A Browder degree theory from the Nagumo degree on the Hilbert space of elliptic super-regularization

Let X be a real reflexive separable locally uniformly convex Banach space with locally uniformly convex dual space X∗. Let Q:H→X be a linear compact injection, according to Browder and Ton, such that Q(H)¯=X, where H is a real separable Hilbert space. A degree mapping d on X is constructed from the Nagumo degree dNA on H by d(T+f,G,0)≔limt→0dNA(I+1tQ∗(Tt+f)Q,Q−1G,0), where G⊂X is open and bounded, Tt is the resolvent (T−1+tJ−1)−1 of a strongly quasibounded maximal monotone operator T:X⊃D(T)→2X∗ with 0∈T(0), and f:G¯→X∗ is demicontinuous, bounded and of type (S+). A “range of sums” result is also given, using the Skrypnik degree theory, in order to further exhibit the methodology of “elliptic super-regularization”.