Some results on the convexity of the closure of the domain of a maximally monotone operator
- 173 Downloads
We provide a concise analysis about what is known regarding when the closure of the domain of a maximally monotone operator on an arbitrary real Banach space is convex. In doing so, we also provide an affirmative answer to a problem posed by Simons.
KeywordsNearly convex set Fitzpatrick function Maximally monotone operator Monotone operator Set-valued operator
Mathematics Subject Classification (2010)Primary 47H05 Secondary 26B25 47A05 47B65
The authors thank an anonymous referee for his/her pertinent comments. Jonathan Borwein and Liangjin Yao were both partially supported by the Australian Research Council.
- 1.Bauschke, H.H., Borwein, J.M., Wang, X., Yao, L.: Construction of pathological maximally monotone operators on non-reflexive Banach spaces. Set-Valued Var. Anal. 20, 387–415 (2012)Google Scholar
- 7.Burachik, R.S., Iusem, A.N.: Set-Valued Mappings and Enlargements of Monotone Operators. Springer, Berlin (2008)Google Scholar
- 8.Fitzpatrick S. Representing monotone operators by convex functions. In: Workshop/Miniconference on Functional Analysis and Optimization (Canberra 1988), Proceedings of the Centre for Mathematical Analysis, vol. 20, pp. 59–65. Australian National University, Canberra (1988)Google Scholar
- 13.Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, 3rd Printing. Springer, Berlin (2009)Google Scholar
- 14.Simons, S.: Minimax and Monotonicity. Springer, Berlin (1998)Google Scholar
- 24.Yao, L.: On Monotone Linear Relations and the Sum Problem in Banach Spaces. Ph.D. thesis, Mathematics, University of British Columbia (Okanagan campus) (2011). http://hdl.handle.net/2429/39970