Abstract and Applied Analysis
Volume 2005 (2005), Issue 2, Pages 121-158

A new topological degree theory for densely defined quasibounded (S˜+)-perturbations of multivalued maximal monotone operators in reflexive Banach spaces

Athanassios G. Kartsatos1 and Igor V. Skrypnik2

1Department of Mathematics, University of South Florida, Tampa 33620-5700, FL, USA
2Institute for Applied Mathematics and Mechanics, National Academy of Science of Ukraine, R. Luxemburg Street 74, Donetsk 83114, Ukraine

Received 24 March 2004

Copyright © 2005 Athanassios G. Kartsatos and Igor V. Skrypnik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let X be an infinite-dimensional real reflexive Banach space with dual space X and GX open and bounded. Assume that X and X are locally uniformly convex. Let T:XD(T)2X be maximal monotone and C:XD(C)X quasibounded and of type (S˜+). Assume that LD(C), where L is a dense subspace of X, and 0T(0). A new topological degree theory is introduced for the sum T+C. Browder's degree theory has thus been extended to densely defined perturbations of maximal monotone operators while results of Browder and Hess have been extended to various classes of single-valued densely defined generalized pseudomonotone perturbations C. Although the main results are of theoretical nature, possible applications of the new degree theory are given for several other theoretical problems in nonlinear functional analysis.