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

Abstract EN

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 MTpu≥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.