Some New Existence Results of Positive Solutions to an Even-Order Boundary Value Problem on Time Scales

Yanbin Sang

doi:10.1155/2013/314382

Abstract

We consider a high-order three-point boundary value problem. Firstly, some new existence results of at least one positive solution for a noneigenvalue problem and an eigenvalue problem are established. Our approach is based on the application of three different fixed point theorems, which have extended and improved the famous Guo-Krasnosel’skii fixed point theorem at different aspects. Secondly, some examples are included to illustrate our results.

Highlights

IntroductionWe are concerned with the following even-order three-point boundary value problem on time scales T:
We note that the conspicuous advantage of Theorem 5 is that the conditions over the set S of the lower solution are deleted
We should point out that the spectrum structure of the corresponding linear problem of (2) is still unknown so far; we can not directly apply the results shown in [32] due to Webb and Infante and in [33] due to Webb and Lan to the problem (2)

Summary

Introduction

We are concerned with the following even-order three-point boundary value problem on time scales T:. They have studied the existence of at least one positive solution to the BVP (3) using the functional-type cone expansion-compression fixed point theorem. In [20], Han and Liu studied the existence and uniqueness of nontrivial solution for the following third-order p-Laplacian m-point eigenvalue problems on time scales:. One of the most frequently used tools for proving the existence of positive solutions to the integral equations is Krasnosel’skii’s theorem on cone expansion and compression and its norm-type version due to Guo and Lakshmikantham (see [25]), and they have been applied extensively to all kinds of problems, such as ordinary differential equations, difference equations, and the general dynamic equations on time scales. The new existence result is obtained by Theorem 3 (see Theorem 11)

Preliminaries

Existence Theorem of Positive Solutions

Some Examples