About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 305279, 14 pages
http://dx.doi.org/10.1155/2012/305279
Research Article

A Generalization of Mahadevan's Version of the Krein-Rutman Theorem and Applications to p-Laplacian Boundary Value Problems

1Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao 266590, China
2Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, China

Received 12 January 2012; Revised 24 March 2012; Accepted 13 July 2012

Academic Editor: Lishan Liu

Copyright © 2012 Yujun Cui and Jingxian Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We will present a generalization of Mahadevan’s version of the Krein-Rutman theorem for a compact, positively 1-homogeneous operator on a Banach space having the properties of being increasing with respect to a cone 𝑃 and such that there is a nonzero π‘’βˆˆπ‘ƒβ§΅{πœƒ}βˆ’π‘ƒ for which 𝑀𝑇𝑝𝑒β‰₯𝑒 for some positive constant 𝑀 and some positive integer p. Moreover, we give some new results on the uniqueness of positive eigenvalue with positive eigenfunction and computation of the fixed point index. As applications, the existence of positive solutions for p-Laplacian boundary-value problems is considered under some conditions concerning the positive eigenvalues corresponding to the relevant positively 1-homogeneous operators.