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The purpose of the present paper is to prove that the topological degree of the subdifferential of a coercive lower semi-continuous function on a sufficiently large ball in a reflexive Banach space is equal to one. A well known result of Krasnoselskii [7] says that the degree of the gradient of a coercive potential mapping on a sufficiently large ball in RN is equal to one. As Nirenberg pointed out in [8], this result seems intuitively clear, but the proof is not. Many authors had new views of this theorem. See [1], [9]-[11]. The aim of this note is to give a generalization of the degree of a maximal monotone mapping and prove that the topological degree of the subdifferential of a coercive convex function is equal to one on a sufficiently large ball. The main result of this paper is Theorem 5. Its proof is very simple and elementary. Let us recall the properties of the degree of mappings of class (S+), as introduced by Browder [4]. See also [3], [6]. Let E be a real reflexive Banach space of dual E* . We can always assume that both E and E* are locally uniformly convex [12]. The duality pairing between E* and E is denoted by (.,.). Let 0* be the zero element of E*, J : E -E* the duality mapping, Q C E an open bounded subset, and T: Q -E* a demi-continuous operator of class (S+) (see [4] for a definition); if 0* 0 T(Q9), then there is a topological degree deg(T, Q, 0*) satisfying Theorem 1. (a) deg(J, Q, 0*) = { 0 ifo* E J(Q). (b) If deg(T, Q, 0*) $7 0, then Tx = 0* has a solution in Q. (c) If Q1, Q2 are two disjoint open subsets of Q, then deg(T, Q, 0*) + deg(T,Q2,0*) = deg(T, Q, U Q2, 0*). (d) If {Tt} o,1] is a homotopy of class (S+), and 0* f TtQ9Q), then deg(T, Q, 0*) doesn't depend on t E [0, 1]. Now, suppose that A: D(A) C E -2 is a maximal monotone operator. Let AA = (AJ-1+A-')-', RA = I-AJ-1AA be its Yosida approximation and resolvent Received by the editors February 25, 1997. 1991 Mathematics Subject Classification. Primary 47H10, 47H15; Secondary 55M25.

Note on the topological degree of the subdifferential of a lower semi-continuous convex function