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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 976914, 4 pages
http://dx.doi.org/10.1155/2013/976914
Research Article

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

School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China

Received 11 July 2013; Accepted 29 August 2013

Academic Editor: Shurong Sun

Copyright © 2013 Kefeng Li and Chao Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper studies the solutions of second-order linear matrix equations on time scales. Firstly, the necessary and sufficient conditions for the existence of a solution of characteristic equation are introduced; then two diverse solutions of characteristic equation are applied to express general solution of the matrix equations on time scales.

1. Introduction

In this paper, we consider the solutions for the following second-order linear matrix equations: where , , , is the delta derivative, is the forward jump operators, , is the graininess function, and is an infinite isolated time scale, which is given by

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 [25]. 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.

2. Preliminaries

In this section, some basic concepts and some fundamental results on time scales are introduced.

Let be a nonempty closed subset. Define the forward and backward jump operators by where , . We put if is unbounded above and otherwise. The graininess functions are defined by Let be a function defined on . is said to be (delta) differentiable at provided that there exists a constant such that, for any , there is a neighborhood of (i.e., for some ) with In this case, denote . If is (delta) differentiable for every , then is said to be (delta) differentiable on . If is differentiable at , then

For convenience, we introduce the following results ([3, Lemma 1] and [4, Chapter 1]), which are useful in this paper.

Lemma 1. Let and . If and are differentiable at , then is differentiable at and

3. Main Results

We assume throughout this paper that is an invertible matrix. It follows from Lemma 1, (6), and (2) that the matrix equation (1) can be written in the form where , , , and is the identity matrix.

Definition 2. The equation where , is called the characteristic equation of (8).

Definition 3. The functions are called the eigenmatrix and eigenpolynomial of (8). Such a which satisfies is called an eigenvalue of .
In order to study the solutions of matrix equation (8), we now introduce some results about the characteristic equation (9).

Theorem 4. If and there exists such that , , then the characteristic equation (9) has solutions.

Proof. Let , . Then and are solutions of (9).

Remark 5. Generally, if and , we cannot easily get .

Example 6. Let Then Obviously, .

So, Huang and Chen [12] and J. Huang and H. Huang [13] obtained the following results.

Definition 7. Let be an eigenvalue of . Then is called a characteristic subspace of corresponding to ; the nonzero vector of is called the eigenvector corresponding to .

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, , where are the distinct eigenvalues of .

Definition 9. Let , be eigenvalues of . The extension vectors set of are as follows: where is the multiplicity of , is called the component of .

Theorem 10. The characteristic equation (9) has solutions if and only if there exist some components of the set in (14), and they are m linearly independent extension vectors.

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 and are two diverse solutions of the characteristic equation (9), which satisfy , and is invertible, then is the general solution of the matrix equation (8).

Proof. Since and are two solutions of (9), then which deduce Hence, by the invertibility of , we have Let . It follows from that Replacing and with and in matrix equation (8), we obtain Let Then (20) can be rewritten as Hence, the general solution of (21) is as follows: where . Putting into (21), we can get the general solution of (8):

Theorem 12. If and are two diverse solutions of the characteristic equation (9) and is invertible, then where and , is the general solution of the matrix equation (8).

Proof. Firstly, we prove (25) is a solution of (8). Putting (25) into (8), we have By using , , and is invertible, then we get Next, we will prove all the solutions of (8) can be written into the form of (25).
Obviously, by and , we have , ; then and are in the form of (25). Assume the solution of (8) can be written in the form of (25) for ; that is, Then, it follows from (8) and ,   being solutions of (9) that By using the method of mathematical induction, we have all the solutions of (8) can be written in the form of (25).

Acknowledgments

This research was supported by the NNSF of China (Grant nos. 11071143 and 11101241), the NNSF of Shandong Province (Grant nos. ZR2009AL003, ZR2010AL016, and ZR2011AL007), and the Scientific Research and Development Project of Shandong Provincial Education Department (J11LA01).

References

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