Continuity of nonlinear monotone operators
HTML articles powered by AMS MathViewer
- by S. P. Fitzpatrick PDF
- Proc. Amer. Math. Soc. 62 (1977), 111-116 Request permission
Abstract:
If a Banach space E has an equivalent norm such that $\text {weak}^\ast$ sequential convergence and norm convergence coincide on the dual unit sphere, then every monotone operator on E is single-valued and norm-norm continuous on a dense ${G_\delta }$ subset of E. In particular, this holds for reflexive spaces.References
- N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), no. 2, 147–190. MR 425608, DOI 10.4064/sm-57-2-147-190
- Edgar Asplund, Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31–47. MR 231199, DOI 10.1007/BF02391908
- Felix E. Browder, Multi-valued monotone nonlinear mappings and duality mappings in Banach spaces, Trans. Amer. Math. Soc. 118 (1965), 338–351. MR 180884, DOI 10.1090/S0002-9947-1965-0180884-9 G. Choquet, Lectures on analysis. Vol. 1, Benjamin, New York, 1969. MR 40 #3252; erratum, 44, p. 1630.
- P. S. Kenderov, The set-valued monotone mappings are almost everywhere single-valued, C. R. Acad. Bulgare Sci. 27 (1974), 1173–1175. MR 358447
- I. Namioka and R. R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J. 42 (1975), no. 4, 735–750. MR 390721
- Raoul Robert, Points de continuité de multi-applications semi-continues supérieurement, C. R. Acad. Sci. Paris Sér. A 278 (1974), 413–415 (French). MR 348497
- Raoul Robert, Une généralisation aux opérateurs monotones des théorèmes de différentiabilité d’Asplund, Analyse convexe et ses applications (Comptes Rendus Conf., Saint Pierre de Chartreuse, 1974) Lecture Notes in Econom. and Math. Systems, Vol. 102, Springer, Berlin, 1974, pp. 168–179 (French). MR 0482398
- R. T. Rockafellar, Local boundedness of nonlinear, monotone operators, Michigan Math. J. 16 (1969), 397–407. MR 253014
- R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216. MR 262827
- Peter D. Taylor, Subgradients of a convex function obtained from a directional derivative, Pacific J. Math. 44 (1973), 739–747. MR 324407
- Eduardo H. Zarantonello, Dense single-valuedness of monotone operators, Israel J. Math. 15 (1973), 158–166. MR 324490, DOI 10.1007/BF02764602
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 62 (1977), 111-116
- MSC: Primary 47H05
- DOI: https://doi.org/10.1090/S0002-9939-1977-0425687-6
- MathSciNet review: 0425687