A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities
Asfaw, Teffera M.
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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.