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 $u=0$.