A note on the duality mapping of a locally uniformly convex Banach space

Let XX be a real locally uniformly convex Banach space with normalized duality mapping J:X→2X∗J:X→2X∗. The purpose of this note is to show that for every R>0R>0 and every x0∈Xx0∈X there exists a function ϕ=ϕ(R,x0):R+→R+ϕ=ϕ(R,x0):R+→R+, which is nondecreasing and such that ϕ(r)>0ϕ(r)>0 for r>0,ϕ(0)=0r>0,ϕ(0)=0 and 〈x∗−x0∗,x−x0〉≥ϕ(‖x−x0‖)‖x−x0‖, for all x∈BR(x0)¯,x∗∈Jx,x0∗∈Jx0. Simply, it is shown that the necessity part of the proof of the original analogous necessary and sufficient condition of Prüß, for real uniformly convex Banach spaces, goes over equally well in the present setting. This is a natural setting for the study of many existence problems in accretive and monotone operator theories.