一类带积分边界条件的三阶边值问题正解的存在唯一性

{\small 本文运用混合单调算子方法研究了带积分边界条件的三阶边值问题 $$\left\{\begin{aligned} &-u''(t)=f(t,u(t),u(\xi t))+g(t,u(t)),\quad~t\in(0,1), \xi\in(0,1),\&u(0)=u''(0)=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\&u''(1)=\int_{0}^{1}q(t)u''(t)dt~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned} \right. $$ 正解的存在唯一性,~其中~$f:0,1]\times0,+\infty)^{2}\rightarrow0,+\infty)$连续,~$g:0,1]\times0,+\infty)\rightarrow0,+\infty)$连续,~$q\in C(0,1],0,+\infty))$. }

The existence and uniqueness of positive solutions for a class of third-order boundary value problems with integral boundary conditions

In this paper,~by using the mixed monotone operator method,~we study the existence and uniqueness of positive solutions for boundary value problems of third-order nonlinear differential equation $$\left\{\begin{aligned} &-u''(t)=f(t,u(t),u(\xi t))+g(t,u(t)),\quad~t\in(0,1), \xi\in(0,1)\&u(0)=u''(0)=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\&u''(1)=\int_{0}^{1}q(t)u''(t)dt,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned} \right. $$ where~$f:0,1]\times0,+\infty)^{2}\rightarrow0,+\infty)$ is continuous,~$g:0,1]\times0,+\infty)\rightarrow0,+\infty)$ is continuous,~$q\in C(0,1],0,+\infty))$.\\