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

Let XX be an infinite dimensional real reflexive Banach space with dual space X∗X∗ and G⊂XG⊂X, open and bounded. Assume that XX and X∗X∗ are locally uniformly convex. Let T:X⊃D(T)→2X∗T:X⊃D(T)→2X∗ be maximal monotone and strongly quasibounded, S:X⊃D(S)→X∗S:X⊃D(S)→X∗ maximal monotone, and C:X⊃D(C)→X∗C:X⊃D(C)→X∗ strongly quasibounded w.r.t. SS and such that it satisfies a generalized (S+)(S+)-condition w.r.t. SS. Assume that D(S)=L⊂D(T)∩D(C)D(S)=L⊂D(T)∩D(C), where LL is a dense subspace of XX, and 0∈T(0),S(0)=00∈T(0),S(0)=0. A new topological degree theory is introduced for the sum T+S+CT+S+C, with degree mapping d(T+S+C,G,0)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+CS+C, as above.