一类四阶变系数常微分系统固结梁边值问题的正解

Positive Solution of a Class of Clamped Beam BVPs for Fourth Order Ordinary Differential Systems with Variable Coefficients

摘要: 运用Leray-Schauder度理论和不动点定理获得了两端固定支撑边界条件下四阶变系数常微分系统固结梁边值问题$\{\begin{array}{l}{u}^{\left(4\right)}\left(x\right)+a\left(x\right)u\left(x\right)=f_1\left(x,v\left(x\right)\right),x\in \left(0,1\right),\\ v^{\left(4\right)}\left(x\right)+b\left(x\right)v\left(x\right)=f_2\left(x,u\left(x\right)\right),x\in \left(0,1\right),\\ u\left(0\right)=u\left(1\right)=u^{\prime }\left(0\right)=u^{\prime }\left(1\right)=0,\\ v\left(0\right)=v\left(1\right)=v^{\prime }\left(0\right)=v^{\prime }\left(1\right)=0\end{array}$ 正解的存在性和唯一性，其中$a,b:\left[0,1\right]\to \left[0,+\infty \right)$ 连续，非线性项$f_i:\left[0,1\right]\times R\to R$ 为连续函数且$f_i\left(x,0\right)\ge 0\left(i=1,2\right)$ 。

Abstract: The existence and uniqueness of positive solution for the boundary value problem of fourth order variable coefficients ordinary differential system$\{\begin{array}{l}{u}^{\left(4\right)}\left(x\right)+a\left(x\right)u\left(x\right)=f_1\left(x,v\left(x\right)\right),x\in \left(0,1\right),\\ v^{\left(4\right)}\left(x\right)+b\left(x\right)v\left(x\right)=f_2\left(x,u\left(x\right)\right),x\in \left(0,1\right),\\ u\left(0\right)=u\left(1\right)=u^{\prime }\left(0\right)=u^{\prime }\left(1\right)=0,\\ v\left(0\right)=v\left(1\right)=v^{\prime }\left(0\right)=v^{\prime }\left(1\right)=0\end{array}$ with clamped beam conditions were obtained using Leray-Schauder degree theory and fixed point theorem, where $a,b:\left[0,1\right]\to \left[0,+\infty \right)$ are continuous, nonlinear term $f_i:\left[0,1\right]\times R\to R$ are continuous and $f_i\left(x,0\right)\ge 0\left(i=1,2\right)$ .