Existence and Multiplicity of Positive Solutions to a Class of Nonlinear Cantilever Beam Equations
一类非线性悬臂梁方程正解的存在性与多解性

The positive solutions are studied for the nonlinear fourth-order ordinary differential equation $u^{(4)}(t)=f(t,u(t),u'(t)),~t\in 0,1]\backslash E$, subject tothe boundary conditions $u(0)=u'(0)=u'(1)=u''(1)=0$, where $E\subset 0,1]$ is a closed set with measure zero and nonlinear term $f(t,u,v)$ may be singular for $t\in E$. With some conditions, by constructing suitable integral equation and applying fixed point theorems on cone, the existence of $n$ positive solutions is proved for the equations, where $n$ is a positive integral number.