Iterative Construction of Fixed Points of Contractive Type Mappings in Banach Spaces

Abstract

Introductory remarks. It is known that the problem of finding solutions of nonlinear functional equations (e.g., nonlinear integral equations, boundary value problems for nonlinear ordinary and partial differential equations, etc.) can be formulated in terms of finding fixed points of a given nonlinear mapping defined on some subset of an infinite dimensional functional space. For compact mappings 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, Poincar#x00E9;, Lefschetz, Schauder, Leray, Tichonoff, Rothe, Krasnoselsky, Altman, and others). More recently, there has begum a systematic study of fixed points (their existence and actual construction) of various classes of noncompact mappings (Brodski-Milman [2], De Marr [28], Browder [3a, 3b, 5, 6, 4], Kirk 23, Kachurovsky [2l], Edelstein [11,12,13], Belluce-Kirk [1], Gohde [l9, 20], De Prima [l0], Lions and Stampacchia [27], Browder-Petryshyn [8, 9], Kaniel [22], Shinbrot [40], Petryshyn [31, 32, 33, 35, 36], Opial [30], Lees-Schultz [26], de Figueiredo [15, 16], Browder-de Figueiredo [7], Petryshyn-Tucker [37], and others).