Local existence of multiple positive solutions to a singular cantilever beam equation

Abstract

By constructing suitable cone and control functions, we prove some local existence theorems of positive solutions for a singular fourth-order two-point boundary value problem. In mechanics, the problem is called cantilever beam equation. Furthermore, we improve a famous method appeared in the studies of singular boundary value problems. The approximation theorem of completely continuous operators and the Guo-Krasnosel'skii fixed point theorem of cone expansion-compression type play important parts in this work.

On the Existence of Positive Solutions for a Fourth-Order Boundary Value Problem

“…In mechanics, BVP (1) is called cantilever beam equation. Owing to its importance in mechanics, the existence of solutions to this problem has been studied by many authors; see [1][2][3][4][5][6][7] and references therein.…”

Positive solutions for singular higher-order semipositone fractional differential equations with conjugate type integral conditions

“…The idea of this paper may trace to Yao [36,37] and our work [38]. As far as we know, there are few papers to consider fractional semipositone integral BVPs, especially for BVPs with singularities on space variable.…”

Positive Solutions of Some Fourth-order Two Point Boundary Value Problem with All Order Derivatives

“…Owing to its significance in physics, the existence of positive solutions for the fourth-order boundary value problem has been studied by many authors using nonlinear alternatives of Leray-Schauder, the fixed point index theory, the Krasnosel'skii's fixed point theorem and the method of upper and lower solutions, in reference [1][2][3][4][5][6][7][8][9] [11]. In recent years, there has been much attention on the fourth-order differential equations with one or two parameters.…”