A new topological degree theory for densely defined (S+)L(S+)L-perturbations of multivalued maximal monotone operators in reflexive separable Banach spaces

Let XX be a real reflexive separable locally uniformly convex Banach space with locally uniformly convex dual space X∗X∗. Let T:X⊃D(T)→2X∗T:X⊃D(T)→2X∗ be maximal monotone, with 0∈D∘(T) and 0∈T(0)0∈T(0), and C:X⊃D(C)→X∗C:X⊃D(C)→X∗. Assume that L⊂D(C)L⊂D(C) is a dense linear subspace of X,CX,C is of class (S+)L(S+)L, and 〈Cx,x〉≥−ψ(‖x‖),x∈D(C)〈Cx,x〉≥−ψ(‖x‖),x∈D(C), where ψ:R+→R+ψ:R+→R+ is nondecreasing. A new topological degree theory is developed for the sum T+CT+C. The current approach utilizes the “approximate” degree d(Tt+C,G,0),t↓0d(Tt+C,G,0),t↓0, (Tt≔(T−1+tJ−1)−1,G⊂XTt≔(T−1+tJ−1)−1,G⊂X open and bounded) of Kartsatos and Skrypnik for the single-valued mapping Tt+CTt+C. The subdifferential ∂φ∂φ, for φφ belonging to a large class of proper convex lower semicontinuous functions, gives rise to operators TT to which this degree theory applies. A theoretical application to an existence problem of nonlinear analysis is included.