Existence and Nonexistence of Positive Solutions for a Higher-Order Three-Point Boundary Value Problem

Abstract

This paper is concerned with the existence and nonexistence of positive solutions for a nonlinear higher-order three-point boundary value problem. The existence results are obtained by applying a fixed point theorem of cone expansion and compression of functional type due to Avery, Henderson, and O’Regan.

Solvability ofNth Order Linear Boundary Value Problems

“…Most of these papers are focused on obtaining comparison theorems for different problems and different boundary conditions, whereas some of them deal with the existence of solutions for specific boundary value problems. This is the case, for instance, of [21,31,35,37,38] or [39]. This paper shares that objective combining some ideas of Keener and Travis with the approach of [1] in order to yield an iterative sequence of necessary and sufficient conditions in integral form, each one more precise than the previous one, and setting a framework that allows covering a range of problems much wider than the ones originally assessed in [1,21].…”

“…One of the most successful techniques introduced in this analysis has been the combination of the modern cone theory with the positivity and monotonicity characteristics displayed by many of the Green functions resulting from (7). In the books of Krasnosel'skii [16] and Deimling [17], one can find the grounds for this approach, which originated in Krasnosel'skii and Krein and Rutman's works (see [16,18]) and was later pursued by Gentry and Travis [19], Schmitt and Smith [20], Keener and Travis [21], Tomastik [22,23], Kreith [24], Hankerson and Peterson [25], Hankerson and Henderson [26], Eloe [27][28][29][30][31][32], and Diaz [33] and more recently by Graef [34,35], Zhang et al [36], Zhang [37], Sun et al [38], or Hao et al [39], among many others. Most of these papers are focused on obtaining comparison theorems for different problems and different boundary conditions, whereas some of them deal with the existence of solutions for specific boundary value problems.…”

Improving Results on Solvability of a Class ofnth-Order Linear Boundary Value Problems

“…From the historical point of view, let us remark that the use of the theory of cones in boundary value problems dates from the works of Krein and Rutman [4] and Krasnosel'skii [5], which were continued by multiple authors. References [3,[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] are a good account of this.…”

Existence of Positive Solutions for a Nonlinear Higher-Order Multipoint Boundary Value Problem

“…The results extended and improved some recent work in the literature. In a recent paper [7], we study problems (1) and (2) with = 1 by a fixed point theorem of cone expansion and compression of functional type according to Avery et al [8]. For other existence results of positive solutions for higherorder multipoint problems, for a small sample of such work, we refer the reader to Ahmad and Ntouyas [9], Anderson et al [10], Davis et al [11], Du et al [12,13], Eloe and Henderson [14], Fu and Du [15], Graef et al [16,17], Henderson and Luca [18], Ji and Guo [19], Jiang [20], Liu et al [21], Liu and Ge [22], Liu et al [23], Palamides [24], Su and Wang [25], Zhang et al [26], and Zhang [27] and the references therein.…”