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

Let X be a real locally uniformly convex Banach space with normalized duality mapping J : X -> 2(x)*. The purpose of this note is to show that for every R > 0 and every x(0) is an element of X there exists a function phi = phi(R, x(0)) : R+ -> R+, which is nondecreasing and such that phi(r) > 0 for r > 0, phi(0) = 0 and (x* - x(0)*, x - x(0)) >= phi(vertical bar vertical bar x - x(0)vertical bar vertical bar)vertical bar vertical bar x - x(0)vertical bar vertical bar, for all x is an element of B-R(x(0)), x* is an element of J(X), x(0)* is an element of Jx(0). Simply, it is shown that the necessity part of the proof of the original analogous necessary and sufficient condition of Pru beta, 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. (C) 2009 Elsevier Ltd. All rights reserved.