Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

Let X be an infinite-dimensional real reflexive Banach space such that X and its dual X * are locally uniformly convex. Suppose that T : X ⊃ D ( T ) → 2 X * is a maximal monotone multi-valued operator and C : X ⊃ D ( C ) → X * is a generalized pseudomonotone quasibounded operator with L ⊂ D ( C ), where L is a dense subspace of X . Applying a recent degree theory of Kartsatos and Skrypnik, we establish the existence of an eigensolution to the nonlinear inclusion 0 ∈ T x + λ C x , with a regularization method by means of the duality operator. Moreover, possible branches of eigensolutions to the above inclusion are discussed. Furthermore, we give a surjectivity result about the operator λT + C when λ is not an eigenvalue for the pair ( T, C ), T being single-valued and densely defined.