Maximality of sums of two maximal monotone operators

Borwein, Jonathan

doi:10.1090/S0002-9939-06-08323-7

Maximality of sums of two maximal monotone operators
HTML articles powered by AMS MathViewer

by Jonathan M. Borwein

Proc. Amer. Math. Soc. 134 (2006), 2951-2955

DOI: https://doi.org/10.1090/S0002-9939-06-08323-7

Published electronically: May 1, 2006

PDF | Request permission

Abstract:

We use methods from convex analysis, relying on an ingenious function of Simon Fitzpatrick, to prove maximality of the sum of two maximal monotone operators on reflexive Banach space under weak transversality conditions.

References

Borwein J. M., “Maximal montonicity via convex analysis," to appear in the Fitzpatrick memorial issue of J. Convex Analysis, 2006.
Jon Borwein and Simon Fitzpatrick, Local boundedness of monotone operators under minimal hypotheses, Bull. Austral. Math. Soc. 39 (1989), no. 3, 439–441. MR 995141, DOI 10.1017/S000497270000335X
Jon Borwein, Simon Fitzpatrick, and Jon Vanderwerff, Examples of convex functions and classifications of normed spaces, J. Convex Anal. 1 (1994), no. 1, 61–73. MR 1326942
Jonathan M. Borwein and Adrian S. Lewis, Convex analysis and nonlinear optimization, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 3, Springer-Verlag, New York, 2000. Theory and examples. MR 1757448, DOI 10.1007/978-1-4757-9859-3
Jonathan M. Borwein and Qiji J. Zhu, A survey of subdifferential calculus with applications, Nonlinear Anal. 38 (1999), no. 6, Ser. A: Theory Methods, 687–773. MR 1710152, DOI 10.1016/S0362-546X(98)00142-4
Jonathan M. Borwein and Qiji J. Zhu, Techniques of variational analysis, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 20, Springer-Verlag, New York, 2005. MR 2144010
Regina Sandra Burachik and B. F. Svaiter, Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2379–2383. MR 1974634, DOI 10.1090/S0002-9939-03-07053-9
Simon Fitzpatrick, Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 20, Austral. Nat. Univ., Canberra, 1988, pp. 59–65. MR 1009594
Jonathan M. Borwein and Qiji J. Zhu, A survey of subdifferential calculus with applications, Nonlinear Anal. 38 (1999), no. 6, Ser. A: Theory Methods, 687–773. MR 1710152, DOI 10.1016/S0362-546X(98)00142-4
Jean-Paul Penot, The relevance of convex analysis for the study of monotonicity, Nonlinear Anal. 58 (2004), no. 7-8, 855–871. MR 2086060, DOI 10.1016/j.na.2004.05.018
R. T. Rockafellar, Local boundedness of nonlinear, monotone operators, Michigan Math. J. 16 (1969), 397–407. MR 253014, DOI 10.1307/mmj/1029000324
R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), 75–88. MR 282272, DOI 10.1090/S0002-9947-1970-0282272-5
R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683, DOI 10.1515/9781400873173
Stephen Simons, Minimax and monotonicity, Lecture Notes in Mathematics, vol. 1693, Springer-Verlag, Berlin, 1998. MR 1723737, DOI 10.1007/BFb0093633
S. Simons, A new version of the Hahn-Banach theorem, Arch. Math. (Basel) 80 (2003), no. 6, 630–646. MR 1997529, DOI 10.1007/s00013-003-0500-2
S. Simons and C. Zălinescu, A new proof for Rockafellar’s characterization of maximal monotone operators, Proc. Amer. Math. Soc. 132 (2004), no. 10, 2969–2972. MR 2063117, DOI 10.1090/S0002-9939-04-07462-3
Andrei Verona and Maria Elena Verona, Regular maximal monotone operators and the sum theorem, J. Convex Anal. 7 (2000), no. 1, 115–128. MR 1773179