A new topological degree theory for perturbations of the sum of two maximal monotone operators

Abstract

Let X be an infinite dimensional real reflexive Banach space with dual space X ∗ and G ⊂ X , open and bounded. Assume that X and X ∗ are locally uniformly convex. Let T : X ⊃ D ( T ) → 2 X ∗ be maximal monotone and strongly quasibounded, S : X ⊃ D ( S ) → X ∗ maximal monotone, and C : X ⊃ D ( C ) → X ∗ strongly quasibounded w.r.t. S and such that it satisfies a generalized ( S + ) -condition w.r.t. S . Assume that D ( S ) = L ⊂ D ( T ) ∩ D ( C ) , where L is a dense subspace of X , and 0 ∈ T ( 0 ) , S ( 0 ) = 0 . A new topological degree theory is introduced for the sum T + S + C , with degree mapping d ( T + S + C , G , 0 ) . The reason for this development is the creation of a useful tool for the study of a class of time-dependent problems involving three operators. This degree theory is based on a degree theory that was recently developed by Kartsatos and Skrypnik just for the single-valued sum S + C , as above.