A note on the degree for maximal monotone mappings in finite dimensional spaces

Let Rⁿ be the n-dimensional Euclidean space, T : D (T) ⊆ Rⁿ → 2^Rn a maximal monotone mapping, and Ω ⊂ Rⁿ an open bounded subset such that Ω ∩ D (T) ≠ 0{combining long solidus overlay} and assume 0 ∉ T (∂ Ω ∩ D (T)). In this note we show an easy way to define the topological degree deg (T, Ω ∩ D (T), 0) of T on Ω ∩ D (T) as the limit of the classical Brouwer degree deg (T_λ, Ω, 0) as λ → 0⁺; here T_λ is the Yosida approximation of T. Furthermore, if T_i : D → 2^Rn, i = 1, 2, are two maximal monotone mappings such that Ω ∩ D ≠ 0{combining long solidus overlay} and 0 ∉ ∪_{t ∈ [0, 1]} [t T₁ + (1 - t) T₂] (∂ Ω ∩ D) and if t T₁ + (1 - t) T₂ is maximal monotone for each t ∈ [0, 1], we give an easy argument to show deg (T₁, D ∩ Ω, 0) = deg (T₂, D Ω, 0).