On the Convexity of the Closure of the Domain/Range of a Maximally Monotone Operator

Maria E. Verona; Andrei Verona

doi:10.68381/jca29045

Abstract

We prove that a sufficient condition (due to Simons) for the convexity of the closure of the domain/range of a monotone operator is also necessary when the operator has bounded domain/range and is maximal.

Author information

Contact details are reproduced from the original publication and may be historical.

Maria E. Verona

Dept. of Mathematics, University of Southern California, Los Angeles, U.S.A.

verona@usc.edu

Andrei Verona

Dept. of Mathematics, California State University, Los Angeles, U.S.A.

averona10@yahoo.com

Keywords

Monotone operator
maximally monotone operator
Fitzpatrick function
Simons' conditions
type (FP)

Mathematics Subject Classification

47H05; 47A65

References

J. M. Borwein, L. Yao: Some results on the convexity of the closure of the domain of a maximally monotone operator, Optimization Letters 8 (2014) 237–246.https://doi.org/10.1007/s11590-012-0564-7 Google Scholar
J. M. Borwein, L. Yao: Recent progress on monotone operator theory, in: Infinite Products of Operators and Their Applications, Haifa 2012, S. Reich et al. (eds.), Contemporary Mathematics 636, American Mathematical Society, Providence (2015) 51–81.https://doi.org/10.1090/conm/636/12728 Google Scholar
S. P. Fitzpatrick: Representing monotone operators by convex functions, in: Workshop/Miniconference on Functional Analysis and Optimization, Canberra 1988, Proc. Centre Math. Analysis Vol. 20, Australian National University, Canberra (1988) 59–65.Google Scholar
R. Phelps: Lectures on Maximal Monotone Operators, 2nd Summer School on Banach Spaces, Related Areas and Applications, Prague and Paseky, August 15-28 (1993).Google Scholar
R. T. Rockafellar: Local boundedness of nonlinear, monotone operators, Michigan Math. J. 16 (1969) 397–407.https://doi.org/10.1307/mmj/1029000324 Google Scholar
S. Simons: From Hahn-Banach to Monotonicity, 2nd ed., Lecture Notes in Mathematics 1693, Springer, Berlin (2008).Google Scholar

Suggested citation

Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.

[1] J. M. Borwein, L. Yao: Some results on the convexity of the closure of the domain of a maximally monotone operator, Optimization Letters 8 (2014) 237–246.https://doi.org/10.1007/s11590-012-0564-7 Google Scholar

[2] J. M. Borwein, L. Yao: Recent progress on monotone operator theory, in: Infinite Products of Operators and Their Applications, Haifa 2012, S. Reich et al. (eds.), Contemporary Mathematics 636, American Mathematical Society, Providence (2015) 51–81.https://doi.org/10.1090/conm/636/12728 Google Scholar

[3] S. P. Fitzpatrick: Representing monotone operators by convex functions, in: Workshop/Miniconference on Functional Analysis and Optimization, Canberra 1988, Proc. Centre Math. Analysis Vol. 20, Australian National University, Canberra (1988) 59–65.Google Scholar

[4] R. Phelps: Lectures on Maximal Monotone Operators, 2nd Summer School on Banach Spaces, Related Areas and Applications, Prague and Paseky, August 15-28 (1993).Google Scholar

[5] R. T. Rockafellar: Local boundedness of nonlinear, monotone operators, Michigan Math. J. 16 (1969) 397–407.https://doi.org/10.1307/mmj/1029000324 Google Scholar

[6] S. Simons: From Hahn-Banach to Monotonicity, 2nd ed., Lecture Notes in Mathematics 1693, Springer, Berlin (2008).Google Scholar