Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700 ; Institute for Applied Mathematics and Mechanics, R. Luxemburg Str. 74, Donetsk 340114, Ukraine
Abstract:
Let be a real reflexive Banach space with dual and open and bounded and such that Let be maximal monotone with and and with and A general and more unified eigenvalue theory is developed for the pair of operators Further conditions are given for the existence of a pair such that
The ``implicit" eigenvalue problem, with in place of is also considered. The existence of continuous branches of eigenvectors of infinite length is investigated, and a Fredholm alternative in the spirit of Necas is given for a pair of homogeneous operators No compactness assumptions have been made in most of the results. The degree theories of Browder and Skrypnik are used, as well as the degree theories of the authors involving densely defined perturbations of maximal monotone operators. Applications to nonlinear partial differential equations are included.