书书书
第
32
卷 第
4
期
2012
年
12
月
数学理论与应用
MATHEMATICAL THEORY AND APPLICATIONS
Vol. 32 No. 4
Dec. 2012
Lyapunov - type Inequality for a Class of Quasilinear Systems
Liu Xinge Tang Meilan
( School of Mathematics and Statistics,Central South University,Changsha,410083,China)
Abstract This paper has considered a class of quasilinear systems. Anti - periodic boundary conditions are intro-
duced instead of boundary conditions. Based on anti - periodic boundary conditions and some techniques of mathemat-
ics analysis
,a new Lyapunov - type inequality is derived.
Key words Lyapunov Inequality Anti - periodic Boundary Condition Quasilinear Systems
一类拟线性系统的
Lyapunov
不等式
*
刘心歌 唐美兰
(
中南大学数学与统计学院
,
长沙
,410083)
摘 要 本文研究了一类拟线性系统
,
引入了反周期边值条件
,
基于反周期边值条件和数学分析的技巧
,
建
立了新的
Lyapunov
不等式
.
关键词
Lyapunov
型不等式 反周期边值条件 拟线性系统
1 Introduction
Lyapunov inequalities have found many applications in the study of various properties of differ-
ential equations
[1 - 6]. In 2006,De Nápoli and Pinasco[7]investigated the quasilinear elliptic
systems of resonant type
,they established the following Lyapunov - type inequality.
Theorem A Consider the following ( p,q) - quasilinear system:
(
φ
p
( u'( t) ) ) ' + f
1
( t) | u( t) |
α
-2
u( t) | v( t) |
β
= 0,
(
φ
q
( v'( t) ) ) ' + f
2
( t) | u( t) |
α
| v ( t) |
β
-2
v( t) = 0
{
,
( 1)
where 1 < p,q < +
∞
;
φ
p
( u) = | u |
p -2
u; f
1
,f
2
are real nonnegative continouous functions and
α
,
β
are nonnegative numbers satisfying
*
中南大学前沿研究计划重点项目
( 2010QZZD015)
和国家自然科学基金
( 61271355)
资助
收稿日期
: 2012
年
11
月
8
日