ with (2.7) or (2.21) in terms of the finite eigenfunctions of (5.1) or (6.1) is another classical result in this context, see [35, Theorem 4.7]. The analysis in Sections 5 and 6 reveals that the result of [35, Theorem 4.7] remains valid also in the more general case of separated or jointly varying endpoints. Remark 6.17. (i) The eigenvalue problem (6.1) contains as a special case the periodic boundary conditions xðaÞ ¼ xðbÞ and uðaÞ ¼ uðbÞ, for which we take

 R¼

I

I

0

0

 ;





R ¼

0 0 I

I

 ;

S ¼ 0;

KðkÞ ¼ I  J

b Xðb; kÞ b Uðb; kÞ

! e ðb; kÞ X ; e Uðb; kÞ

and the antiperiodic boundary conditions xðaÞ ¼ xðbÞ and uðaÞ ¼ uðbÞ, for which we take

 R¼

I

I

0 0

 ;

R ¼



0

0

I

I

 ;

S ¼ 0;

KðkÞ ¼ I þ J

b Xðb; kÞ b Uðb; kÞ

! e ðb; kÞ X : e Uðb; kÞ

In both cases we have Ry ¼ 12 RT and the matrix KðkÞ is calculated from Eq. (6.8). The resulting oscillation theorems and the Rayleigh principle for periodic and antiperiodic endpoints then easily follow from Theorems 6.9, 6.11, and 6.14. (ii) The quadratic functional Gð; 0Þ in (6.2) or F ð; 0Þ in (5.2) can be regarded as the second variation of a nonlinear time scale optimal control problem, see [25, Section 7]. Applications of the oscillation theorems and the Rayleigh principles obtained in this paper to second order optimality conditions in time scale control problems will be addressed in our future work. 7. Conclusion In this paper we extended the known spectral theory (i.e., we derived the oscillation theorems, geometric properties of finite eigenvalues, Rayleigh principle, and coercivity results) of linear Hamiltonian and symplectic systems to cover the problems with the separated and jointly varying endpoints, thus including the problems with the periodic boundary conditions. For the symplectic dynamic systems on time scales, we extended the oscillation theorems and the Rayleigh principle from the Dirichlet endpoints in [31, Theorem 6.2 and Corollary 6.3] and [35, Theorem 4.1] to the separated endpoints (Theorems 5.10, 5.12, and 5.16) and to the jointly varying endpoints (Theorems 6.9, 6.11, and 6.14). As special cases we obtained new results both for the continuous and discrete time. For the continuous time linear Hamiltonian systems, we improved the oscillation theorem in [36, Theorem 1.5] and extended at the same time the oscillation theorem and the Rayleigh principle in [30, Theorems 2.9 and 1.1] from the Dirichlet endpoints to separated endpoints (Theorems 2.3 and 2.5) and to jointly varying endpoints (Theorems 2.9 and 2.11). At the same time we removed the normality and strong observability assumptions in the oscillation theorems and Rayleigh principle in [27, Theorems 7.2.2 and 7.7.1]. Concerning the coercivity of the quadratic functional, we proved that it is equivalent with its positivity (Theorems 4.11, 5.17, and 6.15). In particular, our result (Theorem 2.6) removes the normality assumption in [38, Theorem 4.2] and [37, Theorem 5.5], and simultaneously generalizes [30, Theorem 4.5] from the Dirichlet endpoints to the separated ones. Its extension to the jointly varying endpoints (Theorem 2.12) is also new. For the discrete symplectic systems, we extended the oscillation theorem and Rayleigh principle in [14, Theorems 1 and 2] from the separated endpoints to jointly varying endpoints (Theorems 3.1 and 3.2). Therefore, our paper demonstrates the utility of the analysis on time scales in establishing not only unified results for the continuous and discrete time theories, but as well as in obtaining new results for each of the continuous and discrete cases. Acknowledgements The first author wishes to thank the Michigan State University for the hospitality provided while conducting a part of this research project. The support of the research project ME 891 (program Kontakt) of the Ministry of Education, Youth, and Sports of the Czech Republic and the grant DMS – 0707789 of the National Science Foundation is highly appreciated. The authors are grateful to an anonymous referee for detailed reading of the paper and for the suggested improvements. References [1] C.D. Ahlbrandt, M. Bohner, J. Ridenhour, Hamiltonian systems on time scales, J. Math. Anal. Appl. 250 (2) (2000) 561–578. [2] C.D. Ahlbrandt, A.C. Peterson, Discrete Hamiltonian Systems. Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Texts in the Mathematical Sciences, vol. 16, Kluwer Academic Publishers., Group, Dordrecht, 1996. [3] A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory and Applications, second ed., Springer-Verlag, New York, NY, 2003.

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[4] D.S. Bernstein, Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory, Princeton University Press, Princeton, 2005. [5] M. Bohner, O. Došly´, W. Kratz,Discrete Reid roundabout theorems,in: ‘‘Discrete and Continuous Hamiltonian Systems’’, R.P. Agarwal and M. Bohner, editors, Dynam. Systems Appl. 8(3–4) (1999) 345–352. [6] M. Bohner, O. Došly´, W. Kratz, An oscillation theorem for discrete eigenvalue problems, Rocky Mountain J. Math. 33 (4) (2003) 1233–1260. [7] M. Bohner, O. Došly´, W. Kratz, Sturmian and spectral theory for discrete symplectic systems, Trans. Amer. Math. Soc. 361 (6) (2009) 3109–3123. [8] M. Bohner, R. Hilscher, An eigenvalue problem for linear Hamiltonian dynamic systems, Fasc. Math. 35 (2005) 35–49. [9] M. Bohner, A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser, Boston 2001 [10] M. Bohner, A. Peterson (Eds.), Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. [11] O. Došly´, S. Hilger, R. Hilscher, Symplectic dynamic systems, in: M. Bohner, A. Peterson (Eds.), Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003, pp. 293–334. [12] O. Došly´, R. Hilscher, Disconjugacy transformations and quadratic functionals for symplectic dynamic systems on time scales, J. Difference Equ. Appl. 7 (2001) 265–295. [13] O. Došly´, W. Kratz, Oscillation theorems for symplectic difference systems, J. Difference Equ. Appl. 13 (7) (2007) 585–605. [14] O. Došly´, W. Kratz, Oscillation and spectral theory for symplectic difference systems with separated boundary conditions, J. Difference Equ. Appl. 16 (7) (2010) 831–846. [15] R. Hilscher,Linear Hamiltonian systems on time scales: positivity of quadratic functionals,in: ‘‘Boundary value problems and related topics’’, A.C. Peterson, editor, Math. Comput. Modelling 32(5–6) (2000) 507–527. [16] R. Hilscher, W. Kratz, V. Zeidan, Differentiation of solutions of dynamic equations on time scales with respect to parameters, Adv. Dyn. Syst. Appl. 4 (1) (2009) 35–54. [17] R. Hilscher, V. Ru˚zˇicˇková, Riccati inequality and other results for discrete symplectic systems, J. Math. Anal. Appl. 322 (2) (2006) 1083–1098. [18] R. Hilscher, V. Ru˚zˇicˇková, Implicit Riccati equations and quadratic functionals for discrete symplectic systems, Int. J. Difference Equ. 1 (1) (2006) 135– 154. [19] R. Hilscher, V. Ru˚zˇicˇková, Perturbation of time scale quadratic functionals with variable endpoints, Adv. Dyn. Syst. Appl. 2 (2) (2007) 207–224. [20] R. Hilscher, V. Zeidan, Symplectic difference systems: variable stepsize discretization and discrete quadratic functionals, Linear Algebra Appl. 367 (2003) 67–104. [21] R. Hilscher, V. Zeidan, Time scale symplectic systems without normality, J. Difference Equ. Appl. 230 (1) (2006) 140–173. [22] R. Hilscher, V. Zeidan, Coupled intervals for discrete symplectic systems, Linear Algebra Appl. 419 (2–3) (2006) 750–764. [23] R. Hilscher, V. Zeidan, Applications of time scale symplectic systems without normality, J. Math. Anal. Appl. 340 (1) (2008) 451–465. [24] R. Hilscher, V. Zeidan, Riccati equations for abnormal time scale quadratic functionals, J. Differential Equations 244 (6) (2008) 1410–1447. [25] R. Hilscher, V. Zeidan, Weak maximum principle and accessory problem for control problems on time scales, Nonlinear Anal. 70 (9) (2009) 3209–3226. [26] R. Hilscher, V. Zeidan, Multiplicities of focal points for discrete symplectic systems: revisited, J. Difference Equ. Appl. 15 (10) (2009) 1001–1010. [27] W. Kratz, Quadratic Functionals in Variational Analysis and Control Theory, Akademie Verlag, Berlin, 1995. [28] W. Kratz, Discrete oscillation, J. Difference Equ. Appl. 9 (1) (2003) 135–147. [29] W. Kratz, Definiteness of quadratic functionals, Analysis (Munich) 23 (2) (2003) 163–183. [30] W. Kratz, R. Šimon Hilscher, Rayleigh principle for linear Hamiltonian systems without controllability, ESAIM Control Optim. Calc. Var., (2012). doi:10.1051/cocv/2011104. [31] W. Kratz, R. Šimon Hilscher, V. Zeidan, Eigenvalue and oscillation theorems for time scale symplectic systems, in: Dynamic Equations on Time Scales and Applications, L. Erbe and A. Peterson (Eds.), Int. J. Dyn. Syst. Differ. Equ. 3(1–2) (2011) 84–131. [32] W.T. Reid, Ordinary Differential Equations, Wiley, New York, 1971. [33] G. Stefani, P. Zezza, Constrained regular LQ-control problems, SIAM J. Control Optim. 35 (3) (1997) 876–900. [34] R. Šimon Hilscher, V. Zeidan, Picone type identities and definiteness of quadratic functionals on time scales, Appl. Math. Comput. 215 (7) (2009) 2425– 2437. [35] R. Šimon Hilscher, V. Zeidan, Rayleigh principle for time scale symplectic systems and applications, Electron. J. Qual. Theory Differ. Equ. 26 (83) (2011) (electronic). [36] M. Wahrheit, Eigenvalue problems and oscillation of linear Hamiltonian systems, Int. J. Difference Equ. 2 (2) (2007) 221–244. [37] V. Zeidan, Nonnegativity and positivity of a quadratic functional, in: Discrete and Continuous Hamiltonian Systems, R. P. Agarwal and M. Bohner (Eds.), Dynam. Systems Appl. 8(3–4) (1999) 571–588. [38] V. Zeidan, New second-order optimality conditions for variational problems with C 2 -Hamiltonians, SIAM J. Control Optim. 40 (2) (2001) 577–609.