[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.14403/jcms.2013.26.4.757

A FIXED POINT APPROACH TO THE STABILITY OF QUINTIC MAPPINGS IN QUASI β-NORMED SPACES

Koh, Heejeong (Department of Mathematical Education Dankook University)

Publication Information

Journal of the Chungcheong Mathematical Society / v.26, no.4, 2013 , pp. 757-767 More about this Journal

Abstract

We investigate the general solution of the following functional equation and the generalized Hyers-Ulam-Rassias stability problem in quasi ${\beta}$ -normed spaces and then the stability by using alternative fixed point method for the following quintic function $f:X{\rightarrow}Y$ such that f(3x+y)+f(3x-y)+5[f(x+y)+f(x-y)]=4[f(2x+y)+f(2x-y)]+2f(3x)-246f(x), for all $x,y{\in}X$ .

Keywords

Hyers-Ulam-Rassias stability; functional equation; quintic mapping; quasi ${\beta}$ -mormed space; alternative fixed point;

Citations & Related Records

Reference

1	T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66. DOI
2	J. H. Bae and W. G. Park, On the generalized Hyers-Ulam-Rassias stability in Banach modules over a C*-algebra, J. Math. Anal. Appl. 294 (2004), 196-205. DOI ScienceOn
3	Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, Colloq. Publ. vol.48, Amer. Math. Soc., Providence, (2000).
4	I. G. Cho, D. Kang, and H. Koh, Stability Problems of Quintic Mappings in Quasi-fi-Normed Spaces, Journal of Inequalities and Applications, Article ID 368981, 9 pages, 2010 (2010).
5	S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64. DOI ScienceOn
6	S. Cherwik, Functional Equations and Inequalities in Several Variables, World Scientific Publ.Co., New Jersey, London, Singapore, Hong Kong, (2002).
7	Z. Gajda, On the stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431-434. DOI ScienceOn
8	M. E. Gordji, Y. J. Cho, M. B. Ghaemi, and H. Majani, Approximately quintic and sextic mappings formn r-divisible groups into into Serstnev Probabilistic Banach Spaces: Fixed Point Method , Disc. Dynam. in Natural and society, Article ID 572062, 16 pages, 2011 (2011).
9	D. H. Hyers, On the stability of the linear equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. DOI ScienceOn
10	D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Mathematicae, 44 (1992), 125-153. DOI ScienceOn
11	D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston, Basel, Berlin, 1998.
12	S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press,Inc., Florida, 2001.
13	B. Margolis and J. B. Diaz, A fixed point theorem of the alternative for con-tractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305-309. DOI
14	C. Park, J. L. Cui, and M. E. Gordji, Orthogonality and quintic functional equations, Acta Mathematica Sinica, English Series, 29 (2013), no. 7, 1381-1390.
15	J. M. Rassias and H. M. Kim Generalized Hyers.Ulam stability for general additive functional equations in quasi-fi-normed spaces, J. Math. Anal. Appl. 356 (2009), 302-309. DOI ScienceOn
16	Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. DOI ScienceOn
17	Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Applicandae Mathematicae, 62 (2000), 23-130. DOI ScienceOn
18	Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264-284. DOI ScienceOn
19	Th. M. Rassias and K. Shibata, Variational problem of some quadratic functions in complex analysis, J. Math. Anal. Appl. 228 (1998), 234-253. DOI ScienceOn
20	Th. M. Rassias and P. Semrl On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325-338. DOI ScienceOn
21	S. Rolewicz, Metric Linear Spaces, Reidel/PWN-Polish Sci. Publ., Dordrecht, 1984.
22	I. A. Rus, Principles and Appications of Fixed Point Theory, Ed. Dacia, Cluj-Napoca, 1979 (in Romanian).
23	S. M. Ulam, Problems in Morden Mathematics, Wiley, New York, 1960.
24	T. Z. Xu, J. M. Rassias, and W. X. Xu, A generalized mixed quadratic-quartic functional equation, Bull. Malaysian Math. Scien. Soc. 35 (2012), no. 3, 633-649.