一类三阶两点边值问题的变号解

数学物理学报 ›› 2013, Vol. 33 ›› Issue (2): 216-223.

一类三阶两点边值问题的变号解

张琦

山西大学数学科学学院太原 |030006

收稿日期:2011-04-27 修回日期:2012-09-01 出版日期:2013-04-25 发布日期:2013-04-25
基金资助:
国家自然科学基金 (11071149)和山西省自然科学基金 (2010011001-1, 2012011004-2)资助

Sign-changing Solutions to Third-order Two-point Boundary Value Problem

ZHANG Qi

School of Mathematics, Shanxi University, Taiyuan 030006

Received:2011-04-27 Revised:2012-09-01 Online:2013-04-25 Published:2013-04-25
Supported by:
国家自然科学基金 (11071149)和山西省自然科学基金 (2010011001-1, 2012011004-2)资助

摘要/Abstract

摘要：

利用不动点指数理论和 Leray-Schauder 度理论讨论带有边值u(0)=u'(0)=u''(1)=0 的三阶两点边值问题-u'''(t)=f(t, u(t)), t ∈[0,1], 其中f ∈C([0,1]×R, R). 通过计算相应的线性算子的特征值与代数重数, 获得了一些包括变号解的存在性结果. 如果f 满足一定的条件, 则问题至少存在六个不同的非平凡解, 其中两个正解, 两个负解以及两个变号解. 进一步, 如果f(t, •), t ∈[0,1] 是奇函数, 则问题至少存在八个不同的非平凡解, 其中两个正解, 两个负解以及四个变号解.

关键词: 三阶边值问题, 变号解, 不动点指数, Leray-Schauder 度

Abstract:

In this paper, we use the fixed point index theory and the Leray-Schauder degree theory to discuss the third-order boundary value problem -u'''(t)=f(t, u(t)) for all t ∈[0,1] subject to u(0)=u'(0)=u''(1)=0, where f ∈C([0,1]×R, R). By computing hardly the eigenvalues and their algebraic multiplicities of the associated linear problem, we obtain some new existence results concerning sign-changing solutions to this problem. If f satisfies certain conditions, then the problem has at least six different nontrivial solutions: two positive solutions, two negative solutions and two sign-changing solutions. Moreover, if f(t, •) is odd for all t ∈[0,1], then the problem has at least eight different nontrivial solutions, which are two positive, two negative and four sign-changing solutions.

Key words: Third-order boundary value problem, Sign-changing solutions, Fixed point index, Leray-Schauder degree

中图分类号:

34B15

张琦. 一类三阶两点边值问题的变号解[J]. 数学物理学报, 2013, 33(2): 216-223.

ZHANG Qi. Sign-changing Solutions to Third-order Two-point Boundary Value Problem[J]. Acta mathematica scientia,Series A, 2013, 33(2): 216-223.

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