Abstract
In 1966 R.T. Rockafellar [14] exploring the operator of subdifferential of convex function on a Banach space revealed that the operator is maximal monotone i.e.
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J.-P. Aubin, Optima and equilibria, Springer-Verlag, 1993.
D. Aussel, J.-N. Corvellec, M. Lassonde, Subdifferential characterization of quasiconvexity and convexity, J. Convex Analysis 1:195–201, 1994.
D. Aussel, J.-N. Corvellec, M. Lassonde, Mean value property and subdifferential criteria for lower semicontinuous function, Trans. Amer. Math. Soc. 147: 4147–4161, 1995.
R. Correa, A. Jofre, L. Thibault, Subdifferential monotonicity as characterization of convex function,Numer. Funct. Anal. and Optimiz. 15:531–535, 1994.
K. Deimling, Nonlinear functional analysis, Spriger-Verlag, Berlin,1985.
I. Ekeland Nonconvex minimization problems,Bull, of the AMS 1:443–474, 1979.
S.P. Fitzpatrick and R.R. Phelps, Some properties of maximal monotone operators on nonreflexive Banach spaces, Set—Valued Analysis 3:5169,1995.
G. Kothe, Topological Vector Space I, Springer-Verlag, 1985.
E. Krauss, Maximal monotone operators and saddle functions I. Zeitschrift für Analysis and ihre Anwendungen, 5:333–346, 1986.
D. T. Luc, Characterization of quasiconvex function Bull. Austral. Math. Soc. 48:393–406, 1993.
D. T. Luc, On generalized convex nonsmooth functions,Bull. Ausrtal. Math. Soc. 49:139–149, 1994.
D. T. Luc, A resolution of Simons’ maximal monotonicity problem, Journal of Convex Analysis 3:367–370,1996.
R. R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Math. 1364, Springer-Verlag, 1993 (second edition).
R.T. Rockafellar, Characterization of the subdifferentials of convex functions,Pacific J. Math 17:497–509,1966.
R.T. Rockafellar, Local boundedness of nonlinear,monotone operators. Michigan Math. J., 16:397–407, 1969.
R.T. Rockafellar and R.J.-B. Wets, Variational Analysis, (book to be published by Springer—Verlag), 1995.
S. Simons, Subtangents with controlled slope,Nonlinear Anal. Theory Meth. App. 22:1373–1389, 1994.
S. Simons, Swimming below icebergs, Set-Valued Analysis 2:327–337, 1994.
S. Simons, The range of a monotone operator. J. Math. Anal. Appl., 196:176–201, 1996.
L. Thibault, D. Zagrodny, Integration of subdifferentials of lower semi-continuous functions on Banach spaces, J. Math. Anal. Appl. 189:33–58, 1995.
D. Zagrodny Approximate mean value theorem for upper subderivatives, Nonlinear Anal. Theory Meth. Appl 12:1413–1428, 1988.
D. Zagrodny, The maximal monotonicity of the subdifferentials of convex functions: Simons’ problem, Set-Valued Anal. 4:301–314, 1996.
D. Zagrodny, Answers to the basic questions on maximal monotone operators on nonreflexive Banach spaces (in preparation).
E. Zeidler, Nonlinear functional analysis and its applications II: Ðœono-tone operators; IV: Applications to mathematical physics, Springer- Verlag, New-York, Ð’erlin, 1986.
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© 2001 Kluwer Academic Publishers
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Przeworski, M., Zagrodny, D. (2001). Maximal Monotonicity, Subdifferentials and Generalizations. In: Gilbert, R.P., Panagiotopoulos, P.D., Pardalos, P.M. (eds) From Convexity to Nonconvexity. Nonconvex Optimization and Its Applications, vol 55. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0287-2_26
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DOI: https://doi.org/10.1007/978-1-4613-0287-2_26
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