Abstract
Important properties of maximal monotone operators on reflexive Banach spaces remain open questions in the nonreflexive case. The aim of this paper is to investigate some of these questions for the proper subclass of locally maximal monotone operators. (This coincides with the class of maximal monotone operators in reflexive spaces.) Some relationships are established with the maximal monotone operators of dense type, which were introduced by J.-P. Gossez for the same purpose.
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References
Borwein, J., Fitzpatrick, S. P., and Vanderwerff, J.: Examples of convex functions and classifications of normed spaces,J. Convex Anal., to appear.
Fitzpatrick, S. P. and Phelps, R. R.: Bounded approximations to monotone operators on Banach spaces,Ann. Inst. Henri Poincaré, Analyse non Linéaire 9 (1992), 573–595.
Gossez, J.-P.: Opérateurs monotones non linéaires dans les espaces de Banach non réflexifs,J. Math. Anal. Appl. 34 (1971), 371–395.
Gossez, J.-P.: On the range of a coercive maximal monotone operator in a nonreflexive Banach space,Proc. Amer. Math. Soc. 35 (1972), 88–92.
Gossez, J.-P.: On a convexity property of the range of a maximal monotone operator,Proc. Amer. Math. Soc. 55 (1976), 359–360.
Gossez, J.-P.: On the extensions to the bidual of a maximal monotone operator,Proc. Amer. Math. Soc. 62 (1977) 67–71.
Phelps, R. R.:Convex Functions, Monotone Operators and Differentiability, Lecture Notes Math. 1364, Springer-Verlag, 1989; 2nd edn. 1993.
Phelps, R. R.:Lectures on Maximal Monotone Operators, 2nd Summer School on Banach Spaces, Related Areas and Applications, Prague and Paseky, 15–28 August 1993 (Preprint, 30 pages). TeX file: phelpsmaxmonop. tex, Banach Space Bulletin Board Archive: ftp.math.okstate.edu. Posted Nov. 1993.
Reich, S.: The range of sums of accretive and monotone operators,J. Math. Anal. Appl. 68 (1979), 310–317.
Rockafellar, R. T.: On the maximality of sums of nonlinear monotone operators,Trans. Amer. Math. Soc. 149 (1970), 75–88.
Rockafellar, R. T.: Local boundedness of nonlinear, monotone operators,Mich. Math. J. 16 (1969), 397–407.
Simons, S.: Subdifferentials are locally maximal monotone,Bull. Australian Math. Soc. 47 (1993), 465–471.
Simons, S.: Les dérivées directionelles et la monotonicité des sous-différentiels,Sém. d'Initiation à l'Analyse (Sém. Choquet), 1991/92,Publications Mathématiques de l'Université de Paris 6 (1993).
Verona, A. and Verona, M. E.: Remarks on subgradients and ε-subgradients,Set-Valued Anal. 1 (1993), 261–272.
Zeidler, E.:Nonlinear Functional Analysis and Its Applications, Vol. II/B, Nonlinear Monotone Operators, Springer-Verlag, 1985.
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Fitzpatrick, S.P., Phelps, R.R. Some properties of maximal monotone operators on nonreflexive Banach spaces. Set-Valued Anal 3, 51–69 (1995). https://doi.org/10.1007/BF01033641
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DOI: https://doi.org/10.1007/BF01033641