Ranges of densely defined generalized pseudomonotone perturbations of maximal monotone operators

Abstract

Let X be a real reflexive Banach space and A : X→2 X ∗ be maximal monotone. Let B : X→2 X ∗ 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 S⊂ R(A+B) 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.