# Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators

Radu Ioan Boţ; Sorin-Mihai Grad

Open Mathematics (2011)

- Volume: 9, Issue: 1, page 162-172
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topRadu Ioan Boţ, and Sorin-Mihai Grad. "Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators." Open Mathematics 9.1 (2011): 162-172. <http://eudml.org/doc/269788>.

@article{RaduIoanBoţ2011,

abstract = {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.},

author = {Radu Ioan Boţ, Sorin-Mihai Grad},

journal = {Open Mathematics},

keywords = {Conjugate functions; Subdifferentials; Representative functions; Maximal monotone operators; Surjectivity; conjugate functions; subdifferentials; representative functions; maximal monotone operators; surjectivity},

language = {eng},

number = {1},

pages = {162-172},

title = {Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators},

url = {http://eudml.org/doc/269788},

volume = {9},

year = {2011},

}

TY - JOUR

AU - Radu Ioan Boţ

AU - Sorin-Mihai Grad

TI - Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators

JO - Open Mathematics

PY - 2011

VL - 9

IS - 1

SP - 162

EP - 172

AB - 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.

LA - eng

KW - Conjugate functions; Subdifferentials; Representative functions; Maximal monotone operators; Surjectivity; conjugate functions; subdifferentials; representative functions; maximal monotone operators; surjectivity

UR - http://eudml.org/doc/269788

ER -

## References

top- [1] Attouch H., Baillon J.-B., Théra M., Variational sum of monotone operators, J. Convex Anal., 1994, 1(1), 1–29 Zbl0824.47044
- [2] Attouch H., Théra M., A general duality principle for the sum of two operators, J. Convex Anal., 1996, 3(1), 1–24 Zbl0861.47028
- [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 Zbl1119.47050
- [4] Borwein J.M., Maximal monotonicity via convex analysis, J. Convex Anal., 2006, 13(3–4), 561–586 Zbl1111.47042
- [5] Boţ R.I., Conjugate Duality in Convex Optimization, Lecture Notes in Econom. and Math. Systems, 637, Springer, Berlin, 2010 Zbl1190.90002
- [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 Zbl1119.47051
- [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 Zbl1087.49026
- [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 Zbl1213.47058
- [9] Martínez-Legaz J.-E., Some generalizations of Rockafellar's surjectivity theorem, Pac. J. Optim., 2008, 4(3), 527–535 Zbl1198.47071
- [10] Moudafi A., Oliny M., Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 2003, 155(2), 447–454 Zbl1027.65077
- [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 Zbl0891.49005
- [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) Zbl1217.47094
- [13] Rockafellar R.T., On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 1970, 33(1), 209–216 Zbl0199.47101
- [14] Simons S., From Hahn-Banach to Monotonicity, 2nd ed., Lecture Notes in Math., 1693, Springer, Berlin, 2008 Zbl1131.47050
- [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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.