Positivity 13 (2009), 399–405
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020399-7, published online July 5, 2008
Set-valued mapping monotonicity
as characterization of D.C functions
Lafhim Lahoussine, Ahmed Alaoui Elhilali and N. Gadhi
Abstract. In this paper, using a result of Kung-Ching Chang , we give a
characterization of locally Lipschitz functions which are diﬀerences of convex
functions deﬁned on a Banach space (not necessarily Asplund) in terms on
maximal cyclically monotone set-valued mappings. A subdiﬀerential integra-
tion of locally D.C functions. is also given.
Mathematics Subject Classiﬁcation (2000). 46N10, 26B25, 54B20.
Keywords. Lipschitz functions, D.C-functions, regular functions, maximal monotone
In the literature, there are various results with monotone set-valued operators
characterizing many important classes of functions, see  for convex functions,
 for locally Lipschitz functions on Banach spaces,  for lower semi-continuous
functions on ﬁnite dimensional spaces,  for lower semi-continuous functions on
Banach spaces,  for quasiconvex functions on ﬁnite dimentional spaces and 
for quasiconvex functions on Banach spaces.
A natural question arises : does there exist an analogous characterization for
D.C functions ? In , Elhilali Alaoui proved that a locally Lipschitz function on
an Asplund space is D.C if and only if its subdiﬀerential is a diﬀerence of two
maximal cyclically monotone set-valued mappings. The technique of the proof is
based on integration of subdiﬀerentials of lower semi-continuous functions and the
Preiss theorem [9,12].
The object of this paper is twofold. We extend Elhilali Alaoui’s characteriza-
tion of locally Lipschitz functions to a large class of Banach spaces (not necessarily
Asplund) by using Kung-Ching Chang result . We give also a subdiﬀerential in-
tegration of locally D.C functions.