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 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 zΔ t t z t , t ∈ a,∞ , 3.5 in which the matrix function · satisfies the identity in 1.8 ; see 44–47, 57 , and compare also, for example, with 58–61 . Since for a fixed λ, ν ∈ the matrix S t, λ from 3.3 satisfies S∗ t, λ J JS t, ν μ t S∗ t, λ JS t, ν ( λ − ν )[ I μ t S∗ t ] W̃ t [ I μ t S t ] , 3.6 it follows that the system Sλ is a true time scale symplectic system according to the above terminology only for λ ∈ , while strictly speaking Sλ is not a time scale symplectic system for λ ∈ \ . However, since Sλ is a perturbation of the time scale symplectic system S0 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 Sλ for any λ ∈ . Equation 3.4 represents a fundamental identity for the theory of time scale symplectic systems Sλ . Some important properties of the matrixS 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,∞ , λ ∈ . 3.7 Abstract and Applied Analysis 7 In this case for any λ ∈ we have [ I μ t S∗ t, λ ] J [ I μ t S ( t, λ )] J, t ∈ a,∞ , 3.8 [ I μ t S ( t, λ )] J [ I μ t S∗ t, λ ] J, t ∈ a,∞ , 3.9and Applied Analysis 7 In this case for any λ ∈ we have [ I μ t S∗ t, λ ] J [ I μ t S ( t, λ )] J, t ∈ a,∞ , 3.8 [ I μ t S ( t, λ )] J [ I μ t S∗ t, λ ] J, t ∈ a,∞ , 3.9 and the matrices I μ t S t, λ and I μ t S t, λ are invertible with [ I μ t S t, λ ]−1 −J [ I μ t S∗ ( t, λ )] J, t ∈ a,∞ . 3.10 Proof. Let t ∈ a,∞ 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 − I μ t S t, λ −1 J from the right and by using J2 −I, we get formula 3.9 , which is equivalent to 3.7 due to μ t > 0. Remark 3.3. Equation 3.10 allows writing the system Sλ in the equivalent adjoint form zΔ t, λ JS∗ ( t, λ ) Jz t, λ , t ∈ a,∞ . 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 Sλ . Theorem 3.4 existence and uniqueness theorem . Let λ ∈ , t0 ∈ a,∞ , and z0 ∈ 2n be given. Then the initial value problem (Sλ) with z t0 z0 has a unique solution z ·, λ ∈ Cprd on the interval a,∞ . Proof. The coefficient matrix of system Sλ , or equivalently of system 3.2 , is piecewise rdcontinuous on a,∞ . By Lemma 3.2, the matrix I μ t S t, λ is invertible for all t ∈ a,∞ , which proves that the function S ·, λ is regressive on a,∞ . Hence, the result follows from Remark 2.1. If not specified otherwise, we use a common agreement that 2n-vector solutions of system Sλ and 2n × n-matrix solutions of system Sλ are denoted by small letters and capital letters, respectively, typically by z ·, λ or z̃ ·, λ and Z ·, λ or Z̃ ·, λ . Next we establish several identities involving solutions of system Sλ 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 11, Theorem 4.1 or 8, equation 2.23 , discrete linear Hamiltonian systems in 9, equation 2.55 8 Abstract and Applied Analysis or 14, Lemma 2.2 , discrete symplectic systems in 1, Lemma 2.6 or 2, Lemma 2.3 , and time scale linear Hamiltonian systems in 3, Lemma 3.5 and 5, Theorem 2.2 . Theorem 3.5 Lagrange identity . Let λ, ν ∈ andm ∈ be given. If z ·, λ and z ·, ν are 2n×m solutions of systems (Sλ) and (Sν), respectively, then z∗ t, λ Jz t, ν Δ ( λ − ν ) zσ∗ t, λ W̃ t z t, ν , t ∈ a,∞ . 3.12 Proof. Formula 3.12 follows from the time scales product rule 2.1 by using the relation z 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. Corollary 3.6. Let λ, ν ∈ andm ∈ be given. If z ·, λ and z ·, ν are 2n×m solutions of systems (Sλ) and (Sν), respectively, then for all t ∈ a,∞ we have z∗ t, λ Jz t, ν z∗ a, λ Jz a, ν ( λ − ν )∫ t a zσ∗ s, λ W̃ s z s, ν Δs. 3.13 One can easily see that if z ·, λ is a solution of system Sλ , then z ·, λ is a solution of system Sλ . Therefore, Theorem 3.5with ν λ yields aWronskian-type property of solutions of system Sλ . Corollary 3.7. Let λ ∈ andm ∈ be given. For any 2n ×m solution z ·, λ of systems (Sλ) z∗ t, λ Jz ( t, λ ) ≡ z∗ a, λ Jz ( a, λ ) , is constant on a,∞ . 3.14 The following result gives another interesting property of solutions of system Sλ and Sλ . Lemma 3.8. Let λ ∈ and m ∈ be given. For any 2n ×m solutions z ·, λ and z̃ ·, λ of system (Sλ), the 2n × 2n matrix function K ·, λ defined by K t, λ : z t, λ z̃∗ ( t, λ ) − z̃ t, λ z∗ ( t, λ ) , t ∈ a,∞ , 3.15 satisfies the dynamic equation KΔ t, λ S t, λ K t, λ [ I μ t S t, λ ] K t, λ S∗ ( t, λ ) , t ∈ a,∞ , 3.16 and the identities K∗ t, λ −K t, λ and K t, λ [ I μ t S t, λ ] K t, λ [ I μ t S∗ ( t, λ )] , t ∈ a,∞ . 3.17 Abstract and Applied Analysis 9 Proof. Having that z ·, λ and z̃ ·, λ are solutions of system Sλ , it follows that z ·, λ and z̃ ·, λ are solutions of system Sλ . 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,∞ that S t, λ K λ K λ S∗ ( t, λ ) μ t S t, λ K λ S∗ ( t, λ ) 0. 3.18and Applied Analysis 9 Proof. Having that z ·, λ and z̃ ·, λ are solutions of system Sλ , it follows that z ·, λ and z̃ ·, λ are solutions of system Sλ . 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,∞ 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 ofK λ ≡ J. Next we present properties of certain fundamental matrices Ψ ·, λ of system Sλ , 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 (Sλ) such thatΨ a, λ is symplectic and independent of λ, then for any t ∈ a,∞ we have Ψ∗ t, λ JΨ ( t, λ ) J, Ψ−1 t, λ −JΨ∗ ( t, λ ) J, Ψ t, λ JΨ∗ ( t, λ ) J. 3.19 Proof. Identity 3.19 i is a consequence of Corollary 3.7, in which we use the fact thatΨ a, λ is symplectic and independent of λ. The second identity in 3.19 follows from the first one, while the third identity is obtained from the equation Ψ t, λ Ψ−1 t, λ I. Remark 3.11. If the fundamentalmatrixΨ ·, λ Z ·, λ Z̃ ·, λ in Lemma 3.10 is partitioned into two 2n × n blocks, then 3.19 i and 3.19 iii have, respectively, the form Z∗ t, λ JZ ( t, λ ) 0, Z∗ t, λ JZ̃ ( t, λ ) I, Z̃∗ t, λ JZ̃ ( t, λ ) 0, 3.20 Z t, λ Z̃∗ ( t, λ ) − Z̃ t, λ Z∗ ( t, λ ) J. 3.21 Observe that the matrix on the left-hand side of 3.21 represents a constant matrix K t, λ from Lemma 3.8 and Remark 3.9. Corollary 3.12. Under the conditions of Lemma 3.10, for any t ∈ a,∞ , we have Ψ t, λ JΨ∗ ( t, λ ) [ I μ t S t, λ ] J, 3.22 which in the notation of Remark 3.11 has the form Z t, λ Z̃∗ ( t, λ ) − Z̃ t, λ Z∗ ( t, λ ) [ I μ t S t, λ ] J. 3.23 10 Abstract and Applied Analysis Proof. Identity 3.22 follows from the equation Ψ t, λ I μ t S t, λ Ψ t, λ by applying formula 3.19 ii . 4. M λ -Function for Regular Spectral Problem In this section we consider the regular spectral problem on the time scale interval a, b with some fixed b ∈ a,∞ . We will specify the corresponding boundary conditions in terms of complex n × 2nmatrices 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 ( α∗ −Jα∗ ) ( α αJ ) I ∈ 2n×2n , that is, α∗α − Jα∗αJ I. 4.2 The last equation also implies, compare with 60, Remark 2.1.2 , that