Abstract
In three previous papers, we discussed quasidense multifunctions from a Banach space into its dual, or, equivalently, quasidense subsets of the product of a Banach space and its dual. In this paper, we survey (without proofs) some of the main results about quasidensity, and give some simple limiting examples in Hilbert spaces, reflexive Banach spaces, and nonreflexive Banach spaces.
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References
Alves, M.M., Svaiter, B.F.: A new old class of maximal monotone operators. J. of Convex Anal. 16, 881–890 (2009)
Bauschke, H., Borwein, J.M., Wang, X., Yao, L.: Every maximally monotone operator of Fitzpatrick-Phelps type is actually of dense type. Optim. Lett. 6, 1875–1881 (2012)
Bauschke, H.H., Simons, S.: Stronger maximal monotonicity properties of linear operators. Bull. Austral. Math. Soc. 60, 163–174 (1999)
Brøndsted, A., Rockafellar, R.: On the subdifferentiability of convex functions. Proc. Amer. Math. Soc. 16, 605–611 (1965)
Fitzpatrick, S.: Representing monotone operators by convex functions. In: Functional Analysis and Optimization, vol. 20, pp. 59–65. Austral. Nat. Univ., Canberra (1988)
Fitzpatrick, S.P., Phelps, R.R.: Some properties of maximal monotone operators on nonreflexive Banach spaces. Set–Valued Analysis 3, 51–69 (1995)
Galán, M.R., Simons, S.: A new minimax theorem and a perturbed James’s theorem. Bull. Austral. Math. Soc. 66, 43–56 (2002)
Gossez, J.P.: Opérateurs monotones non linéaires dans les espaces de banach non réflexifs. J. Math. Anal. Appl. 34, 371–395 (1971)
Pryce, J.D.: Weak compactness in locally convex spaces. Proc. Amer. Math. Soc. 17, 148–155 (1966)
Rockafellar, R.T.: Extension of Fenchel’s duality theorem for convex functions. Duke Math. J. 33, 81–89 (1966)
Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math 33, 209–216 (1970)
Rockafellar, R.T.: On the maximal monotonicity of sums of nonlinear monotone operators. Trans. Amer. Math. Soc. 149, 75–88 (1970)
Simons, S.: Subtangents with controlled slope. Nonlinear Analysis 22, 1373–1389 (1994)
Simons, S.: Minimax and monotonicity. Lecture Notes in Mathematics., vol. 1693. Springer–Verlag (1998)
Simons, S.: From Hahn–Banach to monotonicity. Lecture Notes in Mathematics., vol. 1693, 2nd edn. Springer–Verlag (2008)
Simons, S.: “densities” and maximal monotonicity. J. Convex Anal. 23, 1017–1050 (2016)
Simons, S.: Quasidense monotone multifunctions. Set–Valued Var. Anal. 26, 5–26 (2018)
Simons, S., Wang, X.: Ubiquitous subdifferentials, r l-density and maximal monotonicity. Set-Valued Var. Anal. 23, 631–642 (2015)
Voisei, M.D., Zălinescu, C.: Strongly–representable monotone operators. J. of Convex Anal. 16, 1011–1033 (2009)
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Simons, S. (2019). Quasidensity: A Survey and Some Examples. In: Bauschke, H., Burachik, R., Luke, D. (eds) Splitting Algorithms, Modern Operator Theory, and Applications. Springer, Cham. https://doi.org/10.1007/978-3-030-25939-6_14
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DOI: https://doi.org/10.1007/978-3-030-25939-6_14
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