Abstract
The main goal of this article is to present several new results on the maximality of the composition and of the sum of maximal monotone operators in Banach spaces under weak interiority conditions involving their domains. Direct applications of our results to the structure of the range and domain of a maximal monotone operator are discussed. The last section of this note studies continuity properties of the duality product between a Banach space X and its dual X* with respect to topologies compatible with the natural duality (X × X*, X* × X).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Borwein J.M.: Maximal monotonicity via convex analysis. J. Convex Anal. 13, 561–586 (2006)
Borwein J.M.: Maximality of sums of two maximal monotone operators in general Banach space. Proc. Amer. Math. Soc. 135, 3917–3924 (2007)
Burachik R.S., Svaiter B.F.: Maximal monotone operators, convex functions and a special family of enlargements. Set-Valued Anal. 10, 297–316 (2002)
Edwards, R.E.: Functional Analysis. Theory and Applications. Corrected reprint of the 1965 original. Dover Publications, Inc., New York (1995)
Fitzpatrick, S.: Representing monotone operators by convex functions. Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), In: Proceedings of the Centre for Mathematics Anal. Austral. Nat. Univ. 20, Austral. Nat. Univ., Canberra, 1988, pp. 59–65
Fitzpatrick S., Phelps R.R.: Some properties of maximal monotone operators on nonreflexive Banach spaces. Set-Valued Anal. 3, 51–69 (1995)
Gossez J.P.: On a convexity property of the range of a maximal monotone operator. Proc. Am. Math. Soc. 55, 359–360 (1976)
Holmes R.: Geometric Functional Analysis and its Applications. Graduate Texts in Mathematics, No. 24. Springer, New York (1975)
Martínez-Legaz J.E., Théra M.: A convex representation of maximal monotone operators. J. Nonlinear Convex Anal. 2, 243–247 (2001)
Penot J.-P.: Miscellaneous incidences of convergence theories in optimization and nonlinear analysis II: applications in nonsmooth analysis. In: Du, D.-Z. (eds) Recent Advances in Nonsmooth Optimization, pp. 289–321. World Scientific, Singapore (1995)
Penot J.-P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58, 855–871 (2004)
Penot J.-P., Zălinescu C.: Some problems about the representation of monotone operators by convex functions. ANZIAM J. 47, 1–20 (2005)
Penot J.-P., Zălinescu C.: On the convergence of maximal monotone operators. Proc. Am. Math. Soc. 134, 1937–1946 (2006)
Rockafellar R.T.: Local boundedness of nonlinear, monotone operators. Michigan Math. J. 16, 397–407 (1969)
Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Simons S.: Minimax and monotonicity. Lecture Notes in Mathematics, 1693. Springer, Berlin (1998)
Simons S.: Dualized and scaled Fitzpatrick functions. Proc. Am. Math. Soc. 134, 2983–2987 (2006)
Simons S.: From Hahn–Banach to monotonicity. Springer, Berlin (2008)
Simons S., Zălinescu C.: A new proof for Rockafellar’s characterization of maximal monotone operators. Proc. Am. Math. Soc. 132, 2969–2972 (2004)
Simons S., Zălinescu C.: Fenchel duality, Fitzpatrick functions and maximal monotonicity. J. Nonlinear Convex Anal. 6, 1–22 (2005)
Voisei M.D.: A maximality theorem for the sum of maximal monotone operators in non-reflexive Banach spaces. Math. Sci. Res. J. 10(2), 36–41 (2006)
Voisei M.D.: The sum theorem for linear maximal monotone operators. Math. Sci. Res. J. 10(4), 83–85 (2006)
Voisei M.D.: Calculus rules for maximal monotone operators in general Banach space. J. Convex Anal. 15, 73–85 (2008)
Voisei M.D.: The sum and chain rules for maximal monotone operators. Set-Valued Anal. 16, 461– 476 (2008)
Voisei, M.D., Zălinescu, C.: Strongly-representable monotone operators. J. Convex Anal. http://arxiv.org/abs/0802.3640 (2009, to appear)
Zălinescu C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)
Zălinescu C.: A new proof of the maximal monotonicity of the sum using the Fitzpatrick function. In: Giannessi, F., Maugeri, A. (eds) Variational Analysis and Applications, pp. 1159–1172. Springer, New York (2005)
Zălinescu C.: Hahn–Banach extension theorems for multifunctions revisited. Math. Meth. Oper. Res. 68, 493–508 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Voisei, M.D., Zălinescu, C. Maximal monotonicity criteria for the composition and the sum under weak interiority conditions. Math. Program. 123, 265–283 (2010). https://doi.org/10.1007/s10107-009-0314-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-009-0314-5