RAYLEIGH PRINCIPLE FOR TIME SCALE SYMPLECTIC SYSTEMS AND APPLICATIONS

V tomto clanku jsme odvodili Rayleighův princip, tj. variacni charakterizaci vlastnich hodnot, pro obecnou okrajovou ulohu sestavajici ze symplektickeho systemu na casove skale a Dirichletových okrajových podminek. Nepředpokladame žadnou normalitu ci kontrolovatelnost uvažovaneho systemu. Jako aplikace jsme odvodili Sturmovy srovnavaci a oddělovaci věty pro symplekticke systemy na casových skalach. Tento clanek zobecňuje a sjednocuje nedavne výsledky pro diskretni symplekticke systemy a spojite linearni hamiltonovske systemy. Nase výsledky jsou take nove a zejmena zajimave i pro tzv. "specialni" casovou skalu, tj. pro casovou skalu, ktera je sjednocenim konecně mnoha diskunktnich kompaktnich intervalů a/nebo konecně mnoha izolovaných bodů.


Introduction
In this paper we consider the eigenvalue problem where (S λ ) is the time scale symplectic system x ∆ = A(t) x + B(t) u, u ∆ = C(t) x + D(t) u − λ W (t) x σ , t ∈ [a, ρ(b)] T , (S λ ) and λ ∈ R is a spectral parameter.Here we consider a bounded time scale T and with a := min T and b := max T we represent T as the time scale interval [a, b] T .For the theory of dynamic equations on time scales and its basic notation we refer to [7,8,13].The coefficients of system (S λ ) are piecewise rd-continuous (C prd ) n × n matrix functions on [a, ρ(b)] T satisfying S T (t) J + J S(t) + µ(t) S T (t) J S(t) = 0, W (t) symmetric, t ∈ [a, ρ(b)] T , where 0 and I are the zero and identity matrices of appropriate dimensions.The word "symplectic" refers to the fact that under (1.1) the fundamental matrix of system (S λ ) is a symplectic 2n × 2n matrix.In the present paper we require no controllability or normality of system (S λ ).This implies that solutions of (S λ ) may be singular on nontrivial subintervals of [a, b] T or even on the whole interval [a, b] T , see Section 2 for more details.
In the continuous time case, i.e., when T = [a, b] is a real connected interval, system (S λ ) is the linear Hamiltonian system where the coefficients are piecewise continuous (C p ) on [a, b] with B(•) and C(•) symmetric.
In the classical results (under normality), the eigenvalue problem for the system (H λ ) is considered in [17].When the normality assumption is absent, the corresponding oscillation and eigenvalue theory was developed in [25] and more recently in [20,23].In particular, the latter two papers contain respectively the Rayleigh principle and the Sturmian theory for such possibly "abnormal" systems (H λ ).
The above results were motivated by the corresponding discrete time theory in [6].Specifically, in the discrete time setting the system (S λ ) reduces to the discrete symplectic system where [0, N] Z := {0, 1, . . ., N}, see also [1,4,11].Since the interval [0, N] Z contains only finitely many points, the normality assumption is naturally absent in the oscillation and eigenvalue theory for system (1.4).
The time scale eigenvalue problem (E) was introduced in [21].In this reference, the oscillation theorem was proven, which relates the number of eigenvalues (called the finite eigenvalues, see Section 2) of (E) which are less or equal to a given number λ and the number of proper focal points of a special solution of the system (S λ ).In the present paper we first derive the corresponding Rayleigh principle for the eigenvalue problem (E), i.e., we prove the variational characterization of the finite eigenvalues (see Theorem 4.1).This result generalizes the continuous and discrete time statements in [20,Theorem 1.1] and [6,Theorem 4.6] to arbitrary time scales.In the second part of this paper we then apply the oscillation theorem from [21,Corollary 6.4] and the new Rayleigh principle to obtain the Sturmian comparison and separation theorems for time scale symplectic systems, thus generalizing the corresponding results in [6] and [23] to arbitrary time scales.The new results in this paper are important not only on their own.For example, the Rayleigh principle (Theorem 4.1) can be used as a tool for deriving further new results for problems with more general boundary conditions, see e.g.[16, pg. 453] for the description of such a method.We shall proceed in this way in our subsequent work.
Our results in this paper are new and interesting even in the case when the underlying time scale [a, b] T is "special" in the sense that it is the union of finitely many disjoint real intervals and/or finitely many isolated points.In such a case, a certain assumption made for the general time scales reduces to a simple condition (the Legendre condition) on the coefficient B(•) over the continuous parts of [a, b] T .
The paper is organized as follows.In Section 2 we recall the basic properties of the eigenvalue problem (E).In Section 3 we collect some technical calculations related to admissible pairs of functions.In Section 4 we state and prove the Rayleigh principle for problem (E), while in Section 5 we establish the Sturmian comparison and separation theorems for time scale symplectic systems.The final section contains the discussion related to the above mentioned "special" time scales.EJQTDE, 2011 No. 83, p. 2 2. Time scale symplectic systems By using the expansion x σ = x + µx ∆ , the system (S) can be written in the form where we put By a direct calculation it follows that the matrix S(t) − λQ(t) also satisfies condition (1.1)(i) for all t ∈ [a, ρ(b)] T and λ ∈ R. As a consequence we have the coefficient matrix S(•) − λQ(•) regressive on [a, ρ(b)] T and hence, the system (S λ ) possesses unique (vector or matrix) solutions on [a, b] T once the initial conditions are prescribed at any point The solutions of (S λ ) belong to the set C 1 prd of piecewise rd-continuously deltadifferentiable functions on [a, b] T , i.e., they are continuous on [a, b] T and their ∆-derivative is in C prd .We adopt a usual convention that the vector and matrix solutions of (S λ ) or equivalently of system (2.1) will be denoted by small and capital letters, respectively, typically by z( Since the dependence on λ in system (S λ ) is linear, it follows by [14,Corollary 4.5] that the solutions of (S λ ) are entire functions in λ when their initial conditions at some fixed t 0 ∈ [a, b] T are independent of λ.We shall utilize special matrix solutions of (S λ ) which are called the conjoined bases or prepared or isotropic solutions of (S λ ), see [9,12,22].Such a matrix solution Z(•, λ) = (X(•, λ), U(•, λ)) is defined by the symmetry of (X T U)(•, λ) and by rank(X T (•, λ), U T (•, λ)) = n.The principal solution Ẑ(•, λ) = ( X(•, λ), Û(•, λ)) of (S λ ) given by the initial conditions X(a, λ) ≡ 0, Û (a, λ) ≡ I (2.2) will play a prominent role in our investigations.Since the initial conditions in (2.2) do not depend on λ, the functions X(t, •) and Û(t, •) are entire in the argument λ for every t ∈ [a, b] T .This and assumption (1.2) imply that the kernel of X(t, •) is piecewise constant on R with the same values of the subspaces Ker X(t, λ + ) and Ker X(t, λ − ) for every λ ∈ R, see [21,Proposition 4.5] and its proof.Based on the above, the following algebraic definition of (finite) eigenvalues of (E) was given in [21,Definition 2.4].A number λ 0 ∈ R is a finite eigenvalue of the eigenvalue problem (E) if In this case we call θ(λ 0 ) the algebraic multiplicity of the finite eigenvalue λ 0 .By [21, Theorem 5.2], for every finite eigenvalue λ 0 of (E) there is a corresponding finite eigenfunction z(•, λ 0 ) = (x(•, λ 0 ), u(•, λ 0 )) which solves (E) with λ = λ 0 and satisfies Moreover, the geometric multiplicity of the finite eigenvalue λ 0 , i.e., the dimension of the corresponding eigenspace is equal to θ(λ 0 ).Under (1.2) the finite eigenvalues of (E) are real and the finite eigenfunctions corresponding to different finite eigenvalues are orthogonal with respect to the bilinear form where z = (x, u) and z = (x, ũ), see [21,Propositions 5.7 and 5.8].By (2.3) it follows that the number z W := z, z W is positive for every finite eigenfunction z of (E).Hence, the finite eigenfunctions of (E) can be orthonormalized by the standard Gram-Schmidt procedure.
Next we discuss the concept of proper focal points for the conjoined bases of system (S λ ) as it is given in [ where we suppress the arguments t and λ in the conjoined basis and the argument t in the coefficient B, and where X † denotes the Moore-Penrose generalized inverse of X, see [2,3].
while it has a proper focal point of multiplicity m(t 0 ) ≥ 1 in the interval (ρ(t 0 ), t 0 ] T if t 0 is left-scattered and m(t 0 ) := rank M(ρ(t 0 ), λ) + ind P (ρ(t 0 ), λ). (2.6) Here def A and ind A denote the defect and index of a matrix A, i.e., the dimension of its kernel and the number of its negative eigenvalues, respectively.This means that the conjoined basis Z(•, λ) does not have any proper focal points in see also [15,Definition 4.1].In order to avoid infinitely many proper focal points in the interval (a, b] T , the following assumption was introduced in [21, pg.95].
For every λ ∈ R, Assumption (2.9) implies that the number of proper focal points of Z(•, λ) in (a, b] T is finite, because the numbers m(t 0 ) defined in (2.5) and (2.6) can now be positive only at finitely many points.In addition, by [21,Remark 3.4(viii)] we have m(t 0 ) ≤ n.
With the system (S λ ) we associate the quadratic functional where for z = (x, u) and z = (x, ũ) we define (suppressing the argument t) (2.10) The pair z = (x, u) ] T , and it satisfies the first equation in (S λ ) on [a, ρ(b)] T .The functional F λ is positive definite (or shortly positive) if F λ (z) > 0 for every z = (x, u) ∈ A with x(•) ≡ 0, where Since the first equation in the system (S λ ) does not contain λ, the admissible set A is the same for all λ ∈ R. Denote by n 1 (λ) := the number of proper focal points of Ẑ(•, λ) in (a, b] T , n 2 (λ) := the number of finite eigenvalues of (E) which are less or equal to λ, where we recall Ẑ(•, λ) = ( X(•, λ), Û(•, λ)) to be the principal solution of (S λ ).The quantities n 1 (λ) and n 2 (λ) include the multiplicities of proper focal points and finite eigenvalues.The following characterization of the positivity of F λ was proven in [ The relationship between the numbers n 1 (λ) and n 2 (λ) is described in the following result from [21, Corollary 6.4] combined with Corollary 5.2 below.Proposition 2.2 (Oscillation theorem).Assume that the principal solution Ẑ(•, λ) = ( X(•, λ), Û(•, λ)) of (S λ ) satisfies condition (2.9).Then if and only if there exists λ 0 < 0 such that the functional F λ 0 is positive definite.

Technical calculations
In this section we collect some technical results regarding admissible pairs, which are needed in the proofs of the Rayleigh principle in Section 4 and the Sturmian separation and comparison theorems in Section 5. Throughout this section we let Z = (X, U) be a conjoined basis of (S) with finitely many proper focal points in (a, b] T , and recall the definition of Ω(z, ẑ) from (2.10).Lemma 3.1.Let z = (x, u) be admissible and ẑ = (x, û) be such that On the other hand, if ẑ ∈ C prd only, then for every In particular, if Z = Ẑ is the principal solution of (S), then z ∈ A and F 0 (z) = 0.
Lemma 3.3.If there exists a left-scattered point t 0 ∈ (a, b] T such that P (ρ(t 0 )) ≥ 0, then for each vector c ∈ R n with c T P (ρ(t 0 )) c < 0 the pair z = (x, u) defined by ) In particular, if Z = Ẑ is the principal solution of (S), then z ∈ A and F 0 (z) < 0.
Proof.See [15, Proposition 6.2] and Subcases IIa-IIb in its proof.Note that in the latter reference the definitions of the admissible pairs z = (x, u) can be unified to have the form as in (3.4).
Remark 3.4.If z 1 = (x 1 , u 1 ) and z 2 = (x 2 , u 2 ) are two admissible pairs defined by formulas (3.3) and/or (3.4) through vectors d 1 and d 2 , respectively, then the symmetry of X T (a) U(a) implies the identity x 2 (a).Next we calculate the value of the integral b a Ω(z 1 , z 2 )(t) ∆t when the functions z 1 and z 2 in (3.3) and (3.4) correspond to proper focal points of the conjoined basis Z.Following the definition of proper focal points in (2.5)-(2.6),we distinguish the cases when Z has a proper focal point at some point t 0 , meaning that either def X(t 0 ) − def X(t − 0 ) ≥ 1 if t 0 is left-dense or rank M(ρ(t 0 )) ≥ 1 if t 0 is left-scattered, and when Z has a proper focal point in (ρ(t 0 ), t 0 ) T , meaning that ind P (ρ(t 0 )) ≥ 1 if t 0 is left-scattered.Note that as in [18,Lemma 1(ii)] we have at all left-scattered points t 0 ∈ (a, b] T the equivalence M(ρ(t 0 )) = 0 if and only if Ker X(t 0 ) ⊆ Ker X(ρ(t 0 )).Lemma 3.5.Suppose that Z has proper focal points at some (not necessarily distinct) points τ 1 and τ 2 in (a, b] T .Then there are vectors d 1 , d 2 such that d j ∈ Ker X(τ j ) and either In both cases the vectors d 1 , d 2 satisfy the assumption of Lemma 3.2, so that for the admissible pairs z 1 = (x 1 , u 1 ) and z 2 = (x 2 , u 2 ) constructed through formula (3.3) with t 2 := τ j and t Proof.The result follows from identity (3.1) of Lemma 3.
The details of this calculation, as well as of similar calculations below, are here omitted.
Lemma 3.6.Suppose that Z has proper focal points at some point τ 1 (which can be either left-dense or left-scattered) and in (ρ(τ 2 ), τ 2 ) T where τ 2 is left-scattered.Then there is a vector d 1 ∈ R n and an admissible z 1 = (x 1 , u 1 ) defined by (3.3) which satisfies Lemma 3.2 with t 2 := τ 1 and t Also, there are vectors c 2 , d 2 ∈ R n and an admissible z 2 = (x 2 , u 2 ) defined by (3.4) which satisfies Lemma 3.3 with t 0 := τ 2 .And in this case formula (3.5) holds.
Proof.The result is proven by applying identity Proof.The first part is proven by identity The second part, i.e, formula (3.6), follows by identity (3.1) on [a, τ 1 ] T and by identity (3.2) at ρ(τ 1 ).
The next result corresponds to the discrete time case in [10, Lemma 4].

Rayleigh principle
In this section we prove the following variational characterization of the finite eigenvalues of the eigenvalue problem (E).This theorem is a time scale generalization of the continuous and discrete time results in [20, Theorem 1.1] and [6, Theorem 4.6] Theorem 4.1 (Rayleigh principle).Assume that the principal solution Ẑ(•, λ) satisfies condition (2.9), the functional F λ 0 is positive definite for some λ 0 < 0, and The list of finite eigenvalues λ 1 ≤ • • • ≤ λ m ≤ . . . in Theorem 4.1 really makes sense, because by [21, Proposition 4.5 and Corollary 6.3] the finite eigenvalues of (E) are isolated and bounded below.In addition, when there are only finitely many (say p < ∞) finite eigenvalues of (E), then we put λ p+1 := ∞ in (4.1).Before proving Theorem 4.1 we shall develop some necessary auxiliary tools.EJQTDE, 2011 No. 83, p. 8 Lemma 4.2.Let z 1 , . . ., z m be orthonormal finite eigenfunctions of (E) corresponding to the (not necessarily distinct and not necessarily consecutive) finite eigenvalues λ 1 , . . ., λ m .For any β 1 , . . ., β m ∈ R we set ẑ := m i=1 β i z i .Then ẑ = (x, û) ∈ C 1 prd ∩ A and Proof.The identities in (4.2) follow by direct calculations by the aid of Lemma 3.1, compare also with the proof of [20,Lemma 2.13].
Lemma 4.3 (Global Picone formula).Let λ ∈ R be fixed and suppose that Z = (X, U) is a conjoined basis of (S λ ) satisfying conditions (i) and (ii) in (2.9).Then for any admissible z = (x, u) with x(t) ∈ Im X(t) on [a, b] T we have where Proof.The result follows from [24,Theorem 3.19] and its proof.Note that the assumption P (t) ≥ 0 used in the proof of [24,Theorem 3.19] is satisfied under conditions (i) and (ii) in (2.9).Remark 4.4.In the global Picone formula (4.3) we have the equality sign if the kernel of X(•, λ) changes only at isolated points, which is for example the case of discrete time in [5, Proposition 2.1(iv)].
In the next result we extend Lemma 4.3 to include the finite eigenfunctions of (E).This statement is an extension of [6, Theorem 4.2] and [20, Theorem 3.1] to general time scales.
Theorem 4.5 (Extended global Picone formula).Assume (1.2) and fix λ ∈ R. Let Z = (X, U) be a a conjoined basis of (S λ ) satisfying conditions (i) and (ii) in (2.9).Let λ 1 ≤ • • • ≤ λ m be finite eigenvalues of (E) with the corresponding orthonormal finite eigenfunctions z 1 , . . ., z m .For any Then with w := ũ − UX † x on [a, b] T we have the inequality Proof.From z = z + ẑ we have For the last term in (4.5) we have by Lemma 3. We are now ready to establish the Rayleigh principle on time scales.
Let now z = (x, u) ∈ A be such that z ⊥ z 1 , . . ., z m .Then for the function z := z + ẑ the conditions in (4.9)-(4.10)represent a linear system for the coefficients β 1 , . . ., β m (and in general this system may be inhomogeneous) with an invertible coefficient matrix, as we just proved.Therefore, there exist unique β 1 , . . ., β m ∈ R satisfying this system.By the same way as in the previous part of this proof (i.e., by the time scale induction principle) we conclude that the image condition (4.11) is now satisfied for this z = (x, ũ).The extended global Picone formula (Theorem 4.5) then yields since λ > λ i for every i = 1, . . ., m, and since (4.18) holds as a consequence of assumption (2.9) for Ẑ(•, λ) and the construction of w(•) in (4.10)(ii).From (4.19) we get Inequality (4.20) is therefore established for every λ ∈ (λ m , λ m+1 ).If we now take the limit as λ → λ − m+1 , we get from (4.20) the inequality showing that the infimum of the Rayleigh quotient F 0 (z)/ z, z W in (4.1) does not exceed λ m+1 .Since z m+1 = (x m+1 , u m+1 ) is a finite eigenfunction of (E) corresponding to the finite eigenvalue λ m+1 , it follows that z m+1 ∈ A and W (•) x σ m+1 (•) ≡ 0 on [a, ρ(b)] T , and F λ m+1 (z m+1 ) = 0. Hence, Since by the construction of the finite eigenfunctions we have z m+1 ⊥ z 1 , . . ., z m , it follows that the minimum in (4.1) is indeed equal to λ m+1 and this minimum is attained at z = z m+1 .EJQTDE, 2011 No. 83, p. 13 Finally, if λ m+1 = • • • = λ m+p is a multiple finite eigenvalue of (E) with multiplicity p ≥ 2, then any function z ∈ A with z ⊥ z 1 , . . ., z m+q (for any q ∈ {1, . . ., p}) satisfies automatically z ⊥ z 1 , . . ., z m .Therefore, by the previous argument we have for such z This completes the proof of the Rayleigh principle on time scales (Theorem 4.1).
Similarly to [20,Corollary 4.1] we can characterize the existence of finitely or infinitely many finite eigenvalues in terms of the dimension of the space The space W contains all the functions (W x σ i )(•), where z i = (x i , u i ) are the finite eigenfunctions of (E).Consequently, the number of finite eigenvalues cannot be larger than dim W. From Theorem 4.1 we can then conclude the following.
(i) The eigenvalue problem (E) has infinitely many finite eigenvalues In both cases (i) and (ii) in Corollary 4.6 the finite eigenvalues of (E) satisfy (4.1),where in (ii) we put λ p+1 := ∞.The final result of this section is a generalization of [20,Theorem 4.3] and [6,Theorem 4.7] to time scales.Theorem 4.7 (Expansion theorem).Assume that (1.2) holds, the principal solution Ẑ(•, λ) of (S λ ) satisfies condition (2.9), and that F λ is positive definite for some λ < 0. Denote by I the index set which is equal to Proof.The proof is the same as in the continuous and discrete time cases in [20, Theorem 4.3] and [6,Theorem 4.7] and it is therefore omitted.We need to mention that the argument in these proofs yields in the time scale setting that

Sturmian theorems
In this section we consider first the system (S λ ) and another time scale symplectic system of the same form and similarly we define the matrices G and E. Then a simple calculation shows that for an admissible z = (x, u) we have The following results gives a comparison of the definiteness of the functionals F λ and F λ .
Proof.The proof is similar to the proof of [16,Theorem 3.2], so the details are here omitted.
As a consequence we obtain a comparison of the definiteness of the functionals F λ for different values of λ.It allows to replace the condition on the positivity of F λ for all λ ≤ λ 0 used in the oscillation theorem in [21,Corollary 6.4] by the positivity of F λ 0 alone (compare the previous reference with Proposition 2.2).
Corollary 5.2.Suppose that (1.2) holds and let λ 0 ∈ R be fixed.The functional F λ 0 is positive definite (nonnegative) if and only if the functional F λ is positive definite (nonnegative) for every λ ≤ λ 0 .
Proof.We take the coefficients in system (S λ ) to be equal to the coefficients of (S λ ).Then the conditions in (5.2) are satisfied trivially and the result follows from Proposition 5.1.
In the subsequent results we establish much more precise relationship between the numbers of proper focal points of conjoined bases of the two systems of the form (S λ ) and (S λ ).Let us consider two generic time scale symplectic systems S) whose coefficients satisfy the assumptions in (1.1)(i).We shall now derive the Sturmian comparison and separation theorems for these two systems.Accordingly to the matrix P (•) in (2.4), we define the matrix P (•) through a conjoined basis Z = (X, U ) of system (S).And as in (2.9) we utilize similar hypotheses for the conjoined bases Z = (X, U) of (S) and Z = (X, U) of (S).The following result is a generalization of [10, Theorem 1] and [23,Theorem 1.1] to time scales.
Since each of the initial values x (1) (a), . . ., x (r) (a) is an n-vector and since we assume that we have r > n of these initial values, then they must be linearly dependent, i.e., r l=1 α l x (l) (a) = 0 for some coefficients α 1 , . . ., α r ∈ R with some α l = 0. (5.7) We now define the pair z = (x, u) by Then z is admissible, (5.3),(5.4)= 0, (5.9) where α [i] Hence, z ∈ A and with notation (5.6) it follows that The value of each of the above integrals is calculated by the aid of Lemmas 3.5-3.7 depending on the type of the proper focal point to which the admissible functions z (l) and z (m) belong.Denote by J the set of indices i ∈ {1, . . ., k} such that the conjoined basis Z has a proper focal point in the interval (ρ(τ i ), τ i ) T , that is, Then by Lemmas 3.5-3.7 we get where Since for each i ∈ J the vectors c q i are mutually orthogonal unit eigenvectors corresponding to the negative eigenvalues λ q i of the symmetric matrix P (ρ(τ i )), it follows that for all j, s ∈ {1, . . ., q i } we have (c for s = j, Thus, by (5.10), and the inequality in (5.11) is strict if the set J is nonempty.Consequently, the positivity of the functional F 0 implies that x(•) ≡ 0 on [a, b] T .We will show that this necessarily leads to α 1 , . . ., α r being zero.Consider the last proper focal point τ k .If τ k is left-dense, then following the proof of [23,Theorem 1.1], the definition of the admissible functions z (l) (•) in (3.3) yields that EJQTDE, 2011 No. 83, p. 17 for some sufficiently small ε > 0 for every l = 1, . . ., r − m k .Therefore, from equations (5.6) and (5.8) we obtain (5.12) The identity in (5.12) holds also at τ k by the definition of x [k] j (τ k ) or just by the continuity of x ] ⊥ , since the vector e is by (5.13) a linear combination of d m k .On the other hand, formula (5.13) implies that e ∈ Ker X(τ − k ).Consequently, e = 0.The definition of e in (5.13) and the linear independence of d If τ k is left-scattered, then inspired by the proof of [10, Theorem 1] we have from (3.3) and (3.4) that x (l) (ρ(τ k )) = 0 for all l = 1, . . ., r − m k .Hence, by (5.6) and (5.8) we get ( where the vector e is defined by the second formula in (5.13).For brevity we set p := p k and q := q k , and recall that the vectors d p "belong" to the focal point at τ k , while the vectors d p+q "belong" to the focal interval (ρ(τ k ), τ k ) T , where from Lemmas 3.3 and 3.8 the vectors d [k] p+j are given by formula (3.7) with t 0 := τ k and c j := c [k] j for j ∈ {1, . . ., q}.From Im X † = Im X T we can see that d . By splitting the vector e into the sum And since by (5.15) we have f + g = e ∈ Ker X(ρ(τ k )), it follows that e = 0.But from Lemma 3.8 we know that the vectors d p+q are linearly independent, so that e = 0 implies α Repeating the above argument with the proper focal points τ k−1 , . . ., τ 1 we obtain α [i] j = 0 for every i ∈ {1, . . ., k} and j ∈ {1, . . ., m i }, i.e., α 1 = • • • = α r = 0.However, EJQTDE, 2011 No. 83, p. 18 this contradicts condition (5.7),where at least one coefficient α l = 0. Therefore, the conjoined basis Z cannot have more than n proper focal points in (a, b] T and the proof is complete. In the proofs of the subsequent results we will utilize the eigenvalue problems (E) and in which the matrices W (•) and W (•) are given by The following result is a generalization of [23, Theorem 1.2] and [6, Theorem 1.2] to arbitrary time scales.Note that in view of Proposition 2.1 the choice of m = 0 in Theorem 5.4 yields the result of Theorem 5.3.
Theorem 5.4 (Sturmian comparison theorem).Under (5.16), suppose that the principal solution of (S λ ) satisfies (2.9), and conditions (i) and (ii) in (2.9) hold for every conjoined basis of (S).Furthermore, let the functional F λ be positive definite for some λ < 0 and (5.17 Proof.The assumptions imply that there is finitely many proper focal points in (a, b] T for every conjoined basis of (S λ ) and (S).Let Ẑ = ( X, Û) be the principal solution of (S) and suppose that it has m proper focal points in (a, b] T .Let Z = (X, U) be a conjoined basis of (S) and let r be its number of proper focal points in (a, b] T .By Lemmas 3.2 and 3.3, for each proper focal point at τ i and for each proper focal point in (ρ(τ i ), τ i ) T of Z there is an (A, B)-admissible z i = (x i , u i ) such that respectively, where λ i is a negative eigenvalue of the matrix P (ρ(τ i )).By Proposition 2.2, the finite eigenvalues of (E) are bounded below and where Consider the numbers ] T x σ j (t) ∆t for i ∈ {1, . . ., r}, j ∈ {1, . . ., m}, and the vectors where the functions z i = (x i , u i ) are from (5.18) and (5.19).If we assume that r > m + n, then these vectors d 1 , . . ., d r must be linearly dependent, i.e., there are coefficients c 1 , . . ., c r ∈ R with some c i = 0 such that x(t) x ∆ (t) ∆t (5.17) This is a contradiction with the previously computed value F 0 (z) > 0. Hence, we must have r ≤ m + n and the proof is complete.
Let now (S λ )=(S λ ), so that conditions (5.17) are trivially satisfied.This yields a generalization of [23,Theorem 1.4] and [6, Theorem 3.1] to time scales.Theorem 5.7 (Sturmian separation theorem).Under (5.16), suppose that for every conjoined basis of (S λ ) condition (2.9) holds and that F λ is positive definite for some λ < 0. Theorem 5.8 (Sturmian separation theorem).Under (5.16), suppose that for every conjoined basis of (S λ ) condition (2.9) holds and that F λ is positive definite for some λ < 0. Then the difference between the numbers of proper focal points in (a, b] T of any two conjoined bases of (S) is at most n.
Proof.By assumption (2.9), every conjoined basis of (S) has finitely many proper focal points in (a, b] T .Let Ẑ = ( X, Û ) be the principal solution of (S) and let Z = (X, U) and Z = ( X, Ũ) be any two conjoined basis of (S).Denote by m, p, p their numbers of proper focal points in (a, b] T , respectively.Then by Theorem 5.7 we have m ≤ p ≤ m + n and m ≤ p ≤ m + n.Upon subtracting m from both sides of these inequalities we obtain 0 ≤ p − m ≤ n and 0 ≤ p − m ≤ n.Therefore, p − p ≤ n if p ≥ p, or p − p ≤ n if p ≤ p. Combining these two inequalities yields |p − p| ≤ n, which is the statement of this theorem.

Special time scales
In this section we continue the study of the oscillation properties of symplectic systems (S) and (S) on special time scales, which was initiated in [21,Section 9] [t j , t j+1 ] T = [t j , t j+1 ] or [t j , t j+1 ] T = {t j , t j+1 }, ( i.e., for every two consecutive partition points t j and t j+1 the interval [t j , t j+1 ] T is connected or (t j , t j+1 ) T is empty.Already such time scales unify the classical purely continuous and discrete time scales.We shall make the following standing hypothesis T × R, so that the matrix P (t, λ) = T (t, λ) B(t) T (t, λ) at every right-dense point.Therefore, we can see that in this case assumption (6.2) implies condition (2.9).In addition, on special time scales we have the result of [21, Theorem 9.5], saying that under (6.2) and W (t) > 0 on [a, ρ(b)] T , in particular for W (•) ≡ I used in the previous section, there exists λ < 0 such that the functional F λ is positive definite.Therefore, the assumptions in the statements of Sections 4 and 5 significantly simplify for the special time scales.The results below follow from the corresponding ones in Sections 4 and 5 and they are stated without the proofs.Theorem 6.1 (Rayleigh principle).Assume (6.1), (6.2), (1.2), and the functional F λ is positive definite for some λ < 0. Let λ 1 ≤ • • • ≤ λ m ≤ . . .be the finite eigenvalues of the eigenvalue problem (E) with the corresponding orthonormal finite eigenfunctions z 1 , . . ., z m , . . . .Then for each m ∈ N ∪ {0} equation (4.1) holds.Moreover, if W (t) > 0 for all t ∈ [a, ρ(b)] T instead of (1.2), then the assumption on F λ positive definite for some λ < 0 can be dropped.Theorem 6.2 (Sturmian comparison theorem).Assume (6.1), (6.2), and (5.17

c( 5 c
i d i = 0 ∈ R m+n .(5.22) Define now the pair z = (x, u) := r i=1 c i z i .Since each z i is (A, B)-admissible with x i (b) = 0, it follows that z is also (A, B)-admissible and x(b) = 0.Moreover, the definition of the vectors d 1 , . . ., d r implies x(a) = r i=1 c i x i (a) every j ∈ {1, . . ., m} we have0 i z i , z j = r i=1 c i z i , z j = z, z j .(5.23)This yields that z ⊥ z 1 , . . ., z m .The fact that x(•) ≡ 0 follows by the same argument as in the proof of Theorem 5.3, i.e., if x(•) ≡ 0, then all the coefficients c 1 , . . ., c r are zero, which is a contradiction.Moreover, from (5.18) and (5.19) we get similarly to the calculations in (5.10)-(5.11) that F 0 (z) ≤ 0. Define the function x(t) := x(t) on [a, b] T .Then (suppressing the argument t) we havex ∆ − Ax = Bu + (A − A) x ∈ Im A − A , B on [a, ρ(b)] T , so that by condition (5.17)(ii) for each t ∈ [a, ρ(b)] T there exists a value u(t) ∈ R n such that Bu = x ∆ − Ax ∈ C prd on [a, ρ(b)] T .This means that the pair z = (x, u) is (A, B)-admissible, x(a) = x(a) = 0, x(b) = x(b) = 0, and x(•) = x(•) ≡ 0.Moreover, by(5.23)  and the definition of •, • we have z, z j = z, z j = 0 for each j ∈ {1, . . ., m}, that is, z ⊥ z 1 , . . ., z m .Inequality (5.21) then implies that F 0 (z) > 0. On the other hand, by the definition of x, G, and G we have (compare with [16, Theorem 3.2]) ) satisfied.If the principal solution of (S) has m ∈ N∪{0} proper focal points in (a, b] T , then every conjoined basis of (S) has at least m proper focal points in (a, b] T . .26) where w(t) := ũ(t) − U (t, λ) X † (t, λ) x(t) and where τ 1 , . . ., τ l are the proper focal points of Z(•, λ) in (a, b) T whose multiplicities add up to p.The matrices M (•, λ) and P (•, λ) are defined by (2.4) through the conjoined basis Z(•, λ).Set x(t) := x(t) on [a, b] T .Since z is (A, B)-admissible, assumption (5.17)(ii) implies that z := (x, u) is (A, B)-admissible for some u(•).Moreover, x(a) = x(a) = 0, x(b) = EJQTDE, 2011 No. 83, p. 21 x(b) = 0, and for each j ∈ {1, . . ., k} z, z j = z, z j = k i=1 β i z i , z j = β j .By the time scale induction principle, following the proof of Theorem 4.1, one can verify that the image condition x(t) ∈ Im X(t, λ) for all t ∈ [a, b] T holds.Since by (5.1) the values of Ω(z, z) and z, z do not depend on the component ũ, it follows that F λ (z) = F λ (z), where F λ (z) := F 0 (z) − λ z, z .(5.27)We now apply the extended global Picone formula (Theorem 4.5) with z = 0, m = k, ẑ = z to obtain for w : .31) EJQTDE, 2011 No. 83, p. 22 If the principal solution of (S) has m ∈ N ∪ {0} proper focal points in (a, b] T , then any other conjoined basis of (S) has at least m and at most m+n proper focal points in (a, b] T .Proof.By assumption (2.9), every conjoined basis of (S) has finitely many proper focal points in (a, b] T .Let the principal solution of (S) have m proper focal points in (a, b] T .Let Z = (X, U) be any other conjoined basis of (S) and denote by p its number of proper focal points in (a, b] T .Set S(t) := S(t) on [a, ρ(b)] T .Then by Theorem 5.4 we have p ≤ m + n, while by Theorem 5.5 we get p ≥ m.Thus, m ≤ p ≤ m + n and the result is proven.The final result of this section generalizes [23, Theorem 1.5] and [6, Theorem 1.1] to time scales.
. A time scale T = [a, b] T is called special if it consists of a finite union of disjoint closed and bounded EJQTDE, 2011 No. 83, p. 23 real intervals and/or finitely many isolated points.That is, a special time scale [a, b] T can be partitioned as a = t 0 < t 1 < • • • < t N +1 = b, where

B
(t) ≥ 0 on continuous intervals [t j , t j+1 ] ⊆ [a, b] T .(6.2) Then, by [19, Theorem 3], every conjoined basis Z(•, λ) = (X(•, λ), U(•, λ)) of (S λ ) has the kernel of X(•, λ) piecewise constant on the continuous intervals [t j , t j+1 ] ⊆ [a, b] T , hence on [a, b] T .Moreover, by [15, Lemma 3.1] we have P = T [(I + µD T ) B − µB T U σ (X σ ) † B] T on [a, ρ(b)] ).If the principal solution of (S) has m ∈ N∪{0} proper focal points in (a, b] T , then every conjoined basis of (S) has at most m + n proper focal points in (a, b] T .Theorem 6.3 (Sturmian comparison theorem).Assume (6.1), (6.2), and (5.17).If the principal solution of (S) has m ∈ N∪{0} proper focal points in (a, b] T , then every conjoined basis of (S) has at least m proper focal points in (a, b] T .Theorem 6.4 (Sturmian separation theorem).Assume (6.1) and (6.2).If the principal solution of (S) has m ∈ N ∪ {0} proper focal points in (a, b] T , then any other conjoined basis of (S) has at least m and at most m + n proper focal points in (a, b] T .Theorem 6.5 (Sturmian separation theorem).Assume (6.1) and (6.2).Then the difference between the numbers of proper focal points in (a, b] T of any two conjoined bases of (S) is at most n.EJQTDE, 2011 No. 83, p. 24 Proposition 2.1 (Positivity).Let λ ∈ R be fixed.The functional F λ is positive definite if and only if the principal solution of (S λ ) has no proper focal points in (a, b] T , i.e., if and only if n 1 Proof.Identity (3.1) follows by the integration by parts formula using the equality µC T B = A T + D + µA T D, obtained from (1.1).Formula (3.2) is proven in a similar way.
We now evaluate the terms in (4.5) separately.By Lemma 4.3 and z ∈ A we have .5) EJQTDE, 2011 No. 83, p. 9