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

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: 𝑃(𝑡)𝑋^Δ(𝑡)^Δ+𝑄𝑋^Δ(𝑡)+𝑅(𝑡)𝑋^𝜎(𝑡)=0,𝑡∈𝕋, (1) where 𝑃(𝑡)=𝑃𝜇(𝑡), 𝑅(𝑡)=𝑅/𝜇(𝑡), 𝑃,𝑄,𝑅,𝑋(𝑡)∈𝐑^𝐦×𝐦, 𝑋^Δ is the delta derivative, 𝜎(𝑡) is the forward jump operators, 𝑋^𝜎(𝑡)∶=𝑋(𝜎(𝑡)), 𝜇(𝑡) is the graininess function, and 𝕋 is an infinite isolated time scale, which is given by 𝕋∶=𝑡₀,𝑡₁,𝑡₂,…,𝑡_𝑛,…. (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 [2–5]. 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.

2. 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 𝜎,𝜌∶𝕋→𝕋 by 𝜎(𝑡)=inf{𝑠∈𝕋∶𝑠>𝑡},𝜌(𝑡)=sup{𝑠∈𝕋∶𝑠<𝑡}, (3) where inf∅=sup𝕋, sup∅=inf𝕋. We put 𝕋^𝑘=𝕋 if 𝕋 is unbounded above and 𝕋^𝑘=𝕋⧵(𝜌(max𝕋),max𝕋] otherwise. The graininess functions 𝜈,𝜇∶𝕋→[0,∞) are defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡,𝜈(𝑡)=𝑡−𝜌(𝑡). (4) Let 𝑓 be a function defined on 𝕋. 𝑓 is said to be (delta) differentiable at 𝑡∈𝕋^𝑘 provided that there exists a constant 𝑎 such that, for any 𝜀>0, there is a neighborhood 𝑈 of 𝑡 (i.e., 𝑈=(𝑡−𝛿,𝑡+𝛿)∩𝕋 for some 𝛿>0) with 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑎(𝜎(𝑡)−𝑠)≤𝜀|𝜎(𝑡)−𝑠|,∀𝑠∈𝑈. (5) In this case, denote 𝑓^Δ(𝑡)∶=𝑎. If 𝑓 is (delta) differentiable for every 𝑡∈𝕋^𝑘, then 𝑓 is said to be (delta) differentiable on 𝕋. If 𝑓 is differentiable at 𝑡∈𝕋^𝑘, then 𝑓^Δ(𝑡)=lim_{𝑠→𝑡𝑠∈𝕋}𝑓(𝑡)−𝑓(𝑠)(𝑡)𝑓(𝜎(𝑡(𝑡)𝑡−𝑠,if𝜇=0,))−𝑓𝜇(𝑡),if𝜇(𝑡)>0. (6)

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

Lemma 1. Let 𝑓,𝑔∶𝕋→𝐑 and 𝑡∈𝕋^𝑘. If 𝑓 and 𝑔 are differentiable at 𝑡, then 𝑓𝑔 is differentiable at 𝑡 and 𝑓𝑔^Δ(𝑡)=𝑓^𝜎(𝑡)𝑔^Δ(𝑡)+𝑓^Δ(𝑡)𝑔(𝑡)=𝑓^Δ(𝑡)𝑔^𝜎(𝑡)+𝑓(𝑡)𝑔^Δ(𝑡). (7)

3. Main Results

We assume throughout this paper that 𝑃 is an invertible matrix. It follows from Lemma 1, (6), and (2) that the matrix equation (1) can be written in the form 𝑋^𝜎2(𝑡)−𝐵𝑋^𝜎(𝑡)−𝐴𝑋(𝑡)=0,𝑡∈𝕋, (8) where 𝑋^𝜎2(𝑡)=𝑋^𝜎(𝜎(𝑡)), 𝐵=2𝐸_𝑚−𝑃⁻¹𝑄−𝑃⁻¹𝑅, 𝐴=𝑃⁻¹𝑄−𝐸_𝑚, and 𝐸_𝑚 is the identity matrix.

Definition 2. The equation 𝐹(Λ)=Λ²−𝐵Λ−𝐴=0, (9) where Λ∈𝐑^𝐦×𝐦, is called the characteristic equation of (8).

Definition 3. The functions 𝐹(𝜆)=𝜆²−𝐵𝜆−𝐴,ℎ(𝜆)=det𝐹(𝜆) (10) are called the eigenmatrix and eigenpolynomial of (8). Such a 𝜆 which satisfies ℎ(𝜆)=0 is called an eigenvalue of 𝐹(𝜆).
In order to study the solutions of matrix equation (8), we now introduce some results about the characteristic equation (9).

Theorem 4. If 𝐴𝐵=𝐵𝐴 and there exists 𝐶∈𝐑^𝐦×𝐦 such that 𝐶²=𝐵²+4𝐴, 𝐵𝐶=𝐶𝐵, then the characteristic equation (9) has solutions.

Proof. Let Λ₁=(1/2)(𝐵+𝐶), Λ₂=(1/2)(𝐵−𝐶). Then Λ₁ and Λ₂ are solutions of (9).

Remark 5. Generally, if 𝐴𝐵=𝐵𝐴 and 𝐶²=𝐵²+4𝐴, we cannot easily get 𝐵𝐶=𝐶𝐵.

Example 6. Let 1𝐴=201414,𝐵=102−1. (11) Then √𝐶=1301+√31√,𝐵𝐶=√305+231+√3,√−1𝐶𝐵=√303+231+√3.−1 (12) Obviously, 𝐵𝐶≠𝐶𝐵.

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

Definition 7. Let 𝜆₀ be an eigenvalue of 𝐹(𝜆). Then 𝑉_𝜆0=𝑥∣𝜆²₀−𝐵𝜆₀−𝐴𝑥=0,𝑥∈𝐑^𝐦×𝟏 (13) is called a characteristic subspace of 𝐹(𝜆) corresponding to 𝜆₀; the nonzero vector 𝑥 of 𝑉_𝜆0 is called the eigenvector corresponding to 𝜆₀.

Theorem 8. There exist the diagonalizable solutions of characteristic equation (9) if and only if the dimension of the sum of characteristic subspaces 𝑉_𝜆𝑖 is n; that is, dim{∑^ℎ_𝑖=1𝑉_𝜆𝑖}=𝑚, where 𝜆_𝑖(𝑖=1,2,…,ℎ) are the distinct eigenvalues of 𝐹(𝜆).

Definition 9. Let 𝜆_𝑖,𝑖=1,2,…,𝑠, be eigenvalues of 𝐹(𝜆). The extension vectors set of 𝑉_𝜆𝑖are as follows: 𝐸𝑉_𝜆𝑖=𝑥₀,𝑥₁,…,𝑥_𝑘∣𝜆²_𝑖−𝐵𝜆_𝑖−𝐴𝑥₀𝜆=0,_2𝑖−𝐵𝜆_𝑖−𝐴𝑥₁=−2𝜆_𝑖𝐸_𝑚−𝐵𝑥₀,𝑘≤𝑘_𝑖𝜆−1,_2𝑖−𝐵𝜆_𝑖−𝐴𝑥_𝑘=−2𝜆_𝑖𝐸_𝑚−𝐵𝑥_𝑘−1−𝑥_𝑘−2𝐸_𝑚, (14) where 𝑘_𝑖 is the multiplicity of 𝜆_𝑖, (𝑥₀,𝑥₁,…,𝑥_𝑘) is called the component of 𝐸𝑉_𝜆𝑖.

Theorem 10. The characteristic equation (9) has solutions if and only if there exist some components (𝑥₀,𝑥₁,…,𝑥_𝑘),…,(𝑦₀,𝑦₁,…,𝑦_𝑞) of the set ∑^𝑠_𝑖=1𝐸𝑉_𝜆𝑖 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 Λ₁ and Λ₂ are two diverse solutions of the characteristic equation (9), which satisfy Λ₁Λ₂=Λ₂Λ₁, and Λ₁−Λ₂ is invertible, then 𝑋𝑡_𝑛=^𝑛_𝑘=1Λ₁^𝑘−1Λ₂^𝑛−𝑘𝑋𝑡₁−^𝑛−1_𝑘=1Λ₁^𝑛−𝑘Λ^𝑘₂𝑋𝑡₀ (15) is the general solution of the matrix equation (8).

Proof. Since Λ₁ and Λ₂ are two solutions of (9), then Λ²₁−𝐵Λ₁Λ−𝐴=0,₂₂−𝐵Λ₂−𝐴=0, (16) which deduce Λ₁+Λ₂Λ₁−Λ₂=𝐵Λ₁−Λ₂. (17) Hence, by the invertibility of Λ₁−Λ₂, we have 𝐵=Λ₁+Λ₂,−𝐴=Λ₁Λ₂,Λ₁−Λ₂²=𝐵²+4𝐴. (18) Let 𝐶=Λ₁−Λ₂. It follows from Λ₁Λ₂=Λ₂Λ₁ that 𝐴𝐵=𝐵𝐴,𝐴𝐶=𝐶𝐴,𝐵𝐶=𝐶𝐵. (19) Replacing 𝐵 and 𝐴 with (1/2)(𝐵+𝐶)+(1/2)(𝐵−𝐶) and −(1/4)(𝐵+𝐶)(𝐵−𝐶) in matrix equation (8), we obtain 2𝑥^𝜎2(𝑡)−(𝐵−𝐶)𝑋^𝜎(𝑡)=12()×𝐵+𝐶2𝑋^𝜎(𝑡)−(𝐵−𝐶)𝑋(𝑡). (20) Let 𝑌(𝑡)=2𝑋^𝜎(𝑡)−(𝐵−𝐶)𝑋(𝑡). (21) Then (20) can be rewritten as 𝑌^𝜎(𝑡)=12(𝐵+𝐶)𝑌(𝑡). (22) Hence, the general solution of (21) is as follows: 𝑌𝑡_𝑛=12^𝑛(𝐵+𝐶)^𝑛𝑌𝑡₀, (23) where 𝑌(𝑡₀)=2𝑋(𝑡₁)−(𝐵−𝐶)𝑋(𝑡₀). Putting 𝑌(𝑡_𝑛) into (21), we can get the general solution of (8): 𝑥𝑡_𝑛=12^𝑛−1(𝐵−𝐶)^𝑛−1𝑋𝑡₁+^𝑛−1_𝑘=112^𝑘(𝐵−𝐶)^𝑘−1𝑌𝑡_𝑛−𝑘1=−2^𝑛𝑛−1_𝑘=1(𝐵+𝐶)^𝑛−𝑘(𝐵−𝐶)^𝑘𝑋𝑡₀+12^𝑛𝑛−1_𝑘=1(𝐵+𝐶)^𝑘−1(𝐵−𝐶)^𝑛−𝑘𝑋𝑡₁=^𝑛_𝑘=1Λ₁^𝑘−1Λ₂^𝑛−𝑘𝑋𝑡₁−^𝑛−1_𝑘=1Λ₁^𝑛−𝑘Λ^𝑘₂𝑋𝑡₀. (24)

Theorem 12. If Λ₁ and Λ₂ are two diverse solutions of the characteristic equation (9) and Λ₁−Λ₂ is invertible, then 𝑋𝑡_𝑛=Λ^𝑛₁𝐶₁+Λ^𝑛₂𝐶₂, (25) where 𝐶₁=(Λ₂−Λ₁)⁻¹(Λ₂𝑋(𝑡₀)−𝑋(𝑡₁)) and 𝐶₂=(Λ₁−Λ₂)⁻¹(Λ₁𝑋(𝑡₀)−𝑋(𝑡₁)), 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 Λ₁^𝑛+2𝐶₁+Λ₂^𝑛+2𝐶₂−𝐵Λ₁^𝑛+1𝐶₁+Λ₂^𝑛+1𝐶₂−𝐴Λ^𝑛₁𝐶₁+Λ^𝑛₂𝐶₂=Λ²₁−𝐵Λ₁−𝐴Λ^𝑛₁𝐶₁+Λ²₂−𝐵Λ₂−𝐴×Λ^𝑛₂𝐶₂=0. (26) By using 𝑋(𝑡₀)=𝐶₁+𝐶₂, 𝑋(𝑡₁)=Λ₁𝐶₁+Λ₂𝐶₂, and Λ₁−Λ₂ is invertible, then we get 𝐶₁=Λ₂−Λ₁⁻¹Λ₂𝑋𝑡₀−𝑋𝑡₁,𝐶₂=Λ₁−Λ₂⁻¹Λ₁𝑋𝑡₀−𝑋𝑡₁. (27) Next, we will prove all the solutions of (8) can be written into the form of (25).
Obviously, by 𝐶₁=(Λ₂−Λ₁)⁻¹(Λ₂𝑋(𝑡₀)−𝑋(𝑡₁)) and 𝐶₂=(Λ₁−Λ₂)⁻¹(Λ₁𝑋(𝑡₀)−𝑋(𝑡₁)), we have 𝑋(𝑡₀)=𝐶₁+𝐶₂, 𝑋(𝑡₁)Λ₁𝐶₁+Λ₂𝐶₂; then 𝑋(𝑡₀) and 𝑋(𝑡₁) are in the form of (25). Assume the solution of (8) can be written in the form of (25) for 𝑛≤𝑘; that is, 𝑋𝑡_𝑘=Λ^𝑘₁𝐶₁+Λ^𝑘₂𝐶₂for𝑛≤𝑘. (28) Then, it follows from (8) and Λ₁, Λ₂  being solutions of (9) that 𝑥𝑡_𝑘+3=𝑋^𝜎2𝑡_𝑘+1=𝐵𝑋^𝜎𝑡_𝑘+1+𝐴𝑋𝑡_𝑘+1=𝐵𝑋𝑡_𝑘+2+𝐴𝑋𝑡_𝑘+1=𝐵Λ₁^𝑘+2𝐶₁+Λ₂^𝑘+2𝐶₂+𝐴Λ₁^𝑘+1𝐶₁+Λ₂^𝑘+1𝐶₂=𝐵Λ₁+𝐴Λ₁^𝑘+1𝐶₁+𝐵Λ₂+𝐴Λ₂^𝑘+1𝐶₂=Λ²₁Λ₁^𝑘+1𝐶₁+Λ²₂Λ₂^𝑘+1𝐶₂=Λ₁^𝑘+3𝐶₁+Λ₂^𝑘+3𝐶₂. (29) 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).