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.
@article{bwmeta1.element.bwnjournal-article-apmv58z2p185bwm,
author = {Tiziana Cardinali and Francesca Papalini},
title = {Some results on stability and on characterization of K-convexity of set-valued functions},
journal = {Annales Polonici Mathematici},
volume = {58},
year = {1993},
pages = {185-192},
zbl = {0786.26016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv58z2p185bwm}
}
Tiziana Cardinali; Francesca Papalini. Some results on stability and on characterization of K-convexity of set-valued functions. Annales Polonici Mathematici, Tome 58 (1993) pp. 185-192. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv58z2p185bwm/
[000] [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.
[001] [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. | Zbl 0815.39009
[002] [3] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86. | Zbl 0549.39006
[003] [4] Z. Daróczy and Z. Páles, Convexity with given infinite weight sequences, Stochastica 11 (1987), 5-12.
[004] [5] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 411-416. | Zbl 0061.26403
[005] [6] D. H. Hyers and S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc. 3 (1952), 821-828. | Zbl 0047.29505
[006] [7] Z. Kominek, A characterization of convex functions in linear spaces, Zeszyty Nauk. Akad. Górniczo-Hutniczej 1277, Opuscula Math. 5 (1989), 71-74.
[007] [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.
[008] [9] C. T. Ng and K. Nikodem, On approximately convex functions, Proc. Amer. Math. Soc., to appear. | Zbl 0823.26006
[009] [10]₁ K. Nikodem, Approximately quasiconvex functions, C. R. Math. Rep. Acad. Sci. Canada 10 (6) (1988), 291-294. | Zbl 0664.26006
[010] [10]₂ K. Nikodem, On some class of midconvex functions, Ann. Polon. Math. 50 (1989), 145-151. | Zbl 0706.39004
[011] [10]₃ K. Nikodem, K-convex and K-concave set-valued functions, Zeszyty Nauk. Politech. Łódz. 559 (Rozprawy Mat. 114) (1989).
[012] [11] H. Rådström, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165-169.
[013] [12] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1970. | Zbl 0193.18401