Constrained topological degree and positive solutions of fully nonlinear boundary value problems

In the first part of the paper we provide a construction of an abstract homotopy invariant detecting zeros of maps of the form -A + F where A: D(A) -o E is a densely defined m-accretive operator in a Banach space E and F : U -> E is a tangent field defined on an open subset U of a neighborhood retract M being invariant with respect to the resolvents of A. The construction is performed under the assumption that resolvents of A are completely continuous. In the second part we derive index formulae for isolated zeros and apply them to show the existence of nontrivial positive steady state solutions for a class of nonlinear reaction-diffusion equations and equations with one-dimensional p-Laplacian with possibly non-positive perturbations as well as some controlled Neumann-like problems. (C) 2009 Elsevier Inc. All rights reserved.