二阶脉冲微分方程Dirichlet问题非平凡解的存在性及多解性

doi:10.6040/j.issn.1671-9352.0.2016.078

山东大学学报（理学版） ›› 2016, Vol. 51 ›› Issue (12): 29-35.doi: 10.6040/j.issn.1671-9352.0.2016.078

二阶脉冲微分方程Dirichlet问题非平凡解的存在性及多解性

李晓燕,徐嫚^*

西北师范大学数学与统计学院, 甘肃兰州 730070

收稿日期:2016-02-27 出版日期:2016-12-20 发布日期:2016-12-20
通讯作者: 徐嫚(1989— ), 女, 硕士研究生, 研究方向为常微分方程边值问题. E-mail:xmannwnu@126.com E-mail:lixydodo@163.com
作者简介:李晓燕(1979— ), 女, 讲师, 研究方向为常微分方程边值问题. E-mail:lixydodo@163.com
基金资助:
国家自然科学基金资助项目(11361054);甘肃省自然科学基金资助项目(1208RJZA258)

Existence and multiplicity of nontrivial solutions of Dirichlet problems for second-order impulsive differential equation

LI Xiao-yan, XU Man^*

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China

Received:2016-02-27 Online:2016-12-20 Published:2016-12-20

摘要/Abstract

摘要： 研究了二阶脉冲微分方程Dirichlet问题{u″(t)+f(t,u(t))=0, t∈(0,1), t≠t_i,Δu|_{t=t_i}=α_iu(t_i), i=1, 2,…,k,u(0)=u(1)=0非平凡解的存在性及多解性。其中α_i>-1, i=1, 2,…,k 为给定常数, 0=t₀12<…kk+1=1 为给定的脉冲点。Δu|_{t=t_i}=u(t⁺_i)-u(t^-_i), u(t⁺_i), u(t^-_i)分别表示u在t=t_i处的右极限和左极限。 f∈C(［0,1］×R, R)。本文的主要结果推广和改进了一些已有的关于二阶脉冲微分方程Dirichlet问题非平凡解的存在性及多解性的结论。主要结果的证明基于López-Gómez在2001年建立的分歧定理。

关键词: 二阶脉冲微分方程, 非平凡解, 分歧理论

Abstract: In this paper, we study the existence and multiplicity of nontrivial solutions of Dirichlet problems for second-order impulsive differential equation{u″(t)+f(t,u(t))=0, t∈(0,1), t≠t_i,Δu|_{t=t_i}=α_iu(t_i), i=1, 2,…,k,u(0)=u(1)=0,where α_i>-1, i=1, 2,…,k are given constants, 0=t012<…kk+1=1 are given impulsive points. Δu|_{t=t_i}=u(t⁺_i)-u(t^-_i), u(t⁺_i), u(t^-_i) denote the right and left limit of u at t=t_i, respectively. f∈C(［0,1］×R, R). The main results extend and improve some results on existence and multiplicity of nontrivial solutions of Dirichlet problems for second-order impulsive differential equation. The proof of the main results are based on the López-Gómezs bifurcation theory established in 2001.

Key words: second-order impulsive differential equation, bifurcation theory, nontrivial solutions

中图分类号:

O175.8

李晓燕,徐嫚. 二阶脉冲微分方程Dirichlet问题非平凡解的存在性及多解性[J]. 山东大学学报（理学版）, 2016, 51(12): 29-35.

LI Xiao-yan, XU Man. Existence and multiplicity of nontrivial solutions of Dirichlet problems for second-order impulsive differential equation[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(12): 29-35.

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二阶脉冲微分方程Dirichlet问题非平凡解的存在性及多解性

Existence and multiplicity of nontrivial solutions of Dirichlet problems for second-order impulsive differential equation

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