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

Let X be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X⁎ and G be a nonempty, bounded and open subset of X. Let T:X⊇D(T)→2X⁎ and A:X⊇D(A)→2X⁎ be maximal monotone operators. Assume, further, that, for each y∈X, there exists a real number β(y) and there exists a strictly increasing function ϕ:[0,∞)→[0,∞) with ϕ(0)=0, ϕ(t)→∞ as t→∞ satisfying〈w⁎,x−y〉≥−ϕ(‖x‖)‖x‖−β(y) for all x∈D(A), w⁎∈Ax, and S:X→2X⁎ is bounded of type (S+) or bounded pseudomonotone such that 0∉(T+A+S)(D(T)∩D(A)∩∂G) or 0∉(T+A+S)(D(T)∩D(A)∩∂G)‾, respectively. New degree theory is developed for operators of the type T+A+S with degree mapping d(T+A+S,G,0). The degree is shown to be unique invariant under suitable homotopies. The theory developed herein generalizes the Asfaw and Kartsatos degree theory for operators of the type T+S. New results on surjectivity and solvability of variational inequality problems are obtained. The mapping theorems extend the corresponding results for operators of type T+S. The degree theory developed herein is used to show existence of weak solution of nonlinear parabolic problem in appropriate Sobolev spaces.