A class of positive linear operators and applications to nonlinear boundary value problems

Abstract

We discuss the class of $u_0$-positive linear operators relative to two cones and use a comparison theorem for this class to give some short proofs of new fixed point index results for some nonlinear operators that arise from boundary value problems. In particular, for some types of boundary conditions, especially nonlocal ones, we obtain a new existence result for multiple positive solutions under conditions which depend solely on the positive eigenvalue of a linear operator. We also treat some problems where the nonlinearity $f(t,u)$ is singular at $u=0$.