带不跨特征值扰动项的梁方程边值问题解的存在性

Existence Theorem for a Beam Equation Which has Nonlinear Term Between some Two Adjacent Eigenvalues

梁是工程建筑的基本构件之一。对于梁方程的研究,有着非常重大的理论意义和实际意义。笔者将讨论的四阶半线性梁方程的边值问题用于描述弹性梁的形变情况。用压缩映像原理和Schauder不动点定理对带有不跨特征值扰动项的梁方程边值问题进行了研究,在非线性项满足渐进一致性条件下,得到了相应的解的存在唯一性结论。在把非线性项的条件削弱后,又得到了一个存在性结论。

Beam is one of the basic terms of engineering and building. So the research on beam equation has important meaning both in theory and in practice. In this paper, the author discusses the fourth-order semi-linear beam equation boundary value problem, in which the nonlinear term is located between some two adjacent eigenvalues. The BVP describes the bending of an elastic beam. With the Banach contraction mapping principle and Schauder fixed-point theorem, on the consistent condition of nonlinear term, we get a unique theorem. On the more general condition for nonlinear term, another existence theorem is obtained.