A Leray-Schauder theorem for a class of nonlinear operators

Abstract

One of the most useful tools for dealing with the theory of nonlinear operators in concrete analytical problems is the Leray-Schauder fixed point theorem [5]. A disadvantage is that it is only available for the completely continuous (that is, compact and continuous) operators, and moreover the proof of the theorem uses the theory of the topological degree of a mapping. However, in a recent publication, Browder [1] has given an elementary proof without recourse to degree theory, tt was observed by the authors that this method of proof is applicable to a wider class of operators, namely those we shall call strongly P-compact. In this note we shall show that for a restricted class of Banach spaces (including, however, all separable Hilbert spaces) the class of strongly P-compact operators is quite extensive, and shall prove a fixed point theorem like that of Leray-Schauder for this class. Throughout this paper X will denote a real Banach space which has the following property: (re)l: There exists a sequence {F.} of finite dimensional subspaces of X, with F.CF,+I for each n, and U F, dense in X, and a corresponding sequence