Positive solutions of fourth-order superlinear singular boundary value problems

Abstract

This paper investigates fourth-order superlinear singular two-point boundary value problems and obtains necessary and sufficient conditions for existence of C2 or C3 positive solutions on the closed interval.

Positive solutions for higher order singular p-Laplacian boundary value problems

“…The singular or nonsingular fourth-order boundary value problems (1.4) have been extensively studied by many authors [1,2,6,7,10,[13][14][15]. Shi and Chen [10,11] gave the sufficient and necessary conditions for the existence of positive solutions to superlinear problem (1.4) by the fixed point theorem in cones when 1 < λ ≤ μ < +∞, in the case of ϕ p (s) = s. Using a modified upper and lower solution method, Wei [12] obtained necessary and sufficient conditions for the existence of positive solutions to fourth-order problem (1.4) for the sublinear case.…”

“…Shi and Chen [10,11] gave the sufficient and necessary conditions for the existence of positive solutions to superlinear problem (1.4) by the fixed point theorem in cones when 1 < λ ≤ μ < +∞, in the case of ϕ p (s) = s. Using a modified upper and lower solution method, Wei [12] obtained necessary and sufficient conditions for the existence of positive solutions to fourth-order problem (1.4) for the sublinear case. The upper and lower solution method is built in [3,5,6] to treat the singular boundary value problems for p-Laplacian and the higher order nonlinear boundary value problems.…”

“…Our results extend the results of Wei [12] to the case of higher order p-Laplacian BVPs. The necessary and sufficient conditions of existence of positive solutions to (1.4) in [10][11][12] involve integrability conditions in terms of the function f and the Green function for the case of p = 2. We found that this fact is also true for the general p-Laplace BVPs, and that it is interesting that the integrability is not relying on the parameter p. In addition, our methods used in this paper can be used to investigate the higher-order p-Laplacian BVPs whose nonlinear term relies on the higher order derivates of u.…”

Necessary and sufficient conditions for the existence of symmetric positive solutions of singular boundary value problems

“…For a small sample of such work, we refer the reader to the papers of Avery and Henderson [1], Davis et al [2], Davis et al [3], Graef and Kong [4], Graef et al [5], Graef et al [6], and Henderson and Thompson [8]. Efforts to obtain necessary and sufficient conditions for the existence of positive solutions of BVPs can also be found in the literature, for example, in [4,[10][11][12][13][14][15][16]. In particular, the BVP consisting of the equation u, u , .…”

Positive solutions of singular sublinear fourth-order boundary value problems

“…However, for singular fourth-order BVPs, the research has proceeded very slowly. Previous works on singular fourth-order BVPs include Wei [18] and Shi and Chen [16], where results were established for the boundary conditions A natural motivation for studying higher order BVPs exists in their applications. For example, it is well-known that the deformation of an elastic beam in equilibrium state, whose both ends clamped, can be described by a fourth-order BVP y 0000 ðtÞ ¼ gðt, yðtÞÞ, t 2 ð0, 1Þ ð 1:1Þ…”