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

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{ u(4)(x)+a(x)u(x)=f1(x,v(x)), x∈(0,1),v(4)(x)+b(x)v(x)=f2(x,u(x)), x∈(0,1),u(0)=u(1)=u′(0)=u′(1)=0,v(0)=v(1)=v′(0)=v′(1)=0正解的存在性和唯一性,其中a,b:[ 0,1 ]→[ 0,+∞ )连续,非线性项fi:[ 0,1 ]×R→R为连续函数且fi(x,0)≥0 (i=1,2)。The existence and uniqueness of positive solution for the boundary value problem of fourth order variable coefficients ordinary differential system{ u(4)(x)+a(x)u(x)=f1(x,v(x)), x∈(0,1),v(4)(x)+b(x)v(x)=f2(x,u(x)), x∈(0,1),u(0)=u(1)=u′(0)=u′(1)=0,v(0)=v(1)=v′(0)=v′(1)=0with clamped beam conditions were obtained using Leray-Schauder degree theory and fixed point theorem, where a,b:[ 0,1 ]→[ 0,+∞ )are continuous, nonlinear term fi:[ 0,1 ]×R→Rare continuous and fi(x,0)≥0 (i=1,2).