The Leray-Schauder approach to the degree theory for (S+) -perturbations of maximal monotone operators in separable reflexive Banach spaces

The purpose of this paper is to demonstrate the fact that the topological degree theory of Leray and Schauder may be used for the development of the topological degree theory for bounded demicontinuous (S+)-perturbations f of strongly quasibounded maximal monotone operators T in separable reflexive Banach spaces. Certain basic homotopy properties and the extension of this degree theory to (possibly unbounded) strongly quasibounded perturbations f are shown to hold. This work uses the well known embedding of Browder and Ton, and extends the work of Berkovits who developed this theory for the case T = 0. Besides being an interesting mathematical problem, the existence of such a degree theory may, conceivably, become useful in Situations where the use of the Leray-Schauder degree (via infinite dimensional compactness) is necessary. (C) 2008 Elsevier Ltd. All rights reserved.