二阶非线性动力学方程的有界性分析

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"这篇研究论文关注的是一类二阶非线性动力学方程的有界性问题。在数学领域，动力学方程是描述物理系统动态行为的数学模型，通常涉及时间变量的变化。二阶动力学方程是这类模型的一个重要类型，因为它能很好地捕捉物体的运动状态，如速度和加速度。该论文发表于《应用数学通讯》(Applied Mathematics Letters)，作者包括Qing-Hua Ma、Jun-Wei Wang、Xiao-Hua Ke和Josip Pečarić，分别来自中国广东外语外贸大学的信息科技学院和克罗地亚萨格勒布大学的纺织技术学院。文章的核心内容是建立一个关于时间尺度的广义非线性动力学积分不等式，并利用这个不等式来分析二阶非线性动力学方程在时间尺度上的有界性。‘时间尺度’是一个数学概念，它扩展了连续和离散时间理论，可以涵盖微分方程和差分方程等不同类型的方程。有界性问题在动力学系统中至关重要，因为它是确保系统稳定性和可预测性的关键因素。论文中提出的定理不仅适用于特定的二阶微分方程、差分方程，还推广到了q-差分方程。q-差分方程是一种特殊形式的离散时间方程，与传统的差分方程相比，其步长可以是任意正实数的幂次。通过这些定理，作者们为理解和分析这类非线性系统的动态行为提供了新的工具和方法。该论文经过多次修订后于2013年6月被接受发表，关键词包括：时间尺度、动力学方程、积分不等式和有界性。这些标签明确了论文的研究重点和应用范围。这篇论文对理解和控制复杂非线性动力系统的行为，特别是在工程、物理学和生物学等领域，具有重要的理论价值和实际意义。"

Applied Mathematics Letters 26 (2013) 1099–1105

Contents lists available at SciVerse ScienceDirect

Applied Mathematics Letters

journal homepage: www.elsevier.com/locate/aml

On the boundedness of a class of nonlinear dynamic

equations of second order

✩

Qing-Hua Ma

a,∗

, Jun-Wei Wang

, Xiao-Hua Ke

, Josip Pečarić

Faculty of Information Science and Technology, Guangdong University of Foreign Studies, Guangzhou 510420, PR China

Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia

a r t i c l e i n f o

Article history:

Received 9 May 2013

Received in revised form 13 June 2013

Accepted 14 June 2013

Keywords:

Time scales

Dynamic equation

Integral inequality

Boundedness

a b s t r a c t

In this paper, a generalized nonlinear dynamic integral inequality on time scales is

established and then is used to study the boundedness of a class of nonlinear second-

order dynamic equations on time scales. These theorems contain as special cases results

for second-order differential equations, difference equations and q-difference equations.

1. Introduction

To unify and extend continuous and discrete analyses, the theory of time scales was introduced by Hilger [1] in his

Ph.D. Thesis in 1988. Since then, the theory has been evolving, and it has been applied to various fields of mathematics; for

example, see [2,3] and the references therein. It is well known that Gronwall-type integral inequalities and their discrete

analogues play a dominant role in the study of quantitative properties of solutions of differential, integral and difference

equations. During the last few years, some Gronwall-type integral inequalities on time scales and their applications have

been investigated by many authors. For example, we refer readers to [4–10]. In this paper, motivated by the paper [5], we

establish a new nonlinear Gronwall–Bellman type dynamic inequality on time scales, and then using the dynamic inequality

we obtain the bounds on the solutions of a class of nonlinear dynamic equations of second order on time scales, which has

generalized the main result of [5]. For all the detailed definitions, notation and theorems on time scales, we refer the readers

to the excellent monographs [2,3] and references given therein. We also present some preliminary results that are needed

in the remainder of this paper as useful lemmas for the discussion of our proof.

In what follows, R denotes the set of real numbers, R

= [0, +∞); C(M, S) denotes the class of all continuous functions

defined on set M with range in set S, T is an arbitrary time scale, and C

denotes the set of rd-continuous functions.

Throughout this paper, we always assume that t

∈ T, T

= [t

, +∞) ∩ T.

2. A nonlinear Bihari-type dynamic inequality and some lemmas

Lemma 2.1 ([9]). Suppose u(t), a(t) ∈ C

, R

), and a is nondecreasing, f (t, s), f

∆

(t, s) ∈ C

×T

, R

), ω ∈ C(R

, R

)

is nondecreasing. If for t ∈ T

, u(t) satisfies the following inequality

✩

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution,

and reproduction in any medium, provided the original author and source are credited.

∗

Corresponding author. Tel.: +86 020 39328032.

E-mail addresses: gdqhma@21cn.com, mqh@gdufs.edu.cn (Q.-H. Ma), pecaric@element.hr (J. Pečarić).

http://dx.doi.org/10.1016/j.aml.2013.06.007

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