Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-12T01:38:25.087Z Has data issue: false hasContentIssue false
  • English
  • Français

ON HYPERSTABILITY OF GENERALISED LINEAR FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES

Published online by Cambridge University Press:  02 June 2015

DONG ZHANG
DONG ZHANG*
Affiliation:
LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, PR China email dongzhang@pku.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We obtain some results on approximate solutions of the generalised linear functional equation $\sum _{i=1}^{m}L_{i}f(\sum _{j=1}^{n}a_{ij}x_{j})=0$ for functions mapping a normed space into a normed space. We show that, under suitable assumptions, the approximate solutions are in fact exact solutions. The theorems correspond to and complement recent results on the hyperstability of generalised linear functional equations.

Keywords

linear functional equation with multiple variablesapproximate solutionfixed pointCauchy equationFréchet equationJordan–von Neumann functional equation

MSC classification

Primary: 39B82: Stability, separation, extension, and related topics
Secondary: 39B52: Equations for functions with more general domains and/or ranges 39B62: Functional inequalities, including subadditivity, convexity, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 2 , October 2015 , pp. 259 - 267
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Almahalebi, M., Charifi, A. and Kabbaj, S., ‘Hyperstability of a monomial functional equation’, J. Sci. Res. Rep. 3(20) (2014), Article no. JSRR.2014.20.006.Google Scholar
Aoki, T., ‘On the stability of the linear transformation in Banach spaces’, J. Math. Soc. Japan 2 (1950), 64–66.CrossRefGoogle Scholar
Bahyrycz, A., Brzdek, J., Piszczek, M. and Sikorska, J., ‘Hyperstability of the Fréchet equation and a characterization of inner product spaces’, J. Funct. Spaces Appl. 2013 (2013), 496361.CrossRefGoogle Scholar
Bahyrycz, A. and Olko, J., ‘On stability of the general linear equation’, Aequationes Math., doi:10.1007/s00010-014-0317-z.Google Scholar
Bahyrycz, A. and Piszczek, M., ‘Hyperstability of the Jensen functional equation’, Acta Math. Hungar. 142 (2014), 353–365.CrossRefGoogle Scholar
Brzdek, J., ‘Hyperstability of the Cauchy equation on restricted domains’, Acta Math. Hungar. 141 (2013), 58–67.CrossRefGoogle Scholar
Brzdek, J., ‘Stability of the equation of the p-Wright affine functions’, Aequationes Math. 85 (2013), 497–503.CrossRefGoogle Scholar
Brzdek, J., ‘A hyperstability result for the Cauchy equation’, Bull. Aust. Math. Soc. 89 (2014), 33–40.CrossRefGoogle Scholar
Brzdek, J., ‘Remarks on stability of some inhomogeneous functional equations’, Aequationes Math. 89 (2015), 83–96.CrossRefGoogle Scholar
Brzdȩk, J., Chudziak, J. and Páles, Zs., ‘A fixed point approach to stability of functional equations’, Nonlinear Anal. 74 (2011), 6728–6732.CrossRefGoogle Scholar
Brzdek, J. and Cieplinski, K., ‘Hyperstability and superstability’, Abstr. Appl. Anal. 2013 (2013), 401756.CrossRefGoogle Scholar
Fechner, W., ‘On the Hyers–Ulam stability of functional equations connected with additive and quadratic mappings’, J. Math. Anal. Appl. 322 (2006), 774–786.CrossRefGoogle Scholar
Gajda, Z., ‘On stability of additive mappings’, Int. J. Math. Math. Sci. 14 (1991), 431–434.CrossRefGoogle Scholar
Hyers, D. H., ‘On the stability of the linear functional equation’, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224.CrossRefGoogle ScholarPubMed
Jun, K. and Kim, H., ‘The generalized Hyers–Ulam–Rassias stability of a cubic functional equation’, J. Math. Anal. Appl. 274 (2002), 867–878.CrossRefGoogle Scholar
Jung, S.-M., ‘On the Hyers–Ulam stability of the functional equations that have the quadratic property’, J. Math. Anal. Appl. 222 (1998), 126–137.CrossRefGoogle Scholar
Lee, Y.-H., ‘On the stability of the monomial functional equation’, Bull. Korean Math. Soc. 45 (2008), 397–403.CrossRefGoogle Scholar
Piszczek, M., ‘Remark on hyperstability of the general linear equation’, Aequationes Math. 88 (2014), 163–168.CrossRefGoogle Scholar
Pólya, G. and Szegö, G., Aufgaben und Lehrsätze aus der Analysis I (Springer, Berlin, 1925).Google Scholar
Pólya, G. and Szegö, G., Problems and Theorems in Analysis I (Springer, Berlin, 1972).Google Scholar
Rassias, J. M., ‘On approximation of approximately linear mappings by linear mappings’, J. Funct. Anal. 46 (1982), 126–130.CrossRefGoogle Scholar
Rassias, J. M., ‘On a new approximation of approximately linear mappings by linear mappings’, Discuss. Math. 7 (1985), 193–196.Google Scholar
Rassias, J. M., ‘Solution of the Ulam stability problem for quartic mappings’, Glas. Mat. Ser. III 34(54) (1999), 243–252.Google Scholar
Rassias, J. M., ‘Solution of the Ulam problem for cubic mappings’, An. Univ. Vest Timiş. Ser. Mat.-Inform. 38 (2000), 121–132.Google Scholar
Rassias, J. M., ‘Solution of the Ulam stability problem for quartic mappings’, J. Indian Math. Soc. (N.S.) 67 (2000), 169–178.Google Scholar
Rassias, J. M., ‘Solution of the Ulam stability problem for cubic mappings’, Glas. Mat. Ser. III 36(56) (2001), 63–72.Google Scholar
Rassias, Th. M., ‘On a modified Hyers–Ulam sequence’, J. Math. Anal. Appl. 158 (1991), 106–113.CrossRefGoogle Scholar
Skof, F., ‘Proprietà locali e approssimazione di operatori’, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129.CrossRefGoogle Scholar
Ulam, S. M., Problems in Modern Mathematics, Science Editions (John Wiley, New York, 1964).Google Scholar