A Generalization of Mahadevan's Version of the Krein-Rutman Theorem and Applications to p-Laplacian Boundary Value Problems. (28th August 2012)

• Record Type: Journal Article
• Authors: Cui, Yujun; Sun, Jingxian
• Other Names: Liu Lishan Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2012(2012)
• Publication Date: 2012-08-28
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2012/305279
• Languages: English
• ISSNs: 1085-3375

Abstract

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 u ∈ P ∖ { θ } − P for which M T p u ≥ u 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.