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

Let RnRn be the nn-dimensional Euclidean space, T:D(T)⊆Rn→2RnT:D(T)⊆Rn→2Rn a maximal monotone mapping, and Ω⊂RnΩ⊂Rn an open bounded subset such that Ω∩D(T)≠0̸Ω∩D(T)≠0̸ and assume 0∉T(∂Ω∩D(T))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 TT on Ω∩D(T)Ω∩D(T) as the limit of the classical Brouwer degree deg(Tλ,Ω,0) as λ→0+λ→0+; here TλTλ is the Yosida approximation of TT. Furthermore, if Ti:D→2RnTi:D→2Rn, i=1,2i=1,2, are two maximal monotone mappings such that Ω∩D≠0̸Ω∩D≠0̸ and 0∉∪t∈[0,1][tT1+(1−t)T2](∂Ω∩D)0∉∪t∈[0,1][tT1+(1−t)T2](∂Ω∩D) and if tT1+(1−t)T2tT1+(1−t)T2 is maximal monotone for each t∈[0,1]t∈[0,1], we give an easy argument to show deg(T1,D∩Ω,0)=deg(T2,DΩ,0).