Fitzpatrick transform of monotone relations in Hadamard spaces

A. Moslemipour; M. Roohi; M. R. Mardanbeigi; M. Azhini

doi:10.2478/auom-2020-0026

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Fitzpatrick transform of monotone relations in Hadamard spaces

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 28 (2020): Issue 2 (July 2020)

By: A. Moslemipour, M. Roohi, M. R. Mardanbeigi and M. Azhini

Open Access

|Sep 2020

Abstract

In the present paper, monotone relations and maximal monotone relations from an Hadamard space to its linear dual space are investigated. Fitzpatrick transform of monotone relations in Hadamard spaces is introduced. It is shown that Fitzpatrick transform of a special class of monotone relations is proper, convex and lower semi-continuous. Finally, a representation result for monotone relations is given.

References

[1] B.Ahmadi Kakavandi, Weak topologies in complete CAT(0) metric spaces, Proc. Amer. Math. Soc. 141(2013), 1029–1039.10.1090/S0002-9939-2012-11743-5
Search in Google Scholar Back to article
[2] B. Ahmadi Kakavandi and M. Amini, Duality and subdifferential for convex functions on complete CAT(0) metric spaces, Nonlinear Anal. 73(2010), 3450–3455.10.1016/j.na.2010.07.033
Search in Google Scholar Back to article
[3] M. Bačák, Convex Analysis and Optimization in Hadamard Spaces, Walter de Gruyter, Berlin, (2014).10.1515/9783110361629
Search in Google Scholar Back to article
[4] M.Bačák, Old and new challenges in Hadamard spaces, arXiv:1807.01355.
Search in Google Scholar Back to article
[5] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Second Edition, Springer, (2017).10.1007/978-3-319-48311-5_2
Search in Google Scholar Back to article
[6] I. D. Berg and I. G. Nikolaev, Quasilinearization and curvature of Aleksanderov spaces, Geom. Dedicata. 133(2008), 195–218.10.1007/s10711-008-9243-3
Search in Google Scholar Back to article
[7] J.M.Borwein, Fifty years of maximal monotonicity, Optim. Lett. 4(2010), 473–490.10.1007/s11590-010-0178-x
Search in Google Scholar Back to article
[8] M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundelhern Math. Wiss., Springer, (1999).10.1007/978-3-662-12494-9
Search in Google Scholar Back to article
[9] R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10(2002), 297–316.10.1023/A:1020639314056
Search in Google Scholar Back to article
[10] P.Chaipunya and P.Kumam, On the proximal point method in Hadamard spaces, Optimization 66(2017), 1647–1665.10.1080/02331934.2017.1349124
Search in Google Scholar Back to article
[11] Ş. Cobzaş, R. Miculescu and A. Nicolae, Lipschitz Functions, Springer, (2019).10.1007/978-3-030-16489-8
Search in Google Scholar Back to article
[12] L. M. Elias and J.-E. Martinez-Legaz, A generalization of the strong Fitzpatrick inequality, Optimization 66(6)(2017), 917–923.10.1080/02331934.2016.1179738
Search in Google Scholar Back to article
[13] S. Fitzpatrick, Representing monotone operators by convex functions, n Workshop and Miniconference on Functional Analysis and Optimization (Canberra, 1988), (Austral. Nat. Univ., Canberra, 1988) 59–65.
Search in Google Scholar Back to article
[14] H. Khatibzadeh and S. Ranjbar, Monotone operators and the proximal point algorithm in complete CAT(0) metric spaces, J. Aust. Math. Soc. 103(2017), 70–90.10.1017/S1446788716000446
Search in Google Scholar Back to article
[15] E. Krauss, A representation of arbitrary maximal monotone operators via subgradients of skew-symmetric saddle functions, Nonlinear Anal. 9(12)(1985), 1381–1399.10.1016/0362-546X(85)90097-5
Search in Google Scholar Back to article
[16] E. Krauss, A representation of maximal monotone operators by saddle functions, Rev. Roumaine Math. Pures Appl. 30(10)(1985), 823–837.
Search in Google Scholar Back to article
[17] E. Krauss, Maximal monotone operators and saddle functions, I. Z. Anal. Anwendungen, 5(4)(1986), 333–346.10.4171/ZAA/202
Search in Google Scholar Back to article
[18] J.-E. Martinez-Legaz and B. F. Svaiter, Monotone operators representable by l.s.c. functions, Set-Valued Anal. 13(2005), 21–46.10.1007/s11228-004-4170-4
Search in Google Scholar Back to article
[19] J.-E. Martinez-Legaz and M. Théra, A convex representation of maximal monotone operators, J. Nonlinear Convex Anal. 2(2001), 243–247.
Search in Google Scholar Back to article
[20] A. Moslemipour and M. Roohi, Monotone relations in Hadamard spaces, arXiv:1906.00396.
Search in Google Scholar Back to article
[21] A. Moslemipour and M. Roohi, Monotonicity of sets in Hadamard spaces from polarity point of view, arXiv:1909.01376.
Search in Google Scholar Back to article
[22] M.Movahedi, D.Behmardi and M.Soleimani-Damaneh, On subdifferential in Hadamard spaces, Bull. Iranian Math. Soc. 42(2016), 707–717.
Search in Google Scholar Back to article
[23] A. Papadopoulos, Metric Spaces, Convexity and Non-positive Curvature, European Matimatical Society, (2014).10.4171/132
Search in Google Scholar Back to article
[24] G. Zamani Eskandani and M. Raeisi, On the zero point problem of monotone operators in Hadamard spaces, Numer. Algor. 80(2019), 1155–1179.10.1007/s11075-018-0521-3
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/auom-2020-0026 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784

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Language: English

Page range: 173 - 193

Submitted on: Sep 14, 2019

Accepted on: Nov 22, 2019

Published on: Sep 22, 2020

Published by: Ovidius University of Constanta

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

Monotone relations,

Hadamard spaces,

flat Hadamard spaces,

ℝ-tree space,

Fitzpatrick transform

Related subjects:

Mathematics,

General mathematics

© 2020 A. Moslemipour, M. Roohi, M. R. Mardanbeigi, M. Azhini, published by Ovidius University of Constanta
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 28 (2020): Issue 2 (July 2020)