Abstract

Let YY be a real vector metric space and K⊂YK\subset Y be a closed convex cone with K∩(−K)={0}K\cap (-K)=\{0\}. We study properties of set-valued maps F ⁣:R→2Y∖{∅}F\colon\mathbb{R}\to 2^Y\setminus\{\emptyset\} which are additive modulo KK, i.e. F(x+y)+K=F(x)+F(y)+KF(x+y)+K=F(x)+F(y)+K for x,y∈Rx,y\in \mathbb{R}, and satisfy condition F(xy)+K=xF(y)+yF(x)+KF(xy)+K=xF(y)+yF(x)+K for x,y∈[0,∞)x,y\in [0,\infty) (or x,y∈Rx,y\in \mathbb{R}). Such maps are called set-valued derivations modulo KK and generalize the well-known single-valued derivations of R\mathbb{R}.

Author information

Contact details are reproduced from the original publication and may be historical.

Eliza Jabłońska

Faculty of Applied Mathematics, AGH University of Krakow, Kraków, Poland

elizajab@agh.edu.pl

Keywords

  • K-additive set-valued map
  • (strong) set-valued K-derivation
  • K-lower boundedness
  • weak K-upper boundedness
  • K-continuity
  • null-finite set

Mathematics Subject Classification

39B62; 54C60, 26B25

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Suggested citation

E. Jabłońska. “On Set-Valued Derivations Modulo K.” Journal of Convex Analysis 32 (2025), No. 4, 1135–1144. https://doi.org/10.68381/jca32060

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press.