Abstract
We show that the lower limit of a sequence of maximal monotone operators on a reflexive Banach space is a representable monotone operator. As a consequence, we obtain that the variational sum of maximal monotone operators and the variational composition of a maximal monotone operator with a linear continuous operator are both representable monotone operators.
Similar content being viewed by others
References
Asplund, E.: Averaged norms. Isr. J. Math. 5, 227–233 (1967)
Attouch, H.: Familles d’opérateurs maximaux monotones et mesurabilité. Ann. Mat. Pura Appl. (4) 120, 35–111 (1979)
Attouch, H., Baillon, J.-B., Théra, M.: Variational sum of monotone operators. J. Convex Anal. 1(1), 1–29 (1994)
Bauschke, H.H., Wang, X., Yao, L.: Monotone linear relations: maximality and Fitzpatrick functions. J. Convex Anal. 16(3–4), 673–686 (2009)
Bauschke, H.H., Wang, X., Yao, L.: Autoconjugate representers for linear monotone operators. Math. Program. 123(1, Ser. B), 5–24 (2010)
Borwein, J.M.: Maximal monotonicity via convex analysis. J. Convex Anal. 13(3–4), 561–586 (2006)
Fitzpatrick, S.: Representing monotone operators by convex functions. In: Workshop/ Miniconference on Functional Analysis and Optimization (Canberra, 1988), Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 20, pp. 59–65. Austral. Nat. Univ., Canberra (1988)
García, Y.: New properties of the variational sum of maximal monotone operators. J. Convex Anal. 16(3–4), 767–778 (2009)
García, Y., Lassonde, M., Revalski, J.P.: Extended sums and extended compositions of monotone operators. J. Convex Anal. 13(3–4), 721–738 (2006)
Martínez-Legaz, J.-E., Svaiter, B.F.: Monotone operators representable by l.s.c. convex functions. Set-Valued Anal. 13(1), 21–46 (2005)
Pennanen, T., Revalski, J.P., Théra, M.: Variational composition of a monotone operator and a linear mapping with applications to elliptic PDEs with singular coefficients. J. Funct. Anal. 198(1), 84–105 (2003)
Penot, J.-P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58(7–8), 855–871 (2004)
Penot, J.-P., Zălinescu, C.: Some problems about the representation of monotone operators by convex functions. ANZIAM J. 47(1), 1–20 (2005)
Phelps, R.R.: Convex functions, monotone operators and differentiability. In: Lecture Notes in Mathematics, vol. 1364, 2nd edn. Springer, Berlin (1993)
Revalski, J.P.: Regularization procedures for monotone operators: recent advances. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optimization and Its Applications, vol. 49 (2011). doi:10.1007/978-1-4419-9569-816
Revalski, J.P., Théra, M.: Generalized sums of monotone operators. C. R. Acad. Sci. Paris Sér. I Math. 329(11), 979–984 (1999)
Revalski, J.P., Théra, M.: Variational and extended sums of monotone operators. In: Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998). Lecture Notes in Economics and Mathematical Systems, vol. 477, pp. 229–246. Springer, Berlin (1999)
Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)
Simons, S.: From Hahn–Banach to monotonicity. In: Lecture Notes in Mathematics, vol. 1693, 2nd edn. Springer, New York (2008)
Simons, S., Zălinescu, C.: Fenchel duality, Fitzpatrick functions and maximal monotonicity. J. Nonlinear Convex Anal. 6(1), 1–22 (2005)
Svaiter, B.F.: Fixed points in the family of convex representations of a maximal monotone operator. Proc. Am. Math. Soc. 131(12), 3851–3859 (electronic) (2003)
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., Zălinescu, C.: Maximal monotonicity criteria for the composition and the sum under weak interiority conditions. Math. Program. 123(1, Ser. B), 265–283 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
García, Y., Lassonde, M. Representable Monotone Operators and Limits of Sequences of Maximal Monotone Operators. Set-Valued Anal 20, 61–73 (2012). https://doi.org/10.1007/s11228-011-0178-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-011-0178-8