一端固定一端活动的非线性弹性梁方程的存在定理

Existence Theorems of Nonlinear Elastic Beam Equations with Fixed End and Movable End

考察了一类非线性四阶弹性梁方程的解和正解的存在性。在力学上,这一类方程描述了1个端点固定,另1个端点被滑动夹子夹住的弹性梁的形变。利用方程的分解技巧并且构造适当的Ba-nach空间,这类微分方程被转化为Banach空间上的不动点方程。通过使用Leray-Schauder不动点定理对于这类方程建立了4个存在定理。主要结论表明:只要非线性项在某个有界集上的“高度”是适当的,这类方程至少有1个解或者正解。

The existence of solution and positive solution is considered for a class of nonlinear fourth-order elastic beam equations. In mechanics, the class of equations describes deformation of the elastic beam in which an end is fixed and the other end is clamped by sliding clamps. By applying the decomposition technique of equations and constructing suitable Banach space, the class of differential equations is transformed to the fixed point equations in Banach spaces. By using Leray-Schauder fixed point theorem, four existence theorems are established for the class of equations. The main results show that the class of equations has at least one solution or positive solution if the “height” of nonlinear term is appropriate on a bounded set.