## Taiwanese Journal of Mathematics

### JENSEN'S FUNCTIONAL EQUATION IN MULTI-NORMED SPACES

#### Abstract

We investigate the Hyers-Ulam stability of the Jensen functional equation for mappings from linear spaces into multi-normed spaces. We then establish an asymptotic behavior of the Jensen equation in the framework of multi-normed spaces which are somewhat similar to the operator sequence spaces and have some connections with operator spaces and Banach lattices.

#### Article information

Source
Taiwanese J. Math., Volume 14, Number 2 (2010), 453-462.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405801

Digital Object Identifier
doi:10.11650/twjm/1500405801

Mathematical Reviews number (MathSciNet)
MR2655781

Zentralblatt MATH identifier
1204.39030

#### Citation

Moslehian, M. S.; Srivastava, H. M. JENSEN'S FUNCTIONAL EQUATION IN MULTI-NORMED SPACES. Taiwanese J. Math. 14 (2010), no. 2, 453--462. doi:10.11650/twjm/1500405801. https://projecteuclid.org/euclid.twjm/1500405801

#### References

• T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950) 64-66.
• D.-H. Boo, S.-Q. Oh, C.-G. Park and J.-M. Park, Generalized Jensen's equations in Banach modules over a $C^*$-algebra and its unitary group, Taiwanese J. Math., 7 (2003), 641-655.
• S.,Czerwik, Functional, Equations, and, Inequalities, in, Several, Variables, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2002.
• V. Faĭ ziev and P. K. Sahoo, On the stability of Jensen's functional equation on groups, Proc. Indian Acad. Sci. Math. Sci., 117 (2007), 31-48.
• H. G. Dales and M. S. Moslehian, Stability of mappings on multi-normed spaces, Glasgow Math. J., 49 (2007), 321-332.
• H. G. Dales and M. E. Polyakov, Multi-normed spaces and multi-Banach algebras, Preprint 2008.
• D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222-224.
• D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, 1998.
• S.-M. Jung, Hyers-Ulam-Rassias stability of Jensen's equation and its application, Proc. Amer. Math. Soc., 126 (1998), 3137-3143.
• S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Florida, 2001.
• Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math., 22 (1989), 499-507.
• L. Li, J. Chung and D. Kim, Stability of Jensen equations in the space of generalized functions, J. Math. Anal. Appl., 299 (2004), 578-586.
• Y.-H. Lee and K.-W. Jun, A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation, J. Math. Anal. Appl., 238 (1999), 305-315.
• M., S., Moslehian, and, L. Székelyhidi, Stability,of, ternary, homomorphisms, via, generalized Jensen equation, Results Math., 49 (2006), 289-300.
• Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.
• Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (2000), 23-130.
• Th. M. Rassias (Editor), Functional Equations$,$ Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003.
• F. Skof, Sulle approssimazione delle applicazioni localmente $\delta$-additive, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 117 (1983), 377-389.
• S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science Editions, John Wiley and Sons, New York, 1964.