Existence of positive solutions to a class of elastic beam equations

Abstract

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.

Keywords

• nonlinear functional analysis theory
• elastic beam
• positive solution
• Guo-Krasnoselskii fixed-point theorem
• material mechanics