In this note we provide regularity conditions of closedness type which guarantee some surjectivity results concerning the sum of two maximal monotone operators by using representative functions. The first regularity condition we give guarantees the surjectivity of the monotone operator S(· + p) + T(·), where p ɛ X and S and T are maximal monotone operators on the reflexive Banach space X. Then, this is used to obtain sufficient conditions for the surjectivity of S + T and for the situation when 0 belongs to the range of S + T. Several special cases are discussed, some of them delivering interesting byproducts.
@article{bwmeta1.element.doi-10_2478_s11533-010-0083-7,
author = {Radu Ioan Bo\c t and Sorin-Mihai Grad},
title = {Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {162-172},
zbl = {1268.47063},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0083-7}
}
Radu Ioan Boţ; Sorin-Mihai Grad. Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators. Open Mathematics, Tome 9 (2011) pp. 162-172. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0083-7/
[1] Attouch H., Baillon J.-B., Théra M., Variational sum of monotone operators, J. Convex Anal., 1994, 1(1), 1–29 | Zbl 0824.47044
[2] Attouch H., Théra M., A general duality principle for the sum of two operators, J. Convex Anal., 1996, 3(1), 1–24 | Zbl 0861.47028
[3] Bartz S., Bauschke H.H., Borwein J.M., Reich S., Wang X., Fitzpatrick functions, cyclic monotonicity and Rockafellar's antiderivative, Nonlinear Anal., 2007, 66(5), 1198–1223 http://dx.doi.org/10.1016/j.na.2006.01.013 | Zbl 1119.47050
[4] Borwein J.M., Maximal monotonicity via convex analysis, J. Convex Anal., 2006, 13(3–4), 561–586 | Zbl 1111.47042
[5] Boţ R.I., Conjugate Duality in Convex Optimization, Lecture Notes in Econom. and Math. Systems, 637, Springer, Berlin, 2010 | Zbl 1190.90002
[6] Boţ R.I., Grad S.-M., Wanka G., Weaker constraint qualifications in maximal monotonicity, Numer. Funct. Anal. Optim., 2007, 28(1–2), 27–41 | Zbl 1119.47051
[7] Boţ R.I., Wanka G., A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces, Nonlinear Anal., 2006, 64(12), 2787–2804. http://dx.doi.org/10.1016/j.na.2005.09.017 | Zbl 1087.49026
[8] Marques Alves M., Svaiter B.F., On the surjectivity properties of perturbations of maximal monotone operators in non-reflexive Banach spaces, J. Convex Anal., 2011, 18(1), 209–226 | Zbl 1213.47058
[9] Martínez-Legaz J.-E., Some generalizations of Rockafellar's surjectivity theorem, Pac. J. Optim., 2008, 4(3), 527–535 | Zbl 1198.47071
[10] Moudafi A., Oliny M., Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 2003, 155(2), 447–454 | Zbl 1027.65077
[11] Moudafi A., Théra M., Finding a zero of the sum of two maximal monotone operators, J. Optim. Theory Appl., 1997, 94(2), 425–448 http://dx.doi.org/10.1023/A:1022643914538 | Zbl 0891.49005
[12] Rocco M., Martínez-Legaz J.-E., On surjectivity results for maximal monotone operators of type (D), J. Convex Anal., 2011, 18(2) (in press) | Zbl 1217.47094
[13] Rockafellar R.T., On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 1970, 33(1), 209–216 | Zbl 0199.47101
[14] Simons S., From Hahn-Banach to Monotonicity, 2nd ed., Lecture Notes in Math., 1693, Springer, Berlin, 2008 | Zbl 1131.47050
[15] Zăalinescu C., Convex Analysis in General Vector Spaces, World Scientific, River Edge, 2002 http://dx.doi.org/10.1142/9789812777096
[16] Zălinescu C., A new convexity property for monotone operators, J. Convex Anal., 2006, 13(3–4), 883–887