稳态吊桥方程耦合系统正解的存在性

An existence result on positive solutions for a coupled system of steady state suspension bridge equations

摘要:

本文研究了二阶和四阶常微分方程耦合系统 \[ \begin{cases} &~u''''(t)=\lambda f(t,v(t)), \ \ \ \ \ t\in (0,1),\&-v''(t)=\lambda g(t,u(t)), \ \ \ \ \ t\in (0,1),\&~u(0)=u(1)=u''(0)=u''(1)=0,\&~v(0)=v(1)=0\\end{cases} \] 正解的存在性,~其中~$\lambda>0$~为参数,~$f,~g\in C([0,1]\times[0,\infty),~\mathbb{R})$.~当~$f,~g$~满足适当的条件时,~证明了~$\lambda$~充分大时,~一个正解的存在性结果,~主要结果的证明基于~Schauder~不动点定理. }

Abstract:

In this paper, we are concerned with the existence of positive solutions of a coupled system of second-order and fourth-order ordinary differential equations \[ \begin{cases} &~u''''(t)=\lambda f(t,v(t)), \ \ \ \ \ \ t\in (0,1),\&-v''(t)=\lambda g(t,u(t)), \ \ \ \ \ t\in (0,1),\&~u(0)=u(1)=u''(0)=u''(1)=0,\&~v(0)=v(1)=0,\\end{cases} \] where $\lambda$ is a positive parameter, $f,~g\in C([0,1]\times[0,\infty),~\mathbb{R})$. We prove the existence of a large positive solution for $\lambda$ large under suitable assumptions on $f$ and $g$. The proof of our main result is based upon the Schauder's fixed point theorem.