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

Yujun Cui and Jingxian Sun

Abstract and Applied Analysis, 2012, vol. 2012, 1-14

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 𠑃 and such that there is a nonzero ð ‘¢ ∈ 𠑃 ⧵ { 𠜃 } − 𠑃 for which ð ‘€ 𠑇 ð ‘ ð ‘¢ ≥ ð ‘¢ for some positive constant ð ‘€ 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.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:305279

DOI: 10.1155/2012/305279

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