Introduction

This book addresses several pivotal problems in spectral theory and nonlinear functional analysis in connection with the problem of analyzing the structure of the set of zeroes of a general class of nonlinear operators defined between two real Banach spaces for which a smooth curve of solutions is assumed to be known. By a change of variable, one can assume that the curve is a straight line. We are going to work with real spaces and real curves since these are the most common situations arising in the applications of the abstract theory to real world models. Our study should be of interest for a broad audience as it contains many important contributions to linear algebra and linear functional analysis, nonlinear functional analysis, and topology. Moreover, this book provides several applications of its abstract theory to the search for positive solutions in the context of semilinear reaction diffusion equations and systems, which should be of interest in a variety of areas of applied sciences and engineering. To be precise, an acute analysis of the following three issues will be performed.

Characterizing the points of the curve from which a continuum of solutions emanates for any perturbing nonlinearity

Analyzing the topological structure of the component of the solution set emanating from any of those branching points

Combining the local and the global results to study the existence of coexistence states in a very general class of reaction diffusion systems modeling the interactions between two species.