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 → 2R n a maximal monotone mapping, and Ω ⊂ Rn an open bounded subset such that Ω ∩ D(T ) 6= ∅ and assume 0 6∈ 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(Tλ,Ω, 0) as λ → 0; here Tλ is the Yosida approximation of T . Furthermore, if Ti : D → 2R n , i = 1, 2, are two maximal monotone mappings such that Ω ∩ D 6= ∅ and 0 6∈ ∪t∈[0,1][tT1 + (1− t)T2](∂Ω ∩ D) and if tT1 + (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,DΩ, 0). © 2009 Elsevier Ltd. All rights reserved.