Weyl-Titchmarsh Theory for Time Scale Symplectic Systems on Half Line
Roman ล imon Hilscher1and Petr Zemรกnek1
Academic Editor: Miroslava Rลฏลพiฤkovรก
Received08 Oct 2010
Accepted03 Jan 2011
Published09 Mar 2011
Abstract
We develop the Weyl-Titchmarsh theory for time scale symplectic systems. We introduce the -function, study its properties, construct the corresponding Weyl disk and Weyl circle, and establish their geometric structure including the formulas for their center and matrix radii. Similar properties are then derived for the limiting Weyl disk. We discuss the notions of the system being in the limit point or limit circle case and prove several characterizations of the system in the limit point case and one condition for the limit circle case. We also define the Green function for the associated nonhomogeneous system and use its properties for deriving further results for the original system in the limit point or limit circle case. Our work directly generalizes the corresponding discrete time theory obtained recently by S. Clark and P. Zemรกnek (2010). It also unifies the results in many other papers on the Weyl-Titchmarsh theory for linear Hamiltonian differential, difference, and dynamic systems when the spectral parameter appears in the second equation. Some of our results are new even in the case of the second-order Sturm-Liouville equations on time scales.
1. 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 [1โ4] and partly in [5โ14].
Historically, the theory nowadays called by Weyl and Titchmarsh started in [15] by the investigation of the second-order linear differential equation
where are continuous, , and , 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 or there is a unique (up to a multiplicative constant) square-integrable solution of (1.1) on . 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 [18โ20] for an overview of the original contributions to the Weyl-Titchmarsh theory for (1.1); see also [21]. Extensions of the Weyl-Titchmarsh theory to more general equations, namely, to the linear Hamiltonian differential systems
was initiated in [22] and developed further in [6, 8, 10, 11, 23โ38].
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
where , , , , are complex matrices such that and are Hermitian and and 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 , , , , are complex matrices such that is Hermitian and nonnegative definite and the transition matrix in (1.4) is symplectic, that is,
In the unifying theory for differential and difference equationsโthe theory of time scalesโthe classification of second-order Sturm-Liouville dynamic equations
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 is the time scale -derivative and , where is the forward jump at ; 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 , , , , are complex matrix functions on , is Hermitian and nonnegative definite, , and the coefficient matrix in system () satisfies
where is the graininess of the time scale. The spectral parameter is only in the second equation of system (). 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 function, the geometric description of the Weyl disks, and characterizations of the limit point and limit circle cases for the time scale symplectic system (). In addition, we give a formula for the 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 [1โ4] and some results from [5โ14]. The theory of time scale symplectic systems or Hamiltonian systems is a topic with active research in recent years; see, for example, [44โ51]. 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 (). 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. In Section 4 we define the time scale -function for system () 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 () and describe their geometric structure in terms of contractive matrices in . The properties of the limiting Weyl disk and Weyl circle are then studied in Section 6, where we also prove that system () has at least linearly independent solutions in the space (see Theorem 6.7). In Section 7 we define the system () 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 () with a nonhomogeneous term. We construct its Green function, discuss its properties, and characterize the 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 which we call the time scale interval, where and . Similarly, a time scale which is unbounded above has the form . The forward and backward jump operators on a time scale are denoted by and and the graininess function by . If not otherwise stated, all functions in this paper are considered to be complex valued. A function on is called piecewise rd-continuous; we write on if the right-hand limit exists finite at all right-dense points , and the left-hand limit exists finite at all left-dense points and is continuous in the topology of the given time scale at all but possibly finitely many right-dense points . A function on is piecewise rd-continuous; we write on if on for every . An matrix-valued function is called regressive on a given time scale interval if is invertible for all in this interval.
The time scale -derivative of a function at a point is denoted by ; see [52, Definitionโโ1.10]. Whenever exists, the formula holds true. The product rule for the -differentiation of the product of two functions has the form
A function on is called piecewise rd-continuously -differentiable; we write on ; if it is continuous on , then exists at all except for possibly finitely many points , and on . As a consequence we have that the finitely many points at which does not exist belong to and these points are necessarily right-dense and left-dense at the same time. Also, since at those points we know that and exist finite, we replace the quantity by in any formula involving for all . Similarly as above we define on . The time scale integral of a piecewise rd-continuous function over is denoted by and over by provided this integral is convergent in the usual sense; see [52, Definitionsโโ1.71 andโโ1.82].
Remark 2.1. As it is known in [52, Theoremโโ5.8] and discussed in [54, Remarkโโ3.8], for a fixed and a piecewise rd-continuous matrix function on which is regressive on , the initial value problem for with has a unique solution on for any . Similarly, this result holds on .
Let us recall some matrix notations from linear algebra used in this paper. Given a complex square matrix , by , , , , , , , , 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 . Moreover, we will use the notation and for the Hermitian components of the matrix ; see [55, pagesโโ268-269] or [56, Factโโ3.5.24]. This notation will be also used with , and in this case and represent the imaginary and real parts of .
Remark 2.2. If the matrix is positive or negative definite, then the matrix is necessarily invertible. The proof of this fact can be found, for example, in [2, Remarkโโ2.6].
In order to simplify the notation we abbreviate and by . Similarly, instead of and we will use .
3. Time Scale Symplectic Systems
Let , , , , be piecewise rd-continuous functions on such that for all ; that is, is Hermitian and nonnegative definite, satisfying identity (1.8). In this paper we consider the linear system () introduced in the previous section. This system can be written as
where the matrix is defined and has the property
The system () can be written in the equivalent form
where the matrix is defined through the matrices and from (1.8) and (3.1) by
By using the identity in (1.8), a direct calculation shows that the matrix function satisfies
Here , and is the usual conjugate number to .
Remark 3.1. The name time scale symplectic system or Hamiltonian system has been reserved in the literature for the system of the form
in which the matrix function satisfies the identity in (1.8); see [44โ47, 57], and compare also, for example, with [58โ61]. Since for a fixed the matrix from (3.3) satisfies
it follows that the system () is a true time scale symplectic system according to the above terminology only for , while strictly speaking () is not a time scale symplectic system for . However, since () is a perturbation of the time scale symplectic system () 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 () for any .
Equation (3.4) represents a fundamental identity for the theory of time scale symplectic systems (). Some important properties of the matrix are displayed below. Note that formula (3.7) is a generalization of [46, equationโโ(10.4)] to complex values of .
Lemma 3.2. Identity (3.4) is equivalent to the identity
In this case for any we have
and the matrices and are invertible with
Proof. Let and be fixed. If is right-dense, that is, , then identity (3.4) reduces to . Upon multiplying this equation by from the left and right side, we get identity (3.7) with . If is right scattered, that is, , then (3.4) is equivalent to (3.8). It follows that the determinants of and are nonzero proving that these matrices are invertible with the inverse given by (3.10). Upon multiplying (3.8) by the invertible matrices from the left and from the right and by using , we get formula (3.9), which is equivalent to (3.7) due to .
Remark 3.3. Equation (3.10) allows writing the system () in the equivalent adjoint form
System (3.11) can be found, for example, in [47, Remarkโโ3.1(iii)] or [50, equationโโ(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 ().
Theorem 3.4 (existence and uniqueness theorem). Let , , and be given. Then the initial value problem () with has a unique solution on the interval .
Proof. The coefficient matrix of system (), or equivalently of system (3.2), is piecewise rd-continuous on . By Lemma 3.2, the matrix is invertible for all , which proves that the function is regressive on . Hence, the result follows from Remark 2.1.
If not specified otherwise, we use a common agreement that -vector solutions of system () and -matrix solutions of system () are denoted by small letters and capital letters, respectively, typically by or and or .
Next we establish several identities involving solutions of system () 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, Theoremโโ4.1] or [8, equationโโ(2.23)], discrete linear Hamiltonian systems in [9, equationโโ(2.55)] or [14, Lemmaโโ2.2], discrete symplectic systems in [1, Lemmaโโ2.6] or [2, Lemmaโโ2.3], and time scale linear Hamiltonian systems in [3, Lemmaโโ3.5] and [5, Theoremโโ2.2].
Theorem 3.5 (Lagrange identity). Let and be given. If and are solutions of systems () and (), respectively, then
Proof. Formula (3.12) follows from the time scales product rule (2.1) by using the relation and identity (3.6).
As consequences of Theorem 3.5, we obtain the following.
Corollary 3.6. Let and be given. If and are solutions of systems () and (), respectively, then for all we have
One can easily see that if is a solution of system (), then is a solution of system (). Therefore, Theorem 3.5 with yields a Wronskian-type property of solutions of system ().
Corollary 3.7. Let and be given. For any solution of systems ()
The following result gives another interesting property of solutions of system () and ().
Lemma 3.8. Let and be given. For any solutions and of system (), the matrix function defined by
satisfies the dynamic equation
and the identities and
Proof. Having that and are solutions of system (), it follows that and are solutions of system (). 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 is constant, identity (3.17) yields for any right-scattered that
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 .
Next we present properties of certain fundamental matrices of system (), which are generalizations of the corresponding results in [46, Sectionโโ10.2] to complex . Some of these results can be proven under the weaker condition that the initial value of does depend on and satisfies . However, these more general results will not be needed in this paper.
Lemma 3.10. Let be fixed. Ifโโ is a fundamental matrix of system () such that is symplectic and independent of , then for any we have
Proof. Identity (3.19)(i) is a consequence of Corollary 3.7, in which we use the fact that is symplectic and independent of . The second identity in (3.19) follows from the first one, while the third identity is obtained from the equation .
Remark 3.11. If the fundamental matrix in Lemma 3.10 is partitioned into two blocks, then (3.19)(i) and (3.19)(iii) have, respectively, the form
Observe that the matrix on the left-hand side of (3.21) represents a constant matrix from Lemma 3.8 and Remark 3.9.
Corollary 3.12. Under the conditions of Lemma 3.10, for any , we have
which in the notation of Remark 3.11 has the form
Proof. Identity (3.22) follows from the equation by applying formula (3.19)(ii).
4. -Function for Regular Spectral Problem
In this section we consider the regular spectral problem on the time scale interval with some fixed . We will specify the corresponding boundary conditions in terms of complex matrices from the set
The two defining conditions for in (4.1) imply that the matrix is unitary and symplectic. This yields the identity
The last equation also implies, compare with [60, Remarkโโ2.1.2], that
Let be fixed and consider the boundary value problem
Our first result shows that the boundary conditions in (4.4) are equivalent with the boundary conditions phrased in terms of the images of the matrices
which satisfy , , and .
Lemma 4.1. Let and be fixed. A solution of system () satisfies the boundary conditions in (4.4) if and only if there exists a unique vector such that
Proof. Assume that (4.4) holds. Identity (4.3) implies the existence of vectors such that and . It follows that satisfies (4.6) with . It remains to prove that is unique such a vector. If satisfies (4.6) and also and for some , then and . Hence, and . If we multiply the latter two equalities by and , respectively, and use , then we obtain and . This yields , which shows that the vector 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 is an eigenvalue of (4.4) if this boundary value problem has a solution . In this case the function is called the eigenfunction corresponding to the eigenvalue , and the dimension of the space of all eigenfunctions corresponding to (together with the zero function) is called the geometric multiplicity of .
Given , we will utilize from now on the fundamental matrix of system () satisfying the initial condition from (4.4), that is,
Then does not depend on , and it is symplectic and unitary with the inverse . Hence, the properties of fundamental matrices derived earlier in Lemma 3.10, Remark 3.11, and Corollary 3.12 apply for the matrix function .
The following assumption will be imposed in this section when studying the regular spectral problem.
Hypothesis 4.2. For every , we have
Condition (4.8) can be written in the equivalent form as
for every nontrivial solution of system (). Assumptions (4.8) and (4.9) are equivalent by a simple argument using the uniqueness of solutions of system (). The latter form (4.9) has been widely used in the literature, such as in the continuous time case in [8, Hypothesisโโ2.2], [30, equationโโ(1.3)], [26, equationโโ(2.3)], in the discrete time case in [9, Conditionโโ(2.16)], [14, equationโโ(1.7)], [1, Assumptionโโ2.2], [2, Hypothesisโโ2.4], and in the time scale Hamiltonian case in [3, Assumptionโโ3] and [5, Conditionโโ(3.9)].
Following Remark 3.11, we partition the fundamental matrix as
where and are the solutions of system () satisfying and . With the notation
we have the classical characterization of the eigenvalues of (4.4); see, for example, the continuous time in [64, Chapterโโ4], the discrete time in [14, Theoremโโ2.3, Lemmaโโ2.4], [2, Lemmaโโ2.9, Theoremโโ2.11], and the time scale case in [62, Lemmaโโ3], [63, Corollaryโโ1].
Proposition 4.3. For and , we have with notation (4.11) the following. (i)The number is an eigenvalue of (4.4) if and only if . (ii)The algebraic multiplicity of the eigenvalue , that is, the number , is equal to the geometric multiplicity of . (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, Sectionโโ5], [63, Corollaryโโ1], [3, Theoremโโ3.6].
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 which has dimension instead of using the matrix which has dimension . Results of this type can be found in special cases of system () in [8, Lemmaโโ2.5], [11, Theoremโโ4.1], [9, Lemmaโโ2.8], [14, Lemmaโโ3.1], [1, Lemmaโโ2.5], [3, Theoremโโ3.4], and [2, Lemmaโโ3.1].
Lemma 4.4. Let and be fixed. Then is an eigenvalue of (4.4) if and only if . In this case the algebraic and geometric multiplicities of are equal to .
Proof. One can follow the same arguments as in the proof of the corresponding discrete symplectic case in [2, Lemmaโโ3.1]. However, having the result of Proposition 4.3, we can proceed directly by the methods of linear algebra. In this proof we abbreviate and . Assume that is singular, that is, for some vectors , not both zero. Then , which yields that . If , then , which implies upon the multiplication by from the left that . Since not both and can be zero, it follows that and the matrix is singular. Conversely, if for some nonzero vector , then ; that is, is singular, with the vector . Indeed, by using identity (4.2) we have . From the above we can also see that the number of linearly independent vectors in is the same as the number of linearly independent vectors in . Therefore, by Proposition 4.3(ii), the algebraic and geometric multiplicities of as an eigenvalue of (4.4) are equal to .
Since the eigenvalues of (4.4) are real, it follows that the matrix is invertible for every except for at most real numbers. This motivates the definition of the -function for the regular spectral problem.
Definition 4.5 (-function). Let . Whenever the matrix is invertible for some value , we define the Weyl-Titchmarsh -function as the matrix
The above definition of the -function is a generalization of the corresponding definitions for the continuous and discrete linear Hamiltonian and symplectic systems in [8, Definitionโโ2.6], [9, Definitionโโ2.9], [14, equationโโ(3.10)], [1, pageโโ2859], [2, Definitionโโ3.2] and time scale linear Hamiltonian systems in [3, equationโโ(4.1)]. The dependence of the -function on , , and will be suppressed in the notation, and or will be used only in few situations when we emphasize the dependence on (such as at the end of Section 5) or on and (as in Lemma 4.14). By [65, Corollaryโโ4.5], see also [44, Remarkโโ2.2], the -function is an entire function in . Another important property of the -function is established in the following.
Lemma 4.6. Let and . Then
Proof. We abbreviate and . By using the definition of in (4.13) and identity (3.21), we have
because . 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 ().
Definition 4.7 (Weyl solution). For any matrix , we define the so-called Weyl solution of system () by
where and are defined in (4.10).
The function , being a linear combination of two solutions of system (), is also a solution of this system. Moreover, , and, if is invertible, then . Consequently, if we take in Definition 4.7, then ; that is, the Weyl solution satisfies the right endpoint boundary condition in (4.4).
Following the corresponding notions in [8, equationโโ(2.18)], [9, equationโโ(2.51)], [14, pageโโ471], [1, pageโโ2859], [2, equationโโ(3.13)], [3, equationโโ(4.2)], we define the Hermitian matrix function for system ().
Definition 4.8. For a fixed and , we define the matrix function
where .
For brevity we suppress the dependence of the function on and . In few cases we will need depending on (as in Theorem 5.1 and Definition 6.2) and in such situations we will use the notation . Since , it follows that is a Hermitian matrix for any . 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 โdiskโ or โcircleโ will be presented in Section 5.
Definition 4.9 (Weyl disk and Weyl circle). For a fixed and , the set
is called the Weyl disk, and the set
is called the Weyl circle.
The dependence of the Weyl disk and Weyl circle on will be again suppressed. In the following result we show that the Weyl circle consists of precisely those matrices with . This result generalizes the corresponding statements in [8, Lemmaโโ2.8], [9, Lemmaโโ2.13], [14, Lemmaโโ3.3], [1, Theoremโโ3.1], [2, Theoremโโ3.6], and [3, Theoremโโ4.2].
Theorem 4.10. Let , , and . The matrix belongs to the Weyl circle if and only if there exists such that . In this case and under Hypothesis 4.2, we have with such a matrix that as defined in (4.13).
Proof. Assume that , that is, . Then, with the vector
where denotes , we have
Moreover, , because the matrices and are invertible and . In addition, the identity yields
Now, if the condition is not satisfied, then we replace by (note that , so that is well defined), and in this case
Conversely, suppose that for a given there exists such that . Then from (4.3) it follows that for the matrix . Hence,
that is, . Finally, since , then by Proposition 4.3(iii) the number is not an eigenvalue of (4.4), which by Lemma 4.4 shows that the matrix is invertible. The definition of the Weyl solution in (4.16) then yields
which implies that .
Remark 4.11. The matrix from the proof of Theorem 4.10 is invertible. This fact was not needed in that proof. However, we show that is invertible because this argument will be used in the proof of Lemma 4.14. First we prove that . For if for some , then from identity (4.2) we get . Therefore, . The opposite inclusion follows by the definition of . And since, by (4.16), , it follows that . Hence, as well; that is, the matrix is invertible.
The next result contains a characterization of the matrices which lie โinsideโ the Weyl disk . In the previous result (Theorem 4.10) we have characterized the elements of the boundary of the Weyl disk , that is, the elements of the Weyl circle , in terms of the matrices . For such we have , which yields . Comparing with that statement we now utilize the matrices which satisfy . In the special cases of the continuous and discrete time, this result can be found in [8, Lemmaโโ2.13], [9, Lemmaโโ2.18], and [2, Theoremโโ3.13].
Theorem 4.12. Let , , and . The matrix satisfies if and only if there exists such that and . In this case and under Hypothesis 4.2, we have with such a matrix that as defined in (4.13) and may be chosen so that .
Proof. For consider on the Weyl solution
Suppose first that . Then the matrices , , are invertible. Indeed, if one of them is singular, then there exists a nonzero vector such that or . Then
which contradicts . Now we set , , and . Then for this matrix we have and, by a similar calculation as in (4.29),
where we used the equality . Since and is invertible, it follows that . Conversely, assume that for a given matrix there is satisfying and . Condition is equivalent to when and to when . The positive or negative definiteness of implies the invertibility of and ; see Remark 2.2. Therefore, from the equality , we obtain , and so
The matrix is invertible, because if for some nonzero vector , then , showing that . This however contradicts which we have from the definition of the Weyl solution in (4.16). Consequently, (4.31) yields through that . If the matrix does not satisfy , then we modify it according to the procedure described in the proof of Theorem 4.10. Finally, since , we get from Proposition 4.3(iii) and Lemma 4.4 that the matrix is invertible which in turn implies through the calculation in (4.27) that .
In the following lemma we derive some additional properties of the Weyl disk and the -function. Special cases of this statement can be found in [8, Lemmaโโ2.9], [33, Theoremโโ3.1], [9, Lemmaโโ2.14], [14, Lemmaโโ3.2(ii)], [1, Theoremโโ3.7], [2, Lemmaโโ3.7], and [3, Theoremโโ4.13].
Theorem 4.13. Let and . For any matrix we have
In addition, under Hypothesis 4.2, we have .
Proof. By identity (4.18), for any matrix , we have
which yields together with on 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 , which is one of the parameters of the -function and the Weyl solution , when varies within the set . For this purpose we will use the -function with all its arguments in the following two statements.
Lemma 4.14. Let and . Then
Proof. Let and be given via (4.13), and consider the Weyl solutions and defined by (4.16) with and , respectively. First we prove that the two Weyl solutions and differ by a constant nonsingular multiple. By definition, and , which implies through (4.3) that and for some matrices , which are invertible by Remark 4.11. This implies that . Consequently, , where . By the uniqueness of solutions of system (), see Theorem 3.4, we obtain that on . Upon the evaluation at we get
Since the matrices and are unitary, it follows from (4.35) that
The first row above yields that , while the second row is then written as identity (4.34).
Corollary 4.15. Let and . With notation (4.16) and (4.13) we have
Proof. The above identity follows from (4.35) and the formula for the matrix from the end of the proof of Lemma 4.14.
5. Geometric Properties of Weyl Disks
In this section we study the geometric properties of the Weyl disks as the point moves through the interval . Our first result shows that the Weyl disks are nested. This statement generalizes the results in [11, Theoremโโ4.5], [66, Sectionโโ3.2.1], [9, equationโโ(2.70)], [14, Theoremโโ3.1], [3, Theoremโโ4.4], and [5, Theoremโโ3.3(i)].
Theorem 5.1 (nesting property of Weyl disks). Let and . Then
Proof. Let with , and take , that is, . From identity (4.18) with and later with and by using , we have
Therefore, by Definition 4.9, the matrix belongs to , which shows the result.
Similarly for the regular case (Hypothesis 4.2) we now introduce the following assumption.
Hypothesis 5.2. There exists such that Hypothesis 4.2 is satisfied with ; that is, inequality (4.8) holds with for every .
From Hypothesis 5.2 it follows by that inequality (4.8) holds for every .
For the study of the geometric properties of Weyl disks we will use the following representation:
of the matrix , where we define on the matrices
Since is Hermitian, it follows that and are also Hermitian. Moreover, by (4.7), we have . In addition, if , then Corollary 3.7 and Hypothesis 5.2 yield for any
Therefore, is invertible (positive definite) for all and monotone nondecreasing as , with a consequence that is monotone nonincreasing as . The following factorization of holds true; see also [2, equationโโ(4.11)].
Lemma 5.3. Let and . With the notation (5.4), for any and we have
whenever the matrix is invertible.
Proof. The result is shown by a direct calculation.
The following identity is a generalization of its corresponding versions in [11, Lemmaโโ4.3], [1, Lemmaโโ3.3], [14, Propositionโโ3.2], [2, Lemmaโโ4.2], [3, Lemmaโโ4.6], and [5, Theoremโโ5.6].
Lemma 5.4. Let and . With the notation (5.4), for any , we have
whenever the matrices and are invertible.
Proof. In order to simplify and abbreviate the notation we introduce the matrices
and use the notation and for and , respectively. Then, since and , we get the identities
Hence, by using that is Hermitian, we see that
Identity (5.7) is now proven.
Corollary 5.5. Let and . Under Hypothesis 5.2, the matrix is invertible for every , and for these values of we have
Proof. Since , then identity (5.5) yields that and . Consequently, inequality (5.14) follows from (5.7) of Lemma 5.4.
In the next result we justify the terminology for the sets and in Definition 4.9 to be called a โdiskโ and a โcircle.โ It is a generalization of [14, Theoremโโ3.1], [2, Theoremโโ5.4], [5, Theoremโโ3.3(iii)]; see also [66, Theoremโโ3.5], [26, pagesโโ70-71], [8, pageโโ3485], [14, Propositionโโ3.3], [1, Theoremโโ3.3], [3, Theoremโโ4.8]. Consider the sets and of contractive and unitary matrices in , respectively, that is,
The set is known to be closed (in fact compact, since is bounded) and convex.
Theorem 5.6. Let and . Under Hypothesis 5.2, for every , the Weyl disk and Weyl circle have the representations
where, with the notation (5.4),
Consequently, for every , the sets are closed and convex.
The representations of and in (5.16) and (5.17) can be written as and . The importance of the matrices and is justified in the following.
Definition 5.7. For , , and such that and are positive definite, the matrix is called the center of the Weyl disk or the Weyl circle. The matrices and are called the matrix radii of the Weyl disk or the Weyl circle.
Proof of Theorem 5.6. By (5.5) and for any , the matrices and are positive definite, so that the matrices , , and are well defined. By Definition 4.9, for , we have , which in turn with notation (5.8) implies by Lemmas 5.3 and 5.4 that
Therefore, the matrix
satisfies . This relation between the matrices and is bijective (more precisely, it is a homeomorphism), and the inverse to (5.20) is given by . The latter formula proves that the Weyl disk has the representation in (5.16). Moreover, since by the definition means that , it follows that the elements of the Weyl circle are in one-to-one correspondence with the matrices defined in (5.20) which, similarly as in (5.19), now satisfy . Hence, the representation of in (5.17) follows. The fact that for the sets are closed and convex follows from the same properties of the set , being homeomorphic to .
6. 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 5.2 the matrix function is monotone nondecreasing as , it follows from the definition of and in (5.18) that the two matrix functions and are monotone nonincreasing for . Furthermore, since and are Hermitian and positive definite for , the limits
exist and satisfy and . The index โโ 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 also converges to a limiting matrix when . This is a generalization of [11, Theoremโโ4.7], [1, Theoremโโ3.5], [14, Propositionโโ3.5], [2, Theoremโโ4.5], and [3, Theoremโโ4.10].
Theorem 6.1. Let and . Under Hypothesis 5.2, the center converges as to a limiting matrix , that is,
Proof. We prove that the matrix function satisfies the Cauchy convergence criterion. Let be given with . By Theorem 5.1, we have that . Therefore, by (5.16) of Theorem 5.6, for a matrix , there are (unique) matrices such that
Upon subtracting the two equations in (6.3), we get
This equation, when solved for in terms of , has the form
which defines a continuous mapping , . Since is compact, it follows that the mapping has a fixed point in , that is, for some matrix . Equation implies that
Hence, by , we have
Since the functions and are monotone nonincreasing, they are bounded; that is, for some , we have and for all . Let be arbitrary. The convergence of and as yields the existence of such that for every with we have
Using estimate (6.8) in inequality (6.7) we obtain for
This means that the limit in (6.2) exists, which completes the proof.
By Theorems 5.1 and 5.6 we know that the Weyl disks are closed, convex, and nested as . Thereore the limit of as is a closed, convex, and nonempty set. This motivates the following definition, which can be found in the special cases of system () in [26, Theoremโโ3.3], [1, Theoremโโ3.6], [2, Definitionโโ4.7], and [3, Theoremโโ4.12].
Definition 6.2 (limiting Weyl disk). Let and . Then the set
is called the limiting Weyl disk. The matrix from Theorem 6.1 is called the center of and the matrices and from (6.1) its matrix radii.
As a consequence of Theorem 5.6, we obtain the following characterization of the limiting Weyl disk.
Corollary 6.3. Let and . Under Hypothesis 5.2, we have
where 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 and the limiting matrix radii and of are well defined.
Remark 6.4. By means of the nesting property of the disks (Theorem 5.1) and Theorems 4.10 and 4.12, it follows that the elements of the limiting Weyl disk are of the form
where satisfies and for all . Moreover, from Lemma 4.6, we conclude that
A matrix from (6.12) is called a half-line Weyl-Titchmarsh -function. Also, as noted in [2, Sectionโโ4], see also [8, Theoremโโ2.18], the function is a Herglotz function with rank and has a certain integral representation (which will not be needed in this paper).
Our next result shows another characterization of the elements of in terms of the Weyl solution defined in (4.16). This is a generalization of [11, pageโโ671], [26, equationโโ(3.2)], [1, Theoremโโ3.8(i)], [2, Theoremโโ4.8], and [3, Theoremโโ4.15].
Theorem 6.5. Let , , and . The matrix belongs to the limiting Weyl disk if and only if
Proof. By Definition 6.2, we have if and only if , that is, , for all . Therefore, by formula (4.18), we get
for every , which is equivalent to inequality (6.14).
Remark 6.6. In [1, Definitionโโ3.4], the notion of a boundary of the limiting Weyl disk is discussed. This would be a โlimiting Weyl circleโ according to Definitions 4.9 and 6.2. The description of matrices laying on this boundary follows from Theorems 6.5 and 4.10, giving for such matrices the equality
Condition (6.16) is also equivalent to
This is because, by (4.19) and the Lagrange identity (Corollary 3.6),
for every . From this we can see that the integral on the right-hand side above converges for and (6.16) holds if and only if condition (6.17) is satisfied. Characterizations (6.16) and (6.17) of the matrices on the boundary of the limiting Weyl disk generalize the corresponding results in [1, Theoremsโโ3.8(ii) andโโ3.9]; see also [14, Theoremโโ6.3].
Consider the linear space of square integrable functions
where we define
In the following result we prove that the space contains the columns of the Weyl solution when belongs to the limiting Weyl disk . This implies that there are at least linearly independent solutions of system () in . This is a generalization of [11, Theoremโโ5.1], [14, Theoremโโ4.1], [2, Theoremโโ4.10], and [5, pageโโ716].
Theorem 6.7. Let , , and . The columns of form a linearly independent system of solutions of system (), each of which belongs to .
Proof. Let for be the columns of the Weyl solution , where is the th unit vector. We prove that the functions are linearly independent. Assume that on for some . Then , where . It follows by (4.19) that
which implies the equality . Using that for some , we obtain from Theorem 4.13 that the matrix is positive definite. Hence, so that the functions are linearly independent. Finally, for every we get from Theorem 6.5 the inequality
Thus, for every , and the proof is complete.
Denote by the linear space of all square integrable solutions of system (), that is,
Then as a consequence of Theorem 6.7 we obtain the estimate
Next we discuss the situation when for some .
Lemma 6.8. Let , , and . Then the matrix radii of the limiting Weyl disk satisfy . Consequently, the set consists of the single matrix , that is, the center of , which is given by formula (6.2) of Theorem 6.1.
Proof. With the matrix radii and of defined in (6.1) and with the Weyl solution given by a matrix , we observe that the columns of form a basis of the space . Since the columns of the fundamental matrix span all solutions of system (), the definition of yields that the columns of together with the columns of form a basis of all solutions of system (). Hence, from and Theorem 6.7, we get that the columns of do not belong to . Consequently, by formula (5.5), the Hermitian matrix functions and defined in (5.4) are monotone nondecreasing on without any upper bound; that is, their eigenvaluesโbeing realโtend to . Therefore, the functions and as defined in (5.18) have limits at equal to zero; that is, and . The fact that the set then follows from the characterization of in Corollary 6.3.
In the final result of this section, we establish another characterization of the matrices from the limiting Weyl disk . 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, Lemmaโโ2.21] and [2, Theoremโโ4.16].
Hypothesis 6.9. For every with and for every , we have
Under Hypothesis 6.9, the Weyl disks converge to the limiting disk โmonotonicallyโ as ; that is, the limiting Weyl disk is โopenโ in the sense that all of its elements lie inside . This can be interpreted in view of Theorem 4.12 as for all .
Theorem 6.10. Let , , and . Under Hypothesis 6.9, the matrix if and only if
Proof. If condition (6.26) holds, then follows from the definition of . Conversely, suppose that , and let be given. Then for any we have by formula (4.18) that
where we used the property . Since is assumed, we have , that is, , while Hypothesis 6.9 implies the positivity of the integral over in (6.27). Consequently, (6.27) yields that .
Remark 6.11. If we partition the Weyl solution into two blocks and as in (4.28), then condition (6.26) can be written as
Therefore, by Remark 2.2, the matrices and are invertible for all . A standard argument then yields that the quotient satisfies the Riccati matrix equation (suppressing the argument in the coefficients)
see [57, Theoremโโ3], [48, Sectionโโ6], and [49].
7. 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, pageโโ75], [43, Definitionโโ1.2] in the time scales scalar case , with [8, pageโโ3486], [36, pageโโ1668], [30, pageโโ274], [38, Definitionโโ3.1], [37, Definitionโโ1], [67, pageโโ2826] in the continuous case, and with [14, Definitionโโ5.1], [2, Definitionโโ4.12] in the discrete case.
Definition 7.1 (limit point and limit circle case for system ). The system () is said to be in the limit point case at (or of the limit point type) if
The system () is said to be in the limit circle case at (or of the limit circle type) if
Remark 7.2. According to Remark 6.4 (in which ), the center of the limiting Weyl disk can be expressed in the limit point case as
where is arbitrary but fixed.
Next we establish the first main result of this section. Its continuous time version can be found in [30, Theoremโโ2.1], [11, Theoremโโ8.5] and the discrete time version in [9, Lemmaโโ3.2], [2, Theoremโโ4.13].
Theorem 7.3. Let the system () be in the limit point or limit circle case, fix , and let . Then
where and are the Weyl solutions of () and (), respectively, defined by (4.16) through the matrices and , which are determined by the limit in (6.12).
Proof. For every and matrices such that and and for , we define the matrix (compare with Definition 4.5)
Then, by Theorems 4.10 and 4.12, we have . Following the notation in (4.16), we consider the Weyl solutions . Similarly, let be the Weyl solutions corresponding to the matrices from the statement of this theorem. First assume that the system () is of the limit point type. In this case, by Remark 7.2, we may take for all . Hence, from Theorem 4.10, we get that on . By (4.3), for each and , there is a matrix such that on . Hence, we have on
where we define
If we show that
then (7.6) implies the result claimed in (7.4). First we prove the second limit in (7.8). Pick any . By Theorem 5.6, Corollary 6.3, and , we have
where and . Therefore,
Since and are, respectively, solutions of systems () and () which satisfy , it follows from Corollary 3.6 that
Hence, we can write
where we used the Hermitian property of and . Since we now assume that system () is in the limit point case, we know from Lemma 6.8 that and . Therefore, in order to establish (7.8)(ii), it is sufficient to show that
is bounded for . Let be a unit vector, and denote by the th column of for . With the definition of in (5.18) we have
where the last step follows from the Cauchy-Schwarz inequality (C-S) on time scales. From (5.5) we obtain
so that the first term in the product in (7.14) is bounded by . Moreover, from formula (4.18) we get that the second term in the product in (7.14) is bounded by the number . Hence, upon recalling the limit in (6.12), we conclude that the product in (7.14) is bounded by
which is independent of . 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 () is in the limit circle case, then for the columns of and belong to ; hence, they are bounded in the norm. In this case the limits in (7.8) easily follow from the limit (6.12) for , .
In the next result we provide a characterization of the system () being of the limit point type. Special cases of this statement can be found, for example, in [14, Theoremโโ6.12] and [2, Theoremโโ4.14].
Theorem 7.4. Let . The system () is in the limit point case if and only if, for every and every square integrable solutions and of () and (), respectively, we have
Proof. Let () be in the limit point case. Fix any , and suppose that and are solutions of () and (), respectively. Then, by Theorem 6.7 and Remark 6.4, there are vectors such that and on , where are the Weyl solutions corresponding to some matrices for . In fact, by Lemma 6.8, the matrix is equal to the center of the disk . It follows that for any equality
holds, so that (7.17) is established. Conversely, let be arbitrary but fixed, set , and suppose that, for every square integrable solutions and of () and (), condition (7.17) is satisfied. From Theorem 6.7 we know that for the columns , , of the Weyl solution are linearly independent square integrable solutions of (), . Therefore, , and . Moreover, by identity (3.19)(i), we have
Let be any square integrable solution of system (). Then, by our assumption (7.17),
From (7.19) and (7.20) it follows that the vectors , , and are solutions of the linear homogeneous system
Since, by Theorem 6.7, the vectors for represent a basis of the solution space of system (7.21), there exists a vector such that . By the uniqueness of solutions of system () we then get on . Hence, the solution is square integrable and . Since was arbitrary, it follows that the system () 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, Corollaryโโ2.3], [9, Corollaryโโ3.3], [14, Theoremโโ6.14], [2, Corollaryโโ4.15], [1, Theoremโโ3.9], [3, Theoremโโ4.16].
Corollary 7.5. Let . The system () is in the limit point case if and only if, for every and every square integrable solutions and of () and (), respectively, we have
Proof. The necessity follows directly from Theorem 7.3. Conversely, assume that condition (7.22) holds for every and every square integrable solutions and of () and (). Fix , and set . By Corollary 3.7 we know that is constant on . 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 () is of the limit point type.
8. Nonhomogeneous Time Scale Symplectic Systems
In this section we consider the nonhomogeneous time scale symplectic system
where the matrix function and are defined in (3.3) and (3.1), , and where the associated homogeneous system () 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
with . The following is a generalization of Theorem 3.5 to nonhomogeneous systems.
Theorem 8.1 (Lagrange identity). Let and be given. If and are solutions of systems (8.1) and (8.2), respectively, then
Proof. Formula (8.3) follows by the product rule (2.1) with the aid of the relation
and identity (3.6).
For , , and , we define the function
where is the solution of system () given in (4.10), that is, , and is the Weyl solution of () as in (4.16) determined by a matrix . This matrix is arbitrary but fixed throughout this section. By interchanging the order of the arguments and , we have
In the literature the function is called a resolvent kernel, compare with [30, pageโโ283], [32, pageโโ15], [2, equationโโ(5.4)], and in this section it will play a role of the Green function.
Lemma 8.2. Let and . Then
Proof. Identity (8.7) follows by a direct calculation from the definition of via (4.16) with a matrix by using formulas (3.21) and (6.13).
In the next lemma we summarize the properties of the function , which together with Proposition 8.4 and Theorem 8.5 justifies the terminology โGreen functionโ of the system (8.1); compare with [68, Sectionโโ4]. A discrete version of the following result can be found in [2, Lemmaโโ5.1].
Lemma 8.3. Let and . The function has the following properties: (i) for every , , (ii) for every , (iii) for every right-scattered point , (iv)for every such that , the function solves the homogeneous system () on the set , where
(v)the columns of belong to for every , and the columns of belong to for every .
Proof. Condition (i) follows from the definition of in (8.5). Condition (ii) is a consequence of Lemma 8.2. Condition (iii) is proven from the definition of in (8.5) by using Lemma 8.2 and . Concerning condition (iv), the function solves the system () on because solves this system on . If is left-dense, then solves () on , since solves this system on . For the same reason solves () on if is left-scattered. Condition (v) follows from the definition of in (8.5) used with and from the fact that the columns of belong to , by Theorem 6.7. The columns of then belong to by part (i) of this lemma.
Since by Lemma 8.3(v) the columns of belong to , the function
is well defined whenever . Moreover, by using (8.6), we can write as
Proposition 8.4. For , , and , the function defined in (8.9) solves the nonhomogeneous system (8.1) with the initial condition .
Proof. By the time scales product rule (2.1) when we -differentiate expression (8.10), we have for every (suppressing the dependence on in the the following calculation)
This shows that is a solution of system (8.1). From (8.10) with , we get
where we used the initial condition and coming from .
The following theorem provides further properties of the solution of system (8.1). It is a generalization of [10, Lemmaโโ4.2], [11, Theoremโโ7.5], [2, Theoremโโ5.2] to time scales.
Theorem 8.5. Let , , and . Suppose that system () is in the limit point or limit circle case. Then the solution of system (8.1) defined in (8.9) belongs to and satisfies
Proof. To shorten the notation we suppress the dependence on in all quantities appearing in this proof. Assume first that system () is in the limit point case. For every we define the function on and on and the function
For every we have as in (8.10) that
Since by Theorem 6.7 the solution , (8.16) shows that , being a multiple of , also belongs to . Moreover, by Theorem 7.3,
On the other hand, , and for any identity (8.3) implies
Combining (8.18), where , formula (8.17), and the definition on yields
By using the Cauchy-Schwarz inequality (C-S) on time scales and , we then have
Since is finite by , we get from the above calculation that
We will prove that (8.21) implies estimate (8.13) by the convergence argument. For any we observe that
Now we fix . By the definition of in (8.5) we have for every
Since the functions and belong to , it follows that the right-hand side of (8.23) converges to zero as for every . Hence, converges to the function uniformly on . Since on , we have by and (8.21) that
Since 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 () belong to . Now we prove the existence of the limit (8.14). Assume that the system () is in the limit point case, and let be arbitrary. Following the argument in the proof of [30, Lemmaโโ4.1] and [2, Theoremโโ5.2], we have from identity (8.3) that for any
Since for equality (8.16) holds, it follows that
Hence, by (8.25),
By the uniform convergence of to on compact intervals, we get from (8.27) with the equality
On the other hand, by (8.3), we obtain for every
Upon taking the limit in (8.29) as 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 () now belong to . However, in this case, we can apply a direct argument to show that (8.14) holds. By formula (8.10) we get for every
The limit of the first term in (8.30) is zero because tends to zero for by (7.4), and it is multiplied by a convergent integral as . Since the columns of belong to , the function is bounded on , and it is multiplied by an integral converging to zero as . 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 . It can be considered as an extension of Theorem 8.5.
Corollary 8.6. Let and . Assume that () is in the limit point or limit circle case. For and we define
where is given in (8.9). Then solves the system (8.1) with ,
In addition, if the system () is in the limit point case, then is the only solution of (8.1) satisfying .
Proof. As in the previous proof we suppress the dependence on . Since the function solves (), it follows from Proposition 8.4 that solves the system (8.1) and . Next, as a sum of two functions. The limit in (8.33) follows from the limit (8.14) of Theorem 8.5 and from identity (7.4), because
Inequality (8.32) is obtained from estimate (8.13) by the triangle inequality. Now we prove the uniqueness of in the case of () being of the limit point type. If and are two solutions of (8.1) satisfying , then their difference also belongs to and solves system () with . Since for some , the initial condition implies through (4.7) that for some . If , then , because in the limit point case the columns of do not belong to , which is a contradiction. Therefore, and the uniqueness of is established.
Acknowledgments
The research was supported by the Czech Science Foundation under Grant 201/09/J009, by the research project MSM 0021622409 of the Ministry of Education, Youth, and Sports of the Czech Republic, and by the Grant MUNI/A/0964/2009 of Masaryk University.
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