带两个参数的四阶边值问题正解的存在性

Existence of Positive Solutions of a Fourth-Order Boundary Value Problem with Two Parameters

在不要求f非负的条件下,通过将边值问题转化成积分方程系统,并运用锥上的不动点指数理论研究带2个参数的四阶边值问题u(4)+βu″-αu=f(t,u),0<t<1u(0)=u(1)=u″(0)=u″(1)=0正解的存在性,其中:f:[0,1]×[0,+∞)-→R是连续的;α,β∈R满足β≤0,α≥-(β2)/4,α/(π4)+β/(π2)<1或满足β<2π2,α≥0,α/(π4)+β/(π2)<1.

In the case of not requiring f to be nonnegative,by transforming the boundary value problem into the integral equation system and applying the fixed point index theory in cones,the authors study the following fourth-order boundary value problem with two parameters u（4）＋βu″-αu=f（t,u）,0t1 u（0）=u（1）=u″（0）=u″（1）=0 and obtain the results on the existence of its positive solution,where f： ×[0,＋∞）R is continuous,and α,β∈R satisfies β≤0,α≥-（β2）/4,α/（π4）＋β/（π2）1,or β2π2,α≥0,α/（π4）＋β/（π2）1.