Positive solutions for singular third-order nonhomogeneous boundary value problems

Abstract

In this paper, we investigate the existence of positive solutions for singular third-order nonhomogeneous boundary value problems. By using a fixed point theorem of cone expansion-compression type due to Krasnosel'skii, we establish various results on the existence or nonexistence of single and multiple positive solutions to the singular boundary problems in the explicit intervals for the nonhomogeneous term. An example is also given to illustrate some of the main results.

Positive solution for singular third-order BVPs on the half line with first-order derivative dependence

“…Naturally, in such boundary value problems, the nonlinearity may have a singular dependence on time or on the space variable. This was the case in the papers [3,6,7,8,20,21,27,28,29], which motivated this work.…”

“…Study of existence of positive solutions for third-order bvps has received a great deal of attention and was the subject of many articles, see, for instance, [10,11,12,13,14,21,25,27,28,29,30,31], for the case of finite intervals and [1,2,3,4,6,7,8,9,16,19,20,24,26] for the case posed on the halfline. Naturally, in such boundary value problems, the nonlinearity may have a singular dependence on time or on the space variable.…”

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

“…where a < t 1 < t 2 < t 3 < b and y 1 , y 2 , y 3 ∈ R. Third-order ordinary differential equations arise as models for certain natural phenomena such as boundary-layer flow in fluid mechanics involving convection in a porous medium, or a flow along a standing wall or stretched sheet (see [1,4,6,16,[47][48][49]51,52]). Other theoretical works have dealt with third-order equations devoted to, for example, upper and lower solutions, periodic solutions, limit-point and limit-circle criteria and singular boundary-value problems (see [2,3,5,32,36,41,45,50]).…”

"Existence of positive solutions for 3n^th order boundary value problems involving p-Laplacian"

“…The important and significant operator is one-dimensional p-Laplacian operator and is defined by ϕ p (τ ) = |τ | p−2 τ , where p > 1, ϕ −1 p = ϕ q and 1 p + 1 q = 1. By taking n = 1 and p = 2 in (1.1) and (1.2), reduces to third order three-point boundary value problem and studied the existence of positive solutions based on various methods by many researchers, see [7,13,14,15,16,18,19,27,29,31]. However, as per our knowledge, very few works have been found in the literature on the existence of positive solutions of higher order boundary value problems with p-Laplacian, see [5,21,25,26,30].…”