Theory and Modern Applications

# Stability criteria for linear Hamiltonian dynamic systems on time scales

## Abstract

In this article, we establish some stability criteria for the polar linear Hamiltonian dynamic system on time scales

${x}^{\Delta }\left(t\right)=\alpha \left(t\right)x\left(\sigma \left(t\right)\right)+\beta \left(t\right)y\left(t\right),\phantom{\rule{1em}{0ex}}{y}^{\Delta }\left(t\right)=-\gamma \left(t\right)x\left(\sigma \left(t\right)\right)-\alpha \left(t\right)y\left(t\right),\phantom{\rule{1em}{0ex}}t\in \mathbb{T}$

by using Floquet theory and Lyapunov-type inequalities.

2000 Mathematics Subject Classification: 39A10.

## 1 Introduction

A time scale is an arbitrary nonempty closed subset of the real numbers . We assume that $\mathbb{T}$ is a time scale. For $t\in \mathbb{T}$, the forward jump operator $\sigma :\mathbb{T}\to \mathbb{T}$ is defined by $\sigma \left(t\right):inf\left\{s\in \mathbb{T}:s>t\right\}$, the backward jump operator $\rho :\mathbb{T}\to \mathbb{T}$ is defined by $\rho \left(t\right):sup\left\{s\in \mathbb{T}:s, and the graininess function $\mu :\mathbb{T}\to \left[0,\infty \right\}$ is defined by μ(t) = σ(t) - t. For other related basic concepts of time scales, we refer the reader to the original studies by Hilger , and for further details, we refer the reader to the books of Bohner and Peterson [4, 5] and Kaymakcalan et al. .

Definition 1.1. If there exists a positive number ω such that $t+n\omega \in \mathbb{T}$ for all $t\in \mathbb{T}$ and n , then we call $\mathbb{T}$ a periodic time scale with period ω.

Suppose $\mathbb{T}$ is a ω-periodic time scale and $0\in \mathbb{T}$. Consider the polar linear Hamiltonian dynamic system on time scale $\mathbb{T}$

${x}^{\Delta }\left(t\right)=\alpha \left(t\right)x\left(\sigma \left(t\right)\right)+\beta \left(t\right)y\left(t\right),\phantom{\rule{1em}{0ex}}{y}^{\Delta }\left(t\right)=-\gamma \left(t\right)x\left(\sigma \left(t\right)\right)-\alpha \left(t\right)y\left(t\right),\phantom{\rule{1em}{0ex}}t\in \mathbb{T},$
(1.1)

where α(t), β(t) and γ(t) are real-valued rd-continuous functions defined on $\mathbb{T}$. Throughout this article, we always assume that

$1-\mu \left(t\right)\alpha \left(t\right)>0,\phantom{\rule{1em}{0ex}}\forall \phantom{\rule{2.77695pt}{0ex}}t\in \mathbb{T}$
(1.2)

and

$\beta \left(t\right)\ge 0,\phantom{\rule{1em}{0ex}}\forall \phantom{\rule{2.77695pt}{0ex}}t\in \mathbb{T}.$
(1.3)

For the second-order linear dynamic equation

${\left[p\left(t\right){x}^{\Delta }\left(t\right)\right]}^{\Delta }+q\left(t\right)x\left(\sigma \left(t\right)\right)=0,\phantom{\rule{1em}{0ex}}t\in \mathbb{T},$
(1.4)

if let y(t) = p(t)x Δ (t), then we can rewrite (1.4) as an equivalent polar linear Hamiltonian dynamic system of type (1.1):

${x}^{\Delta }\left(t\right)=\frac{1}{p\left(t\right)}y\left(t\right),\phantom{\rule{1em}{0ex}}{y}^{\Delta }\left(t\right)=-q\left(t\right)x\left(\sigma \left(t\right)\right),\phantom{\rule{1em}{0ex}}t\in \mathbb{T},$
(1.5)

where p(t) and q(t) are real-valued rd-continuous functions defined on $\mathbb{T}$ with p(t) > 0, and

$\alpha \left(t\right)=0,\phantom{\rule{1em}{0ex}}\beta \left(t\right)=\frac{1}{p\left(t\right)},\phantom{\rule{1em}{0ex}}\gamma \left(t\right)=q\left(t\right).$

Recently, Agarwal et al. , Jiang and Zhou , Wong et al.  and He et al.  established some Lyapunov-type inequalities for dynamic equations on time scales, which generalize the corresponding results on differential and difference equations. Lyapunov-type inequalities are very useful in oscillation theory, stability, disconjugacy, eigenvalue problems and numerous other applications in the theory of differential and difference equations. In particular, the stability criteria for the polar continuous and discrete Hamiltonian systems can be obtained by Lyapunov-type inequalities and Floquet theory, see . In 2000, Atici et al.  established the following stablity criterion for the second-order linear dynamic equation (1.4):

Theorem 1.2 . Assume p(t) > 0 for $t\in \mathbb{T}$ , and that

$p\left(t+\omega \right)=p\left(t\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}q\left(t+\omega \right)=q\left(t\right),\phantom{\rule{1em}{0ex}}\forall \phantom{\rule{2.77695pt}{0ex}}t\in \mathbb{T}.$
(1.6)

If

${\int }_{0}^{\omega }q\left(t\right)\Delta t\ge 0,\phantom{\rule{1em}{0ex}}q\left(t\right)\not\equiv 0$
(1.7)

and

$\left[{p}_{0}+{\int }_{0}^{\omega }\frac{1}{p\left(t\right)}\phantom{\rule{2.77695pt}{0ex}}\Delta t\right]\phantom{\rule{2.77695pt}{0ex}}{\int }_{0}^{\omega }{q}^{+}\left(t\right)\Delta t\le 4,$
(1.8)

then equation (1.4 ) is stable, where

${p}_{0}=\underset{t\in \left[0,\rho \left(\omega \right)\right]}{max}\frac{\sigma \left(t\right)-t}{p\left(t\right)},\phantom{\rule{1em}{0ex}}{q}^{+}\left(t\right)=max\left\{q\left(t\right),0\right\},$
(1.9)

where and in the sequel, system (1.1) or Equation ( 1.4 ) is said to be unstable if all nontrivial solutions are unbounded on $\mathbb{T}$ ; conditionally stable if there exist a nontrivial solution which is bounded on $\mathbb{T}$ ; and stable if all solutions are bounded on $\mathbb{T}$.

In this article, we will use the Floquet theory in [18, 19] and the Lyapunov-type inequalities in  to establish two stability criteria for system (1.1) and equation (1.4). Our main results are the following two theorems.

Theorem 1.3. Suppose (1.2) and (1.3) hold and

$\alpha \left(t+\omega \right)=\alpha \left(t\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\beta \left(t+\omega \right)=\beta \left(t\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\gamma \left(t+\omega \right)=\gamma \left(t\right),\phantom{\rule{1em}{0ex}}\forall \phantom{\rule{2.77695pt}{0ex}}t\in \mathbb{T}.$
(1.10)

Assume that there exists a non-negative rd-continuous function θ (t) defined on $\mathbb{T}$ such that

$|\alpha \left(t\right)|\phantom{\rule{2.77695pt}{0ex}}\le \phantom{\rule{2.77695pt}{0ex}}\theta \left(t\right)\beta \left(t\right),\phantom{\rule{1em}{0ex}}\forall \phantom{\rule{2.77695pt}{0ex}}t\in \mathbb{T}\left[0,\omega \right]=\left[0,\omega \right]\cap \mathbb{T},$
(1.11)
${\int }_{0}^{\omega }\left[\gamma \left(t\right)-{\theta }^{2}\left(t\right)\beta \left(t\right)\right]\phantom{\rule{2.77695pt}{0ex}}\Delta t>0,$
(1.12)

and

${\int }_{0}^{\omega }|\alpha \left(t\right)|\Delta t+{\left[{\int }_{0}^{\omega }\beta \left(t\right)\Delta t{\int }_{0}^{\omega }{\gamma }^{+}\left(t\right)\Delta t\right]}^{1∕2}<2.$
(1.13)

Then system (1.1) is stable.

Theorem 1.4. Assume that (1.6) and (1.7) hold, and that

${\int }_{0}^{\omega }\frac{1}{p\left(t\right)}\Delta t{\int }_{0}^{\omega }{q}^{+}\left(t\right)\phantom{\rule{0.3em}{0ex}}\Delta t\le 4.$
(1.14)

Then equation ( 1.4 ) is stable.

Remark 1.5. Clearly, condition (1.14) improves (1.8) by removing term p 0.

We dwell on the three special cases as follows:

1. 1.

If $\mathbb{T}=ℝ$, system (1.1) takes the form:

${x}^{\prime }\left(t\right)=\alpha \left(t\right)x\left(t\right)+\beta \left(t\right)y\left(t\right),\phantom{\rule{1em}{0ex}}{y}^{\prime }\left(t\right)=-\gamma \left(t\right)x\left(t\right)-\alpha \left(t\right)y\left(t\right),\phantom{\rule{1em}{0ex}}t\in ℝ.$
(1.15)

In this case, the conditions (1.12) and (1.13) of Theorem 1.3 can be transformed into

${\int }_{0}^{\omega }\left[\gamma \left(t\right)-{\theta }^{2}\left(t\right)\beta \left(t\right)\right]\phantom{\rule{2.77695pt}{0ex}}dt>0,$
(1.16)

and

${\int }_{0}^{\omega }|\alpha \left(t\right)|dt+\phantom{\rule{2.77695pt}{0ex}}{\left[{\int }_{0}^{\omega }\beta \left(t\right)dt{\int }_{0}^{\omega }{\gamma }^{+}\left(t\right)dt\right]}^{1∕2}<2.$
(1.17)

Condition (1.17) is the same as (3.10) in , but (1.11) and (1.16) are better than (3.9) in  by taking θ (t) = |α (t)|/β (t). A better condition than (1.17) can be found in [14, 15].

2. 2.

If $\mathbb{T}=ℤ$, system (1.1) takes the form:

$\Delta x\left(n\right)=\alpha \left(n\right)x\left(n+1\right)+\beta \left(n\right)y\left(n\right),\phantom{\rule{1em}{0ex}}\Delta y\left(n\right)=-\gamma \left(n\right)x\left(n+1\right)-\alpha \left(n\right)y\left(n\right),\phantom{\rule{1em}{0ex}}n\in ℤ.$
(1.18)

In this case, the conditions (1.11), (1.12), and (1.13) of Theorem 1.3 can be transformed into

$|\alpha \left(n\right)|\le \theta \left(n\right)\beta \left(n\right),\phantom{\rule{1em}{0ex}}\forall \phantom{\rule{2.77695pt}{0ex}}n\in \left\{0,1,\dots ,\omega -1\right\},$
(1.19)
$\sum _{n=0}^{\omega -1}\left[\gamma \left(n\right)-{\theta }^{2}\left(n\right)\beta \left(n\right)\right]>0,$
(1.20)

and

$\sum _{n=0}^{\omega -1}|\alpha \left(n\right)|+{\left[\sum _{n=0}^{\omega -1}\beta \left(n\right)\sum _{n=0}^{\omega -1}{\gamma }^{+}\left(n\right)\right]}^{1∕2}<2.$
(1.21)

Conditions (1.19), (1.20), and (1.21) are the same as (1.17), (1.18) and (1.19) in , i.e., Theorem 1.3 coincides with Theorem 3.4 in . However, when p(n) and q(n) are ω-periodic functions defined on $ℤ$, the stability conditions

$0\le \sum _{n=0}^{\omega -1}q\left(n\right)\le \sum _{n=0}^{\omega -1}{q}^{+}\left(n\right)\le \frac{4}{{\sum }_{n=0}^{\omega -1}\frac{1}{p\left(n\right)}},\phantom{\rule{1em}{0ex}}q\left(n\right)\not\equiv 0,\phantom{\rule{1em}{0ex}}\forall n\in \left\{0,1,\dots ,\omega -1\right\}$
(1.22)

in Theorem 1.4 are better than the one

$0<\sum _{n=0}^{\omega -1}q\left(n\right)\le \sum _{n=0}^{\omega -1}{q}^{+}\left(n\right)<\frac{4}{{\sum }_{n=0}^{\omega -1}\frac{1}{p\left(n\right)}}$
(1.23)

in [16, Corollary 3.4]. More related results on stability for discrete linear Hamiltonian systems can be found in .

3. 3.

Let δ > 0 and N {2, 3, 4, ...}. Set ω = δ + N, define the time scale $\mathbb{T}$ as follows:

$\mathbb{T}=\bigcup _{k\in ℤ}\left[k\omega ,k\omega +\delta \right]\cup \left\{k\omega +\delta +n:n=1,2,\dots ,N-1\right\}.$
(1.24)

Then system (1.1) takes the form:

${x}^{\prime }\left(t\right)=\alpha \left(t\right)x\left(t\right)+\beta \left(t\right)y\left(t\right),\phantom{\rule{1em}{0ex}}{y}^{\prime }\left(t\right)=-\gamma \left(t\right)x\left(t\right)-\alpha \left(t\right)y\left(t\right),\phantom{\rule{1em}{0ex}}t\in \bigcup _{k\in ℤ}\left[k\omega ,k\omega +\delta \right),$
(1.25)

and

$\begin{array}{c}\Delta x\left(t\right)=\alpha \left(t\right)x\left(t+1\right)+\beta \left(t\right)y\left(t\right),\phantom{\rule{1em}{0ex}}\Delta y\left(t\right)=-\gamma \left(t\right)x\left(t+1\right)-\alpha \left(t\right)y\left(t\right),\\ \phantom{\rule{1em}{0ex}}t\in \bigcup _{k\in ℤ}\left\{k\omega +\delta +n:n=0,1,\dots ,N-2\right\}.\end{array}$
(1.26)

In this case, the conditions (1.11), (1.12), and (1.13) of Theorem 1.3 can be transformed into

$|\alpha \left(t\right)|\le \theta \left(t\right)\beta \left(t\right),\phantom{\rule{1em}{0ex}}\forall t\in \left[0,\delta \right]\cup \left\{\delta +1,\delta +2,\dots ,\delta +N-1\right\},$
(1.27)
${\int }_{0}^{\delta }\left[\gamma \left(t\right)-{\theta }^{2}\left(t\right)\beta \left(t\right)\right]dt+\sum _{n=0}^{N-1}\left[\gamma \left(\delta +n\right)-{\theta }^{2}\left(\delta +n\right)\beta \left(\delta +n\right)\right]>0,$
(1.28)

and

$\begin{array}{c}\phantom{\rule{2em}{0ex}}\left({\int }_{0}^{\delta }|\alpha \left(t\right)|dt+\sum _{n=0}^{N-1}|\alpha \left(\delta +n\right)|\right)\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}+{\left[\left({\int }_{0}^{\delta }\beta \left(t\right)dt+\sum _{n=0}^{N-1}|\beta \left(\delta +n\right)|\right)\left({\int }_{0}^{\delta }{\gamma }^{+}\left(t\right)dt+\sum _{n=0}^{N-1}|{\gamma }^{+}\left(\delta +n\right)|\right)\right]}^{1∕2}<2.\hfill \end{array}$
(1.29)

2 Proofs of theorems

Let u(t) = (x(t), y(t)), u σ(t) = (x(σ(t)), y(t)) and

$A\left(t\right)=\left(\begin{array}{cc}\hfill \alpha \left(t\right)\hfill & \hfill \beta \left(t\right)\hfill \\ \hfill -\gamma \left(t\right)\hfill & \hfill -\alpha \left(t\right)\hfill \end{array}\right).$

Then, we can rewrite (1.1) as a standard linear Hamiltonian dynamic system

${u}^{\Delta }\left(t\right)=A\left(t\right){u}^{\sigma }\left(t\right),\phantom{\rule{1em}{0ex}}t\in \mathbb{T}.$
(2.1)

Let u 1(t) = (x 10(t), y 10(t)) and u 2(t) = (x 20(t), y 20(t)) be two solutions of system (1.1) with (u 1(0), u 2(0)) = I 2. Denote by Φ(t) = (u 1(t), u 2(t)). Then Φ(t) is a fundamental matrix solution for (1.1) and satisfies Φ(0) = I 2. Suppose that α(t), β(t) and γ(t) are ω-periodic functions defined on $\mathbb{T}$ (i.e. (1.10) holds), then Φ(t + ω) is also a fundamental matrix solution for (1.1) ( see ). Therefore, it follows from the uniqueness of solutions of system (1.1) with initial condition ( see [9, 18, 19]) that

$\Phi \left(t+\omega \right)=\Phi \left(t\right)\Phi \left(\omega \right),\phantom{\rule{1em}{0ex}}\forall \phantom{\rule{2.77695pt}{0ex}}t\in \mathbb{T}.$
(2.2)

From (1.1), we have

${\left(det\Phi \left(t\right)\right)}^{\Delta }=\left|\begin{array}{cc}\hfill {x}_{10}^{\Delta }\left(t\right)\hfill & \hfill {x}_{20}^{\Delta }\left(t\right)\hfill \\ \hfill {y}_{10}\left(t\right)\hfill & \hfill {y}_{20}\left(t\right)\hfill \end{array}\right|+\left|\begin{array}{cc}\hfill {x}_{10}\left(\sigma \left(t\right)\right)\hfill & \hfill {x}_{20}\left(\sigma \left(t\right)\right)\hfill \\ \hfill {y}_{10}^{\Delta }\left(t\right)\hfill & \hfill {y}_{20}^{\Delta }\left(t\right)\hfill \end{array}\right|=0,\phantom{\rule{1em}{0ex}}\forall t\in \mathbb{T}.$
(2.3)

It follows that det Φ(t) = det Φ(0) = 1 for all $t\in \mathbb{T}$. Let λ1 and λ2 be the roots (real or complex) of the characteristic equation of Φ(ω)

$\mathsf{\text{det}}\left(\lambda {I}_{2}-\Phi \left(\omega \right)\right)=0,$

which is equivalent to

${\lambda }^{2}-H\lambda +1=0,$
(2.4)

where

$H={x}_{10}\left(\omega \right)+{y}_{20}\left(\omega \right).$

Hence

${\lambda }_{1}+{\lambda }_{2}=H,\phantom{\rule{1em}{0ex}}{\lambda }_{1}{\lambda }_{2}=1.$
(2.5)

Let v 1 = (c 11, c 21) and v 2 = (c 12, c 22) be the characteristic vectors associated with the characteristic roots λ1 and λ2 of Φ(ω), respectively, i.e.

$\Phi \left(\omega \right){v}_{j}={\lambda }_{j}{v}_{j},\phantom{\rule{1em}{0ex}}j=1,2.$
(2.6)

Let v j (t) = Φ(t)v j , j = 1, 2. Then it follows from (2.2) and (2.6) that

${v}_{j}\left(t+\omega \right)={\lambda }_{j}{v}_{j}\left(t\right),\phantom{\rule{1em}{0ex}}\forall t\in \mathbb{T},\phantom{\rule{1em}{0ex}}j=1,2.$
(2.7)

On the other hand, it follows from (2.1) that

$\begin{array}{cc}\hfill {v}_{j}^{\Delta }\left(t\right)& ={\Phi }^{\Delta }\left(t\right){v}_{j}\hfill \\ =\left({u}_{1}^{\Delta }\left(t\right),{u}_{2}^{\Delta }\left(t\right)\right){v}_{j}\hfill \\ =A\left(t\right)\left({u}_{1}^{\sigma }\left(t\right),{u}_{2}^{\sigma }\left(t\right)\right){v}_{j}\hfill \\ =A\left(t\right){v}_{j}^{\sigma }\left(t\right),\phantom{\rule{1em}{0ex}}j=1,2.\hfill \end{array}$
(2.8)

This shows that v 1(t) and v 2(t) are two solutions of system (1.1) which satisfy (2.7). Hence, we obtain the following lemma.

Lemma 2.1. Let Φ(t) be a fundamental matrix solution for (1.1) with Φ(0) = I 2 , and let λ 1 and λ 2 be the roots (real or complex) of the characteristic equation ( 2.4 ) of Φ(ω). Then system (1.1) has two solutions v 1(t) and v 2(t) which satisfy (2.7).

Similar to the continuous case, we have the following lemma.

Lemma 2.2. System (1.1) is unstable if |H| > 2, and stable if |H| < 2.

Instead of the usual zero, we adopt the following concept of generalized zero on time scales.

Definition 2.3. A function $f:\mathbb{T}\to ℝ$ is said to have a generalized zero at ${t}_{0}\in \mathbb{T}$ provided either f(t 0) = 0 or f(t 0)f(σ(t 0)) < 0.

Lemma 2.4.  Assume $f,g:\mathbb{T}\to ℝ$ are differential at $t\in {\mathbb{T}}^{k}$ . If f Δ(t) exists, then f(σ(t)) = f(t) + μ(t)f Δ(t).

Lemma 2.5.  (Cauchy-Schwarz inequality). Let $a,b\in \mathbb{T}$ . For f,g C rd we have

${\int }_{a}^{b}\left|f\left(t\right)g\left(t\right)\right|\Delta t\le {\left[{\int }_{a}^{b}{f}^{2}\left(t\right)\Delta t\cdot {\int }_{a}^{b}{g}^{2}\left(t\right)\Delta t\right]}^{\frac{1}{2}}.$

The above inequality can be equality only if there exists a constant c such that f(t) = cg(t) for $t\in \mathbb{T}\left[a,b\right]$.

Lemma 2.6. Let v 1(t) = (x 1(t), y 1(t)) and v 2(t) = (x 2(t), y 2(t)) be two solutions of system (1.1) which satisfy (2.7). Assume that (1.2), (1.3) and (1.10) hold, and that exists a non-negative function θ(t) such that (1.11) and (1.12) hold. If H 2 ≥ 4, then both x 1(t) and x 2(t) have generalized zeros in $\mathbb{T}\left[0,\omega \right]$.

Proof. Since |H| ≥ 2, then λ1 and λ2 are real numbers, and v 1(t) and v 2(t) are also real functions. We only prove that x 1(t) must have at least one generalized zero in $\mathbb{T}\left[0,\omega \right]$. Otherwise, we assume that x 1(t) > 0 for $t\in \mathbb{T}\left[0,\omega \right]$ and so (2.7) implies that x 1(t) > 0 for $t\in \mathbb{T}$. Define z(t): = y 1(t)/x 1(t). Due to (2.7), one sees that z(t) is ω-periodic, i.e. z(t + ω) = z(t), $\forall t\in \mathbb{T}$. From (1.1), we have

$\begin{array}{cc}\hfill {z}^{\Delta }\left(t\right)\phantom{\rule{1em}{0ex}}& =\phantom{\rule{1em}{0ex}}\frac{{x}_{1}\left(t\right){y}_{1}^{\Delta }\left(t\right)-{x}_{1}^{\Delta }\left(t\right){y}_{1}\left(t\right)}{{x}_{1}\left(t\right){x}_{1}\left(\sigma \left(t\right)\right)}\hfill \\ =\phantom{\rule{1em}{0ex}}\frac{-\gamma \left(t\right){x}_{1}\left(t\right){x}_{1}\left(\sigma \left(t\right)\right)-\alpha \left(t\right)\left[{x}_{1}\left(t\right)+{x}_{1}\left(\sigma \left(t\right)\right)\right]{y}_{1}\left(t\right)-\beta \left(t\right){y}_{1}^{2}\left(t\right)}{{x}_{1}\left(t\right){x}_{1}\left(\sigma \left(t\right)\right)}\hfill \\ =\phantom{\rule{1em}{0ex}}-\gamma \left(t\right)-\alpha \left(t\right)\left[\frac{{y}_{1}\left(t\right)}{{x}_{1}\left(t\right)}+\frac{{y}_{1}\left(t\right)}{{x}_{1}\left(\sigma \left(t\right)\right)}\right]-\beta \left(t\right)\left[\frac{{y}_{1}\left(t\right)}{{x}_{1}\left(t\right)}\phantom{\rule{2.77695pt}{0ex}}\frac{{y}_{1}\left(t\right)}{{x}_{1}\left(\sigma \left(t\right)\right)}\right]\hfill \\ =\phantom{\rule{1em}{0ex}}-\gamma \left(t\right)-\alpha \left(t\right)\left[z\left(t\right)+\frac{{y}_{1}\left(t\right)}{{x}_{1}\left(\sigma \left(t\right)\right)}\right]-\beta \left(t\right)z\left(t\right)\left[\frac{{y}_{1}\left(t\right)}{{x}_{1}\left(\sigma \left(t\right)\right)}\right].\hfill \end{array}$
(2.9)

From the first equation of (1.1), and using Lemma 2.4, we have

$\left[1-\mu \left(t\right)\alpha \left(t\right)\right]{x}_{1}\left(\sigma \left(t\right)\right)={x}_{1}\left(t\right)+\mu \left(t\right)\beta \left(t\right){y}_{1}\left(t\right),\phantom{\rule{1em}{0ex}}t\in \mathbb{T}.$
(2.10)

Since x 1(t) > 0 for all $t\in \mathbb{T}$, it follows from (1.2) and (2.10) that

$1+\mu \left(t\right)\beta \left(t\right)z\left(t\right)=1+\mu \left(t\right)\beta \left(t\right)\frac{{y}_{1}\left(t\right)}{{x}_{1}\left(t\right)}=\left[1-\mu \left(t\right)\alpha \left(t\right)\right]\frac{{x}_{1}\left(\sigma \left(t\right)\right)}{{x}_{1}\left(t\right)}>0,$
(2.11)

which yields

$\frac{{y}_{1}\left(t\right)}{{x}_{1}\left(\sigma \left(t\right)\right)}=\frac{\left[1-\mu \left(t\right)\alpha \left(t\right)\right]z\left(t\right)}{1+\mu \left(t\right)\beta \left(t\right)z\left(t\right)}.$
(2.12)

Substituting (2.12) into (2.9), we obtain

${z}^{\Delta }\left(t\right)=-\gamma \left(t\right)+\frac{\left[-2\alpha \left(t\right)+\mu \left(t\right){\alpha }^{2}\left(t\right)\right]z\left(t\right)-\beta \left(t\right){z}^{2}\left(t\right)}{1+\mu \left(t\right)\beta \left(t\right)z\left(t\right)}.$
(2.13)

If β(t) > 0, together with (1.11), it is easy to verify that

$\frac{\left[-2\alpha \left(t\right)+\mu \left(t\right){\alpha }^{2}\left(t\right)\right]z\left(t\right)-\beta \left(t\right){z}^{2}\left(t\right)}{1+\mu \left(t\right)\beta \left(t\right)z\left(t\right)}\le \frac{{\alpha }^{2}\left(t\right)}{\beta \left(t\right)}\le {\theta }^{2}\left(t\right)\beta \left(t\right);$
(2.14)

If β(t) = 0, it follows from (1.11) that α(t) = 0, hence

$\frac{\left[-2\alpha \left(t\right)+\mu \left(t\right){\alpha }^{2}\left(t\right)\right]z\left(t\right)-\beta \left(t\right){z}^{2}\left(t\right)}{1+\mu \left(t\right)\beta \left(t\right)z\left(t\right)}=0={\theta }^{2}\left(t\right)\beta \left(t\right).$
(2.15)

Combining (2.14) with (2.15), we have

$\frac{\left[-2\alpha \left(t\right)+\mu \left(t\right){\alpha }^{2}\left(t\right)\right]z\left(t\right)-\beta \left(t\right){z}^{2}\left(t\right)}{1+\mu \left(t\right)\beta \left(t\right)z\left(t\right)}\le {\theta }^{2}\left(t\right)\beta \left(t\right).$
(2.16)

Substituting (2.16) into (2.13), we obtain

${z}^{\Delta }\left(t\right)\le -\gamma \left(t\right)+{\theta }^{2}\left(t\right)\beta \left(t\right).$
(2.17)

Integrating equation (2.17) from 0 to ω, and noticing that z(t) is ω-periodic, we obtain

$0\le -{\int }_{0}^{\omega }\left[\gamma \left(t\right)-{\theta }^{2}\left(t\right)\beta \left(t\right)\right]\Delta t,$

Lemma 2.7. Let v 1(t) = (x 1(t), y 1(t)) and v 2(t) = (x 2(t), y 2(t)) be two solutions of system (1.1) which satisfy (2.7). Assume that

$\alpha \left(t\right)=0,\phantom{\rule{1em}{0ex}}\beta \left(t\right)>0,\phantom{\rule{1em}{0ex}}\gamma \left(t\right)\not\equiv 0,\phantom{\rule{1em}{0ex}}\forall t\in \mathbb{T},$
(2.18)
$\beta \left(t+\omega \right)=\beta \left(t\right),\phantom{\rule{1em}{0ex}}\gamma \left(t+\omega \right)=\gamma \left(t\right),\phantom{\rule{1em}{0ex}}\forall t\in \mathbb{T},$
(2.19)

and

${\int }_{0}^{\omega }\gamma \left(t\right)\Delta t\ge 0.$
(2.20)

If H 2 ≥ 4, then both x 1(t) and x 2(t) have generalized zeros in $\mathbb{T}\left[0,\omega \right]$.

Proof. Except (1.12), (2.18), and (2.19) imply all assumptions in Lemma 2.6 hold. In view of the proof of Lemma 2.6, it is sufficient to derive an inequality which contradicts (2.20) instead of (1.12). From (2.11), (2.13), and (2.18), we have

$1+\mu \left(t\right)\beta \left(t\right)z\left(t\right)=1+\mu \left(t\right)\beta \left(t\right)\frac{{y}_{1}\left(t\right)}{{x}_{1}\left(t\right)}=\frac{{x}_{1}\left(\sigma \left(t\right)\right)}{{x}_{1}\left(t\right)}>0$
(2.21)

and

${z}^{\Delta }\left(t\right)=-\gamma \left(t\right)-\frac{\beta \left(t\right){z}^{2}\left(t\right)}{1+\mu \left(t\right)\beta \left(t\right)z\left(t\right)}.$
(2.22)

Since z(t) is ω-periodic and γ(t) 0,, it follows from (2.22) that z 2(t) 0 on $\mathbb{T}\left[0,\omega \right]$. Integrating equation (2.22) from 0 to ω, we obtain

$0=-{\int }_{0}^{\omega }\left[\gamma \left(t\right)+\frac{\beta \left(t\right){z}^{2}\left(t\right)}{1+\mu \left(t\right)\beta \left(t\right)z\left(t\right)}\right]\Delta t<-{\int }_{0}^{\omega }\gamma \left(t\right)\Delta t,$

Lemma 2.8.  Suppose that (1.2) and (1.3) hold and let $a,b\in {\mathbb{T}}^{k}$ with σ(a) ≤ b. Assume (1.1) has a real solution (x(t), y(t)) such that x(t) has a generalized zero at end-point a and (x(b), y(b)) = (κ 1 x(a), κ 2 y(a)) with $0<{\kappa }_{1}^{2}\le {\kappa }_{1}{\kappa }_{2}\le 1$ and x(t) 0 on $\mathbb{T}\left[a,b\right]$. Then one has the following inequality

${\int }_{a}^{b}\left|\alpha \left(t\right)\right|\Delta t+{\left[{\int }_{a}^{b}\beta \left(t\right)\Delta t{\int }_{a}^{b}{\gamma }^{+}\left(t\right)\Delta t\right]}^{1∕2}\ge 2.$
(2.23)

Lemma 2.9. Suppose that (2.18) holds and let $a,b\in {\mathbb{T}}^{k}$ with σ(a) ≤ b. Assume (1.1) has a real solution (x(t), y(t)) such that x(t) has a generalized zero at end-point a and (x(b), y(b)) = (κx(a), κy(a)) with 0 < κ 2 ≤ 1 and x(t) is not identically zero on $\mathbb{T}\left[a,b\right]$ . Then one has the following inequality

${\int }_{a}^{b}\beta \left(t\right)\Delta t{\int }_{a}^{b}{\gamma }^{+}\left(t\right)\Delta t>4.$
(2.24)

Proof. In view of the proof of [10, Theorem 3.5] (see (3.8), (3.29)-(3.34) in ), we have

$x\left(a\right)=-\xi \mu \left(a\right)\beta \left(a\right)y\left(a\right),$
(2.25)
$x\left(\tau \right)=\left(1-\xi \right)\mu \left(a\right)\beta \left(a\right)y\left(a\right)+{\int }_{\sigma \left(a\right)}^{\tau }\beta \left(t\right)y\left(t\right)\Delta t,\phantom{\rule{1em}{0ex}}\sigma \left(a\right)\le \tau \le b,$
(2.26)
${\vartheta }_{1}\mu \left(a\right)\beta \left(a\right){y}^{2}\left(a\right)+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right){y}^{2}\left(t\right)\Delta t={\int }_{a}^{b}\gamma \left(t\right){x}^{2}\left(\sigma \left(t\right)\right)\Delta t,$
(2.27)

and

$2|x\left(\tau \right)|\le {\vartheta }_{2}\mu \left(a\right)\beta \left(a\right)|y\left(a\right)|\phantom{\rule{2.77695pt}{0ex}}+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right)|y\left(t\right)|\Delta t,\phantom{\rule{1em}{0ex}}\sigma \left(a\right)\le \tau \le b,$
(2.28)

where ξ [0, 1), and

${\vartheta }_{1}=1-\xi +{\kappa }^{2}\xi ,\phantom{\rule{1em}{0ex}}{\vartheta }_{2}=1-\xi \phantom{\rule{2.77695pt}{0ex}}+\phantom{\rule{2.77695pt}{0ex}}|\kappa |\xi .$
(2.29)

Let |x(τ*)| = max σ(a)≤τb |x(τ)|. There are three possible cases:

1. (1)

y(t) ≡ y(a) ≠ 0, $\forall \phantom{\rule{2.77695pt}{0ex}}t\in \mathbb{T}\left[a,b\right]$;

2. (2)

y(t) y(a), |y(t)| ≡ |y(a)|, $\forall \phantom{\rule{2.77695pt}{0ex}}t\in \mathbb{T}\left[a,b\right]$;

3. (3)

|y(t)| |y(a)|, $\forall \phantom{\rule{2.77695pt}{0ex}}t\in \mathbb{T}\left[a,b\right]$.

Case (1). In this case, κ = 1. It follows from (2.25) and (2.26) that

$\begin{array}{cc}\hfill x\left(b\right)\phantom{\rule{1em}{0ex}}& =\phantom{\rule{1em}{0ex}}\left(1-\xi \right)\mu \left(a\right)\beta \left(a\right)y\left(a\right)+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right)y\left(t\right)\Delta t\hfill \\ =\phantom{\rule{1em}{0ex}}y\left(a\right)\left[\left(1-\xi \right)\mu \left(a\right)\beta \left(a\right)+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right)\Delta t\right]\hfill \\ =\phantom{\rule{1em}{0ex}}x\left(a\right)+y\left(a\right){\int }_{a}^{b}\beta \left(t\right)\Delta t\hfill \\ \ne \phantom{\rule{1em}{0ex}}x\left(a\right),\hfill \end{array}$

which contradicts the assumption that x(b) = κx(a) = x(a).

Case (2). In this case, we have

$2|x\left(\tau \right)|<{\vartheta }_{2}\mu \left(a\right)\beta \left(a\right)|y\left(a\right)|\phantom{\rule{2.77695pt}{0ex}}+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right)|y\left(t\right)|\Delta t,\phantom{\rule{1em}{0ex}}\sigma \left(a\right)\le \tau \le b$
(2.30)

instead of (2.28). Applying Lemma 2.5 and using (2.27) and (2.30), we have

$\begin{array}{c}\phantom{\rule{2em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}2|x\left({\tau }^{*}\right)|\hfill \\ <\phantom{\rule{1em}{0ex}}{\vartheta }_{2}\mu \left(a\right)\beta \left(a\right)|y\left(a\right)|\phantom{\rule{2.77695pt}{0ex}}+\phantom{\rule{2.77695pt}{0ex}}{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right)|y\left(t\right)|\Delta t\hfill \\ \le \phantom{\rule{1em}{0ex}}{\left\{\left[\frac{{\vartheta }_{2}^{2}}{{\vartheta }_{1}}\mu \left(a\right)\beta \left(a\right)+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right)\Delta t\right]\left[{\vartheta }_{1}\mu \left(a\right)\beta \left(a\right){y}^{2}\left(a\right)+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right){y}^{2}\left(t\right)\Delta t\right]\right\}}^{1∕2}\hfill \\ =\phantom{\rule{1em}{0ex}}{\left\{\left[\frac{{\vartheta }_{2}^{2}}{{\vartheta }_{1}}\mu \left(a\right)\beta \left(a\right)+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right)\Delta t\right]{\int }_{a}^{b}\gamma \left(t\right){x}^{2}\left(\sigma \left(t\right)\right)\Delta t\right\}}^{1∕2}\hfill \\ \le \phantom{\rule{1em}{0ex}}|x\left({\tau }^{*}\right)|{\left[\left(\frac{{\vartheta }_{2}^{2}}{{\vartheta }_{1}}\mu \left(a\right)\beta \left(a\right)+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right)\Delta t\right){\int }_{a}^{b}{\gamma }^{+}\left(t\right)\Delta t\right]}^{1∕2}.\hfill \end{array}$
(2.31)

Dividing the latter inequality of (2.31) by |x(τ*)|, we obtain

${\left[\left(\frac{{\vartheta }_{2}^{2}}{{\vartheta }_{1}}\mu \left(a\right)\beta \left(a\right)+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right)\Delta t\right){\int }_{a}^{b}{\gamma }^{+}\left(t\right)\Delta t\right]}^{1∕2}>2.$
(2.32)

Case (3). In this case, applying Lemma 2.5 and using (2.27) and (2.28), we have

$\begin{array}{c}\phantom{\rule{1.5em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}2|x\left({\tau }^{*}\right)|\hfill \\ \le \phantom{\rule{1em}{0ex}}{\vartheta }_{2}\mu \left(a\right)\beta \left(a\right)|y\left(a\right)|+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right)|y\left(t\right)|\Delta t\hfill \\ <\phantom{\rule{1em}{0ex}}{\left\{\left[\frac{{\vartheta }_{2}^{2}}{{\vartheta }_{1}}\mu \left(a\right)\beta \left(a\right)+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right)\Delta t\right]\left[{\vartheta }_{1}\mu \left(a\right)\beta \left(a\right){y}^{2}\left(a\right)+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right){y}^{2}\left(t\right)\Delta t\right]\right\}}^{1∕2}\hfill \\ =\phantom{\rule{1em}{0ex}}{\left\{\left[\frac{{\vartheta }_{2}^{2}}{{\vartheta }_{1}}\mu \left(a\right)\beta \left(a\right)+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right)\Delta t\right]{\int }_{a}^{b}\gamma \left(t\right){x}^{2}\left(\sigma \left(t\right)\right)\Delta t\right\}}^{1∕2}\hfill \\ \le \phantom{\rule{1em}{0ex}}|x\left({\tau }^{*}\right)|{\left[\left(\frac{{\vartheta }_{2}^{2}}{{\vartheta }_{1}}\mu \left(a\right)\beta \left(a\right)+{\int }_{\sigma \left(a\right)}^{b}\beta \left(t\right)\Delta t\right){\int }_{a}^{b}{\gamma }^{+}\left(t\right)\Delta t\right]}^{1∕2}.\hfill \end{array}$
(2.33)

Dividing the latter inequality of (2.33) by |x(τ*)|, we also obtain (2.32). It is easy to verify that

$\frac{{\vartheta }_{2}^{2}}{{\vartheta }_{1}}=\frac{{\left[1-\xi +|\kappa |\xi \right]}^{2}}{1-\xi +{\kappa }^{2}\xi }\le 1.$

Substituting this into (2.32), we obtain (2.24). □

Proof of Theorem 1.3. If |H| ≥ 2, then λ1 and λ2 are real numbers and λ1λ2 = 1, it follows that $0. Suppose ${\lambda }_{1}^{2}\le 1$. By Lemma 2.6, system (1.1) has a non-zero solution v 1(t) = (x 1(t), y 1(t)) such that (2.7) holds and x 1(t) has a generalized zero in $\mathbb{T}\left[0,\omega \right]$, say t 1. It follows from (2.7) that (x 1(t 1 + ω), y 1(t 1 + ω)) = λ1(x 1(t 1), y 1(t 1)). Applying Lemma 2.8 to the solution (x 1(t), y 1(t)) with a = t 1, b = t 1 + ω and κ 1 = κ 2 = λ1, we get

${\int }_{{t}_{1}}^{{t}_{1}+\omega }|\alpha \left(t\right)|\Delta t+{\left[{\int }_{{t}_{1}}^{{t}_{1}+\omega }\beta \left(t\right)\Delta t{\int }_{{t}_{1}}^{{t}_{1}+\omega }{\gamma }^{+}\left(t\right)\Delta t\right]}^{1∕2}\ge 2.$
(2.34)

Next, noticing that for any ω-periodic function f(t) on $\mathbb{T}$, the equality

${\int }_{{t}_{0}}^{{t}_{0}+\omega }f\left(t\right)\Delta t={\int }_{0}^{\omega }f\left(t\right)\Delta t$

holds for all ${t}_{0}\in \mathbb{T}$. It follows from (3.1) that

${\int }_{0}^{\omega }|\alpha \left(t\right)|\Delta t+{\left[{\int }_{0}^{\omega }\beta \left(t\right)\Delta t{\int }_{0}^{\omega }{\gamma }^{+}\left(t\right)\Delta t\right]}^{1∕2}\ge 2.$
(2.35)

which contradicts condition (1.13). Thus |H| < 2 and hence system (1.1) is stable. □

Proof of Theorem 1.4. By using Lemmas 2.7 and 2.9 instead of Lemmas 2.6 and 2.8, respectively, we can prove Theorem 1.4 in a similar fashion as the proof of Theorem 1.3. So, we omit the proof here. □

## References

1. Hilger S: Einßmakettenkalk ü l mit Anwendung auf Zentrumsmannigfaltigkeiten. Ph.D. Thesis, Universität Würzburg in German; 1988.

2. Hilger S: Analysis on measure chain--A unified approach to continuous and discrete calculus. Results Math 1990, 18: 18-56.

3. Hilger S: Differential and difference calculus-unified. Nonlinear Anal 1997, 30: 2683-2694. 10.1016/S0362-546X(96)00204-0

4. Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston; 2001.

5. Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Inc., Boston. MA; 2003.

6. Kaymakcalan B, Lakshimikantham V, Sivasundaram S: Dynamic System on Measure Chains. Kluwer Academic Publishers, Dordrecht; 1996.

7. Agarwal R, Bohner M, Rehak P: Half-linear dynamic equations. Nonlinear Anal Appl 2003, 1: 1-56.

8. Jiang LQ, Zhou Z: Lyapunov inequality for linear Hamiltonian systems on time scales. J Math Anal Appl 310: 579-593.

9. Wong F, Yu S, Yeh C, Lian W: Lyapunov's inequality on timesscales. Appl Math Lett 2006, 19: 1293-1299. 10.1016/j.aml.2005.06.006

10. He X, Zhang Q, Tang XH: On inequalities of Lyapunov for linear Hamiltonian systems on time scales. J Math Anal Appl 2011, 381: 695-705. 10.1016/j.jmaa.2011.03.036

11. Guseinov GSh, Kaymakcalan B: Lyapunov inequalities for discrete linear Hamiltonian systems. Comput Math Appl 2003, 45: 1399-1416. 10.1016/S0898-1221(03)00095-6

12. Guseinov GSh, Zafer A: Stability criteria for linear periodic impulsive Hamiltonian systems. J Math Anal Appl 2007, 335: 1195-1206. 10.1016/j.jmaa.2007.01.095

13. Krein MG: Foundations of the theory of λ-zones of stability of canonical system of linear differential equations with periodic coefficients. memory of A.A. Andronov, Izdat. Acad. Nauk SSSR, Moscow; 1955:413-498.

14. Wang X: Stability criteria for linear periodic Hamiltonian systems. J Math Anal Appl 2010, 367: 329-336. 10.1016/j.jmaa.2010.01.027

15. Tang XH, Zhang M: Lyapunov inequalities and stability for linear Hamiltonian systems. J Diff Equ 2012, 252: 358-381. 10.1016/j.jde.2011.08.002

16. Zhang Q, Tang XH: Lyapunov inequalities and stability for discrete linear Hamiltonian system. Appl Math Comput 2011, 218: 574-582. 10.1016/j.amc.2011.05.101

17. Atici FM, Guseinov GSh, Kaymakcalan B: On Lyapunov inequality in stability theory for Hill's equation on time scales. J Inequal Appl 2000, 5: 603-620.

18. Ahlbrandt CD, Ridenhour J: Floquet theory for time scales and Putzer representations of matrix logarithms. J Diff Equ Appl 2003, 9: 77-92.

19. DaCunha JJ: Lyapunov stability and floquet theory for nonautonomous linear dynamic systems on time scales. Ph.D. dissertation, Baylor University, Waco, Tex, USA; 2004.

20. Halanay A, Răsvan Vl: Stability and boundary value problems, for discrete-time linear Hamiltonian systems. In Dyn Syst Appl. Volume 8. Edited by: Agarwal RP, Bohner M. Special Issue on "Discrete and Continuous Hamiltonian Systems"; 1999:439-459.

21. Răsvan Vl: Stability zones for discrete time Hamiltonian systems. Archivum Mathematicum Tomus 2000, 36: 563-573. (CDDE2000 issue)

22. Răsvan Vl: Krein-type results for λ-zones of stability in the discrete-time case for 2-nd order Hamiltonian systems. Folia FSN Universitatis Masarykianae Brunensis, Mathematica 2002, 10: 1-12. (CDDE2002 issue)

23. Răsvan Vl: On central λ-stability zone for linear discrete time Hamiltonian systems. Proc. fourth Int. Conf. on Dynamical Systems and Differential Equations, Wilmington NC; 2002.

24. Răsvan Vl: On stability zones for discrete time periodic linear Hamiltonian systems. Adv Diff Equ ID80757, pp. 1-13. doi:10.1155/ADE/2006/80757, e-ISSN 1687-1847, 2006

## Acknowledgements

The authors thank the referees for valuable comments and suggestions. This project is supported by Scientific Research Fund of Hunan Provincial Education Department (No. 11A095) and partially supported by the NNSF (No: 11171351) of China.

## Author information

Authors

### Corresponding author

Correspondence to Xianhua Tang.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

XH carried out the theoretical proof and drafted the manuscript. Both XT and QZ participated in the design and coordination. All authors read and approved the final manuscript.

## Rights and permissions

Reprints and Permissions

He, X., Tang, X. & Zhang, QM. Stability criteria for linear Hamiltonian dynamic systems on time scales. Adv Differ Equ 2011, 63 (2011). https://doi.org/10.1186/1687-1847-2011-63

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1687-1847-2011-63

### Keywords

• Hamiltonian dynamic system
• Lyapunov-type inequality
• Floquet theory
• stability
• time scales 