×

zbMATH — the first resource for mathematics

Set-valued mapping monotonicity as characterization of D.C functions. (English) Zbl 1222.46005
Summary: Using a result of K.-C. Chang [J. Math. Anal. Appl. 80, 102–129 (1981; Zbl 0487.49027)], 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 integration of locally D.C functions is also given.

MSC:
46A40 Ordered topological linear spaces, vector lattices
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
46B40 Ordered normed spaces
46B42 Banach lattices
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] D. Aussel, J.N. Corvellec, M. Lassonde, Subdifferential characterization of quasiconvexity and convexity, J. Convex Anal. 1(2) (1994), 195–201. · Zbl 0832.49010
[2] F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley, New York (1983). · Zbl 0582.49001
[3] R. Correa, A. Jofré, L. Thibault, Subdifferential monotonicity as characterization of convex functions, Numer. Anal. Optimiz. 15 (1994), 531–536. · Zbl 0807.49015
[4] A.A. Elhilali, Caractérisation des fonctions D.C., Annales des sciences Mathématiques du Québec 20(1) (1996), 1–15. · Zbl 0915.49014
[5] R. Ellaia, A. Hassouni, Characterization of nonsmooth functions through their generalized gradient, Optimization 22(3) (1991), 401–416. · Zbl 0734.49005
[6] K.-C. Chang, Variational methods for non-differentiable functionals and their application to partial differential equations, J. Math. Anal. Appl. 80, (1981), 102–129. · Zbl 0487.49027
[7] R.R. Phelps, Convex functions, monotone operators and differentiability, Lecture Note in Mathematics, Vol. 1364. Springer, Berlin (1989). · Zbl 0658.46035
[8] R.A. Poliquin, Integration of subdifferential of nonconvex functions, Nonlinear Anal., Theory Methods Appl. 17 (1991), 385–389.
[9] D. Preiss, Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal. 91 (1990), 312–345. · Zbl 0711.46036
[10] R.T. Rockafellar, Convex Analysis, University Press, Princeton (1970). · Zbl 0193.18401
[11] R.T. Rockafellar, Characterization of the subdifferentials of convex functions, Pac. J. Math. 33 (1966), 497510 · Zbl 0145.15901
[12] L. Thibault, D. Zagrodny, Integration of subdifferentials of lower semi-continuous functions on Banach spaces, J. Math. Anal. Appl. 189 (1995), 33–58. · Zbl 0826.49009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.