Some results on stability and on characterization of K-convexity of set-valued functions

Tiziana Cardinali; Francesca Papalini

Annales Polonici Mathematici (1993)

  • Volume: 58, Issue: 2, page 185-192
  • ISSN: 0066-2216

Abstract

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We present a stability theorem of Ulam-Hyers type for K-convex set-valued functions, and prove that a set-valued function is K-convex if and only if it is K-midconvex and K-quasiconvex.

How to cite

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Tiziana Cardinali, and Francesca Papalini. "Some results on stability and on characterization of K-convexity of set-valued functions." Annales Polonici Mathematici 58.2 (1993): 185-192. <http://eudml.org/doc/262414>.

@article{TizianaCardinali1993,
abstract = {We present a stability theorem of Ulam-Hyers type for K-convex set-valued functions, and prove that a set-valued function is K-convex if and only if it is K-midconvex and K-quasiconvex.},
author = {Tiziana Cardinali, Francesca Papalini},
journal = {Annales Polonici Mathematici},
keywords = {set-valued functions; K-convex (K-midconvex, K-quasiconvex) set-valued functions; Ulam-Hyers stability; -convexity; stability of convex functions; - quasiconvexity},
language = {eng},
number = {2},
pages = {185-192},
title = {Some results on stability and on characterization of K-convexity of set-valued functions},
url = {http://eudml.org/doc/262414},
volume = {58},
year = {1993},
}

TY - JOUR
AU - Tiziana Cardinali
AU - Francesca Papalini
TI - Some results on stability and on characterization of K-convexity of set-valued functions
JO - Annales Polonici Mathematici
PY - 1993
VL - 58
IS - 2
SP - 185
EP - 192
AB - We present a stability theorem of Ulam-Hyers type for K-convex set-valued functions, and prove that a set-valued function is K-convex if and only if it is K-midconvex and K-quasiconvex.
LA - eng
KW - set-valued functions; K-convex (K-midconvex, K-quasiconvex) set-valued functions; Ulam-Hyers stability; -convexity; stability of convex functions; - quasiconvexity
UR - http://eudml.org/doc/262414
ER -

References

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  1. [1] A. Averna e T. Cardinali, Sui concetti di K-convessità (K-concavità) e di K-convessità* (K-concavità*), Riv. Mat. Univ. Parma (4) 16 (1990), 311-330. 
  2. [2] F. A. Behringer, Convexity is equivalent to midpoint convexity combined with strict quasiconvexity, Optimization (ed. K.-H. Elster, Ilmenau, Germany), 24 (1992), 219-228. Zbl0815.39009
  3. [3] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86. Zbl0549.39006
  4. [4] Z. Daróczy and Z. Páles, Convexity with given infinite weight sequences, Stochastica 11 (1987), 5-12. 
  5. [5] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 411-416. Zbl0061.26403
  6. [6] D. H. Hyers and S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc. 3 (1952), 821-828. Zbl0047.29505
  7. [7] Z. Kominek, A characterization of convex functions in linear spaces, Zeszyty Nauk. Akad. Górniczo-Hutniczej 1277, Opuscula Math. 5 (1989), 71-74. 
  8. [8] N. Kuhn, A note on t-convex functions, in: General Inequalities 4 (Proc. Oberwolfach 1983), Internat. Ser. Numer. Math. 71, Birkhäuser, 1984, 269-276. 
  9. [9] C. T. Ng and K. Nikodem, On approximately convex functions, Proc. Amer. Math. Soc., to appear. Zbl0823.26006
  10. [10]₁ K. Nikodem, Approximately quasiconvex functions, C. R. Math. Rep. Acad. Sci. Canada 10 (6) (1988), 291-294. Zbl0664.26006
  11. [10]₂ K. Nikodem, On some class of midconvex functions, Ann. Polon. Math. 50 (1989), 145-151. Zbl0706.39004
  12. [10]₃ K. Nikodem, K-convex and K-concave set-valued functions, Zeszyty Nauk. Politech. Łódz. 559 (Rozprawy Mat. 114) (1989). 
  13. [11] H. Rådström, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165-169. 
  14. [12] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1970. Zbl0193.18401

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