Abstract
The purpose of this work is to obtain several Lyapunov inequalities for the nonlinear dynamic systems
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 \([a,b]_{\mathbb{T}}\).
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1 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 [4–7].
In 2002, Bohner et al. [8] investigated the second-order Sturm-Liouville dynamic equation
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
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
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:
This was later strengthened with \(|u(t)|\) replaced by \(u^{+}(t)=\max\{u(t),0\}\) by Wintner [10] and thereafter by some other authors:
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
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
The purpose of this paper is to establish several Lyapunov inequalities for the nonlinear dynamic system
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
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
then we have the following inequality:
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
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)
$$\int_{a}^{b}{\frac{[\int_{a}^{\sigma(t)}|B(s)||e_{\Theta A}(\sigma(t),s)|^{2}\Delta s ][\int_{\sigma(t)}^{b}|B(s)||e_{\Theta A}(\sigma(t),s)|^{2}\Delta s]}{\int_{a}^{b}|B(s)||e_{\Theta A}(\sigma(t),s)|^{2}\Delta s }}\bigl\vert C_{1}(t)\bigr\vert \Delta t\geq1, $$
-
(2)
$$\int_{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \biggl\{ \int_{a}^{b}\bigl\vert B(s)\bigr\vert \bigl\vert e_{\Theta A}\bigl(\sigma(t),s\bigr)\bigr\vert ^{2}\Delta s \biggr\} \Delta t\geq 4, $$
-
(3)
$$\int_{a}^{b}\bigl\vert A(t)\bigr\vert \Delta t+ \biggl( \int_{a}^{b}\bigl\vert \sqrt {B(t)}\bigr\vert ^{2}\Delta t \biggr)^{1/2} \biggl( \int _{a}^{b}\bigl\vert C_{1}(t)\bigr\vert \Delta t \biggr)^{1/2}\geq2. $$
For some other related results on Lyapunov-type inequalities, see, for example, [14–23].
2 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
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
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
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
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 \(U,V\in{\mathbb{R}}^{n\times n}\),
Let \(\mathbb{R}^{n\times n}_{s}\) be the set of all symmetric real \(n\times n\) matrices. We can show that, for any \(U\in{\mathbb{R}}_{s}^{n\times n}\),
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
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 \(f,g\in C_{\mathrm{rd}}([a,b]_{ \mathbb{T}},{\mathbb{R}})\),
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 \(f,g\in C_{\mathrm{rd}}([a,b]_{\mathbb{T}},(-\infty,0)\cup(0,\infty))\),
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
This completes the proof of Lemma 2.2. □
Remark 2.3
Let \(\gamma=0\) in Lemma 2.2. Then we obtain that, for any \(f,g\in C_{\mathrm{rd}}([a,b]_{ \mathbb{T}},(-\infty, 0)\cup(0,\infty))\),
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
is the unique solution of the initial value problem
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
Lemma 2.5
([2])
Let \(A,B\in C_{\mathrm{rd}}({\mathbb{T}, \mathbb{R}^{n\times n}})\) be Δ-differentiable. Then
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
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 \(x \in\mathbb{R}^{n}\),
3 Main results and proofs
In this section, we assume that \(\alpha,\beta\in\mathbb{R}\) and \(p,q\in(1,+\infty)\) satisfy
For any \(t,\tau\in [a,b]_{\mathbb{T}}\), write
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
Proof
Since \((x(t),y(t))\) is a solution of (1.4), we have
Integrating (3.2) from a to b and noting that \(x(a)=x(b)=0\), we obtain
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
which implies \(B(t)y(t)\equiv0\) (\(t\in[a,b]_{\mathbb{T}}\)). Thus, the first equation of (1.4) reduces to
By Lemma 2.4 it follows
which is a contradiction to (2.1). Hence, we obtain that
and it follows from Lemma 2.4 that, for \(t\in[a,b]_{\mathbb{T}}\),
which implies that, for \(t\in[a,b)_{\mathbb{T}}\),
Note that, for \(a\leq\sigma(t)\leq b\),
Then by Remark 2.3 and Lemma 2.6 we obtain
that is,
Similarly, for \(a\leq\sigma(t) \leq b\), we have
It follows from (3.4) and (3.5) that
Then by (3.3) and Lemma 2.7 we have
Since
we get
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
Proof
Note that
It follows from (3.1) that
that is,
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
Proof
Note that
It follows from (3.1) that
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
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
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,
and
By (3.4) we have that
Integrating (3.11) from a to \(\sigma(c)\), we obtain
Similarly, we obtain from (3.4) and (3.10) that
This yields
Since
we have \(\int_{a}^{\sigma(c)}U(t)|C_{1}(t)|\Delta t\geq1\).
Next, we obtain from (3.5) that
Integrating (3.12) from c to b, we have
Similarly, we obtain
This yields
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
Proof
Since \(x(a)=x(b)=0\), we have
It follows from the first equation of (1.4) that, for all \(a\leq t \leq b\),
Thus, we have
Similarly, we have
Then we obtain
Denote \(M=\max_{a\leq t\leq b}|x(t)|>0\). Then
Thus,
This completes the proof of Theorem 3.5. □
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This project is supported by NNSF of China (11461003).
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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