含各阶导数的非线性弹性梁方程的一个存在定理

摘要

By choosing suitable Banach space and applying Leray-Schauder Nonlinear Alternative, an existence theorem of solutions is established for the nonlinear elastic beam equation with all order derivatives{u(4)(t)=f(t,u(t),u'(t),un(t),um(t),0≤t≤1,u(0)=u'(1)=u"(0)=u(''')(1)=0.In the material mechanics, the equation describes the deformation of an elastic beam whose one end is simply supported and the other is clamped by sliding clamps. The existence theorem shows that the equation has at least one solution provided the nonlinear term satisfies a linear growth condition.%通过选择适当的Banach空间并利用Leray-Schauder非线性抉择对于含各阶导数的非线性弹性梁方程u(4)(t)=f(t,u(t),u′(t),u″(t),u(t)), 0t1,u(0)=u′(1)=u″(0)=u(''')(1)=0.建立了一个解的存在定理. 在材料力学中,该方程描述了一端简单支撑,另一端被滑动夹子夹住的弹性梁的形变. 这个存在定理说明只要非线性项满足某种线性增长条件该方程至少有一个解.