Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T06:21:57.830Z Has data issue: false hasContentIssue false

STABILITY OF MAPPINGS ON MULTI-NORMED SPACES

Published online by Cambridge University Press:  09 August 2007

H. G. DALES
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom e-mail: garth@maths.leeds.ac.uk
MOHAMMAD SAL MOSLEHIAN
Affiliation:
Department of Mathematics, Ferdowsi University, P.O. Box 1159, Mashhad 91775, Iran. e-mail: moslehian@ferdowsi.um.ac.ir
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.

In this paper, we define multi-normed spaces, and investigate some properties of multi-bounded mappings on multi-normed spaces. Moreover, we prove a generalized Hyers–Ulam–Rassias stability theorem associated to the Cauchy additive equation for mappings from linear spaces into multi-normed spaces.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

REFERENCES

1. Aczél, J., A short course on functional equations (D. Reidel Publ. Co., Dordrecht, 1987).CrossRefGoogle Scholar
2. Amyari, M. and Moslehian, M. S., Approximately ternary semigroup homomorphisms, Lett. Math. Phys. 77 (2006), 19.CrossRefGoogle Scholar
3. Baak, C. and Moslehian, M. S., Stability of J*-homomorphisms, Nonlinear Anal. 63 (2005), 4248.CrossRefGoogle Scholar
4. adariu, L. Cu and Radu, V., Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math. 4 (2003), no. 1, art. 4, 7 pp.Google Scholar
5. Czerwik, S., Functional equations and inequalities in several variables (World Scientific, 2002).CrossRefGoogle Scholar
6. Dales, H. G. and Polyakov, M. E., Multi-normed spaces and multi-Banach algebras (preprint).Google Scholar
7. Gajda, Z., On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431434.CrossRefGoogle Scholar
8. avruta, P. Gu, A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431436.CrossRefGoogle Scholar
9. Hyers, D. H., On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222224.CrossRefGoogle ScholarPubMed
10. Hyers, D. H., Isac, G. and Rassias, Th. M., Stability of functional equations in several variables (Birkhäuser, Basel, 1998).CrossRefGoogle Scholar
11. Isac, G. and Rassias, Th. M., Stability of psi-additive mappings: Applications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219228.CrossRefGoogle Scholar
12. Jarosz, K., Perturbations of Banach algebras, Lecture Notes in Mathematics, No. 1120 (Springer–Verlag, 1985).CrossRefGoogle Scholar
13. Johnson, B. E., Approximately multiplicative functionals, J. London Math. Soc. (2) 34 (1986), 489510.CrossRefGoogle Scholar
14. Johnson, B. E., Approximately multiplicative maps between Banach algebras, J. London Math. Soc. (2) 37 (1988), 294316.CrossRefGoogle Scholar
15. Jung, S.-M., Hyers–Ulam–Rassias stability of functional equations in mathematical analysis (Hadronic Press, Palm Harbor, Florida, 2001).Google Scholar
16. Margolis, B. and Diaz, J. B., A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 126 (1968), 305309.Google Scholar
17. Mirzavaziri, M. and Moslehian, M. S., A fixed point approach to stability of a quadratic equation, Bull. Brazilian Math. Soc. 37 (2006), 361376.CrossRefGoogle Scholar
18. Moslehian, M. S., Approximately vanishing of topological cohomology groups, J. Math. Anal. Appl. 318 (2006), 758771.CrossRefGoogle Scholar
19. Rassias, Th. M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297300.CrossRefGoogle Scholar
20. Rassias, Th. M., Problem 16, Second report of the 27th International Symposium on Functional Equations, Aequationes Math. 39 (1990), 292293.Google Scholar
21. Rassias, Th. M. and Semrl, P., On the behaviour of mappings which do not satisfy Hyers–Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989993.CrossRefGoogle Scholar
22. Ulam, S. M., Problems in modern mathematics, Chapter VI (Wiley, 1964).Google Scholar