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http://dx.doi.org/10.4134/BKMS.2013.50.6.2061

GENERALIZED HYERS-ULAM-RASSIAS STABILITY FOR A GENERAL ADDITIVE FUNCTIONAL EQUATION IN QUASI-β-NORMED SPACES  

Moradlou, Fridoun (Department of Mathematics Sahand University of Technology)
Rassias, Themistocles M. (Department of Mathematics National Technical University of Athens Zografou Campus)
Publication Information
Bulletin of the Korean Mathematical Society / v.50, no.6, 2013 , pp. 2061-2070 More about this Journal
Abstract
In this paper, we investigate the generalized HyersUlam-Rassias stability of the following additive functional equation $$2\sum_{j=1}^{n}f(\frac{x_j}{2}+\sum_{i=1,i{\neq}j}^{n}\;x_i)+\sum_{j=1}^{n}f(x_j)=2nf(\sum_{j=1}^{n}x_j)$$, in quasi-${\beta}$-normed spaces.
Keywords
generalized Hyers-Ulam stability; contractively subadditive; expansively superadditive; quasi-${\beta}$-normed space; (${\beta}$, p)-Banach space;
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1 F. Moradlou, A. Najati, and H. Vaezi, Stability of homomorphisms and derivations on C*-ternary rings associated to an Euler-Lagrange type additive mapping, Results Math. 55 (2009), no. 3-4, 469-486.   DOI
2 F. Moradlou, H. Vaezi, and G. Z. Eskandani, Hyers-Ulam-Rassias stability of a quadratic and additive functional equation in quasi-Banach spaces, Mediterr. J. Math. 6 (2009), no. 2, 233-248.   DOI
3 F. Moradlou, H. Vaezi, and C. Park, Fixed points and stability of an additive functional equation of n-Apollonius type in C*-algebras, Abstr. Appl. Anal. 2008 (2008), Art. ID 672618, 13 pp.
4 F. Moradlou, H. Vaezi, and C. Park, Approximate Euler-Lagrange-Jensen type additive mapping in multi-Banach spaces, A Fixed Point Approach, Commun. Korean Math. Soc. 28 (2013), no. 2, 319-333.   과학기술학회마을   DOI   ScienceOn
5 P. M. Pardalos, P. G. Georgiev, and H. M. Srivastava (eds.), Nonlinear Analysis, Stability, Approximation and Inequalities. In honor of Themistocles M. Rassias on the occasion of his 60th birthday, Springer, New York, 2012.
6 C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), no. 2, 711-720.   DOI   ScienceOn
7 C. Park, On an approximate automorphism on a C*-algebra, Proc. Amer. Math. Soc. 132 (2004), no. 6, 1739-1745.   DOI   ScienceOn
8 C. Park, Homomorphisms between Poisson JC*-algebras, Bull. Braz. Math. Soc. 36 (2005), no. 1, 79-97.   DOI   ScienceOn
9 C. Park and Th. M. Rassias, Hyers-Ulam stability of a generalized Apollonius type quadratic mapping, J. Math. Anal. Appl. 322 (2006), no. 1, 371-381.   DOI   ScienceOn
10 J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, 1989.
11 T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66.   DOI
12 Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, Colloq. Publ., vol. 48, Amer. Math. Soc., Providence, RI, 2000.
13 C. Borelli and G. L. Forti, On a general Hyers-Ulam stability result, Internat. J. Math. Math. Sci. 18 (1995), no. 2, 229-236.   DOI   ScienceOn
14 D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223-237.   DOI
15 L. Cadariu and V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math. 4 (2003), no. 1, Article 4, 7 pp.
16 L. Cadariu and V. Radu, Fixed points and the stability of quadratic functional equations, An. Univ. Timisoara, Ser. Mat.-Inform. 41 (2003), no. 1, 25-48.
17 L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Iteration theory (ECIT '02), 43-52, Grazer Math. Ber., 346, Karl-Franzens-Univ. Graz, Graz, 2004.
18 J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), no. 1, 126-130.   DOI
19 C. Park and Th. M. Rassias, Homomorphisms in C*-ternary algebras and JB*-triples, J. Math. Anal. Appl. 337 (2008), no. 1, 13-20.   DOI   ScienceOn
20 V. Radu, The fixed point alternative and stability of functional equations, Fixed Point Theory IV(1) (2003), no. 1, 91-96.
21 J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), no. 4, 445-446.
22 J. M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989), no. 3, 268-273.   DOI
23 J. M. Rassias, Alternative contraction principle and Ulam stability problem, Math. Sci. Res. J. 9 (2005), no. 7, 190-199.
24 J. M. Rassias and H. M. Kim, Generalized Hyers-Ulam stability for general additive functional equations in quasi-$\beta$-normed spaces, J. Math. Anal. Appl. 356 (2009), no. 1, 302-309.   DOI   ScienceOn
25 Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300.   DOI   ScienceOn
26 Th. M. Rassias, The problem of S. M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), no. 2, 352-378.   DOI   ScienceOn
27 Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), no. 1, 264-284.   DOI   ScienceOn
28 Th. M. Rassias and J. Brzdek (eds.), Functional Equations in Mathematical Analysis, Springer, New York, 2012.
29 P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436.   DOI   ScienceOn
30 S. Czerwik, The stability of the quadratic functional equation, In: Th. M. Rassias, J. Tabor (Eds.), Stability of Mappings of Hyers-Ulam Type, pp. 81-91, Hadronic Press, Florida, 1994.
31 P. Gavruta and L. Gavruta, A new method for the generalized Hyers-Ulam-Rassias stability, Int. J. Nonlinear Anal. Appl. 1 (2010), no. 2, 11-18.
32 D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.   DOI   ScienceOn
33 D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston,Basel, Berlin, 1998.
34 K. Jun and Y. Lee, On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality, Math. Inequal. Appl. 4 (2001), no. 1, 93-118.
35 S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011.
36 S.-M. Jung and J. M. Rassias, A fixed point approach to the stability of a functional equation of the spiral of Theodorus, Fixed Point Theory Appl. 2008 (2008), Art. ID 945010, 7 pp.
37 Pl. Kannappan,Functional Equations and Inequalities with Applications, Springer, New York, 2009.
38 G. H. Kim, On the stability of the quadratic mapping in normed spaces, Int. J. Math. Math. Sci. 25 (2001), no. 4, 217-229.   DOI
39 F. Moradlou, Additive functional inequalities and derivations on Hilbert C*-modules, Glasg. Math. J. 55 (2013), 341-348.   DOI
40 S. Rolewicz, Metric Linear Spaces, PWN-Polish Sci. Publ., Warszawa, Reidel, Dordrecht, 1984.
41 F. Skof, Local properties and approximations of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129.   DOI   ScienceOn
42 S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.