The sum theorem for maximal monotone operators in reflexive Banach spaces revisited

Authors

  • Mircea D. VOISEI Baltimore, MD-21252, U.S.A. e-mail: mdvoisei@yahoo.com

DOI:

https://doi.org/10.24193/subbmath.2021.1.11

Keywords:

Maximal monotone operator, Minkowski sum.

Abstract

The goal of this note is to present a new shorter proof for the maximal monotonicity of the Minkowski sum of two maximal monotone multi-valued operators defined in a reflexive Banach space under the classical interiority condition involving their domains.

Mathematics Subject Classification (2010): 47H05, 46N10.

References

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Chu, L.-J., On the sum of monotone operators, Michigan Math. J., 43(1996), no. 2, 273-289.

Debrunner, H., Flor, P., Ein Erweiterungssatz fur monotone Mengen, Arch. Math., 15(1964), 445-447.

Fitzpatrick, S., Representing monotone operators by convex functions, In ”Workshop/Miniconference on Functional Analysis and Optimization” (Canberra, 1988), vol. 20 of Proc. Centre Math. Anal. Austral. Nat. Univ., Austral. Nat. Univ., Canberra, 1988, 59-65.

Rockafellar, R.T., On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149(1970), 75-88.

Voisei, M.D., The sum and chain rules for maximal monotone operators, Set-Valued Anal., 16(2008), no. 4, 461-476.

Voisei, M.D., Maximal monotone normal cones in locally convex spaces, J. Math. Anal. Appl., 476(2019), no. 2, 811-818.

Voisei, M.D., Zalinescu, C., Maximal monotonicity criteria for the composition and the sum under weak interiority conditions, Math. Program., 123(2010), no. 1, Ser. B, 265- 283.

Zalinescu, C., Convex Analysis in General Vector Spaces, World Scientific Publishing Co. Inc., River Edge, NJ, 2002.

Zalinescu, C., A new proof of the maximal monotonicity of the sum using the Fitzpatrick function, In: Variational analysis and applications, vol. 79 of Nonconvex Optim. Appl., Springer, New York, 2005, 1159-1172.

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Published

2021-03-30

How to Cite

VOISEI, M. D. (2021). The sum theorem for maximal monotone operators in reflexive Banach spaces revisited. Studia Universitatis Babeș-Bolyai Mathematica, 66(1), 123–126. https://doi.org/10.24193/subbmath.2021.1.11

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