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Maximality of sums of two maximal monotone operators in general Banach space


Author: Jonathan M. Borwein
Journal: Proc. Amer. Math. Soc. 135 (2007), 3917-3924
MSC (2000): Primary 47H05, 46N10, 46A22
DOI: https://doi.org/10.1090/S0002-9939-07-08960-5
Published electronically: September 12, 2007
MathSciNet review: 2341941
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Abstract | References | Similar Articles | Additional Information

Abstract: We combine methods from convex analysis, based on a function of Simon Fitzpatrick, with a fine recent idea due to Voisei, to prove maximality of the sum of two maximal monotone operators in Banach space under various natural domain and transversality conditions.


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  • 1. BORWEIN, J. M., ``Maximality of sums of two maximal monotone operators,'' Proc. Amer. Math. Soc., 134 (2006), 2951-2956. MR 2231619
  • 2. BORWEIN, J. M., ``Maximal montonicity via convex analysis,'' Fitzpatrick Memorial Issue of J. Convex Analysis, 13/14 (2006), 561-586.
  • 3. BORWEIN, J. M. & LEWIS, A. S., Convex Analysis and Nonlinear Optimization: Theory and Examples, Enlarged Edition, Springer-Verlag, New York, 2005. MR 1757448 (2001h:49001)
  • 4. BORWEIN, J. M. & ZHU, Q. J., Techniques of Variational Analysis: An Introduction, CMS Books, Springer-Verlag, 2005. MR 2144010 (2006h:49002)
  • 5. BURACHIK, R.S. & SVAITER, B.F., Maximal monotonicity, conjugation and the duality product,'' Proc AMS, 131 (2003), 2379-2383. MR 1974634 (2004a:49037)
  • 6. FITZPATRICK, S. ``Representing monotone operators by convex functions,'' Workshop/ Miniconference on Functional Analysis and Optimization (Canberra, 1988), 59-65, Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra, 1988. MR 1009594 (90i:47054)
  • 7. FITZPATRICK, S. & PHELPS, R.R. ``Some properties of maximal monotone operators on nonreflexive Banach spaces,'' Set-valued Analysis, 3 (1995), 51-69. MR 1333366 (96c:47076)
  • 8. PENOT, J.P., ``The relevance of convex analysis for the study of monotonicity,'' Nonlinear Anal, Theory, Methods, Appl., 58 (2004),855-871. MR 2086060 (2005g:49026)
  • 9. ROCKAFELLAR, R. T., ``On the maximality of sums of nonlinear monotone operators,'' Trans. Amer. Math. Soc., 149 (1970), 75-88. MR 0282272 (43:7984)
  • 10. SIMONS, S., Minimax and Monotonicity, Lecture Notes in Mathematics, 1693, Springer-Verlag, Berlin, 1998. MR 1723737 (2001h:49002)
  • 11. VERONA, A. & VERONA, M. E., ``Regular maximal monotone operators and the sum rule,'' J. Convex Analysis, 7 (2000), 115-128. MR 1773179 (2001h:47087)
  • 12. VOISEI, M.D., ``A maximality theorem for the sum of maximal monotone operators in non-reflexive Banach Spaces,'' Math. Sci. Res. J., 10 (2) (2006), 36-41. MR 2207807 (2007a:47054)
  • 13. VOISEI, M.D., ``Calculus rules for maximal monotone operators in general Banach space,'' preprint, 2006.

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Additional Information

Jonathan M. Borwein
Affiliation: Faculty of Computer Science, Dalhousie University, Halifax, Nova Scotia, Canada
Email: jborwein@cs.dal.ca

DOI: https://doi.org/10.1090/S0002-9939-07-08960-5
Keywords: Maximal monotone operators, convex analysis, Fitzpatrick function, Fenchel duality, sum theorem
Received by editor(s): May 3, 2006
Received by editor(s) in revised form: May 10, 2006, and September 27, 2006
Published electronically: September 12, 2007
Additional Notes: This author’s research was supported by NSERC and by the Canada Research Chair Program.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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