一类弹性梁方程正解的存在性

弹性梁是弹力力学和工程物理中一种比较常见的数学模型,为了将此模型更准确地应用于工程领域中,在对一端固定,一端滑动支撑的弹性梁方程研究的基础上,研究了此类弹性梁方程的多解性。通过将此类边值问题转化为积分方程后,进而等价于算子的不动点问题,结合其Green函数的性质与Guo-Krasnoselskii锥拉伸与压缩不动点定理,讨论了此类弹性梁方程正解的存在性问题。在非线性项满足适当条件下建立参数的取值范围,获得了此类边值问题至少有1个正解,2个正解的存在性结果与正解的不存在性结果。结论上获得了关于此类问题至少有1个正解,2个正解及没有正解的存在的特征值区间。研究结果有助于弹性梁的稳定性分析,丰富了材料力学的相关理论。

Existence of positive solutions to a class of elastic beam equations

Elastic beam is a kind of mathematical model in elastic mechanics and engineering physics. For now, this type of model is often used in real life. On the basis of the relative research on the elastic beam equations with one end fixed and one end sliding support, and the multiple solutions of the elastic beam equation are researched. In this paper, through putting this problem into an integral equation, which is equivalent to an operator fixed-point problem, and combining with the properties of Green function and Guo- Krasnoselskii fixed point theorem of cone expansion and compression, the existence of positive solutions of this kind of elastic beam equations is discussed. Under various assumptions on nonlinear terms, the intervals of the parameters are established, and the existence of one positive solution, two positive solutions or nonexistence of positive solutions for this elastic beam equations are obtained. In conclusion, the intervals of eigenvalue about this problem for at least one positive solution, two positive solutions and nonexistence of positive solutions are obtained. The study of the existence of such solution can not only contribute to the stability analysis of elastic beams, but also enrich the theory of material mechanics.