Browse > Article
http://dx.doi.org/10.14317/jami.2011.29.5_6.1421

GENERALIZED HYERES{ULAM STABILITY OF A QUADRATIC FUNCTIONAL EQUATION WITH INVOLUTION IN QUASI-${\beta}$-NORMED SPACES  

Janfada, Mohammad (Department of Mathematics, Sabzevar Tarbiat Moallem University)
Sadeghi, Ghadir (Department of Mathematics, Sabzevar Tarbiat Moallem University)
Publication Information
Journal of applied mathematics & informatics / v.29, no.5_6, 2011 , pp. 1421-1433 More about this Journal
Abstract
In this paper, using a fixed point approach, the generalized Hyeres-Ulam stability of the following quadratic functional equation $f(x+y+z)+f(x+{\sigma}(y))+f(y+{\sigma}(z))+f(x+{\sigma}(z))=3(f(x)+f(y)+f(z))$ will be studied, where f is a function from abelian group G into a quasi-${\beta}$-normed space and ${\sigma}$ is an involution on the group G. Next, we consider its pexiderized equation of the form $f(x+y+z)+f(x+{\sigma}(y))+f(y+{\sigma}(z))+f(x+{\sigma}(z))=g(x)+g(y)+g(z)$ and its generalized Hyeres-Ulam stability.
Keywords
Hyers-Ulam-Rassias stability; quadratic equation; quasi-${\beta}$-normed spaces;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 G.L. Forti, Hyers-Ulam stability of functional equations in several variables Aequationes Math., Vol. 50, (1995), No. 1-2, 143-190.   DOI   ScienceOn
2 D.H. Hyers and Th.M. Rassias, Approximate homomorphisms, Aequationes Math., Vol. 44, (1992), 125-153.   DOI   ScienceOn
3 D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
4 K.-W. Jun, H.-M. Kim, On the stability of Euler-Lagrange type cubic mappings in quasiBanach spaces, J. Math. Anal. Appl., Vol. 332, (2007), No. 15(2), 1335-1350.
5 S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001.
6 S.-M. Jung, Z.-H. Lee, A Fixed Point Approach to the Stability of Quadratic Functional Equation with Involution, Fixed Point Theory and Applications, Volume 2008, Article ID 732086, 11 pages. doi:10.1155/2008/732086
7 B. Margolis and J. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., Vol. 74, (1968), 305-309.   DOI
8 M.S. Moslehian, Approximately vanishing of topological cohomology groups, J. Math. Anal. Appl., Vol. 318, (2006), No. 2, 758- 771.   DOI   ScienceOn
9 M.S. Moslehian and Gh. Sadeghi, Stability of linear mappings in quasi-Banach modules, Math. Inequal. Appl., Vol. 11, (2008), No. 3, 549557.
10 A. Najati, C. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation, J. Math. Anal. and Appl., Vol. 335, (2007), No. 15(2), 763-778.
11 A. Najati, G. Zamani Eskandani Stability of a mixed additive and cubic functional equation in quasi-Banach spaces, J. Math. Anal. Appl., Vol. 342, (2008), No. 15(2), 1318-1331.
12 S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, Vol. 62, (1992), 59-64.   DOI   ScienceOn
13 S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Palm Harbor, Florida, 2003.
14 M. Eshaghi Gordji, H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, Vol. 71 (2009), No 11-1, 5629-5643.
15 J. Michael Rassias, H.-M. Kim, Generalized HyersUlam stability for general additive functional equations in quasi-$\beta$-normed spaces, J. Math. Anal. Appl., Vol. 356, (2009), No. 1(1), 302-309.   DOI
16 H. Stetker, Functional equations on abelian groups with involution, Aequationes Mathematicae, Vol. 54, (1997), No. 1-2, 144-172.   DOI
17 J. Tabor, , Stability of the Cauchy functional equations in quasi-Banach spaces, Ann. Polon. Math., Vol. 50 (2004), 243-255.
18 G. Zamani Eskandani, On the Hyers-Ulam-Rassias stability of an additive functional equa- tion in quasi-Banach spaces, J. Math. Anal. Appl., Vol. 345, (2008), No. 1(1), 405-409.   DOI
19 Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., Vol. 62, (2000), No. 1, 23-130.   DOI   ScienceOn
20 Th.M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003.
21 J.-H. Bae, K.-W. Jun. On the generalized Hyers-Ulam-Rassias stability of a quadratic func- tional equation, Bull. Korean Math. Soc., Vol. 38 (2001), 325-336.
22 J.-H. Bae and I.-S. Chang, On the Ulam stability problem of a quadratic functional equation, Korean. J. Comput. & Appl. Math. (Series A), Vol. 8, (2001), 561-567.
23 J.-H. Bae and Y.-S. Jung, THE Hyers-Ulam stability of the quadratic functional equations on abelian groups, Bull. Korean Math. Soc. 39, Vol. (2002), No.2, 199-209.   DOI
24 B. Belaid, E. Elhoucien, and Th. M. Rassias, On the genaralized Hyers-Ulam stability of the quadratic functional equation with a general involution, Nonlinear Funct. Anal. Appl., Vol. 12, (2007), 247-262.
25 Y. Benyamini, J. Lindenstraauss, Geometric Nonlinear Analysis, Vol. 1 ,Colloq. Publ., Vol. 48, Amer. Math. Soc., Providence, 2000.
26 L. Cadariu and V. Radu, Fixed points and the stability of Jensens functional equation, J. Ineq. in Pure and Appl. Math., Vol. 4, (2003), No. 1, article 4, 7 pages.
27 J.-H. Bae, On the stability of 3-dimensional quadratic functional equations, Bull. Korean Math. Soc., Vol. 37 (2000), 477-486.