# Lyapunov inequalities for a class of nonlinear dynamic systems on time scales

## Abstract

The purpose of this work is to obtain several Lyapunov inequalities for the nonlinear dynamic systems

$$\left \{ \textstyle\begin{array}{l} x^{\Delta}(t)= -A(t)x(\sigma(t))-B(t)y(t)|\sqrt{B(t)}y(t)|^{p-2}, \\ y^{\Delta}(t)= C(t)x(\sigma(t))|x(\sigma(t))|^{q-2}+A^{T}(t)y(t), \end{array}\displaystyle \right .$$

on a given time scale interval $$[a,b]_{\mathbb{T}}$$ ($$a,b\in{\mathbb{T}}$$ with $$\sigma(a)< b$$), where $$p,q\in (1,+\infty)$$ satisfy $$1/p+1/q=1$$, $$A(t)$$ is a real $$n\times n$$ matrix-valued function on $$[a,b]_{\mathbb{T}}$$ such that $$I+\mu(t)A(t)$$ is invertible, $$B(t)$$ and $$C(t)$$ are two real $$n\times n$$ symmetric matrix-valued functions on $$[a,b]_{ \mathbb{T}}$$, $$B(t)$$ is positive definite, and $$x(t)$$, $$y(t)$$ are two real n-dimensional vector-valued functions on $$[a,b]_{\mathbb{T}}$$.

## Introduction

The theory of dynamic equations on time scales, which follows Hilger’s landmark paper [1], is a new study area of mathematics that has received a lot of attention. For example, we refer the reader to monographs [2, 3] and the references therein. During the last few years, some Lyapunov inequalities for dynamic equations on time scales have been obtained by many authors [47].

In 2002, Bohner et al. [8] investigated the second-order Sturm-Liouville dynamic equation

$$x^{\Delta^{2}}(t)+q(t)x^{\sigma}(t)=0$$
(1.1)

on time scale $${\mathbb{T}}$$ under the conditions $$x(a)=x(b)=0$$ ($$a,b\in{\mathbb{T}}$$ with $$a< b$$) and $$q\in C_{\mathrm{rd}}({\mathbb{T}}, (0,\infty))$$ and showed that if $$x(t)$$ is a solution of (1.1) with $$\max_{t\in[a,b]_{ \mathbb{T}}}|x(t)|>0$$, then

$$\int_{a}^{b}q(t)\Delta t\geq \frac{b-a}{C},$$

where $$[a,b]_{\mathbb{T}}\equiv\{t\in \mathbb{T}:a\leq t\leq b\}$$ and $$C=\max\{(t-a)(b-t):t\in [a,b]_{\mathbb{T}}\}$$.

When $${\mathbb{T}}= \mathbb{R}$$, (1.1) reduces to the Hills equation

$$x''(t) + u(t)x(t) = 0.$$
(1.2)

In 1907, Lyapunov [9] showed that if $$u\in C([a, b], {\mathbb{R}})$$ and $$x(t)$$ is a solution of (1.2) satisfying $$x(a) = x(b) = 0$$ and $$\max_{t\in[a,b]}|x(t)|>0$$, then the following classical Lyapunov inequality holds:

$$\int_{a}^{b}\bigl\vert u(t)\bigr\vert \, dt> \frac{4}{b-a}.$$

This was later strengthened with $$|u(t)|$$ replaced by $$u^{+}(t)=\max\{u(t),0\}$$ by Wintner [10] and thereafter by some other authors:

$$\int_{a}^{b}u^{+}(t)\, dt>\frac{4}{b-a}.$$

Moreover, the last inequality is optimal.

When $${\mathbb{T}}$$ is the set $${\mathbb{Z}}$$ of the integers, (1.1) reduces to the linear difference equation

$$\Delta^{2}x(n)+u(n)x(n+1)=0.$$
(1.3)

In 1983, Cheng [11] showed that if $$a,b\in{\mathbb{Z}}$$ with $$0< a< b$$ and $$x(n)$$ is a solution of (1.3) satisfying $$x(a)=x(b)=0$$ and $$\max_{n\in\{a,a+1,\ldots,b\}}|x(n)|>0$$, then

$$\sum_{n=a}^{b-2}\bigl\vert u(n)\bigr\vert \geq \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{4(b-a)}{(b-a)^{2}-1} & \mbox{if } b-a-1 \mbox{ is even}, \\ \frac{4}{b-a} & \mbox{if } b-a-1 \mbox{ is odd} . \end{array}\displaystyle \right .$$

The purpose of this paper is to establish several Lyapunov inequalities for the nonlinear dynamic system

$$\left \{ \textstyle\begin{array}{l} x^{\Delta}(t)= -A(t)x(\sigma(t))-B(t)y(t)|\sqrt{B(t)}y(t)|^{p-2}, \\ y^{\Delta}(t)= C(t)x(\sigma(t))|x(\sigma(t))|^{q-2}+A^{T}(t)y(t), \end{array}\displaystyle \right .$$
(1.4)

on a given time scale interval $$[a,b]_{\mathbb{T}}$$ ($$a,b\in{\mathbb{T}}$$ with $$\sigma(a)< b$$), where $$p,q\in (1,+\infty)$$ satisfy $$1/p+1/q=1$$, $$A(t)$$ is a real $$n\times n$$ matrix-valued function on $$[a,b]_{\mathbb{T}}$$ such that $$I+\mu(t)A(t)$$ is invertible, $$B(t)$$ and $$C(t)$$ are two real $$n\times n$$ symmetric matrix-valued functions on $$[a,b]_{ \mathbb{T}}$$, $$B(t)$$ being positive definite, $$A^{T}(t)$$ is the transpose of $$A(t)$$, and $$x(t)$$, $$y(t)$$ are two real n-dimensional vector-valued functions on $$[a,b]_{\mathbb{T}}$$.

When $$n=1$$ and $$p=q=2$$, (1.4) reduces to

$$\left \{ \textstyle\begin{array}{l} x^{\Delta}(t)=u(t)x(\sigma(t))+v(t)y(t), \\ y^{\Delta}(t)=-w(t)x(\sigma(t))-u(t)y(t), \end{array}\displaystyle \right .$$
(1.5)

where $$u(t)$$, $$v(t)$$, and $$w(t)$$ are real-valued rd-continuous functions on $$\mathbb{T}$$ satisfying $$v(t)\geq0$$ for any $$t\in {\mathbb{T}}$$.

In 2011, He et al. [12] obtained the following result.

### Theorem 1.1

([12])

Let $$1-\mu(t)u(t)>0$$ for any $$t\in\mathbb{T}$$ and $$a, b \in\mathbb{T}^{k}$$ with $$\sigma(a)\leq b$$. If (1.5) has a real solution $$(x(t),y(t))$$ such that

\begin{aligned}& x(a)=0 \quad \textit{or}\quad x(a)x\bigl(\sigma(a)\bigr)< 0 ; \\& x(b)=0\quad \textit{or}\quad x(b)x\bigl(\sigma(b)\bigr)< 0 ; \qquad \max_{t\in[a,b]_{ \mathbb{T}}} \bigl\vert x(t)\bigr\vert >0, \end{aligned}

then we have the following inequality:

$$\int_{a}^{b}\bigl\vert u(t)\bigr\vert \Delta (t)+ \biggl[ \int_{a}^{\sigma (b)}v(t)\Delta (t) \int_{a}^{b}w^{+}(t)\Delta (t) \biggr]^{1/2}\geq 2,$$

where $$w^{+}(t)=\max\{w(t),0\}$$.

In 2016, Liu et al. [13] obtained the following theorem.

### Theorem 1.2

Let $$p=q=2$$ and $$a,b\in \mathbb{T}$$ with $$\sigma(a)< b$$. If (1.4) has a solution $$(x(t),y(t))$$ such that

$$x(a)=x(b)=0\quad \textit{and}\quad \max_{t\in[a,b]_{ \mathbb{T}}}x^{T}(t)x(t)>0 ,$$
(1.6)

then for any $$n\times n$$ symmetric matrix-valued function $$C_{1}(t)$$ with $$C_{1}(t)- C(t)\geq0$$, we have the following inequalities:

1. (1)
$$\int_{a}^{b}{\frac{[\int_{a}^{\sigma(t)}|B(s)||e_{\Theta A}(\sigma(t),s)|^{2}\Delta s ][\int_{\sigma(t)}^{b}|B(s)||e_{\Theta A}(\sigma(t),s)|^{2}\Delta s]}{\int_{a}^{b}|B(s)||e_{\Theta A}(\sigma(t),s)|^{2}\Delta s }}\bigl\vert C_{1}(t)\bigr\vert \Delta t\geq1,$$
2. (2)
$$\int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \biggl\{ \int_{a}^{b}\bigl\vert B(s)\bigr\vert \bigl\vert e_{\Theta A}\bigl(\sigma(t),s\bigr)\bigr\vert ^{2}\Delta s \biggr\} \Delta t\geq 4,$$
3. (3)
$$\int_{a}^{b}\bigl\vert A(t)\bigr\vert \Delta t+ \biggl( \int_{a}^{b}\bigl\vert \sqrt {B(t)}\bigr\vert ^{2}\Delta t \biggr)^{1/2} \biggl( \int _{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \Delta t \biggr)^{1/2}\geq2.$$

For some other related results on Lyapunov-type inequalities, see, for example, [1423].

## Preliminaries and some lemmas

Throughout this paper, we adopt basic definitions and notation of monograph [2]. A time scale $$\mathbb{T}$$ is a nonempty closed subset of the real numbers $$\mathbb{R}$$. On a time scale $$\mathbb{T}$$, the forward jump operator, the backward jump operator, and the graininess function are defined as

$$\sigma(t)=\inf\{s\in\mathbb{T}:s>t\},\qquad \rho(t)=\sup\{s\in \mathbb{ T}:s< t \}, \quad \mbox{and} \quad \mu(t)=\sigma(t)-t,$$

respectively.

The point $$t \in\mathbb{T}$$ is said to be left-dense (resp. left-scattered) if $$\rho(t) = t$$ (resp. $$\rho(t) < t$$). The point $$t\in\mathbb{T}$$ is said to be right-dense (resp. right-scattered) if $$\sigma(t) = t$$ (resp. $$\sigma(t) > t$$). If $$\mathbb{T}$$ has a left-scattered maximum M, then we define $$\mathbb{T}^{k} = \mathbb{T}- \{M\}$$, otherwise $$\mathbb{T}^{k} = \mathbb{T}$$.

A function $$f : \mathbb{T}\rightarrow \mathbb{R}$$ is said to be rd-continuous if f is continuous at right-dense points and has finite left-sided limits at left-dense points in $$\mathbb{T}$$. The set of all rd-continuous functions from $$\mathbb{T}$$ to $$\mathbb{R}$$ is denoted by $$C_{\mathrm{rd}}(\mathbb{T},\mathbb{R})$$. For a function $$f : \mathbb{T} \rightarrow \mathbb{R}$$, the notation $$f^{\sigma}$$ means the composition $$f\circ\sigma$$.

For a function $$f : \mathbb{T}\rightarrow \mathbb{R}$$, the (delta) derivative $$f^{\Delta}(t)$$ at $$t\in\mathbb{T}$$ is defined as the number (if it exists) such that for given any $$\varepsilon> 0$$, there is a neighborhood U of t with

$$\bigl\vert f\bigl(\sigma(t)\bigr)-f(s)-f^{\Delta}(t) \bigl(\sigma(t)-s \bigr)\bigr\vert \leq\varepsilon \bigl\vert \sigma(t)-s\bigr\vert$$

for all $$s\in U$$. If the (delta) derivative $$f^{\Delta}(t)$$ exists for every $$t\in\mathbb{T}^{k}$$, then we say that f is Δ-differentiable on $$\mathbb{T}$$.

Let $$F,f\in C_{\mathrm{rd}}(\mathbb{T}, \mathbb{R})$$ satisfy $$F^{\Delta }(t)=f(t)$$ for all $$t\in \mathbb{T}^{k}$$. Then, for any $$c,d\in\mathbb{T}$$, the Cauchy integral of f is defined as

$$\int_{c}^{d}f(t)\Delta t=F(d)-F(c).$$

For any $$z\in{\mathbb{R}}^{n}$$ and any $$S\in{ \mathbb{R}}^{n\times n}$$ (the space of real $$n\times n$$ matrices), write

$$|z|=\sqrt{z^{T}z} \quad \mbox{and} \quad |S|=\max_{z\in\mathbb {R}^{n},z\neq0} \frac{ |Sz|}{|z|},$$

which are called the Euclidean norm of z and the matrix norm of S, respectively. It is obvious that, for any $$z\in{ \mathbb{R}}^{n}$$ and $$U,V\in{\mathbb{R}}^{n\times n}$$,

$$|Uz|\leq|U||z| \quad \mbox{and} \quad |UV|\leq|U||V|.$$

Let $$\mathbb{R}^{n\times n}_{s}$$ be the set of all symmetric real $$n\times n$$ matrices. We can show that, for any $$U\in{\mathbb{R}}_{s}^{n\times n}$$,

$$|U|=\max_{|\lambda I-U|=0}|\lambda| \quad \mbox{and} \quad |U^{2}|=|U|^{2}.$$

A matrix $$S\in {\mathbb{R}}_{s}^{n\times n}$$ is said to be positive definite (resp. semipositive definite), written as $$S> 0$$ (resp. $$S \geq 0$$), if $$y^{T}Sy> 0$$ (resp. $$y^{T}Sy\geq0$$) for any $$y\in{ \mathbb{R}}^{n}$$ with $$y\neq0$$. If S is positive definite (resp. semipositive definite), then there exists a unique positive definite matrix (resp. semipositive definite matrix), written as $$\sqrt{S}$$, satisfying $$[\sqrt{S}]^{2}=S$$.

In this paper, we establish Lyapunov inequalities for (1.4) that has a solution $$(x(t),y(t))$$ satisfying

$$x(a)=x(b)=0 \quad \mbox{and}\quad \max_{t\in[a,b]_{ \mathbb{T}}}x^{T}(t)x(t)>0.$$
(2.1)

We first introduce the following lemmas.

### Lemma 2.1

([2])

Let $$1/p+1/q=1$$ ($$p,q\in(1,+\infty)$$) and $$a,b\in{\mathbb{T}}$$ ($$a< b$$). Then, for any $$f,g\in C_{\mathrm{rd}}([a,b]_{ \mathbb{T}},{\mathbb{R}})$$,

$$\int_{a}^{b}\bigl\vert f(t)g(t)\bigr\vert \Delta t\leq \biggl( \int _{a}^{b}\bigl\vert f(t)\bigr\vert ^{p}\Delta t \biggr)^{\frac{1}{p}} \biggl( \int_{a}^{b}\bigl\vert g(t)\bigr\vert ^{q}\Delta t \biggr)^{\frac{1}{q}}.$$

### Lemma 2.2

Let $$a,b\in{\mathbb{T}}$$ with $$a< b$$. Suppose that $$\alpha,\beta,\gamma,\delta\in \mathbb{R}$$ and $$p,q\in (1,+\infty)$$ with $$\alpha/p+\beta/q=\gamma/p+\delta/q=1/p+1/q=1$$. Then, for any $$f,g\in C_{\mathrm{rd}}([a,b]_{\mathbb{T}},(-\infty,0)\cup(0,\infty))$$,

$$\int_{a}^{b}\bigl\vert f(t)g(t)\bigr\vert \Delta t\leq \biggl( \int_{a}^{b}\bigl\vert f(t)\bigr\vert ^{\alpha}\bigl\vert g(t)\bigr\vert ^{\gamma} \Delta t \biggr)^{\frac{1}{p}} \biggl( \int_{a}^{b}\bigl\vert f(t)\bigr\vert ^{\beta}\bigl\vert g(t)\bigr\vert ^{\delta} \Delta t \biggr)^{\frac{1}{q}}.$$

### Proof

Let $$M(t)=(|f(t)|^{\alpha}|g(t)|^{\gamma})^{\frac{1}{p}}$$ and $$N(t)=(|f(t)|^{\beta}|g(t)|^{\delta})^{\frac{1}{q}}$$. Then by Lemma 2.1 we have

\begin{aligned} \int^{b}_{a}\bigl\vert f(t)g(t)\bigr\vert \Delta t =& \int ^{b}_{a}M(t)N(t)\Delta t \\ \leq& \biggl( \int^{b}_{a}M^{p}(t)\Delta t \biggr)^{\frac{1}{p}} \biggl( \int ^{b}_{a}N^{q}(t)\Delta t \biggr)^{\frac{1}{q}} \\ =& \biggl( \int_{a}^{b}\bigl\vert f(t)\bigr\vert ^{\alpha}\bigl\vert g(t)\bigr\vert ^{\gamma} \Delta t \biggr)^{\frac{1}{p}} \biggl( \int_{a}^{b}\bigl\vert f(t)\bigr\vert ^{\beta}\bigl\vert g(t)\bigr\vert ^{\delta} \Delta t \biggr)^{\frac{1}{q}}. \end{aligned}

This completes the proof of Lemma 2.2. □

### Remark 2.3

Let $$\gamma=0$$ in Lemma 2.2. Then we obtain that, for any $$f,g\in C_{\mathrm{rd}}([a,b]_{ \mathbb{T}},(-\infty, 0)\cup(0,\infty))$$,

$$\int_{a}^{b}\bigl\vert f(t)g(t)\bigr\vert \Delta t\leq \Bigl\{ \max_{t\in[a,b]_{ \mathbb{T}}}\bigl\vert f(t)\bigr\vert ^{\beta}\Bigr\} ^{\frac{1}{q}} \biggl( \int _{a}^{b}\bigl\vert f(t)\bigr\vert ^{\alpha} \Delta t \biggr)^{\frac{1}{p}} \biggl( \int_{a}^{b}\bigl\vert g(t)\bigr\vert ^{q}\Delta t \biggr)^{\frac{1}{q}}.$$

### Lemma 2.4

([2])

If $$A\in C_{\mathrm{rd}}({\mathbb{T}, \mathbb{R}^{n\times n}})$$ with invertible $$I+\mu(t)A(t)$$, $$f\in C_{\mathrm{rd}}(\mathbb{T},\mathbb{R}^{n})$$, $$t_{0} \in\mathbb{T}$$, and $$a \in\mathbb{R}^{n}$$, then

$$x(t)=e_{\Theta A}(t,t_{0})a+ \int_{t_{0}}^{t}e_{\Theta A}(t,\tau)f(\tau)\Delta{ \tau}$$

is the unique solution of the initial value problem

$$\left \{ \textstyle\begin{array}{l} x^{\Delta}(t)= -A(t)x(\sigma(t))+f(t), \\ x(t_{0})= a, \end{array}\displaystyle \right .$$

where $$(\Theta A)(t)=-[I+\mu(t)A(t)]^{-1}A(t)$$ for any $$t\in\mathbb{T}^{k}$$, and $$e_{\Theta A}(t,t_{0})$$ is the unique matrix-valued solution of the initial value problem

$$\left \{ \textstyle\begin{array}{l} Y^{\Delta}(t)= (\Theta A)(t)Y(t), \\ Y(t_{0})= I. \end{array}\displaystyle \right .$$

### Lemma 2.5

([2])

Let $$A,B\in C_{\mathrm{rd}}({\mathbb{T}, \mathbb{R}^{n\times n}})$$ be Δ-differentiable. Then

$$\bigl(A(t)B(t)\bigr)^{\Delta }=A^{\sigma}(t) B^{\Delta }(t)+A^{\Delta }(t) B(t)=A^{\Delta }(t) B^{\sigma}(t)+A(t)B^{\Delta }(t).$$

### Lemma 2.6

([13])

If $$f_{1}(t),f_{2}(t),\ldots,f_{n}(t)$$ are Δ-integrable on $${[a,b]_{\mathbb{T}}}$$ and $$x(t)=(f_{1}(t),f_{2}(t), \ldots,f_{n}(t))$$, then

$$\biggl\vert \int_{a}^{b}x(t)\Delta t\biggr\vert = \Biggl\{ \sum_{i=1}^{n} \biggl( \int _{a}^{b}f_{i}(t)\Delta t \biggr)^{2} \Biggr\} ^{\frac{1}{2}}\leq \int_{a}^{b} \Biggl\{ \sum _{i=1}^{n}f^{2}_{i}(t) \Biggr\} ^{\frac{1}{2}}\Delta t= \int _{a}^{b}\bigl\vert x(t)\bigr\vert \Delta t.$$

### Lemma 2.7

([13])

If $$A_{1},A_{2}\in \mathbb{R}^{n\times n}_{s}$$ and $$A_{1}-A_{2}\geq0$$, then, for any $$x \in\mathbb{R}^{n}$$,

$$\bigl(x^{\sigma}\bigr)^{T}A_{2}x^{\sigma}\leq|A_{1}|\bigl\vert x^{\sigma}\bigr\vert ^{2} .$$

## Main results and proofs

In this section, we assume that $$\alpha,\beta\in\mathbb{R}$$ and $$p,q\in(1,+\infty)$$ satisfy

$$\alpha/p+\beta/q=1/p+1/q=1.$$

For any $$t,\tau\in [a,b]_{\mathbb{T}}$$, write

\begin{aligned}& F(t,\tau) = \bigl\vert e_{\Theta A}\bigl(\sigma(t),\tau\bigr)\bigr\vert \bigl\vert \sqrt{B(\tau)}\bigr\vert , \\& G(t) = \bigl\vert \sqrt{B(t)}y(t)\bigr\vert ^{p-2}y^{T}(t)B(t)y(t)= \bigl\vert \sqrt{B(t)}y(t)\bigr\vert ^{p}, \\& \Phi\bigl(\sigma(t)\bigr) = \biggl( \int_{a}^{\sigma(t)}F^{\alpha}(t,s)\Delta s \biggr)^{\frac{q}{p}}, \\& \Psi\bigl(\sigma(t)\bigr) = \biggl( \int_{\sigma(t)}^{b}F^{\alpha}(t,s)\Delta s \biggr)^{\frac{q}{p}}, \\& P(t) = \Phi\bigl(\sigma(t)\bigr)\Psi\bigl(\sigma(t)\bigr)\max _{a\leq\tau\leq \sigma(t)}F^{\beta}(t,\tau) \max_{\sigma(t)\leq\tau\leq b}F^{\beta}(t, \tau), \\& Q(t) = \Phi\bigl(\sigma(t)\bigr)\max_{a\leq\tau\leq \sigma(t)}F^{\beta}(t, \tau)+\Psi\bigl(\sigma(t)\bigr)\max_{\sigma(t)\leq \tau\leq b}F^{\beta}(t, \tau). \end{aligned}

### Theorem 3.1

Let $$a,b\in\mathbb{T}$$ with $$\sigma(a)< b$$ and $$C_{1}\in\mathbb{R}^{n\times n}_{s}$$ with $$C_{1}(t)-C(t)\geq0$$. If (1.4) has a solution $$(x(t),y(t))$$ with $$x(t),y(t)\in C_{\mathrm{rd}}(\mathbb{T},\mathbb{R}^{n})$$ satisfying (2.1) on the interval $$[a,b]_{\mathbb{T}}$$, then

$$\int_{a}^{b}{\frac{P(t)}{Q(t)}}\bigl\vert C_{1}(t)\bigr\vert \Delta t\geq1.$$
(3.1)

### Proof

Since $$(x(t),y(t))$$ is a solution of (1.4), we have

$$\bigl(y^{T}(t)x(t)\bigr)^{\Delta}=\bigl(x^{\sigma}(t) \bigr)^{T}C(t)x^{\sigma}(t)\bigl\vert x^{\sigma}(t)\bigr\vert ^{q-2}-G(t).$$
(3.2)

Integrating (3.2) from a to b and noting that $$x(a)=x(b)=0$$, we obtain

$$\int_{a}^{b}G(t)\Delta t= \int_{a}^{b}\bigl\vert x^{\sigma}(t)\bigr\vert ^{q-2}\bigl(x^{\sigma}(t)\bigr)^{T}C(t)x^{\sigma}(t) \Delta t.$$

Noting that $$B(t)>0$$, we know that $$y^{T}(t)B(t)y(t)\geq0$$, $$t\in [a,b]_{\mathbb{T}}$$.

We claim that $$y^{T}(t)B(t)y(t)\not\equiv0$$ ($$t\in [a,b]_{\mathbb{T}}$$). Indeed, if $$y^{T}(t)B(t)y(t)\equiv0$$ ($$t\in [a,b]_{\mathbb{T}}$$), then

$$\bigl\vert \sqrt{B(t)}y(t)\bigr\vert ^{2}=y^{T}(t)B(t)y(t) \equiv0,$$

which implies $$B(t)y(t)\equiv0$$ ($$t\in[a,b]_{\mathbb{T}}$$). Thus, the first equation of (1.4) reduces to

$$x^{\Delta}(t)= -A(t)x\bigl(\sigma(t)\bigr),\qquad x(a)=0.$$

By Lemma 2.4 it follows

$$x(t)=e_{\Theta A}(t,a)\cdot0=0,$$

which is a contradiction to (2.1). Hence, we obtain that

$$\int_{a}^{b}\bigl\vert x^{\sigma}(t)\bigr\vert ^{q-2}\bigl(x^{\sigma}(t)\bigr)^{T}C(t)x^{\sigma }(t) \Delta t= \int_{a}^{b}G(t)\Delta t>0,$$
(3.3)

and it follows from Lemma 2.4 that, for $$t\in[a,b]_{\mathbb{T}}$$,

\begin{aligned} x(t) =& - \int_{a}^{t}e_{\Theta A}(t,\tau)B(\tau)y(\tau ) \bigl\vert \sqrt{B(\tau)}y(\tau)\bigr\vert ^{p-2}\Delta{\tau} \\ =& - \int_{b}^{t}e_{\Theta A} (t,\tau)B(\tau)y(\tau) \bigl\vert \sqrt{B(\tau)}y(\tau)\bigr\vert ^{p-2}\Delta{\tau}, \end{aligned}

which implies that, for $$t\in[a,b)_{\mathbb{T}}$$,

\begin{aligned} x^{\sigma}(t) =&- \int_{a}^{\sigma(t)}e_{\Theta A}\bigl(\sigma (t),\tau \bigr)B(\tau)y(\tau)\bigl\vert \sqrt{B(\tau)}y(\tau)\bigr\vert ^{p-2} \Delta {\tau} \\ =&+ \int_{\sigma(t)}^{b}e_{\Theta A}\bigl(\sigma(t),\tau \bigr)B(\tau)y(\tau)\bigl\vert \sqrt{B(\tau)}y(\tau)\bigr\vert ^{p-2} \Delta {\tau}. \end{aligned}

Note that, for $$a\leq\sigma(t)\leq b$$,

\begin{aligned}& \bigl\vert e_{\Theta A}\bigl(\sigma(t),\tau\bigr)B(\tau)y(\tau)\bigr\vert \sqrt {B(\tau)}y(\tau)\bigl\vert ^{p-2}\bigr\vert \\& \quad \leq \bigl\vert e_{\Theta A}\bigl(\sigma(t),\tau\bigr)\bigr\vert \bigl\vert B(\tau)y(\tau)\bigr\vert \bigl\vert \sqrt{B(\tau)}y(\tau)\bigr\vert ^{p-2} \\& \quad \leq F(t,\tau)\bigl\vert \sqrt{B(\tau)}y(\tau)\bigr\vert \bigl\vert \sqrt{B(\tau)}y(\tau)\bigr\vert ^{p-2} \\& \quad = F(t,\tau)G^{\frac{1}{q}}(\tau). \end{aligned}

Then by Remark 2.3 and Lemma 2.6 we obtain

\begin{aligned} \bigl\vert x^{\sigma}(t)\bigr\vert ^{q} =&\biggl\vert \int_{a}^{\sigma(t)}e_{\Theta A}\bigl(\sigma(t),\tau \bigr)B(\tau)y(\tau)\bigl\vert \sqrt{B(\tau)}y(\tau)\bigr\vert ^{p-2} \Delta {\tau}\biggr\vert ^{q} \\ \leq& \biggl[ \int_{a}^{\sigma(t)}\bigl\vert e_{\Theta A}\bigl( \sigma(t),\tau\bigr)B(\tau )y(\tau)\bigl\vert \sqrt{B(\tau)}y(\tau)\bigr\vert ^{p-2}\bigr\vert \Delta{\tau} \biggr]^{q} \\ \leq& \biggl[ \int_{a}^{\sigma(t)}F(t,\tau)G^{\frac{1}{q}}(\tau)\Delta { \tau} \biggr]^{q} \\ \leq& \biggl( \int_{a}^{\sigma(t)}F^{\alpha}(t,\tau)\Delta { \tau} \biggr)^{\frac{q}{p}} \int_{a}^{\sigma(t)}F^{\beta}(t,\tau)G(\tau )\Delta {\tau} \\ \leq& \max_{a\leq\tau\leq\sigma(t)}F^{\beta}(t,\tau) \biggl( \int _{a}^{\sigma(t)}F^{\alpha}(t,\tau)\Delta { \tau} \biggr)^{\frac{q}{p}} \int_{a}^{\sigma(t)}G(\tau)\Delta {\tau}, \end{aligned}

that is,

$$\bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\leq\max _{a\leq\tau\leq\sigma(t)}F^{\beta}(t,\tau)\Phi \bigl(\sigma(t)\bigr) \int_{a}^{\sigma(t)}G(\tau)\Delta{\tau}.$$
(3.4)

Similarly, for $$a\leq\sigma(t) \leq b$$, we have

$$\bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\leq\max _{\sigma(t)\leq\tau\leq b}F^{\beta}(t,\tau)\Psi \bigl(\sigma(t)\bigr) \int_{\sigma(t)}^{b}G(\tau)\Delta{\tau}.$$
(3.5)

It follows from (3.4) and (3.5) that

$$\bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\leq\frac{P(t)}{Q(t)} \int_{a}^{b}G(\tau)\Delta{\tau}.$$

Then by (3.3) and Lemma 2.7 we have

\begin{aligned}& \int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\Delta t \\& \quad \leq \int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \frac{P(t)}{Q(t)}\Delta t \int _{a}^{b}G(t)\Delta t \\& \quad = \int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \frac{P(t)}{Q(t)}\Delta t \int_{a}^{b}\bigl\vert x^{\sigma}(t)\bigr\vert ^{q-2}\bigl(x^{\sigma}(t)\bigr)^{T}C(t)x^{\sigma }(t) \Delta t \\& \quad \leq \int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \frac{P(t)}{Q(t)}\Delta t \int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\Delta t. \end{aligned}

Since

$$\int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\Delta t\geq \int_{a}^{b}\bigl\vert x^{\sigma}(t)\bigr\vert ^{q-2}\bigl(x^{\sigma}(t)\bigr)^{T}C(t)x^{\sigma}(t) \Delta t= \int_{a}^{b} G(t)\Delta {t}>0,$$

we get

$$\int_{a}^{b}\frac{P(t)}{Q(t)}\bigl\vert C_{1}(t)\bigr\vert \Delta t\geq1.$$

This completes the proof of Theorem 3.1. □

### Corollary 3.2

Let $$a,b\in\mathbb{T}$$ with $$\sigma(a)< b$$ and $$C_{1}\in\mathbb{R}^{n\times n}_{s}$$ with $$C_{1}(t)-C(t)\geq0$$. If (1.4) has a solution $$(x(t),y(t))$$ with $$x(t),y(t)\in C_{\mathrm{rd}}(\mathbb{T},\mathbb{R}^{n})$$ satisfying (2.1) on the interval $$[a,b]_{\mathbb{T}}$$, then

$$\int_{a}^{b}Q(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \geq4.$$
(3.6)

### Proof

Note that

$$\frac{P(t)}{Q(t)}\leq \frac{Q(t)}{4}.$$

It follows from (3.1) that

$$\int_{a}^{b} \frac{Q(t)}{4}\bigl\vert C_{1}(t)\bigr\vert \Delta t\geq1,$$

that is,

$$\int_{a}^{b}Q(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \geq4.$$

This completes the proof of Corollary 3.2. □

### Corollary 3.3

Let $$a,b\in\mathbb{T}$$ with $$\sigma(a)< b$$ and $$C_{1}\in\mathbb{R}^{n\times n}_{s}$$ with $$C_{1}(t)-C(t)\geq0$$. If (1.4) has a solution $$(x(t),y(t))$$ with $$x(t),y(t)\in C_{\mathrm{rd}}(\mathbb{T},\mathbb{R}^{n})$$ satisfying (2.1) on the interval $$[a,b]_{\mathbb{T}}$$, then

$$\int_{a}^{b}\sqrt{P(t)}\bigl\vert C_{1}(t)\bigr\vert \Delta t \geq2.$$
(3.7)

### Proof

Note that

$$Q(t)\geq2\sqrt{P(t)}.$$

It follows from (3.1) that

$$\int_{a}^{b}\sqrt{P(t)}\bigl\vert C_{1}(t)\bigr\vert \Delta t\geq \int_{a}^{b}2\frac {P(t)}{Q(t)}\bigl\vert C_{1}(t)\bigr\vert \Delta t\geq2.$$

This completes the proof of Corollary 3.3. □

### Theorem 3.4

Let $$a,b\in\mathbb{T}$$ with $$\sigma(a)< b$$ and $$C_{1}\in\mathbb{R}^{n\times n}_{s}$$ with $$C_{1}(t)-C(t)\geq0$$. If (1.4) has a solution $$(x(t),y(t))$$ with $$x(t),y(t)\in C_{\mathrm{rd}}(\mathbb{T},\mathbb{R}^{n})$$ satisfying (2.1) on the interval $$[a,b]_{\mathbb{T}}$$, then there exists $$c\in(a,b)$$ such that

$$\left \{ \textstyle\begin{array}{l} {\int}_{a}^{\sigma(c)}\Phi(\sigma(t))\max_{a\leq\tau\leq \sigma(t)}F^{\beta}(t,\tau)|C_{1}(t)|\Delta t\geq1, \\ \int_{c}^{b} \Psi(\sigma(t))\max_{\sigma(t)\leq\tau\leq b}F^{\beta}(t,\tau)|C_{1}(t)|\Delta t\geq1. \end{array}\displaystyle \right .$$
(3.8)

### Proof

Set $$U(t)=\Phi(\sigma(t))\max_{a\leq \tau\leq\sigma(t)}F^{\beta}(t,\tau)$$ and $$V(t)= \Psi(\sigma(t))\max_{\sigma(t)\leq\tau\leq b}F^{\beta}(t,\tau)$$. Let

$$f(t)= \int_{a}^{t}U(s)\bigl\vert C_{1}(s) \bigr\vert \Delta s- \int_{t}^{b}V(s)\bigl\vert C_{1}(s) \bigr\vert \Delta s.$$

Then we have $$f(a)<0$$ and $$f(b)>0$$. Hence, we can choose $$c\in(a,b)$$ such that $$f(c)\leq0$$ and $$f(\sigma(c))\geq0$$, that is,

$$\int_{a}^{c}U(s)\bigl\vert C_{1}(s) \bigr\vert \Delta s\leq \int _{c}^{b}V(s)\bigl\vert C_{1}(s) \bigr\vert \Delta s$$
(3.9)

and

$$\int_{a}^{\sigma(c)}U(s)\bigl\vert C_{1}(s) \bigr\vert \Delta s\geq \int_{\sigma(c)}^{b}V(s)\bigl\vert C_{1}(s) \bigr\vert \Delta s.$$
(3.10)

By (3.4) we have that

$$\bigl\vert C_{1}(t)\bigr\vert \bigl\vert x^{\sigma}(t) \bigr\vert ^{q}\leq U(t)\bigl\vert C_{1}(t)\bigr\vert \int_{a}^{\sigma(t)}G(\tau )\Delta{\tau}.$$
(3.11)

Integrating (3.11) from a to $$\sigma(c)$$, we obtain

\begin{aligned} \int_{a}^{\sigma(c)}\bigl\vert C_{1}(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\Delta t \leq& \int _{a}^{\sigma(c)}U(t)\bigl\vert C_{1}(t) \bigr\vert \biggl( \int_{a}^{\sigma(t)} G(\tau)\Delta{\tau} \biggr) \Delta t \\ \leq& \int_{a}^{c}U(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \int_{a}^{\sigma(c)}G(\tau)\Delta {\tau} \\ &{}+U(c)\bigl\vert C_{1}(c)\bigr\vert \bigl(\sigma(c)-c\bigr) \int_{a}^{\sigma(c)}G(\tau)\Delta {\tau} \\ =& \int_{a}^{\sigma(c)}U(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \int_{a}^{\sigma(c)}G(\tau)\Delta{\tau}. \end{aligned}

Similarly, we obtain from (3.4) and (3.10) that

\begin{aligned} \int_{\sigma(c)}^{b}\bigl\vert C_{1}(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\Delta t \leq& \int _{\sigma(c)}^{b}V(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \int_{\sigma(c)}^{b}G(\tau )\Delta{\tau} \\ \leq& \int _{a}^{\sigma(c)}U(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \int_{\sigma(c)}^{b}G(\tau )\Delta{\tau}. \end{aligned}

This yields

\begin{aligned} \int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\Delta t \leq& \int _{a}^{\sigma(c)}U(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \int_{a}^{b}G(t)\Delta t \\ =& \int_{a}^{\sigma (c)}U(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \int_{a}^{b}\bigl\vert x^{\sigma}(t)\bigr\vert ^{q-2}\bigl(x^{\sigma}(t)\bigr)^{T}C(t)x^{\sigma}(t) \Delta t \\ \leq& \int_{a}^{\sigma(c)}U(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\Delta t. \end{aligned}

Since

\begin{aligned} \int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\Delta t \geq& \int_{a}^{b}\bigl\vert x^{\sigma}(t)\bigr\vert ^{q-2}\bigl(x^{\sigma}(t)\bigr)^{T}C(t)x^{\sigma}(t) \Delta t \\ =& \int _{a}^{b}G(t)\Delta {t}>0, \end{aligned}

we have $$\int_{a}^{\sigma(c)}U(t)|C_{1}(t)|\Delta t\geq1$$.

Next, we obtain from (3.5) that

$$\bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\bigl\vert C_{1}(t)\bigr\vert \leq V(t)\bigl\vert C_{1}(t)\bigr\vert \int_{\sigma(t)}^{b}G(\tau )\Delta{\tau}.$$
(3.12)

Integrating (3.12) from c to b, we have

\begin{aligned} \int_{c}^{b}\bigl\vert C_{1}(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\Delta t \leq& \int _{c}^{b}V(t)\bigl\vert C_{1}(t) \bigr\vert \biggl( \int_{\sigma(t)}^{b}G(\tau)\Delta{\tau} \biggr) \Delta t \\ \leq& \int_{c}^{b}V(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \int_{\sigma (c)}^{b}G(\tau)\Delta{\tau}. \end{aligned}

Similarly, we obtain

\begin{aligned} \begin{aligned} \int_{a}^{c}\bigl\vert C_{1}(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\Delta t&\leq \int _{a}^{c}U(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \int_{a}^{\sigma(c)}G(\tau)\Delta {\tau} \\ &\leq \int _{c}^{b}V(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \int_{a}^{\sigma(c)}G(\tau)\Delta {\tau}. \end{aligned} \end{aligned}

This yields

\begin{aligned} \int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\Delta t \leq& \int _{c}^{b}V(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \int_{a}^{b}G(t)\Delta t \\ =& \int_{c}^{b}V(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \int_{a}^{b}\bigl\vert x^{\sigma}(t)\bigr\vert ^{q-2}\bigl(x^{\sigma}(t)\bigr)^{T}C(t)x^{\sigma }(t) \Delta t \\ \leq& \int_{c}^{b}V(t)\bigl\vert C_{1}(t) \bigr\vert \Delta t \int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\Delta t. \end{aligned}

Thus, we have $$\int_{c}^{b}V(t)|C_{1}(t)|\Delta t\geq1$$. This completes the proof of Theorem 3.4. □

### Theorem 3.5

Let $$a,b\in\mathbb{T}$$ with $$\sigma(a)< b$$ and $$C_{1}\in\mathbb{R}^{n\times n}_{s}$$ with $$C_{1}(t)-C(t)\geq0$$. If (1.4) has a solution $$(x(t),y(t))$$ with $$x(t),y(t)\in C_{\mathrm{rd}}(\mathbb{T},\mathbb{R}^{n})$$ satisfying (2.1) on the interval $$[a,b]_{\mathbb{T}}$$, then

$$\int_{a}^{b}\bigl\vert A(t)\bigr\vert \Delta t+\Bigl\{ \max_{a\leq t\leq b}\bigl\vert \sqrt {B(t)}\bigr\vert ^{\beta}\Bigr\} ^{\frac{1}{q}} \biggl( \int_{a}^{b}\bigl\vert \sqrt{B(t)}\bigr\vert ^{\alpha} \Delta t \biggr)^{\frac{1}{p}} \biggl( \int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \Delta t \biggr)^{\frac{1}{q}}\geq2.$$

### Proof

Since $$x(a)=x(b)=0$$, we have

$$\int_{a}^{b}G(t)\Delta t= \int_{a}^{b}\bigl\vert x^{\sigma}(t)\bigr\vert ^{q-2}\bigl(x^{\sigma}(t)\bigr)^{T}C(t)x^{\sigma}(t) \Delta t.$$

It follows from the first equation of (1.4) that, for all $$a\leq t \leq b$$,

\begin{aligned} x(t) =& \int_{a}^{t}\bigl(-A(\tau)x^{\sigma}(\tau)-B( \tau)\bigl\vert \sqrt{B(\tau)}y(\tau )\bigr\vert ^{p-2}y(\tau)\bigr) \Delta \tau \\ =& \int_{t}^{b}\bigl(A(\tau)x^{\sigma}(\tau)+B( \tau)\bigl\vert \sqrt{B(\tau )}y(\tau)\bigr\vert ^{p-2}y(\tau)\bigr) \Delta \tau. \end{aligned}

Thus, we have

\begin{aligned} \bigl\vert x(t)\bigr\vert =&\biggl\vert \int_{a}^{t}\bigl(-A(\tau)x^{\sigma}(\tau)-B( \tau )y(\tau)\bigr)\bigl\vert \sqrt{B(\tau)}y(\tau)\bigr\vert ^{p-2} \Delta \tau\biggr\vert \\ \leq& \int_{a}^{t}\bigl\vert A(\tau)x^{\sigma}( \tau)+B(\tau)y(\tau)\bigr\vert \sqrt{B(\tau )}y(\tau)\bigl\vert ^{p-2} \bigr\vert \Delta \tau \\ \leq& \int_{a}^{t}\bigl\vert A(\tau)x^{\sigma}( \tau)\bigr\vert \Delta \tau+ \int _{a}^{t}\bigl\vert B(\tau)y(\tau)\bigr\vert \bigl\vert \sqrt{B(\tau)}y(\tau)\bigr\vert ^{p-2} \Delta \tau \\ \leq& \int_{a}^{t}\bigl\vert A(\tau)\bigr\vert \bigl\vert x^{\sigma}(\tau)\bigr\vert \Delta \tau+ \int_{a}^{t}\bigl\vert \sqrt{B(\tau)}\bigr\vert G^{\frac{1}{q}}(\tau)\Delta \tau. \end{aligned}

Similarly, we have

$$\bigl\vert x(t)\bigr\vert \leq \int_{t}^{b}\bigl\vert A(\tau)\bigr\vert \bigl\vert x^{\sigma}(\tau)\bigr\vert \Delta \tau + \int_{t}^{b}\bigl\vert \sqrt{B(\tau)}\bigr\vert G^{\frac{1}{q}}(\tau)\Delta \tau.$$

Then we obtain

\begin{aligned} \bigl\vert x(t)\bigr\vert \leq&\frac{1}{2} \biggl[ \int _{a}^{b}\bigl\vert A(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert \Delta t+ \int_{a}^{b}\bigl\vert \sqrt{B(t )}\bigr\vert G^{\frac{1}{q}}(t) \Delta t \biggr] \\ \leq& \frac{1}{2} \biggl[ \int_{a}^{b}\bigl\vert A(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert \Delta t+\Bigl\{ \max _{a\leq t\leq b}\bigl\vert \sqrt{B(t)}\bigr\vert ^{\beta}\Bigr\} ^{\frac{1}{q}} \\ &{}\times\biggl( \int_{a}^{b} \bigl\vert \sqrt{B(t)}\bigr\vert ^{\alpha} \Delta t \biggr)^{\frac{1}{p}} \biggl( \int _{a}^{b}G(t)\Delta t \biggr)^{\frac{1}{q}} \biggr] \\ =& \frac{1}{2} \biggl[ \int_{a}^{b}\bigl\vert A(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert \Delta t+\Bigl\{ \max _{a\leq t\leq b}\bigl\vert \sqrt{B(t)}\bigr\vert ^{\beta}\Bigr\} ^{\frac{1}{q}} \biggl( \int _{a}^{b}\bigl\vert \sqrt{B(t)}\bigr\vert ^{\alpha} \Delta t \biggr)^{\frac{1}{p}} \\ &{}\times \biggl( \int_{a}^{b}\bigl\vert x^{\sigma}(t)\bigr\vert ^{q-2}\bigl(x^{\sigma }(t)\bigr)^{T}C(t)x^{\sigma}(t) \Delta t \biggr)^{\frac{1}{q}} \biggr] \\ \leq&\frac{1}{2} \biggl[ \int_{a}^{b}\bigl\vert A(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert \Delta t+\Bigl\{ \max _{a\leq t\leq b}\bigl\vert \sqrt{B(t)}\bigr\vert ^{\beta}\Bigr\} ^{\frac{1}{q}} \\ &{}\times\biggl( \int_{a}^{b}\bigl\vert \sqrt {B(t)}\bigr\vert ^{\alpha} \Delta t \biggr)^{\frac{1}{p}} \biggl( \int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \bigl\vert x^{\sigma}(t)\bigr\vert ^{q}\Delta t \biggr)^{\frac{1}{q}} \biggr]. \end{aligned}

Denote $$M=\max_{a\leq t\leq b}|x(t)|>0$$. Then

$$M\leq \frac{1}{2} \biggl[ \int_{a}^{b}\bigl\vert A(t)\bigr\vert M \Delta t+\Bigl\{ \max_{a\leq t\leq b}\bigl\vert \sqrt{B(t)}\bigr\vert ^{\beta}\Bigr\} ^{\frac{1}{q}} \biggl( \int _{a}^{b}\bigl\vert \sqrt{B(t)}\bigr\vert ^{\alpha} \Delta t \biggr)^{\frac{1}{p}} \biggl( \int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert M^{q}\Delta t \biggr)^{\frac{1}{q}} \biggr].$$

Thus,

$$\int_{a}^{b}\bigl\vert A(t)\bigr\vert \Delta t+\Bigl\{ \max_{a\leq t\leq b}\bigl\vert \sqrt {B(t)}\bigr\vert ^{\beta}\Bigr\} ^{\frac{1}{q}} \biggl( \int_{a}^{b}\bigl\vert \sqrt{B(t)}\bigr\vert ^{\alpha} \Delta t \biggr)^{\frac{1}{p}} \biggl( \int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \Delta t \biggr)^{\frac{1}{q}}\geq2.$$

This completes the proof of Theorem 3.5. □

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## Acknowledgements

This project is supported by NNSF of China (11461003).

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Correspondence to Taixiang Sun.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Sun, T., Xi, H., Liu, J. et al. Lyapunov inequalities for a class of nonlinear dynamic systems on time scales. J Inequal Appl 2016, 80 (2016). https://doi.org/10.1186/s13660-016-1022-3

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• DOI: https://doi.org/10.1186/s13660-016-1022-3

• 34K11
• 39A10
• 39A99

### Keywords

• Lyapunov inequality
• nonlinear dynamic system
• time scale