STABILITY OF QUARTIC SET-VALUED FUNCTIONAL EQUATIONS

Title & Authors
STABILITY OF QUARTIC SET-VALUED FUNCTIONAL EQUATIONS
Koh, Heejeong;

Abstract
We will show the general solution of the functional equation \begin{eqnarray}f(x+ay)+f(x-ay)+2(a^2-1)f(x)\\
Keywords
Hyers-Ulam-Rassias stability;quartic mapping;set-valued functional equation;closed and convex subset;cone;fixed point;
Language
English
Cited by
References
1.
K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy, Econometrica 22 (1954), 265-290.

2.
T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66.

3.
J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston (1990).

4.
R. J. Aumann,Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1-12.

5.
N. Brillouet-Belluot, J. Brzdek, and K. Cieplinski, Fixed Point Theory and the Ulam Stability, Abstract and Applied Analysis 2014, Article ID 829419, 16pages (2014).

6.
J. Brzdek, L. Cadariu and K. Cieplinski, On Some Recent Developments in Ulam's Type Stability, Abstract and Applied Analysis 2012, Article ID 716936, 41 pages (2012).

7.
C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, in: Lect. Notes in Math. 580, Springer, Berlin (1977).

8.
J. K. Chung and P. K. Sahoo, On the general solution of a quartic functional equation, Bulletin of the Korean Mathematical Society 40 (2003), no. 4, 565-576.

9.
G. Debreu, Integration of correspondences, in: Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. II, 1966, 351-372, Part I.

10.
C. Hess, Set-valued Integration and Set-valued Probability Theory: an Overview, in: Handbook of Measure Theory, vols. I, II, North-Holland, Amsterdam (2002).

11.
W. Hindenbrand, Core and Equilibria of a Large Economy, Princeton Univ. Press, Princeton (1974).

12.
D. H. Hyers, On the stability of the linear equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.

13.
D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston, Mass, USA (1998).

14.
S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, (2011).

15.
S.-M. Jung and Z.-H. Lee, A xed point approach to the stability of quadratic functional equation with involution, Fixed Point Theory Appl. 2008 Article ID 732086, 11 pages (2008).

16.
E. Klein and A. Thompson, Theory of Correspondence, Wiley, New York (1984).

17.
Y.-S. Lee and S.-Y. Chung, Stability of quartic functional equations in the spaces of generalized functions, Adv. Di . Equa. Article ID 838347, doi:10.1155/2009/838347 (2009).

18.
L. W. McKenzie, On the existence of general equilibrium for a competitive market, Econometrica 27 (1959), 54-71.

19.
B. Margolis and J. B. Diaz, A xed point theorem of the alternative for con- tractions on a generalized complete metric space, Bull. Amer. Math. Soc. 126 74 (1968), 305-309.

20.
K. Nikodem, K-Convex and K-Concave Set-Valued Functions, Zeszyty Naukowe Nr., 559, Lodz (1989).

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

22.
J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glasnik Matematicki Series III 34 (1999), no. 2, 243-252.

23.
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

24.
I. A. Rus, Principles and Applications of Fixed Point Theory, Ed. Dacia, Cluj-Napoca, 1979 (in Romanian).

25.
S. M. Ulam, Problems in Morden Mathematics, Wiley, New York (1960).