On the convergence of maximal monotone operators

Penot, Jean-Paul; Zǎlinescu, Constantin

doi:10.1090/S0002-9939-05-08275-4

On the convergence of maximal monotone operators
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Proc. Amer. Math. Soc. 134 (2006), 1937-1946 Request permission

Abstract:

We study the convergence of maximal monotone operators with the help of representations by convex functions. In particular, we prove the convergence of a sequence of sums of maximal monotone operators under a general qualification condition of the Attouch–Brezis type.

References

H. Attouch, A. Moudafi, and H. Riahi, Quantitative stability analysis for maximal monotone operators and semi-groups of contractions, Nonlinear Anal. 21 (1993), no. 9, 697–723. MR 1246288, DOI 10.1016/0362-546X(93)90065-Z
Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843, DOI 10.1007/978-94-010-1537-0
Gerald Beer, Topologies on closed and closed convex sets, Mathematics and its Applications, vol. 268, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1269778, DOI 10.1007/978-94-015-8149-3
H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland Mathematics Studies. 5. Notas de matematica (50). North-Holland, Amsterdam (1973).
Felix E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1976, pp. 1–308. MR 0405188
Regina Sandra Burachik and B. F. Svaiter, Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10 (2002), no. 4, 297–316. MR 1934748, DOI 10.1023/A:1020639314056
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
Juan-Enrique Martinez-Legaz and Michel Théra, A convex representation of maximal monotone operators, J. Nonlinear Convex Anal. 2 (2001), no. 2, 243–247. Special issue for Professor Ky Fan. MR 1848704
Teemu Pennanen, Dualization of generalized equations of maximal monotone type, SIAM J. Optim. 10 (2000), no. 3, 809–835. MR 1774774, DOI 10.1137/S1052623498340448
Teemu Pennanen, Julian P. Revalski, and Michel Théra, Graph-distance convergence and uniform local boundedness of monotone mappings, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3721–3729. MR 1998179, DOI 10.1090/S0002-9939-03-07179-X
T. Pennanen, R. T. Rockafellar, and M. Théra, Graphical convergence of sums of monotone mappings, Proc. Amer. Math. Soc. 130 (2002), no. 8, 2261–2269. MR 1896407, DOI 10.1090/S0002-9939-02-06450-X
Jean-Paul Penot, A representation of maximal monotone operators by closed convex functions and its impact on calculus rules, C. R. Math. Acad. Sci. Paris 338 (2004), no. 11, 853–858 (English, with English and French summaries). MR 2059661, DOI 10.1016/j.crma.2004.03.017
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
Jean-Paul Penot and Constantin Zălinescu, Continuity of usual operations and variational convergences, Set-Valued Anal. 11 (2003), no. 3, 225–256. MR 1992063, DOI 10.1023/A:1024432532388
J.-P. Penot and C. Zălinescu, Bounded (Hausdorff) convergence: basic facts and applications, preprint, October 2003; in Variational Analysis and Applications, F. Giannessi and A. Maugeri, Eds., Kluwer Acad. Publ., Dordrecht, 2005, 827–854.
Jean-Paul Penot and Constantin Zălinescu, Continuity of the Legendre-Fenchel transform for some variational convergences, Optimization 53 (2004), no. 5-6, 549–562. MR 2115270, DOI 10.1080/02331930412331329533
Jean-Paul Penot and Constantin Zălinescu, Bounded convergence for perturbed minimization problems, Optimization 53 (2004), no. 5-6, 625–640. MR 2115995, DOI 10.1080/02331930412331328570
Jean-Paul Penot and Constantin Zălinescu, Some problems about the representation of monotone operators by convex functions, ANZIAM J. 47 (2005), no. 1, 1–20. MR 2159848, DOI 10.1017/S1446181100009731
Stephen Simons, Minimax and monotonicity, Lecture Notes in Mathematics, vol. 1693, Springer-Verlag, Berlin, 1998. MR 1723737, DOI 10.1007/BFb0093633
S. Simons and C. Zălinescu, Fenchel duality, Fitzpatrick functions and maximal monotonicity, J. Nonlinear Convex Anal. 6 (2005), 1–22.
C. Zălinescu, A new proof of the maximal monotonicity of the sum using the Fitzpatrick function, in Variational Analysis and Applications, F. Giannessi and A. Maugeri, Eds., Kluwer Acad. Publ., Dordrecht, 2005, 1159–1172.
Eberhard Zeidler, Nonlinear functional analysis and its applications. II/B, Springer-Verlag, New York, 1990. Nonlinear monotone operators; Translated from the German by the author and Leo F. Boron. MR 1033498, DOI 10.1007/978-1-4612-0985-0