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

Abstract

We will present a generalization of Mahadevan’s version of theKrein-Rutman theorem for a compact, positively 1-homogeneous operator on aBanach space having the properties of being increasing with respect to a cone $P$ and such that there is a nonzero $u\in P\setminus \{\theta \}-P$ for which $M{T}^{p}u\ge u$ for somepositive constant $M$ and some positive integer p. Moreover, we give some newresults on the uniqueness of positive eigenvalue with positive eigenfunction andcomputation of the fixed point index. As applications, the existence of positivesolutions for p-Laplacian boundary-value problems is considered under someconditions concerning the positive eigenvalues corresponding to the relevantpositively 1-homogeneous operators.