Ranges of densely defined generalized pseudomonotone perturbations of maximal monotone operators

Let X be a real reflexive Banach space and A : X→2X* be maximal monotone. Let B : X→2X* be quasibounded, finitely continuous and generalized pseudomonotone with X′ D(B), where X′ is a dense subspace of X such that X′∩D(A)≠ ︀. Let S X*. Conditions are given under which and int S int R(A+B). Results of Browder concerning everywhere defined continuous and bounded operators B are improved. Extensions of this theory are also given using the degree theory of the last two authors concerning densely defined perturbations of nonlinear maximal monotone operators which satisfy a generalized (S+)-condition. Applications of this extended theory are given involving nonlinear parabolic problems on cylindrical domains.