Nonlinear Functional Analysis and Nonlinear Integral Equations of Hammerstein and Urysohn Type

Abstract

This chapter discusses nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type. The theory of nonlinear integral equations of Hammerstein type has been, since its inception in the paper of Hammerstein, one of the most important domains of application of the ideas and techniques of nonlinear functional analysis, second only to the theory of solutions of boundary value problems for nonlinear partial differential equations. The development of the fixed point and degree theory for compact nonlinear mappings in Banach spaces was strongly influenced, in its form, by the theory of nonlinear integral equations and was directly applied to this domain and many others. The chapter presents a unified development of the theory of the Hammerstein equation using the theory of the topological degree for mappings of the form I - C with C compact as well as the basic theory of monotone nonlinear mappings from X to X*.