Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces

This paper gives a survey over some of the most important methods and results of nonlinear functional analysis in ordered Banach spaces. By means of iterative techniques and by using topological tools, fixed point theorems for completely continuous maps in ordered Banach spaces are deduced, and particular attention is paid to the derivation of multiplicity results. Moreover, solvability and bifurcation problems for fixed point equations depending nonlinearly on a real parameter are investigated. In order to demonstrate the importance of the abstract results, there are given some nontrivial applications to nonlinear elliptic boundary value problems. But, of course, the abstract techniques and results of this paper apply also to a variety of other problems which are not considered here. This paper presents in a unified manner most of the recent work in this field. In addition, by making consequent use of the fixed point index for compact maps, short and simple proofs are obtained for most of the “classical” results contained in M. A. Krasnosel’skii’s book 11.