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浙江大学学报（理学版）
浙江大学学报（理学版）  2016, Vol. 43 Issue (5): 525-531    DOI: 10.3785/j.issn.1008-9497.2016.05.005
数学与计算机科学     
非线性项在零点非渐进增长的四阶边值问题单侧全局分歧
沈文国
兰州工业学院, 基础学科部, 甘肃 兰州 730050
Unilateral global bifurcation for fourth-order boundary value problem with non-asymptotic nonlinearity at 0
SHEN Wenguo
Department of Basic Courses, Lanzhou Institute of Technology, Lanzhou 730050, China
 全文: PDF(350 KB)  
摘要： 建立一类四阶两点边值问题x""+kx"+lx=λh（t）x+g（t，x，λ），0< t< 1，x（0）=x（1）=x'（0）=x'（1）=0的Dancer-型单侧全局分歧结果.当扰动函数g:（0，1）×R2→R满足一些自然条件时，可以得到（λk，0）是所研究问题的一个分歧点，并且存在从（λk，0）发出的2个不同的连通分支Ck+和Ck-，其中λk是对应于上述线性特征值问题的第k个特征值.作为其应用，进一步研究了一类含参数的四阶两点边值问题x""+kx"+lx=rh（t）f（x），0< t< 1，x（0）=x（1）=x'（0）=x'（1）=0结点解的存在性.当非线性项满足ff0=∞，f∞ ∈（0，∞），f0=f（s）/s，f∞=f（s）/s时，获得有多个结点解存在这一结论.
关键词： 四阶问题;  单侧全局分歧;  结点解;  非线性项在零点非渐进增长    
Abstract: We present a Dancer-type unilateral global bifurcation result for a class of fourth-order two-point boundary value problem x""+kx"+lx=λh(t)x+g(t, x,λ), 0< t< 1,x(0)=x(1)=x'(0)=x'(1)=0. Under some natural hypotheses on the perturbation function g:(0,1)×R2→R, we show that (λk, 0) is a bifurcation point of the above problem. And there are two distinct unbounded continuas, Ck+ and Ck-, consisting of the bifurcation branch Ck from (λk, 0), where λk is the k-th eigenvalue of the linear problem corresponding to the above problems. As an application of the above result, the global behavior of the components of nodal solutions of the following problem x""+kx"+lx=rh(t)f(x), 0< t< 1, x(0)=x(1)=x'(0)=x'(1)=0 is studied. We obtain the existence of multiple nodal solutions for the problem if f0=∞, f∞ ∈ (0, ∞), f0=f(s)/s, f∞=f(s)/s.
Key words: fourth-order problems    unilateral global bifurcation    nodal solutions    non-asymptotic non-linearity at 0
收稿日期: 2015-08-01 出版日期: 2016-05-01
CLC:  O175.8  
基金资助: Supported by the National Natural Science Foundation of China (11561038); the Gansu Provincial Natural Science Foundation(145RJZA087).
作者简介: SHEN Wenguo(1963-),ORCID:http://orcid.org/0000-0001-7323-1887,Doctor,Professor,the field of interest is nonlinear functional differential equations, E-mail:shenwg369@163.com.
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沈文国

引用本文:

沈文国. 非线性项在零点非渐进增长的四阶边值问题单侧全局分歧[J]. 浙江大学学报（理学版）, 2016, 43(5): 525-531.

SHEN Wenguo. Unilateral global bifurcation for fourth-order boundary value problem with non-asymptotic nonlinearity at 0. Journal of ZheJIang University(Science Edition), 2016, 43(5): 525-531.

链接本文:

https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2016.05.005    或    https://www.zjujournals.com/sci/CN/Y2016/V43/I5/525

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