On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces

Let X be a real reflexive Banach space with dual X* and G subset of X open and bounded and such that 0 is an element of G. Let T : X superset of D(T). 2(X*) be maximal monotone with 0 is an element of D(T) and 0 is an element of T(0), and C : X superset of D(C) -> X* with 0 is an element of D(C) and C(0) not equal 0. A general and more unified eigenvalue theory is developed for the pair of operators (T, C). Further conditions are given for the existence of a pair (lambda, x). (0,8) x (D( T + C) n boolean AND partial derivative G) such that (**) Tx+lambda Cx there exists 0. The " implicit" eigenvalue problem, with C(lambda, x) in place of lambda Cx, is also considered. The existence of continuous branches of eigenvectors of in. nite length is investigated, and a Fredholm alternative in the spirit of Necas is given for a pair of homogeneous operators T, C. 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 de. ned perturbations of maximal monotone operators. Applications to nonlinear partial di. erential equations are included.