Locally efficient monotone operators

Authors:
Andrei Verona and Maria Elena Verona

Journal:
Proc. Amer. Math. Soc. **109** (1990), 195-204

MSC:
Primary 47H05; Secondary 49J52, 58C07

DOI:
https://doi.org/10.1090/S0002-9939-1990-1012939-0

MathSciNet review:
1012939

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study monotone operators on quasi open or convex subsets of a real Banach space (quasi open means that the contingent cone at each point equals ). Among others we characterize the maximality of such an operator in terms of its -upper semicontinuity properties and, in the case of a convex domain, also in terms of its behavior at the support points. We next give sufficient conditions for such an operator to be generically single valued, extending Kenderov's theorems. As an application we reobtain generic Gâteaux and Fréchet differentiability results for convex functions defined on not necessarily open convex sets.

**[1]**J. M. Borwein and S. P. Fitzpatrick,*Local boundedness of monotone operators under minimal hypotheses*, Bull. Austral. Math. Soc. (to appear). MR**995141 (90c:47093)****[2]**P. S. Kenderov,*The set-valued monotone mappings are almost everywhere single-valued*, C. R. Acad. Bulgare Sci.**27**(1974), 1173-1175. MR**0358447 (50:10913)****[3]**-,*Monotone operators in Asplund spaces*, C. R. Acad. Bulgare Sci.**30**(1977), 963-964. MR**0463981 (57:3919)****[4]**D. Noll,*Generic Fréchet differentiability of convex functions on small sets*, preprint (1987). MR**1049204 (91i:46046)****[5]**R. R. Phelps,*Convex functions, monotone operators and differentiability*, Lecture Notes in Math., vol. 1364, Springer-Verlag, 1989. MR**984602 (90g:46063)****[6]**J. Rainwater,*Yet more on the differentiability of convex functions*, Proc. Amer. Math. Soc.**103**(1988), 773-778. MR**947656 (89m:46081)****[7]**R. T. Rockafellar,*Local boundedness of nonlinear monotone operators*, Mich. Math. J.**16**(1969), 397-407. MR**0253014 (40:6229)****[8]**C. Stegall,*A class of topological spaces and differentiation of functions on Banach spaces*, Proc. Conf. on Vector Measures and Integral Representations of Operators, Vorlesungen aus dem Fachbereich Math., Heft**10**(W. Ruess, ed.), Essen, 1983. MR**730947 (85j:46026)****[9]**M. E. Verona,*More on the differentiability of convex functions*, Proc. Amer. Math. Soc.**103**(1988), 137-140. MR**938657 (89f:58016)****[10]**-,*On the differentiability of convex functions*, Proceedings of the Centre of Mathematical Analysis, vol. 20, Canberra, 1988, 195-202. MR**1009606 (90j:58008)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47H05,
49J52,
58C07

Retrieve articles in all journals with MSC: 47H05, 49J52, 58C07

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-1012939-0

Article copyright:
© Copyright 1990
American Mathematical Society