STABILITY OF MAPPINGS ON MULTI-NORMED SPACES

DALES, H. G. ; MOSLEHIAN, MOHAMMAD SAL

Glasgow mathematical journal, Tome 49 (2007) no. 2, pp. 321-332

Résumé

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.

Détail

DOI : 10.1017/S0017089507003552

Mots-clés : Primary 39B82, Secondary 39B52, 46B99, 47A99

Bibliographie

[1] 1. Aczél, J., A short course on functional equations (D. Reidel Publ. Co., Dordrecht, 1987). Google Scholar | DOI

[2] 2. Amyari, M. and Moslehian, M. S., Approximately ternary semigroup homomorphisms, Lett. Math. Phys. 77 (2006), 1–9. Google Scholar | DOI

[3] 3. Baak, C. and Moslehian, M. S., Stability of J*-homomorphisms, Nonlinear Anal. 63 (2005), 42–48. Google Scholar | DOI

[4] 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] 5. Czerwik, S., Functional equations and inequalities in several variables (World Scientific, 2002). Google Scholar | DOI

[6] 6. Dales, H. G. and Polyakov, M. E., Multi-normed spaces and multi-Banach algebras (preprint). Google Scholar

[7] 7. Gajda, Z., On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. Google Scholar | DOI

[8] 8. avruta, P. Gu, A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. Google Scholar | DOI

[9] 9. Hyers, D. H., On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. Google Scholar PubMed | DOI

[10] 10. Hyers, D. H., Isac, G. and Rassias, Th. M., Stability of functional equations in several variables (Birkhäuser, Basel, 1998). Google Scholar | DOI

[11] 11. Isac, G. and Rassias, Th. M., Stability of psi-additive mappings: Applications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. Google Scholar | DOI

[12] 12. Jarosz, K., Perturbations of Banach algebras, Lecture Notes in Mathematics, No. 1120 (Springer–Verlag, 1985). Google Scholar | DOI

[13] 13. Johnson, B. E., Approximately multiplicative functionals, J. London Math. Soc. (2) 34 (1986), 489–510. Google Scholar | DOI

[14] 14. Johnson, B. E., Approximately multiplicative maps between Banach algebras, J. London Math. Soc. (2) 37 (1988), 294–316. Google Scholar | DOI

[15] 15. Jung, S.-M., Hyers–Ulam–Rassias stability of functional equations in mathematical analysis (Hadronic Press, Palm Harbor, Florida, 2001). Google Scholar

[16] 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), 305–309. Google Scholar

[17] 17. Mirzavaziri, M. and Moslehian, M. S., A fixed point approach to stability of a quadratic equation, Bull. Brazilian Math. Soc. 37 (2006), 361–376. Google Scholar | DOI

[18] 18. Moslehian, M. S., Approximately vanishing of topological cohomology groups, J. Math. Anal. Appl. 318 (2006), 758–771. Google Scholar | DOI

[19] 19. Rassias, Th. M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. Google Scholar | DOI

[20] 20. Rassias, Th. M., Problem 16, Second report of the 27th International Symposium on Functional Equations, Aequationes Math. 39 (1990), 292–293. Google Scholar

[21] 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), 989–993. Google Scholar | DOI

[22] 22. Ulam, S. M., Problems in modern mathematics, Chapter VI (Wiley, 1964). Google Scholar