摘 要:通过引入参数函数H(t,s)及h(t,s),利用积分平均技巧,积分变换和广义Riccati变换给出了一类二阶微分方程的振动准则.
关 键 词 :振动性;微分方程;广义Riccati变换
中图分类号:O1758文献标识码:A
[WT]文章编号:1672-1098(2011)02-0061-05
收稿日期:2011-03-15
基金项目:安徽理工大学青年教师科学研究基金资助项目(2010)
作者简介:唐楠(1981-),女,河北邢台人,助教,硕士,主要从事微分方程定性与稳定性理论的教学和研究工作.
[JZ(〗[WT3BZ]Oscillation Criteria of A Class of Second Order Differential Equation
TANG Nan
(School of Mathematics, Anhui University of Science and Technology, Huainan Anhui 232001, China)
Abstract:By introducing H(t,s),h(t,s),using the iterated integral tranormations and generalized Riccati tranormation, some oscillation criteria of a class ofsecond order differential equation were given.
Key words:oscillation, differential equation, generalized Riccati tranormation.
近年来,微分方程解的振动性问题引起了广泛关注.文献[1-4]分别讨论了二阶非线性方程解的振动性问题.目前二阶半线性微分方程已有较多研究成果[5-8],但对于具有特殊形式的二阶半线性微分方程的结果并不多见.
考虑二阶微分方程
引理1 如果A,B是非负数,那么Aλ+(λ-1)Bλ-λABλ-1≥0,λ>1,
等号成立当且仅当A等于B[9].
通过引入参数函数H(t,s)及h(t,s),下面给出式(1)的解振动的充分条件.
定理1 令D等于{(t,s)|t≥s≥t0},D0等于{(t, s)|t>s≥t0}; 若ddtg(t, a)存在, 并且存在函数H(t,s)∈C(D,R),h(t,s)∈C(D0,R+),满足以下条件
H(t,t)等于0,t≥t0;H(t,s)s≤0;H(t,s)>0,(t,s)∈D0(2)
h(t,s)等于-H(t,s)s,(t,s)∈D0(3)
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(责任编辑:何学华)