Some eigenvalue results for perturbations of maximal monotone operators

We study the solvability of a nonlinear eigenvalue problem for maximal monotone operators under a normalization observation. The investigation is based on degree theories for appropriate classes of operators, and a regularization method by the duality operator is used. Let X be a real reflexive Banach space with its dual X ∗ Open image in new window and Ω be a bounded open set in X. Suppose that T : D ( T ) ⊂ X → X ∗ Open image in new window is a maximal monotone operator and C : ( 0 , ∞ ) × Ω ¯ → X ∗ Open image in new window is a bounded demicontinuous operator satisfying condition ( S + Open image in new window). Applying the Browder degree theory, we solve a nonlinear eigenvalue problem of the form T x + C ( λ , x ) = 0 Open image in new window. In the case where T x + λ C x = 0 Open image in new window, an eigenvalue result for generalized pseudomonotone densely defined perturbations is obtained by the Kartsatos-Skrypnik degree theory.