含有各阶导数的非线性4阶边值问题的正解

Positive Solutions of a Fourth-order Boundary Value Problem Involving all Derivatives

主要研究如下非线性4阶常微分方程边值问题的正解:{u^(4)=f(t,u,u′,-u″,-u′″),u(0)=u′(1)=u″(0)=u′″(1)=0,其中f∈C([0,1]×R_+~4,R+)(R+=[0,+∞)).为了克服各阶导数带来的困难,首先把上述问题转化成一个二阶积分-常微分方程的边值问题.然后,结合先验估计,运用不动点指数理论,证明了该问题正解的存在性,多重正解的存在性和正解的唯一性的几个结果.最后,把主要结果应用于建立Dirichlet问题对称正解的存在性,多重对称正解的存在性和对称正解的唯一性.

This paper is mainly concerned with the positive solutions of the fourth-order boundary value problem of ordinary differential equations：{u^（4）=f（t,u,u＇,-u＇＇,-u＇＇＇）,u（0）=u＇（1）=u＇＇（0）=u＇＇＇（1）=0,where f∈C （[0,1]×R＋^4,R＋）（R＋=[0,＋∞））.To overcome the difficulty resulting from all derivatives,we first transform the above problem into a boundary value problem for an associated second-order integro-ordinary differential equation.Then,using fixed point theory,combined with a priori estimates of positive solutions,we prove some results results on the existence of positive solutions,the existence of multiple positive solutions and the uniqueness of solutions for the derived problem.Finally,the main results obtained are applied to establish some results of symmetric positive solutions for the Dirichlet problem.