Construction of fixed points of nonlinear mappings in Hilbert space

Abstract

Many of the most important nonlinear problems of applied mathematics reduce to finding solutions of nonlinear functional equations (e.g., nonlinear integral equations, boundary value problems for nonlinear ordinary or partial differential equations, the existence of periodic solutions of nonlinear partial differential equations) which can be formulated in terms of finding the fixed points of a given nonlinear mapping of an infinite dimensional function space X into itself. For mappings satisfying compactness conditions, a general existence theory of fixed points based upon topological arguments has been constructed over a number of decades (associated with the names of Brouwer, Poincare, Lefschetz, Schauder, Leray, and others). More recently, there has begun the systematic study of fixed points of various classes of noncompact mappings, some of which are described in the Discussion below. It is our object in the present paper to survey, systematize, and extend a number of recent results concerning the existence of fixed points of noncompact mappings of a subset C of a Hilbert space H into H. From the point of view of application, it is essential not only to show the existence of fixed points of such mappings under suitable hypotheses, but also to develop systematic techniques for the construction or calculation of such fixed points. The results presented below for various classes of nonlinear mappings (contractive, strictly pseudocontractive, pseudocontractive) may be considered as somewhat sophisticated, sharpened forms of the classical iteration scheme (the method of successive approximations) of Picard-BanachCacciopoli et al. that work in contexts in which the classical iteration scheme no longer applies (and in particular, outside the class of strictly contractive mappings). We have restricted the discussion to the case of mappings defined in Hilbert space both to avoid technical complications in the presentation of the results and proofs, and also, more essentially, because many of the 197