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.
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.
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.
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2016, Vol. 43 
f（s）/s，f∞=
f（s）/s时，获得有多个结点解存在这一结论.