Existence of Positive Solution for a Third-Order BVP with Advanced Arguments and Stieltjes Integral Boundary Conditions

Jian-Ping Sun,Fang-Di Kong,Ya-Hong Zhao,Ping Yan

doi:10.1155/2013/157849

Jian-Ping Sun, Fang-Di Kong + Show 2 more

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https://doi.org/10.1155/2013/157849

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Abstract

A class of third-order boundary value problems with advanced arguments and Stieltjes integral boundary conditions is discussed. Some existence criteria of at least one positive solution are established. The main tool used is the Guo-Krasnoselskii fixed point theorem.

Highlights

Third-order differential equations arise in a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross-section, a three-layer beam, electromagnetic waves or gravity-driven flows, and so on [1]
In 2012, by using a fixed point theorem due to Avery and Peterson [7], Jankowski [4] established the existence of at least three nonnegative solutions to the following BVP: u󸀠󸀠󸀠 (t) + h (t) f (t, u (α (t))) = 0, t ∈ (0, 1), u (0) = u󸀠󸀠 (0) = 0, (1) u (1) = βu (η) + λ [u], where λ denoted a linear functional on C[0, 1] given by λ [u] = ∫ u (t) dΛ (t) involving a Stieltjes integral with a suitable function Λ of bounded variation
It follows from (56), (61), and Theorem 1 that the operator S has one fixed point u ∈ K ∩ (Ω2 \ Ω1), which is a positive solution of the BVP (3)

Summary

Introduction

Third-order differential equations arise in a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross-section, a three-layer beam, electromagnetic waves or gravity-driven flows, and so on [1].Recently, third-order boundary value problems (BVPs for short) with integral boundary conditions, which cover thirdorder multipoint BVPs as special cases, have attracted much attention from many authors; see [2,3,4,5,6] and the references therein. In 2012, by using a fixed point theorem due to Avery and Peterson [7], Jankowski [4] established the existence of at least three nonnegative solutions to the following BVP: u󸀠󸀠󸀠 (t) + h (t) f (t, u (α (t))) = 0, t ∈ (0, 1) , u (0) = u󸀠󸀠 (0) = 0, (1) u (1) = βu (η) + λ [u] , where λ denoted a linear functional on C[0, 1] given by λ [u] = ∫ u (t) dΛ (t) involving a Stieltjes integral with a suitable function Λ of bounded variation.

Results

Conclusion