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

Let R n be the n -dimensional Euclidean space, T : D ( T ) ⊆ R n → 2 R n a maximal monotone mapping, and Ω ⊂ R n an open bounded subset such that Ω ∩ D ( T ) ≠ 0̸ 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 R n ,  i = 1 , 2 , are two maximal monotone mappings such that Ω ∩ D ≠ 0̸ and 0 ∉ ∪ t ∈ [ 0 , 1 ] [ t T 1 + ( 1 − t ) T 2 ] ( ∂ Ω ∩ D ) and if t T 1 + ( 1 − t ) T 2 is maximal monotone for each t ∈ [ 0 , 1 ] , we give an easy argument to show deg ( T 1 , D ∩ Ω , 0 ) = deg ( T 2 , D Ω , 0 ) .