Weyl-Titchmarsh Theory for Time Scale Symplectic Systems on Half Line

and Applied Analysis 3 to be of the limit point or limit circle type is given in 4, 43 . These two papers seem to be the only ones on time scales which are devoted to the Weyl-Titchmarsh theory for the second order dynamic equations. Another way of generalizing the Weyl-Titchmarsh theory for continuous and discrete Hamiltonian systems was presented in 3, 5 . In these references the authors consider the linear Hamiltonian system xΔ t A t x t B t λW2 t u t , uΔ t C t − λW1 t x t −A∗ t u t , t ∈ a,∞ , 1.7 on the so-called Sturmian or general time scales, respectively. Here fΔ t is the time scale Δ-derivative and f t : f σ t , where σ t is the forward jump at t; see the time scale notation in Section 2. In the present paper we develop the Weyl-Titchmarsh theory for more general linear dynamic systems, namely, the time scale symplectic systems xΔ t A t x t B t u t , uΔ t C t x t D t u t − λW t x t , t ∈ a,∞ , Sλ where A, B, C, D, W are complex n × n matrix functions on a,∞ , W t is Hermitian and nonnegative definite, λ ∈ , and the 2n × 2n coefficient matrix in system Sλ satisfies S t : ( A t B t C t D t ) , S∗ t J JS t μ t S∗ t JS t 0, t ∈ a,∞ , 1.8 where μ t : σ t − t is the graininess of the time scale. The spectral parameter λ is only in the second equation of system Sλ . This system was introduced in 44 , and it naturally unifies the previously mentioned continuous, discrete, and time scale linear Hamiltonian systems having the spectral parameter in the second equation only and discrete symplectic systems into one framework. Our main results are the properties of the M λ function, the geometric description of theWeyl disks, and characterizations of the limit point and limit circle cases for the time scale symplectic system Sλ . In addition, we give a formula for the LW solutions of a nonhomogeneous time scale symplectic system in terms of its Green function. These results generalize and unify in particular all the results in 1–4 and some results from 5–14 . The theory of time scale symplectic systems or Hamiltonian systems is a topic with active research in recent years; see, for example, 44–51 . This paper can be regarded not only as a completion of these papers by establishing the Weyl-Titchmarsh theory for time scale symplectic systems but also as a comparison of the corresponding continuous and discrete time results. The references to particular statements in the literature are displayed throughout the text. Many results of this paper are new even for 1.6 , being a special case of system Sλ . An overview of these new results for 1.6 will be presented in our subsequent work. This paper is organized as follows. In the next section we recall some basic notions from the theory of time scales and linear algebra. In Section 3 we present fundamental properties of time scale symplectic systems with complex coefficients, including the important Lagrange identity Theorem 3.5 and other formulas involving their solutions. 4 Abstract and Applied Analysis In Section 4 we define the time scale M λ -function for system Sλ and establish its basic properties in the case of the regular spectral problem. In Section 5 we introduce the Weyl disks and circles for system Sλ and describe their geometric structure in terms of contractive matrices in n×n . The properties of the limiting Weyl disk andWeyl circle are then studied in Section 6, where we also prove that system Sλ has at least n linearly independent solutions in the space LW see Theorem 6.7 . In Section 7 we define the system Sλ to be in the limit point and limit circle case and prove several characterizations of these properties. In the final section we consider the system Sλ with a nonhomogeneous term. We construct its Green function, discuss its properties, and characterize the LW solutions of this nonhomogeneous system in terms of the Green function Theorem 8.5 . A certain uniqueness result is also proven for the limit point case. 2. Time Scales Following 52, 53 , a time scale is any nonempty and closed subset of . A bounded time scale can be therefore identified as a, b : a, b ∩ which we call the time scale interval, where a : min and b : max . Similarly, a time scale which is unbounded above has the form a,∞ : a,∞ ∩ . The forward and backward jump operators on a time scale are denoted by σ t and ρ t and the graininess function by μ t : σ t − t. If not otherwise stated, all functions in this paper are considered to be complex valued. A function f on a, b is called piecewise rd-continuous; we write f ∈ Cprd on a, b if the right-hand limit f t exists finite at all right-dense points t ∈ a, b , and the left-hand limit f t− exists finite at all leftdense points t ∈ a, b and f is continuous in the topology of the given time scale at all but possibly finitely many right-dense points t ∈ a, b . A function f on a,∞ is piecewise rd-continuous; we write f ∈ Cprd on a,∞ if f ∈ Cprd on a, b for every b ∈ a,∞ . An n × nmatrix-valued function f is called regressive on a given time scale interval if I μ t f t is invertible for all t in this interval. The time scale Δ-derivative of a function f at a point t is denoted by fΔ t ; see 52, Definition 1.10 . Whenever fΔ t exists, the formula f t f t μ t fΔ t holds true. The product rule for the Δ-differentiation of the product of two functions has the form ( fg )Δ t fΔ t g t f t gΔ t fΔ t g t f t gΔ t . 2.1 A function f on a, b is called piecewise rd-continuously Δ-differentiable; we write f ∈ Cprd on a, b ; if it is continuous on a, b , then fΔ t exists at all except for possibly finitely many points t ∈ a, ρ b , and fΔ ∈ Cprd on a, ρ b . As a consequence we have that the finitely many points ti at which fΔ ti does not exist belong to a, b and these points ti are necessarily right-dense and left-dense at the same time. Also, since at those points we know that fΔ t i and f Δ ti exist finite, we replace the quantity f Δ ti by fΔ ti in any formula involving fΔ t for all t ∈ a, ρ b . Similarly as above we define f ∈ Cprd on a,∞ . The time scale integral of a piecewise rd-continuous function f over a, b is denoted by ∫b a f t Δt and over a,∞ by ∫∞ a f t Δt provided this integral is convergent in the usual sense; see 52, Definitions 1.71 and 1.82 . Abstract and Applied Analysis 5 Remark 2.1. As it is known in 52, Theorem 5.8 and discussed in 54, Remark 3.8 , for a fixed t0 ∈ a, b and a piecewise rd-continuous n × n matrix function A · on a, b which is regressive on a, t0 , the initial value problem y Δ t A t y t for t ∈ a, ρ b with y t0 y0 has a unique solution y · ∈ Cprd on a, b for any y0 ∈ n . Similarly, this result holds on a,∞ . Let us recall some matrix notations from linear algebra used in this paper. Given a complex square matrix M, by M∗, M > 0, M ≥ 0, M < 0, M ≤ 0, rankM, KerM, defM, we denote, respectively, the conjugate transpose, positive definiteness, positive semidefiniteness, negative definiteness, negative semidefiniteness, rank, kernel, and the defect i.e., the dimension of the kernel of the matrix M. Moreover, we will use the notation Im M : M − M∗ / 2i and Re M : M M∗ /2 for the Hermitian components of the matrix M; see 55, pages 268-269 or 56, Fact 3.5.24 . This notation will be also used with λ ∈ , and in this case Im λ and Re λ represent the imaginary and real parts of λ. Remark 2.2. If the matrix Im M is positive or negative definite, then the matrix M is necessarily invertible. The proof of this fact can be found, for example, in 2, Remark 2.6 . In order to simplify the notation we abbreviate f t ∗ and f∗ t σ by fσ∗ t . Similarly, instead of fΔ t ∗ and f∗ t Δ we will use fΔ∗ t .and Applied Analysis 5 Remark 2.1. As it is known in 52, Theorem 5.8 and discussed in 54, Remark 3.8 , for a fixed t0 ∈ a, b and a piecewise rd-continuous n × n matrix function A · on a, b which is regressive on a, t0 , the initial value problem y Δ t A t y t for t ∈ a, ρ b with y t0 y0 has a unique solution y · ∈ Cprd on a, b for any y0 ∈ n . Similarly, this result holds on a,∞ . Let us recall some matrix notations from linear algebra used in this paper. Given a complex square matrix M, by M∗, M > 0, M ≥ 0, M < 0, M ≤ 0, rankM, KerM, defM, we denote, respectively, the conjugate transpose, positive definiteness, positive semidefiniteness, negative definiteness, negative semidefiniteness, rank, kernel, and the defect i.e., the dimension of the kernel of the matrix M. Moreover, we will use the notation Im M : M − M∗ / 2i and Re M : M M∗ /2 for the Hermitian components of the matrix M; see 55, pages 268-269 or 56, Fact 3.5.24 . This notation will be also used with λ ∈ , and in this case Im λ and Re λ represent the imaginary and real parts of λ. Remark 2.2. If the matrix Im M is positive or negative definite, then the matrix M is necessarily invertible. The proof of this fact can be found, for example, in 2, Remark 2.6 . In order to simplify the notation we abbreviate f t ∗ and f∗ t σ by fσ∗ t . Similarly, instead of fΔ t ∗ and f∗ t Δ we will use fΔ∗ t . 3. Time Scale Symplectic Systems LetA · , B · , C · ,D · ,W · be n×n piecewise rd-continuous functions on a,∞ such that W t ≥ 0 for all t ∈ a,∞ ; that is, W t is Hermitian and nonnegative definite, satisfying identity 1.8 . In this paper we consider the linear system Sλ introduced in the previous section. This system can be written as zΔ t, λ S t z t, λ λJW̃ t z t, λ , t ∈ a,∞ , Sλ where the 2n × 2nmatrix W̃ t is defined and has the property W̃ t : ( W t 0 0 0 ) , JW̃ t ( 0 0 −W t 0 ) . 3.1 The system Sλ can be written in the equivalent form zΔ t, λ S t, λ z t, λ , t ∈ a,∞ , 3.2 6 Abstract and Applied Analysis where the matrix S t, λ is defined through the matrices S t and W̃ t from 1.8 and 3.1 by S t, λ : S t λJW̃ t [ I μ t S t ] ( A t B t C t − λW t [ I μ t A t ] D t − λμ t W t B t ) . 3.3 By using the identity in 1.8 , a direct calculation shows that the matrix function S ·, · satisfies S∗ t, λ J JS ( t, λ ) μ t S∗ t, λ JS ( t, λ ) 0, t ∈ a,∞ , λ ∈ . 3.4 Here S∗ t, λ S t, λ ∗, and λ is the usua


Introduction
In this paper we develop systematically the Weyl-Titchmarsh theory for time scale symplectic systems.Such systems unify and extend the classical linear Hamiltonian differential systems and discrete symplectic and Hamiltonian systems, including the Sturm-Liouville differential and difference equations of arbitrary even order.As the research in the Weyl-Titchmarsh theory has been very active in the last years, we contribute to this development by presenting a theory which directly generalizes and unifies the results in several recent papers, such as 1-4 and partly in 5-14 .
Historically, the theory nowadays called by Weyl and Titchmarsh started in 15 by the investigation of the second-order linear differential equation r t z t q t z t λz t , t ∈ 0, ∞ , 1.1 where r, q : 0, ∞ → Ê are continuous, r t > 0, and λ ∈ , is a spectral parameter.By using a geometrical approach it was showed that 1.1 can be divided into two classes called the limit circle and limit point meaning that either all solutions of 1.1 are square integrable for all λ ∈ \ Ê or there is a unique up to a multiplicative constant square-integrable solution of 1.1 on 0, ∞ .Analytic methods for the investigation of 1.1 have been introduced in a series of papers starting with 16 ; see also 17 .We refer to 18-20 for an overview of the original contributions to the Weyl-Titchmarsh theory for 1.1 ; see also 21  According to 19 , the first paper dealing with the parallel discrete time Weyl theory for second-order difference equations appears to be the work mentioned in 39 .Since then a long time elapsed until the theory of difference equations attracted more attention.The Weyl-Titchmarsh theory for the second-order Sturm-Liouville difference equations was developed in 22,40,41 ; see also the references in 19 .For higher-order Sturm-Liouville difference equations and linear Hamiltonian difference systems, such as  k are Hermitian and nonnegative definite, the Weyl-Titchmarsh theory was studied in 9, 14, 42 .Recently, the results for linear Hamiltonian difference systems were generalized in 1, 2 to discrete symplectic systems where A k , B k , C k , D k , W k are complex n × n matrices such that W k is Hermitian and nonnegative definite and the 2n × 2n transition matrix in 1.4 is symplectic, that is, In the unifying theory for differential and difference equations-the theory of time scales-the classification of second-order Sturm-Liouville dynamic equations y ΔΔ t q t y σ t λy σ t , t ∈ a, ∞ Ì ,

1.6
Abstract and Applied Analysis 3 to be of the limit point or limit circle type is given in 4, 43 .These two papers seem to be the only ones on time scales which are devoted to the Weyl-Titchmarsh theory for the second order dynamic equations.Another way of generalizing the Weyl-Titchmarsh theory for continuous and discrete Hamiltonian systems was presented in 3, 5 .In these references the authors consider the linear Hamiltonian system on the so-called Sturmian or general time scales, respectively.Here f Δ t is the time scale Δ-derivative and f σ t : f σ t , where σ t is the forward jump at t; see the time scale notation in Section 2.
In the present paper we develop the Weyl-Titchmarsh theory for more general linear dynamic systems, namely, the time scale symplectic systems where A, B, C, D, W are complex n × n matrix functions on a, ∞ Ì , W t is Hermitian and nonnegative definite, λ ∈ , and the 2n × 2n coefficient matrix in system S λ satisfies S t : A t B t C t D t , S * t J JS t μ t S * t JS t 0, t ∈ a, ∞ Ì , 1.8 where μ t : σ t − t is the graininess of the time scale.The spectral parameter λ is only in the second equation of system S λ .This system was introduced in 44 , and it naturally unifies the previously mentioned continuous, discrete, and time scale linear Hamiltonian systems having the spectral parameter in the second equation only and discrete symplectic systems into one framework.Our main results are the properties of the M λ function, the geometric description of the Weyl disks, and characterizations of the limit point and limit circle cases for the time scale symplectic system S λ .In addition, we give a formula for the L 2 W solutions of a nonhomogeneous time scale symplectic system in terms of its Green function.These results generalize and unify in particular all the results in 1-4 and some results from 5-14 .The theory of time scale symplectic systems or Hamiltonian systems is a topic with active research in recent years; see, for example, 44-51 .This paper can be regarded not only as a completion of these papers by establishing the Weyl-Titchmarsh theory for time scale symplectic systems but also as a comparison of the corresponding continuous and discrete time results.The references to particular statements in the literature are displayed throughout the text.Many results of this paper are new even for 1.6 , being a special case of system S λ .An overview of these new results for 1.6 will be presented in our subsequent work.
This paper is organized as follows.In the next section we recall some basic notions from the theory of time scales and linear algebra.In Section 3 we present fundamental properties of time scale symplectic systems with complex coefficients, including the important Lagrange identity Theorem 3.5 and other formulas involving their solutions.

Time Scales
Following 52, 53 , a time scale Ì is any nonempty and closed subset of Ê .A bounded time scale can be therefore identified as a, b Ì : a, b ∩ Ì which we call the time scale interval, where a : min Ì and b : max Ì.Similarly, a time scale which is unbounded above has the form a, ∞ Ì : a, ∞ ∩ Ì.The forward and backward jump operators on a time scale are denoted by σ t and ρ t and the graininess function by μ t : σ t − t.If not otherwise stated, all functions in this paper are considered to be complex valued.A function f on a, b Ì is called piecewise rd-continuous; we write f ∈ C prd on a, b Ì if the right-hand limit f t exists finite at all right-dense points t ∈ a, b Ì , and the left-hand limit f t − exists finite at all left- dense points t ∈ a, b Ì and f is continuous in the topology of the given time scale at all but possibly finitely many right-dense points t ∈ a, b Ì .A function f on a, ∞ Ì is piecewise rd-continuous; we write f ∈ C prd on a, ∞ Ì if f ∈ C prd on a, b Ì for every b ∈ a, ∞ Ì .An n × n matrix-valued function f is called regressive on a given time scale interval if I μ t f t is invertible for all t in this interval.
The time scale Δ-derivative of a function f at a point t is denoted by f Δ t ; see 52, Definition 1.10 .Whenever f Δ t exists, the formula f σ t f t μ t f Δ t holds true.The product rule for the Δ-differentiation of the product of two functions has the form exists at all except for possibly finitely many points t ∈ a, ρ b Ì , and f Δ ∈ C prd on a, ρ b Ì .As a consequence we have that the finitely many points t i at which f Δ t i does not exist belong to a, b Ì and these points t i are necessarily right-dense and left-dense at the same time.Also, since at those points we know that f Δ t i and f Δ t − i exist finite, we replace the quantity f Δ t i by f Δ t ± i in any formula involving f Δ t for all t ∈ a, ρ b Ì .Similarly as above we define f ∈ C 1 prd on a, ∞ Ì .The time scale integral of a piecewise rd-continuous function f over a, b Ì is denoted by b a f t Δt and over a, ∞ Ì by ∞ a f t Δt provided this integral is convergent in the usual sense; see 52, Definitions 1.71 and 1.82 .Remark 2.1.As it is known in 52, Theorem 5.8 and discussed in 54, Remark 3.8 , for a fixed t 0 ∈ a, b Ì and a piecewise rd-continuous n × n matrix function A • on a, b Ì which is regressive on a, t 0 Ì , the initial value problem y Δ t A t y t for t ∈ a, ρ b Ì with y t 0 y 0 has a unique solution y • ∈ C 1 prd on a, b Ì for any y 0 ∈ n .Similarly, this result holds on a, ∞ Ì .
Let us recall some matrix notations from linear algebra used in this paper.Given a complex square matrix M, by M * , M > 0, M ≥ 0, M < 0, M ≤ 0, rank M, Ker M, def M, we denote, respectively, the conjugate transpose, positive definiteness, positive semidefiniteness, negative definiteness, negative semidefiniteness, rank, kernel, and the defect i.e., the dimension of the kernel of the matrix M.Moreover, we will use the notation Im M : M − M * / 2i and Re M : M M * /2 for the Hermitian components of the matrix M; see 55, pages 268-269 or 56, Fact 3.5.24 .This notation will be also used with λ ∈ , and in this case Im λ and Re λ represent the imaginary and real parts of λ.
Remark 2.2.If the matrix Im M is positive or negative definite, then the matrix M is necessarily invertible.The proof of this fact can be found, for example, in 2, Remark 2.6 .
In order to simplify the notation we abbreviate f σ t * and f * t σ by f σ * t .Similarly, instead of f Δ t * and f * t Δ we will use f Δ * t .

Time Scale Symplectic Systems
Let A • , B • , C • , D • , W • be n × n piecewise rd-continuous functions on a, ∞ Ì such that W t ≥ 0 for all t ∈ a, ∞ Ì ; that is, W t is Hermitian and nonnegative definite, satisfying identity 1.8 .In this paper we consider the linear system S λ introduced in the previous section.This system can be written as where the 2n × 2n matrix W t is defined and has the property The system S λ can be written in the equivalent form where the matrix S t, λ is defined through the matrices S t and W t from 1.8 and 3.1 by

3.3
By using the identity in 1.8 , a direct calculation shows that the matrix function S •, • satisfies

3.4
Here S * t, λ S t, λ * , and λ is the usual conjugate number to λ.
Remark 3.1.The name time scale symplectic system or Hamiltonian system has been reserved in the literature for the system of the form it follows that the system S λ is a true time scale symplectic system according to the above terminology only for λ ∈ Ê , while strictly speaking S λ is not a time scale symplectic system for λ ∈ \ Ê .However, since S λ is a perturbation of the time scale symplectic system S 0 and since the important properties of time scale symplectic systems needed in the presented Weyl-Titchmarsh theory, such as 3.4 or 3.8 , are satisfied in an appropriate modification, we accept with the above understanding the same terminology for the system S λ for any λ ∈ .Equation 3.4 represents a fundamental identity for the theory of time scale symplectic systems S λ .Some important properties of the matrix S t, λ are displayed below.Note that formula 3.7 is a generalization of 46, equation 10.4 to complex values of λ.Lemma 3.2.Identity 3.4 is equivalent to the identity S t, λ J JS * t, λ μ t S t, λ JS * t, λ 0, t ∈ a, ∞ Ì , λ ∈ .

Abstract and Applied Analysis 7
In this case for any λ ∈ we have and the matrices I μ t S t, λ and I μ t S t, λ are invertible with

3.10
Proof.Let t ∈ a, ∞ Ì and λ ∈ be fixed.If t is right-dense, that is, μ t 0, then identity 3.4 reduces to S * t, λ J JS t, λ 0. Upon multiplying this equation by J from the left and right side, we get identity 3.7 with μ t 0. If t is right scattered, that is, μ t > 0, then 3.4 is equivalent to 3.8 .It follows that the determinants of I μ t S t, λ and I μ t S t, λ are nonzero proving that these matrices are invertible with the inverse given by 3.10 .Upon multiplying 3.8 by the invertible matrices I μ t S t, λ J from the left and J from the right and by using J 2 −I, we get formula 3.9 , which is equivalent to 3.7 due to μ t > 0.

3.11
System 3.11 can be found, for example, in 47, Remark 3.1 iii or 50, equation 3.2 in the connection with optimality conditions for variational problems over time scales.
In the following result we show that 3.4 guarantees, among other properties, the existence and uniqueness of solutions of the initial value problems associated with S λ .Theorem 3.4 existence and uniqueness theorem .Let λ ∈ , t 0 ∈ a, ∞ Ì , and z 0 ∈ 2n be given.Then the initial value problem (S λ ) with z t 0 z 0 has a unique solution z •, λ ∈ C 1 prd on the interval a, ∞ Ì .
Proof.The coefficient matrix of system S λ , or equivalently of system 3.2 , is piecewise rdcontinuous on a, ∞ Ì .By Lemma 3.2, the matrix I μ t S t, λ is invertible for all t ∈ a, ∞ Ì , which proves that the function S •, λ is regressive on a, ∞ Ì .Hence, the result follows from Remark 2.1.
If not specified otherwise, we use a common agreement that 2n-vector solutions of system S λ and 2n × n-matrix solutions of system S λ are denoted by small letters and capital letters, respectively, typically by z •, λ or z •, λ and Z •, λ or Z •, λ .
Next we establish several identities involving solutions of system S λ or solutions of two such systems with different spectral parameters.The first result is the Lagrange identity known in the special cases of continuous time linear Hamiltonian systems in

3.12
Proof.Formula 3.12 follows from the time scales product rule 2.1 by using the relation z σ t, λ I μ t S t, λ z t, λ and identity 3.6 .
As consequences of Theorem 3.5, we obtain the following.

3.13
One can easily see that if z •, λ is a solution of system S λ , then z •, λ is a solution of system S λ .Therefore, Theorem 3.5 with ν λ yields a Wronskian-type property of solutions of system S λ .Corollary 3.7.Let λ ∈ and m ∈ AE be given.For any 2n × m solution z •, λ of systems (S λ ) z * t, λ Jz t, λ ≡ z * a, λ Jz a, λ , is constant on a, ∞ Ì .

3.14
The following result gives another interesting property of solutions of system S λ and S λ .Lemma 3.8.Let λ ∈ and m ∈ AE be given.For any 2n × m solutions z •, λ and z •, λ of system (S λ ), the 2n × 2n matrix function K •, λ defined by

3.17
Proof.Having that z •, λ and z •, λ are solutions of system S λ , it follows that z •, λ and z •, λ are solutions of system S λ .The results then follow by direct calculations.
Remark 3.9.The content of Lemma 3.8 appears to be new both in the continuous and discrete time cases.Moreover, when the matrix function K •, λ ≡ K λ is constant, identity 3.17 yields for any right-scattered t ∈ a, ∞ Ì that S t, λ K λ K λ S * t, λ μ t S t, λ K λ S * t, λ 0.

3.18
It is interesting to note that this formula is very much like 3.7 .More precisely, identity 3.7 is a consequence of 3.18 for the case of K λ ≡ J.
Next we present properties of certain fundamental matrices Ψ •, λ of system S λ , which are generalizations of the corresponding results in 46, Section 10.2 to complex λ.Some of these results can be proven under the weaker condition that the initial value of Ψ a, λ does depend on λ and satisfies Ψ * a, λ JΨ a, λ J.However, these more general results will not be needed in this paper.Lemma 3.10.Let λ ∈ be fixed.If Ψ •, λ is a fundamental matrix of system (S λ ) such that Ψ a, λ is symplectic and independent of λ, then for any t ∈ a, ∞ Ì we have

M λ -Function for Regular Spectral Problem
In this section we consider the regular spectral problem on the time scale interval a, b Ì with some fixed b ∈ a, ∞ Ì .We will specify the corresponding boundary conditions in terms of complex n × 2n matrices from the set Γ : α ∈ n×2n , αα * I, αJα * 0 . 4.1 The two defining conditions for α ∈ n×2n in 4.1 imply that the 2n × 2n matrix α * − Jα * is unitary and symplectic.This yields the identity The last equation also implies, compare with 60, Remark 2. This yields ξ − ζ 0, which shows that the vector ξ in 4.6 is unique.The opposite direction, that is, that 4.6 implies 4.4 , is trivial.
Following the standard terminology, see, for example, 62, 63 , a number λ ∈ is an eigenvalue of 4.4 if this boundary value problem has a solution z •, λ / ≡ 0. In this case the function z •, λ is called the eigenfunction corresponding to the eigenvalue λ, and the dimension of the space of all eigenfunctions corresponding to λ together with the zero function is called the geometric multiplicity of λ.
Given α ∈ Γ, we will utilize from now on the fundamental matrix Ψ •, λ, α of system S λ satisfying the initial condition from 4.4 , that is, Then Ψ a, λ, α does not depend on λ, and it is symplectic and unitary with the inverse Ψ −1 a, λ, α Ψ * a, λ, α .Hence, the properties of fundamental matrices derived earlier in Lemma 3.10, Remark 3.11, and Corollary 3.12 apply for the matrix function Ψ •, λ, α .
The following assumption will be imposed in this section when studying the regular spectral problem. 4.8 Condition 4.8 can be written in the equivalent form as b a z σ * t, λ W t z σ t, λ Δt > 0, 4.9 for every nontrivial solution z •, λ of system S λ .Assumptions 4.8 and 4.9 are equivalent by a simple argument using the uniqueness of solutions of system S λ .The latter form 4.9 has been widely used in the literature, such as in the continuous time case in where Z •, λ, α and Z •, λ, α are the 2n × n solutions of system S λ satisfying Z a, λ, α α * and Z a, λ, α −Jα * .With the notation ii The algebraic multiplicity of the eigenvalue λ, that is, the number def Λ λ, α, β , is equal to the geometric multiplicity of λ.
iii Under Hypothesis 4.2, the eigenvalues of 4.4 are real, and the eigenfunctions corresponding to different eigenvalues are orthogonal with respect to the semi-inner product Proof.The arguments are here standard, and we refer to 44, Section 5 , 63, Corollary The next algebraic characterization of the eigenvalues of 4.4 is more appropriate for the development of the Weyl-Titchmarsh theory for 4.4 , since it uses the matrix β Z b, λ, α which has dimension n instead of using the matrix Λ λ, α, β which has dimension 2n.Results of this type can be found in special cases of system S λ in 8, Lemma 2. Proof.One can follow the same arguments as in the proof of the corresponding discrete symplectic case in 2, Lemma 3.1 .However, having the result of Proposition 4.3, we can proceed directly by the methods of linear algebra.In this proof we abbreviate Λ : Λ λ, α, β and Z : Z b, λ, α .Assume that Λ is singular, that is, − Zc Jβ * d 0 for some vectors c, d ∈ n , not both zero.Then Zc Jβ * d, which yields that β Zc 0. If c 0, then Jβ * d 0, which implies upon the multiplication by βJ from the left that d 0. Since not both c and d can be zero, it follows that c / 0 and the matrix β Z is singular.Conversely, if β Zc 0 for some nonzero vector c ∈ n , then − Zc Jβ * d 0; that is, Λ is singular, with the vector d : −βJ Zc.Indeed, by using identity 4.2 we have Jβ * d −Jβ * βJ Zc I − β * β Zc Zc.From the above we can also see that the number of linearly independent vectors in Ker β Z is the same as the number of linearly independent vectors in Ker Λ.Therefore, by Proposition 4.3 ii , the algebraic and geometric multiplicities of λ as an eigenvalue of 4.4 are equal to def β Z.
Since the eigenvalues of 4.4 are real, it follows that the matrix β Z b, λ, α is invertible for every λ ∈ except for at most n real numbers.This motivates the definition of the M λfunction for the regular spectral problem.Proof.We abbreviate Z λ : Z b, λ, α and Z λ : Z b, λ, α .By using the definition of M λ in 4.13 and identity 3.21 , we have because β ∈ Γ.Hence, equality 4.14 holds true.
The following solution plays an important role in particular in the results concerning the square integrable solutions of system S λ .Definition 4. 7 Weyl solution .For any matrix M ∈ n×n , we define the so-called Weyl solution of system S λ by where Z •, λ, α and Z •, λ, α are defined in 4.10 .
The function X •, λ, α, M , being a linear combination of two solutions of system S λ , is also a solution of this system.Moreover, αX a, λ, α, M I, and, if where δ λ : sgn Im λ .
For brevity we suppress the dependence of the function E • on b and λ.In few cases we will need E M depending on b as in Theorem 5.1 and Definition 6.2 and in such situations we will use the notation E M, b .Since iJ * iJ, it follows that E M is a Hermitian matrix for any M ∈ n×n .Moreover, from Corollary 3.6, we obtain the identity where we used the fact that Next we define the Weyl disk and Weyl circle for the regular spectral problem.The geometric characterizations of the Weyl disk and Weyl circle in terms of the contractive or unitary matrices which justify the terminology "disk" or "circle" will be presented in Section 5. Definition 4.9 Weyl disk and Weyl circle .For a fixed α ∈ Γ and λ ∈ \ Ê , the set is called the Weyl disk, and the set The dependence of the Weyl disk and Weyl circle on b will be again suppressed.In the following result we show that the Weyl circle consists of precisely those matrices M λ with β ∈ Γ.This result generalizes the corresponding statements in 8, Lemma 2.8 , 9, Lemma 2.13 , 14, Lemma 3. Proof.Assume that M ∈ C λ , that is, E M 0.Then, with the vector where X b denotes X b, λ, α, M , we have Moreover, rank β n, because the matrices Ψ b, λ, α and J are invertible and rank I M * n.In addition, the identity J * J −1 yields Now, if the condition ββ * I is not satisfied, then we replace β by β : ββ * −1/2 β note that ββ * > 0, so that ββ * −1/2 is well defined , and in this case Remark 4.11.The matrix P : −βJX b, λ, α, M ∈ n×n from the proof of Theorem 4.10 is invertible.This fact was not needed in that proof.However, we show that P is invertible because this argument will be used in the proof of Lemma 4.14.First we prove that Ker P Ker X b, λ, α, M .For if Pd 0 for some d ∈ n , then from identity 4.2 Jβ * Pd 0. Therefore, Ker P ⊆ Ker X b, λ, α, M .The opposite inclusion follows by the definition of P .And since, by 4.16 , rank X b, λ, α, M rank I M * * n, it follows that Ker X b, λ, α, M {0}.Hence, Ker P {0} as well; that is, the matrix P is invertible.
The next result contains a characterization of the matrices M ∈ n×n which lie "inside" the Weyl disk D λ .In the previous result Theorem 4.10 we have characterized the elements of the boundary of the Weyl disk D λ , that is, the elements of the Weyl circle C λ , in terms of the matrices β ∈ Γ.For such β we have βJβ * 0, which yields iδ λ βJβ * 0. Comparing with that statement we now utilize the matrices β ∈ n×2n which satisfy iδ λ βJβ * > 0. In the special cases of the continuous and discrete time, this result can be found in 8, Lemma 2.13 , 9, Lemma 2.18 , and 2, Theorem 3.13 .
Theorem 4.12.Let α ∈ Γ, λ ∈ \ Ê , and M ∈ n×n .The matrix M satisfies E M < 0 if and only if there exists β ∈ n×2n such that iδ λ βJβ * > 0 and βX b, λ, α, M 0. In this case and under Hypothesis 4.2, we have with such a matrix β that M M λ as defined in 4.13 and β may be chosen so that ββ * I.
Proof.For M ∈ n×n consider on a, b Ì the Weyl solution Suppose first that E M < 0. Then the matrices X j b , j ∈ {1, 2}, are invertible.Indeed, if one of them is singular, then there exists a nonzero vector v ∈ n such that which contradicts E M < 0. Now we set β 1 : I, β 2 : −X 1 b X −1 2 b , and β : β 1 β 2 .Then for this 2n × n matrix β we have βX b 0 and, by a similar calculation as in 4.29 , where we used the equality βJβ * 2i Im β 1 β * 2 .Since E M < 0 and X 2 b is invertible, it follows that iδ λ βJβ * > 0. Conversely, assume that for a given matrix M ∈ n×n there is 2 < 0 when Im λ > 0 and to Im β 1 β * 2 > 0 when Im λ < 0. The positive or negative definiteness of Im β 1 β * 2 implies the invertibility of β 1 and β 2 ; see Remark 2.2.Therefore, from the equality

4.31
The matrix In the following lemma we derive some additional properties of the Weyl disk and the M λ -function.Special cases of this statement can be found in 8, Lemma 2.9 , 33, Theorem 3.1 , 9, Lemma 2.14 , 14, Lemma 3.2 ii , 1, Theorem 3.7 , 2, Lemma 3.7 , and 3, Theorem 4.13 .
Proof.By identity 4.18 , for any matrix M ∈ D λ , we have

4.33
which yields together with W t ≥ 0 on a, ρ b Ì the inequalities in 4.32 .The last assertion in Theorem 4.13 is a simple consequence of Hypothesis 4.2.
In the last part of this section we wish to study the effect of changing α, which is one of the parameters of the M λ -function and the Weyl solution X •, λ, α, M , when α varies within the set Γ.For this purpose we will use the M λ -function with all its arguments in the following two statements.
Proof.The above identity follows from 4.35 and the formula for the matrix P from the end of the proof of Lemma 4.14.

Geometric Properties of Weyl Disks
In

5.2
Therefore, by Definition 4.9, the matrix M belongs to D λ, b 1 , which shows the result.
Similarly for the regular case Hypothesis 4.2 we now introduce the following assumption.we have
Proof.The result is shown by a direct calculation.
Proof.In order to simplify and abbreviate the notation we introduce the matrices and use the notation Z λ and Z λ for Z b, λ, α and Z b, λ, α , respectively.Then, since F * F and δ λ δ λ −1, we get the identities Hence, by using that H is Hermitian, we see that

5.13
Identity 5.7 is now proven.
In the next result we justify the terminology for the sets D λ, b and C λ, b in Definition 4.9 to be called a "disk" and a "circle."It is a generalization of 14, Theorem

5.15
The set V is known to be closed in fact compact, since V is bounded and convex.

5.19
Therefore, the matrix

Limiting Weyl Disk and Weyl Circle
In this section we study the limiting properties of the Weyl disk and Weyl circle and their center and matrix radii.Since under Hypothesis 5. Therefore, by 5.16 of Theorem 5.6, for a matrix M ∈ D λ, b 2 , there are unique matrices Upon subtracting the two equations in 6.3 , we get This equation, when solved for V 1 in terms of V 2 , has the form which defines a continuous mapping T :

6.7
Since the functions R λ, • and R λ, • are monotone nonincreasing, they are bounded; that is, Using estimate 6.8 in inequality 6.7 we obtain for b This means that the limit P λ ∈ n×n in 6.2 exists, which completes the proof.
By Theorems 5.1 and 5.6 we know that the Weyl disks D λ, b are closed, convex, and nested as b → ∞.Thereore the limit of D λ, b as b → ∞ is a closed, convex, and nonempty set.This motivates the following definition, which can be found in the special cases of system S λ in 26, Theorem 3.3 , 1, Theorem 3.6 , 2, Definition 4.7 , and 3, Theorem 4.12 .As a consequence of Theorem 5.6, we obtain the following characterization of the limiting Weyl disk.Corollary 6.3.Let α ∈ Γ and λ ∈ \ Ê .Under Hypothesis 5.2, we have where V is the set of all contractive matrices defined in 5.15 .
From now on we assume that Hypothesis 5.2 holds, so that the limiting center P λ and the limiting matrix radii R λ and R λ of D λ are well defined.A matrix M λ from 6.12 is called a half-line Weyl-Titchmarsh M λ -function.Also, as noted in 2, Section 4 , see also 8, Theorem 2.18 , the function M λ is a Herglotz function with rank n and has a certain integral representation which will not be needed in this paper .
Our next result shows another characterization of the elements of D λ in terms of the Weyl solution X •, α, λ, M defined in 4. 16 for all b ∈ a, ∞ Ì .Therefore, by formula 4.18 , we get b a for every b ∈ a, ∞ Ì , which is equivalent to inequality 6.
for every t ∈ a, ∞ Ì .From this we can see that the integral on the right-hand side above converges for t → ∞ and 6.16 holds if and only if condition 6.17 is satisfied.Characterizations 6.16 and 6.17 of the matrices M on the boundary of the limiting Weyl disk D λ generalize the corresponding results in 1, Theorems 3.8 ii and 3.9 ; see also 14, Theorem 6.3 .
Consider the linear space of square integrable C 1 prd functions where we define In the following result we prove that the space L 2 W contains the columns of the Weyl solution X •, λ, α, M when M belongs to the limiting Weyl disk D λ .This implies that there are at least n linearly independent solutions of system S λ in L 2 W .This is a generalization of 11, Theorem 5.1 , 14, Theorem 4.1 , 2, Theorem 4.10 , and 5, page 716 .Theorem 6.7.Let α ∈ Γ, λ ∈ \ Ê , and M ∈ D λ .The columns of X •, λ, α, M form a linearly independent system of solutions of system (S λ ), each of which belongs to L 2 W .
Denote by N λ the linear space of all square integrable solutions of system S λ , that is, Then as a consequence of Theorem 6.7 we obtain the estimate dim N λ ≥ n, for each λ ∈ \ Ê .

6.24
Next we discuss the situation when dim N λ n for some λ ∈ \ Ê .Lemma 6.8.Let α ∈ Γ, λ ∈ \ Ê , and dim N λ n.Then the matrix radii of the limiting Weyl disk D λ satisfy R λ 0 R λ .Consequently, the set D λ consists of the single matrix M P λ , that is, the center of D λ , which is given by formula 6.2 of Theorem 6.1.
Proof.With the matrix radii R λ and R λ of D λ defined in 6.1 and with the Weyl solution X •, λ, α, M given by a matrix M ∈ D λ , we observe that the columns of X •, λ, α, M form a basis of the space N λ .Since the columns of the fundamental matrix Ψ •, λ, α Z •, λ, α Z •, λ, α span all solutions of system S λ , the definition of X •, λ, α, M Z •, λ, α Z •, λ, α M yields that the columns of Z •, λ, α together with the columns of X •, λ, α, M form a basis of all solutions of system S λ .Hence, from dim N λ n and Theorem 6.7, we get that the columns of Z •, λ, α do not belong to L 2 W . Consequently, by formula 5.5 , the Hermitian matrix functions H •, λ, α and H •, λ, α defined in 5.4 are monotone nondecreasing on a, ∞ Ì without any upper bound; that is, their eigenvalues- being real-tend to ∞.Therefore, the functions R λ, • and R λ, • as defined in 5.18 have limits at ∞ equal to zero; that is, R λ 0 and R λ 0. The fact that the set D λ {P λ } then follows from the characterization of D λ in Corollary 6.3.
In the final result of this section, we establish another characterization of the matrices M from the limiting Weyl disk D λ .In comparison with Theorem 6.5, we now use a similar condition to the one in Theorem 4.12 for the regular spectral problem.However, a stronger assumption than Hypothesis 5.2 is now required for this result to hold; compare with 9, Lemma 2.21 and 2, Theorem 4.16 .Hypothesis 6.9.For every a 0 , b 0 ∈ a, ∞ Ì with a 0 < b 0 and for every λ ∈ , we have 6.25 Under Hypothesis 6.9, the Weyl disks D λ, b converge to the limiting disk "monotonically" as b → ∞; that is, the limiting Weyl disk D λ is "open" in the sense that all of its elements lie inside D λ .This can be interpreted in view of Theorem 4.12 as E M, t < 0 for all t ∈ a, ∞ Ì .Theorem 6.10.Let α ∈ Γ, λ ∈ \ Ê , and M ∈ n×n .Under Hypothesis 6.9, the matrix M ∈ D λ if and only if E M, t < 0, ∀t ∈ a, ∞ Ì .

6.26
Proof.If condition 6.26 holds, then M ∈ D λ follows from the definition of D λ .
Conversely, suppose that M ∈ D λ , and let t ∈ a, ∞ Ì be given.Then for any b ∈ t, ∞ Ì we have by formula 4.18 that where we used the property b ≤ 0, while Hypothesis 6.9 implies the positivity of the integral over t, b Ì in 6.27 .Consequently, 6.27 yields that E M, t < 0. Remark 6.11.If we partition the Weyl solution X •, λ : X •, λ, α, M into two n × n blocks X 1 •, λ and X 2 •, λ as in 4.28 , then condition 6.26 can be written as

6.28
Therefore, by Remark 2.2, the matrices X 1 t, λ and X 2 t, λ are invertible for all t ∈ a, ∞ Ì .A standard argument then yields that the quotient Q •, λ : X 2 •, λ X −1 1 •, λ satisfies the Riccati matrix equation suppressing the argument t in the coefficients

Limit Point and Limit Circle Criteria
Throughout this section we assume that Hypothesis 5.2 is satisfied.The results from Theorem 6.7 and Lemma 6.8 motivate the following terminology; compare with 4, page 75 , 43, Definition 1.2 in the time scales scalar case n 1, with 8, page 3486 , 36, page 1668 , 30, page 274 , 38, Definition 3.1 , 37, Definition 1 , 67, page 2826 in the continuous case, and with 14, Definition 5.1 , 2, Definition 4.12 in the discrete case.Definition 7.1 limit point and limit circle case for system S λ .The system S λ is said to be in the limit point case at ∞ or of the limit point type if dim N λ n, ∀λ ∈ \ Ê .

7.1
The system S λ is said to be in the limit circle case at ∞ or of the limit circle type if dim N λ 2n, ∀λ ∈ \ Ê .

7.2
Remark 7.2.According to Remark 6.4 in which β b ≡ β , the center P λ of the limiting Weyl disk D λ can be expressed in the limit point case as where β ∈ Γ is arbitrary but fixed.
Next we establish the first main result of this section.Its continuous time version can be found in 30, Theorem 2.1 , 11, Theorem 8.5 and the discrete time version in 9, Lemma 3.2 , 2, Theorem 4.13 .Theorem 7.3.Let the system (S λ ) be in the limit point or limit circle case, fix α ∈ Γ, and let λ, ν ∈ \ Ê .Then where X •, λ, α, M λ and X •, ν, α, M ν are the Weyl solutions of (S λ ) and (S ν ), respectively, defined by 4.16 through the matrices M λ and M ν , which are determined by the limit in 6.12 .
Proof.For every t ∈ a, ∞ Ì and matrices β t ∈ n×2n such that β t β * t I and iδ λ β t Jβ * t ≥ 0 and for κ ∈ {λ, ν}, we define the matrix compare with Definition 4.5 Then, by Theorems 4.10 and 4.12, we have M κ, t, α, β t ∈ D κ, t .Following the notation in 4.16 , we consider the Weyl solutions X •, κ : X •, κ, α, M κ, t, α, β • .Similarly, let X •, κ : X •, κ, α, M κ be the Weyl solutions corresponding to the matrices M κ ∈ D κ from the statement of this theorem.First assume that the system S λ is of the limit point type.In this case, by Remark 7.2, we may take β t ∈ Γ for all t ∈ a, ∞ Ì .Hence, from Theorem 4.10, we get that β • X •, κ 0 on a, ∞ Ì .By 4.3 , for each t ∈ a, ∞ Ì and κ ∈ {λ, ν}, there is a matrix where we used the Hermitian property of R λ, t and R λ, t .Since we now assume that system S λ is in the limit point case, we know from Lemma 6.8 that lim t → ∞ R λ, t 0 and lim t → ∞ R λ, t 0. Therefore, in order to establish 7.8 ii , it is sufficient to show that R λ, t t a Z σ * s, λ, α W s X σ s, ν Δs, 7.13 is bounded for t ∈ b 0 , ∞ Ì .Let η ∈ n be a unit vector, and denote by X j •, ν : X •, ν e j the jth column of X •, ν for j ∈ {1, . . ., n}.With the definition of R λ, • in 5.18 we have which is independent of t.Consequently, the second limit in 7.8 is established.The first limit in 7.8 is then proven in a similar manner.The proof for the limit point case is finished.
If the system S λ is in the limit circle case, then for κ ∈ {λ, ν} the columns of Z •, κ, α and X •, κ belong to L 2 W ; hence, they are bounded in the L 2 W norm.In this case the limits in 7.8 easily follow from the limit 6.12 for M κ , κ ∈ {λ, ν}.
In the next result we provide a characterization of the system S λ being of the limit point type.Special cases of this statement can be found, for example, in 14, Theorem 6.12 and 2, Theorem 4.14 .

7.19
Let z •, ν be any square integrable solution of system S ν .Then, by our assumption 7.17 ,

7.20
From 7.19 and 7.20 it follows that the vectors X j a, ν , j ∈ {1, . . ., n}, and z a, ν are solutions of the linear homogeneous system X * a, λ Jη 0.

7.21
Since, by Theorem 6.7, the vectors X j a, ν for j ∈ {1, . . ., n} represent a basis of the solution space of system 7.21 , there exists a vector ξ ∈ n such that z a, ν X a, ν ξ.By the uniqueness of solutions of system S ν we then get z •, ν X •, ν ξ on a, ∞ Ì .Hence, the solution z •, ν is square integrable and dim N ν n.Since ν ∈ \ Ê was arbitrary, it follows that the system S λ is in the limit point case.
As a consequence of the above result, we obtain a characterization of the limit point case in terms of a condition similar to 7.17 , but using a limit.This statement is a generalization of 30, Corollary 2.3 , 9, Corollary 3.3 , 14, Theorem 6.14 , 2, Corollary 4.15 , 1, Theorem 3.9 , 3, Theorem 4.16 .
Corollary 7.5.Let α ∈ Γ.The system (S λ ) is in the limit point case if and only if, for every λ, ν ∈ \Ê and every square integrable solutions z 1 •, λ and z 2 •, ν of (S λ ) and (S ν ), respectively, we have Proof.The necessity follows directly from Theorem 7.3.Conversely, assume that condition 7.22 holds for every λ, ν ∈ \ Ê and every square integrable solutions z 1 •, λ and z 2 •, ν of S λ and S ν .Fix λ ∈ \ Ê , and set ν : λ.By Corollary 3.7 we know that z * 1 •, λ Jz 2 •, ν is constant on a, ∞ Ì .Therefore, by using condition 7.22 , we can see that identity 7.17 must be satisfied, which yields by Theorem 7.4 that the system S λ is of the limit point type.

Nonhomogeneous Time Scale Symplectic Systems
In this section we consider the nonhomogeneous time scale symplectic system where the matrix function S •, λ and W • are defined in 3.3 and 3.1 , f ∈ L 2 W , and where the associated homogeneous system S λ is either of the limit point or limit circle type at ∞. Together with system 8.1 we consider a second system of the same form but with a different spectral parameter and a different nonhomogeneous term W .The following is a generalization of Theorem 3.5 to nonhomogeneous systems.

8.6
In the literature the function G •, •, λ, α is called a resolvent kernel, compare with 30, page 283 , 32, page 15 , 2, equation 5.4 , and in this section it will play a role of the Green function.

8.7
Proof.Identity 8.7 follows by a direct calculation from the definition of X •, λ, α via 4.16 with a matrix M λ ∈ D λ by using formulas 3.21 and 6.13 .
In the next lemma we summarize the properties of the function G •, •, λ, α , which together with Proposition 8.4 and Theorem 8.5 justifies the terminology "Green function" of the system 8.
Proof.Condition i follows from the definition of G •, s, λ, α in 8.5 .Condition ii is a consequence of Lemma 8.2.Condition iii is proven from the definition of G σ t , σ t , λ, α in 8.5 by using Lemma 8.2 and Z t, λ, α Z σ t, λ, α − μ t S t, λ Z t, λ, α .Concerning condition iv , the function G •, s, λ, α solves the system S λ on s, ∞ Ì because X •, λ, α solves this system on s, ∞ Ì .If s ∈ a, ∞ Ì is left-dense, then G •, s, λ, α solves S λ on a, s Ì , since Z •, λ, α solves this system on a, s Ì .For the same reason G •, s, λ, α solves S λ on a, ρ s Ì if s ∈ a, ∞ Ì is left-scattered.Condition v follows from the definition of G •, s, λ, α in 8.5 used with t ≥ s and from the fact that the columns of X •, λ, α belong to L 2 W , by Theorem 6.7.The columns of G t, •, λ, α then belong to L 2 W by part i of this lemma.
Since by Lemma 8.3 v the columns of G t, •, λ, α belong to L 2 W , the function is well defined whenever f ∈ L 2 W .Moreover, by using 8.6 , we can write z t, λ, α as W , the function z •, λ, α defined in 8.9 solves the nonhomogeneous system 8.1 with the initial condition α z a, λ, α 0.
Proof.By the time scales product rule 2.1 when we Δ-differentiate expression 8.10 , we have for every t ∈ a, ∞ Ì suppressing the dependence on α in the the following calculation

8.11
This shows that z •, λ, α is a solution of system 8.1 .From 8.10 with t a, we get where we used the initial condition Z a, λ, α −Jα * and αJα * 0 coming from α ∈ Γ.
The following theorem provides further properties of the solution z •, λ, α of system 8.1 .It is a generalization of 10, Lemma 4.2 , 11, Theorem 7.5 , 2, Theorem 5.2 to time scales.Theorem 8.5.Let α ∈ Γ, λ ∈ \ Ê , and f ∈ L 2 W . Suppose that system (S λ ) is in the limit point or limit circle case.Then the solution z •, λ, α of system 8.1 defined in 8.9 belongs to L 2 W and satisfies 0, for every ν ∈ \ Ê .

8.14
Proof.To shorten the notation we suppress the dependence on α in all quantities appearing in this proof.Assume first that system S λ is in the limit point case.For every r ∈ a, ∞ Ì we define the function f r • : f • on a, r Ì and f r • : 0 on r, ∞ Ì and the function

8.20
Since z r •, λ W is finite by z r •, λ ∈ L 2 W , we get from the above calculation that We will prove that 8.21 implies estimate 8.13 by the convergence argument.For any t, r ∈ a, ∞ Ì we observe that z t, λ − z r t, λ − ∞ r G t, σ s , λ W s f σ s Δs.

8.22
Now we fix q ∈ a, r Ì .By the definition of G •, •, λ in 8.5 we have for every t ∈ a, q Ì z t, λ − z r t, λ − Z t, λ ∞ r X * σ s , λ W s f σ s Δs.

8.23
Since the functions X •, λ and f • belong to L 2 W , it follows that the right-hand side of 8.23 converges to zero as r → ∞ for every t ∈ a, q Ì .Hence, z r •, λ converges to the function z •, λ uniformly on a, q Ì .Since z •, λ z r •, λ on a, q Ì , we have by W • ≥ 0 and 8.21 Since q ∈ a, ∞ Ì was arbitrary, inequality 8.24 implies the result in 8.13 .In the limit circle case inequality 8.13 follows by the same argument by using the fact that all solutions of system S λ belong to L 2 W . Now we prove the existence of the limit 8.14 .Assume that the system S λ is in the limit point case, and let ν ∈ \ Ê be arbitrary.Following the argument in the proof of 30, Lemma 4.1 and 2, Theorem 5.

8.29
Upon taking the limit in 8.29 as t → ∞ and using equality 8.28 , we conclude that the limit in 8.14 holds true.
In the limit circle case, the limit in 8.14 can be proved similarly as above, because all the solutions of system S λ now belong to L 2 W .However, in this case, we can apply a direct argument to show that 8.14 holds.By formula 8.10 we get for every t ∈ a, ∞ Ì X * t, ν J z t, λ −X * t, ν JX t, λ t a Z σ * s, λ W s f σ s Δs − X * t, ν J Z t, λ ∞ t X σ * s, λ W s f σ s Δs.

8.30
The limit of the first term in 8.30 is zero because X * t, ν JX t, λ tends to zero for t → ∞ by 7.4 , and it is multiplied by a convergent integral as t → ∞.Since the columns of Z •, λ belong to L 2 W , the function X * •, ν J Z •, λ is bounded on a, ∞ Ì , and it is multiplied by an integral converging to zero as t → ∞.Therefore, formula 8.14 follows.
In the last result of this paper we construct another solution of the nonhomogeneous system 8.1 satisfying condition 8.14 and such that it starts with a possibly nonzero initial condition at t a.It can be considered as an extension of Theorem 8.5.

8.33
In addition, if the system (S λ ) is in the limit point case, then z •, λ, α is the only L 2 W solution of 8.1 satisfying α z a, λ, α v.
Proof.As in the previous proof we suppress the dependence on α.Since the function X •, λ v solves S λ , it follows from Proposition 8.4 that z •, λ, α solves the system 8.1 and α z a, λ αX a, λ v v. Next, z •, λ ∈ L 2 W as a sum of two L 2 W functions.The limit in 8.33 follows from the limit 8.14 of Theorem 8. 5

2 k
are complex n × n matrices such that B k and C k are Hermitian and W 1 k and W

Hypothesis 4 . 2 .
For every λ ∈ , we have b a

Definition 4 . 5
M λ -function .Let α, β ∈ Γ. Whenever the matrix β Z b, λ, α is invertible for some value λ ∈ , we define the Weyl-Titchmarsh M λ -function as the n × n matrixM λ M λ, b M λ, b, α, β : − β Z b, λ, α −1 βZ b, λ, α .4.13The above definition of the M λ -function is a generalization of the corresponding definitions for the continuous and discrete linear Hamiltonian and symplectic systems in 8, Definition 2.6 , 9, Definition 2.9 , 14, equation 3.10 , 1, page 2859 , 2, Definition 3.2 and time scale linear Hamiltonian systems in 3, equation 4.1 .The dependence of the M λ -function on b, α, and β will be suppressed in the notation, and M λ, b or M λ, b, α, β will be used only in few situations when we emphasize the dependence on b such as at the end of Section 5 or on α and β as in Lemma 4.14 .By 65, Corollary 4.5 , see also 44, Remark 2.2 , the M • -function is an entire function in λ.Another important property of the M λ -function is established in the following.Lemma 4.6.Let α, β ∈ Γ and λ ∈ \ Ê .Then M * λ M λ .4.14 3 , 1, Theorem 3.1 , 2, Theorem 3.6 , and 3, Theorem 4.2 .Theorem 4.10.Let α ∈ Γ, λ ∈ \ Ê , and M ∈ n×n .The matrix M belongs to the Weyl circle C λ if and only if there exists β ∈ Γ such that βX b, λ, α, M 0. In this case and under Hypothesis 4.2, we have with such a matrix β that M M λ as defined in 4.13 .

4 . 25 Conversely
, suppose that for a given M ∈ n×n there exists β ∈ Γ such that βX b 0. Then from 4.3 it follows that X b Jβ * P for the matrix P : −βJX b ∈ n×n .Hence, E M iδ λ P * βJ * JJβ * P iδ λ P * βJβ * P 0, 4.26 that is, M ∈ C λ .Finally, since λ ∈ \ Ê , then by Proposition 4.3 iii the number λ is not an eigenvalue of 4.4 , which by Lemma 4.4 shows that the matrix β Z b, λ, α is invertible.The definition of the Weyl solution in 4.16 then yields βZ b, λ, α β Z b, λ, α M βX b, λ, α, M 0, 4.27 showing that rank X b < n.This however contradicts rank X b n which we have from the definition of the Weyl solution X • in 4.16 .Consequently, 4.31 yields through iδ λ βJβ * > 0 that E M < 0. If the matrix β does not satisfy ββ * I, then we modify it according to the procedure described in the proof of Theorem 4.10.Finally, since λ ∈ \ Ê , we get from Proposition 4.3 iii and Lemma 4.4 that the matrix β Z b, λ, α is invertible which in turn implies through the calculation in 4.27 that M − β Z b, λ, α −1 βZ b, λ, α M λ .

Hypothesis 5 . 2 .
There exists b 0 ∈ a, ∞ Ì such that Hypothesis 4.2 is satisfied with b b 0 ; that is, inequality 4.8 holds with b b 0 for every λ ∈ .From Hypothesis 5.2 it follows by W • ≥ 0 that inequality 4.8 holds for every b ∈ b 0 , ∞ Ì .For the study of the geometric properties of Weyl disks we will use the following representation: E M, b iδ λ X * b, λ, α, M JX b, λ, α, M I M * F b, λ, α G * b, λ, α G b, λ, α H b, λ, matrix E M, b , where we define on a, ∞ Ì the n × n matrices

Theorem 5 . 6 .Definition 5 . 7 .
Let α ∈ Γ and λ ∈ \ Ê .Under Hypothesis 5.2, for every b ∈ b 0 , ∞ Ì , the Weyl disk and Weyl circle have the representations D λ, b P λ, b R λ, b V R λ, b , V ∈ V , 5.16 C λ, b P λ, b R λ, b UR λ, b , U ∈ U , 5.17where, with the notation 5.4 , P λ, b : −H −1 λ, b, α G λ, b, α , R λ, b : H −1/2 λ, b, α .5.18 Consequently, for every b ∈ b 0 , ∞ Ì , the sets D λ, b are closed and convex.The representations of D λ, b and C λ, b in 5.16 and 5.17 can be written as D λ, b P λ, b R λ, b VR λ, b and C λ, b P λ, b R λ, b UR λ, b .The importance of the matrices P λ, b and R λ, b is justified in the following.For α ∈ Γ, λ ∈ \ Ê , and b ∈ a, ∞ Ì such that H λ, b, α and H λ, b, α are positive definite, the matrix P λ, b is called the center of the Weyl disk or the Weyl circle.The matrices R λ, b and R λ, b are called the matrix radii of the Weyl disk or the Weyl circle.Proof of Theorem 5.6.By 5.5 and for any b ∈ b 0 , ∞ Ì , the matrices H : H λ, b, α and H : H λ, b, α are positive definite, so that the matrices P : P λ, b , R λ : R λ, b , and R λ : R λ, b are well defined.By Definition 4.9, for M ∈ D λ, b , we have E M, b ≤ 0, which in turn with notation 5.8 implies by Lemmas 5.3 and 5.4 that

20 satisfiesV
* V ≤ I.This relation between the matrices M ∈ D λ, b and V ∈ V is bijective more precisely, it is a homeomorphism , and the inverse to 5.20 is given by M P R λ V R λ .The latter formula proves that the Weyl disk D λ, b has the representation in 5.16 .Moreover, since by the definition M ∈ C λ, b means that E M, b 0, it follows that the elements of the Weyl circle C λ, b are in one-to-one correspondence with the matrices V defined in 5.20 which, similarly as in 5.19 , now satisfy V * V I. Hence, the representation of C λ, b in 5.17 follows.The fact that for b ∈ b 0 , ∞ Ì the sets D λ, b are closed and convex follows from the same properties of the set V, being homeomorphic to D λ, b .

Theorem 6 . 1 .Proof.
2 the matrix function H •, λ, α is monotone nondecreasing as b → ∞, it follows from the definition of R λ, b and R λ, b in 5.18 that the two matrix functions R λ, • and R λ, • are monotone nonincreasing for b → ∞.Furthermore, since R λ, b and R λ, b are Hermitian and positive definite for b∈ b 0 , ∞ Ì , the limits R λ : lim b → ∞ R λ, b , R λ : lim b → ∞ R λ, b , 6.1exist and satisfy R λ ≥ 0 and R λ ≥ 0. The index " " in the above notation as well as in Definition 6.2 refers to the limiting disk at ∞.In the following result we will see that the center P λ, b also converges to a limiting matrix when b → ∞.This is a generalization of 11, Theorem 4.7 , 1, Theorem 3.5 , 14, Proposition 3.5 , 2, Theorem 4.5 , and 3, Theorem 4.10 .Let α ∈ Γ and λ ∈ \ Ê .Under Hypothesis 5.2, the center P λ, b converges as b → ∞ to a limiting matrix P λ ∈ n×n , that is, We prove that the matrix function P λ, • satisfies the Cauchy convergence criterion.Let b 1 , b 2 ∈ b 0 , ∞ Ì be given with b 1 < b 2 .By Theorem 5.1, we have that D λ, b 2 ⊆ D λ, b 1 .

Definition 6 . 2
limiting Weyl disk .Let α ∈ Γ and λ ∈ \ Ê .Then the set D λ : b∈ a,∞ Ì D λ, b , 6.10 is called the limiting Weyl disk.The matrix P λ from Theorem 6.1 is called the center of D λ and the matrices R λ and R λ from 6.1 its matrix radii.
. Extensions of the Weyl-Titchmarsh theory to more general equations, namely, to the linear Hamiltonian differential systems Lagrange identity .Let λ, ν ∈ and m ∈ AE be given.If z •, λ and z •, ν are 2n×m

14 .
6emark 6.6.In 1, Definition 3.4 , the notion of a boundary of the limiting Weyl disk D λ is discussed.This would be a "limiting Weyl circle" according to Definitions 4.9 and 6.2.The description of matrices M ∈ n×n laying on this boundary follows from Theorems 6.5 and 4.10, giving for such matrices M the equality ∞ a X σ * t, λ, α, M W t X σ t, λ, α, M Δt we have on a, ∞ Ì ν Δs first term in the product in 7.14 is bounded by 1/ 2| Im λ |.Moreover, from formula 4.18 we get that the second term in the product in 7.14 is bounded by the number e * j Im M ν, t, α, β t e j / Im ν .Hence, upon recalling the limit in 6.12 , we conclude that the product in 7.14 is bounded by •, λ and z 2 •, λ are solutions of S λ and S λ , respectively.Then, by Theorem 6.7 and Remark 6.4, there are vectorsξ 1 , ξ 2 ∈ n such that z 1 •, λ X •, λ ξ 1 and z 2 •, λ X •, λ ξ 2 on a, ∞ Ì ,where X •, κ : X •, κ, α, M κ are the Weyl solutions corresponding to some matrices M κ ∈ D κ for κ ∈ {λ, λ}.In fact, by Lemma 6.8, the matrix M κ is equal to the center of the disk D κ .It follows that for any t ∈ b 0 , ∞ Ì equality and suppose that, for every square integrable solutions z 1 •, λ and z 2 •, ν of S λ and S ν , condition 7.17 is satisfied.From Theorem 6.7 we know that for M κ ∈ D κ the columns X j •, κ , j ∈ {1, . . ., n}, of the Weyl solution X •, κ are linearly independent square integrable solutions of S κ , κ ∈ {λ, ν}.Therefore, dim N λ ≥ n, and dim N ν ≥ n.
and from identity 7.4 , because lim t, ν JX t, λ v X * t, ν J z t, λ } 0. 8.34 Inequality 8.32 is obtained from estimate 8.13 by the triangle inequality. *