A Generalization of Mahadevan′s Version of the Krein‐Rutman Theorem and Applications to p ‐Laplacian Boundary Value Problems
Yujun Cui, Jingxian Sun · Abstract and Applied Analysis · 2012
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.