The Solutions of Second-Order Linear Matrix Equations on Time Scales

As a tool for establishing a unified framework for continuous and discrete analysis, a theory of dynamic equations on measure chains was introduced by Hilger in his Ph.D. thesis [1] in 1988. In many cases, it is necessary to study a special case of measure chains-time scales. In the last decade the investigation of dynamic systems on time scales has involved much interesting, including quite a few fields, such as the theory of calculus, the oscillation of the dynamic systems, the eigenvalue problems and boundary value problems, and partial differential equations on time scales, and so forth [2– 5]. Up to now, there are few results about matrix equations on time scales. In 1998, Agarwal and Bohner [6] studied the quadratic functionals for second order matrix equations on time scales; in 2002, Erbe and Peterson [7] obtained oscillation criteria for a second-order self-adjoint matrix differential equation on time scales in terms of the eigenvalues of the coefficient matrices and the graininess function. The theory of dynamic systems on time scales is of very important theoretical significance and has a wide range of applications. Based on the related literatures, the researches of solutions of matrix difference or differential equations have few results. In 1999, Barkatou [8] proposed an algorithm for the rational solutions of special matrix difference equations and discussed their applications; in 2003, Freiling and Hochhaus [9] investigated some properties of the solution for rational matrix difference equations; in 2004, Xu and Zhang [10] studied the representation of general solution for second order homogeneous matrix difference equations; in 2011, Wu and Zhou [11] obtained the particular solutions of one kind of second order matrix differential equations. Since the continuous case and the discrete case are two special cases of time scales, so, we study the solutions of second order linear matrix equations on time scales. This paper is organized as follows. Section 2 introduces some basic concepts and fundamental theory about time scales. By using the characteristic equation of the matrix equation the solutions of (1) are obtained, which will be given in Section 3.


Introduction
In this paper, we consider the solutions for the following second-order linear matrix equations: [ ()  Δ ()] Δ +  Δ () +  ()   () = 0,  ∈ T, (1) where () = (), () = /(), , , , () ∈ R m×m ,  Δ is the delta derivative, () is the forward jump operators,   () := (()), () is the graininess function, and T is an infinite isolated time scale, which is given by T := { 0 ,  1 ,  2 , . . .,   , . ..} . ( As a tool for establishing a unified framework for continuous and discrete analysis, a theory of dynamic equations on measure chains was introduced by Hilger in his Ph.D. thesis [1] in 1988.In many cases, it is necessary to study a special case of measure chains-time scales.In the last decade the investigation of dynamic systems on time scales has involved much interesting, including quite a few fields, such as the theory of calculus, the oscillation of the dynamic systems, the eigenvalue problems and boundary value problems, and partial differential equations on time scales, and so forth [2][3][4][5].Up to now, there are few results about matrix equations on time scales.In 1998, Agarwal and Bohner [6] studied the quadratic functionals for second order matrix equations on time scales; in 2002, Erbe and Peterson [7] obtained oscillation criteria for a second-order self-adjoint matrix differential equation on time scales in terms of the eigenvalues of the coefficient matrices and the graininess function.The theory of dynamic systems on time scales is of very important theoretical significance and has a wide range of applications. Based on the related literatures, the researches of solutions of matrix difference or differential equations have few results.In 1999, Barkatou [8] proposed an algorithm for the rational solutions of special matrix difference equations and discussed their applications; in 2003, Freiling and Hochhaus [9] investigated some properties of the solution for rational matrix difference equations; in 2004, Xu and Zhang [10] studied the representation of general solution for second order homogeneous matrix difference equations; in 2011, Wu and Zhou [11] obtained the particular solutions of one kind of second order matrix differential equations.Since the continuous case and the discrete case are two special cases of time scales, so, we study the solutions of second order linear matrix equations on time scales.This paper is organized as follows.Section 2 introduces some basic concepts and fundamental theory about time scales.By using the characteristic equation of the matrix equation the solutions of (1) are obtained, which will be given in Section 3.

Preliminaries
In this section, some basic concepts and some fundamental results on time scales are introduced.
Let T ⊂ R be a nonempty closed subset.Define the forward and backward jump operators ,  : T → T by where inf 0 = sup T, sup 0 = inf T. We put T  = T if T is unbounded above and T  = T \((max T), max T] otherwise.The graininess functions ],  : T → [0, ∞) are defined by Let  be a function defined on T.  is said to be (delta) differentiable at  ∈ T  provided that there exists a constant  such that, for any  > 0, there is a neighborhood  of  (i.e.,  = ( − ,  + ) ∩ T for some  > 0) with In this case, denote  Δ () := .If  is (delta) differentiable for every  ∈ T  , then  is said to be (delta) differentiable on T.
For convenience, we introduce the following results ([3, Lemma 1] and [4, Chapter 1]), which are useful in this paper.Lemma 1.Let ,  : T → R and  ∈ T  .If  and  are differentiable at , then  is differentiable at  and

Definition 2. The equation
where Λ ∈ R m×m , is called the characteristic equation of ( 8).
Definition 3. The functions are called the eigenmatrix and eigenpolynomial of (8).Such a  which satisfies ℎ() = 0 is called an eigenvalue of ().
So, Huang and Chen [12] and J. Huang and H. Huang [13] obtained the following results.Definition 7. Let  0 be an eigenvalue of ().Then is called a characteristic subspace of () corresponding to  0 ; the nonzero vector  of   0 is called the eigenvector corresponding to  0 .
Theorem 8.There exist the diagonalizable solutions of characteristic equation (9) if and only if the dimension of the sum of characteristic subspaces    is n; that is, dim{∑ ℎ =1    } = , where   ( = 1, 2, . . ., ℎ) are the distinct eigenvalues of ().Definition 9. Let   ,  = 1, 2, . . ., , be eigenvalues of ().The extension vectors set of    are as follows: where   is the multiplicity of   , ( 0 ,  In the following, we discuss the general solutions of the matrix equation ( 8) by using the solutions of the characteristic equation (9).Theorem 11.If Λ 1 and Λ 2 are two diverse solutions of the characteristic equation (9), which satisfy is the general solution of the matrix equation (8).