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Journal of Function Spaces
Volume 2016, Article ID 3970621, 15 pages
Research Article

A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

Received 30 June 2016; Accepted 18 October 2016

Academic Editor: Adrian Petrusel

Copyright © 2016 Teffera M. Asfaw. 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 be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space . Let be maximal monotone of type (i.e., there exist and a nondecreasing function with such that for all , , and be linear, surjective, and closed such that is compact, and be a bounded demicontinuous operator. A new degree theory is developed for operators of the type . The surjectivity of can be omitted provided that is closed, is densely defined and self-adjoint, and , a real Hilbert space. The theory improves the degree theory of Berkovits and Mustonen for , where is bounded demicontinuous pseudomonotone. New existence theorems are provided. In the case when is monotone, a maximality result is included for and . The theory is applied to prove existence of weak solutions in of the nonlinear equation given by ,  ;  ,  ; and ,  , where , , , , , is a nonempty, bounded, and open subset of with smooth boundary, and satisfy suitable growth conditions. In addition, a new existence result is given concerning existence of weak solutions for nonlinear wave equation with nonmonotone nonlinearity.