ON CHARACTERIZATIONS OF SET-VALUED DYNAMICS

Title & Authors
ON CHARACTERIZATIONS OF SET-VALUED DYNAMICS
Chu, Hahng-Yun; Yoo, Seung Ki;

Abstract
In this paper, we generalize the stability for an n-dimensional cubic functional equation in Banach space to set-valued dynamics. Let $\small{n{\geq}2}$ be an integer. We define the n-dimensional cubic set-valued functional equation given by f(2{{\sum}_{i
Keywords
Hyers-Ulam stability;n-dimensional cubic set-valued functional equation;
Language
English
Cited by
References
1.
T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66.

2.
J. Brzdek, Stability of additivity and fixed point methods, Fixed Point Theory Appl. 2013 (2013), 285, 9 pp.

3.
J. Brzdek, Note on stability of the Cauchy equation - an answer to a problem of Th. M. Rassias, Carpathian J. Math. 30 (2014), no. 1, 47-54.

4.
J. Brzdek and M. Piszczek, Selections of set-valued maps satisfying some inclusions and the Hyers-Ulam stability, Handbook of functional equations, 83-100, Springer Optim. Appl. 96, Springer, New York, 2014.

5.
C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lec. Notes in Math., 580, Springer, Berlin, 1977.

6.
H.-Y. Chu, A. Kim, and S. K. Yoo, On the stability of the generalized cubic set-valued functional equation, Appl. Math. Lett. 37 (2014), 7-14.

7.
P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436.

8.
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27 (1941), 222-224.

9.
K.-W. Jun and H.-M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), no. 2, 267-278.

10.
Y.-S. Jung and I.-S. Chang, The stability of a cubic type functional equation with the fixed point alternative, J. Math. Anal. Appl. 306 (2005), no. 2, 752-760.

11.
S.-M. Jung and Z.-H. Lee, A fixed point approach to the stability of quadratic functional equation with involution, Fixed Point Theory Appl. 2008 (2008), Article ID 732086, 11 pages.

12.
D. S. Kang and H.-Y. Chu, Stability problem of Hyers-Ulam-Rassias for generalized forms of cubic functional equation, Acta Math. Sin. (Eng;. Ser.) 24 (2008), no. 3, 491-502.

13.
H. A. Kenary, H. Rezaei, Y. Gheisari, and C. Park, On the stability of set-valued functional equations with the fixed point alternative, Fixed Point Theory Appl. 2012 (2012), 81, 17 pp.

14.
G. Lu and C. Park, Hyers-Ulam stability of additive set-valued functional euqations, Appl. Math. Lett. 24 (2011), no. 8, 1312-1316.

15.
B. Margolis and J. B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 126 (1968), 305-309.

16.
C. Park, D. O'Regan, and R. Saadati, Stabiltiy of some set-valued functional equations, Appl. Math. Lett. 24 (2011), 1910-1914.

17.
M. Piszczek, The properties of functional inclusions and Hyers-Ulam stability, Aequationes Math. 85 (2013), no. 1-2, 111-118.

18.
M. Piszczek, On selections of set-valued inclusions in a single variable with applications to several variables, Results Math. 64 (2013), no. 1-2, 1-12.

19.
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300.

20.
Th.M. Rassias(Ed.), Handbook of Functional Equations, Springer Optim. Appl. 96, Springer, New York, 2014.

21.
H. Radstrom, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165-169.

22.
S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.