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

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• 作者：Taixiang Sun ; Hongjian Xi ; Jing Liu ; Qiuli He
• 关键词：Lyapunov inequality ; nonlinear dynamic system ; time scale
• 刊名：Journal of Inequalities and Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：2016
• 期：1
• 全文大小：1,504 KB
• 刊物主题：Analysis; Applications of Mathematics; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1029-242X

文摘

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 \([a,b]_{\mathbb{T}}\). Keywords Lyapunov inequality nonlinear dynamic system time scale MSC 34K11 39A10 39A99 1 IntroductionThe 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].