Implicit eigenvalue problems for maximal monotone operators

We study the implicit eigenvalue problem of the form 0 ∈ T x + C ( λ , x ) , where T is a maximal monotone multi-valued operator and the operator C satisfies condition ( S + ) or ( S ˜ + ). In a regularization method by the duality operator, we use the degree theories of Kartsatos and Skrypnik upon conditions of C as well as Browder’s degree. There are two cases to consider: One is that C is demicontinuous and bounded with condition ( S + ); and the other is that C is quasibounded and densely defined with condition ( S ˜ + ). Moreover, the eigenvalue problem 0 ∈ T x + λ C x is also discussed.