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Accessible Unlicensed Requires Authentication Published by De Gruyter October 20, 2018

Some results on abstract convexity of functions

Hossein Mohebi and Hasan Barsam
From the journal Mathematica Slovaca


In this paper, we study abstract convexity topical and sub-topical functions. We obtain some results on abstract convexity such as support set and subdifferential in view of new elementary functions. Indeed, first we show that the ICR and IR functions are dense in IPH functions then the topical function and sub-topical function on topological vector space by another elementary function is studied. These elementary functions lead to obtain similar results with easier proofs.

  1. Communicated by Ján Borsík


[1] Barsam, H.—Mohebi, H.: Characterizations of upward and downward sets in semimodules by using topical functions, Numer. Funct. Anal. Optim. 37(11) (2016), 1354–1377.Search in Google Scholar

[2] Doagooei, A. R.: Sub-topical functions and plus-co-radiant set, Optimization 65 (2014), 107–119.Search in Google Scholar

[3] Doagooei, A. R.—Mohebi, H.: Optimization of the difference of topical functions, J. Global Optim. 57 (2013), 1349–1358.Search in Google Scholar

[4] Mohebi, H.: Topical functions and their properties in a class of ordered Banach spaces, Contin. Optim. 99 (2005), 343–360.Search in Google Scholar

[5] Mohebi, H.—Sadeghi, H.: Monotonic analysis over ordered topological vector spaces: I, Optimization 56 (3) (2007), 305–321.Search in Google Scholar

[6] Mohebi, H.—Samet, M.: Abstract convexity of topical functions, J. Global Optim. 58 (2014), 365–375.Search in Google Scholar

[7] Rubinov, A. M.: Abstract Convexity and Global Optimization, Kluwer Acad. Publ./Ister, Boston, Dordrecht, London, 2000.Search in Google Scholar

[8] Rubinov, A. M.—Singer, I.: Topical and sub-topical functions, downward sets and abstract convexity, Optimization 50 (2001), 307–351.Search in Google Scholar

[9] Singer, I.: On radiant sets, downward sets, topical functions and sub-topical functions in lattice ordered groups, Optimization 53 (2004), 393–428.Search in Google Scholar

Received: 2016-11-07
Accepted: 2017-08-08
Published Online: 2018-10-20
Published in Print: 2018-10-25

© 2018 Mathematical Institute Slovak Academy of Sciences