Asymptotic aspect of Jensen and Jensen type functional equations in multi-normed spaces

Volume 8, Issue 4, pp 402--411 http://dx.doi.org/10.22436/jnsa.008.04.12

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Authors

Zhihua Wang - School of Science, Hubei University of Technology, Wuhan, Hubei 430068, P. R. China.

Abstract

In this paper, we investigate the Hyers-Ulam stability of additive functional equations of two forms: of ''Jensen'' and ''Jensen type'' in the framework of multi-normed spaces. We therefore provide a link between multi-normed spaces and functional equations. More precisely, we establish the Hyers-Ulam stability of functional equations of these types for mappings from Abelian groups into multi-normed spaces. We also prove the stability on a restricted domain and discuss an asymptotic behavior of functional equations of these types in the framework of multi-normed spaces.

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Keywords

• Hyers-Ulam stability
• Jensen and Jensen type functional equations
• Multi-normed spaces
• Asymptotic behavior.

MSC

•  39B52
•  58K55

References

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


• [2] R. P. Agarwal, B. Xu, W. Zhang, Stability of functional equations in single variable, J. Math. Anal. Appl., 288 (2003), 852-869.


• [3] M. Amyari, M. S. Moslehian, Approximately ternary semigroup homomorphisms, Lett. Math. Phys., 77 (2006), 1-9.


• [4] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66.


• [5] J. Brzdȩk, K. Ciepliński , Remarks on the Hyers-Ulam stability of some systems of functional equations, Appl. Math. Comput., 219 (2012), 4096-4105.


• [6] J. Brzdȩk, K. Ciepliński , A fixed point theorem and the Hyers-Ulam stability in non-Archimedean spaces, J. Math. Anal. Appl., 400 (2013), 68-75.


• [7] H. G. Dales, M. S. Moslehian, Stability of mappings on multi-normed spaces, Glasg. Math. J., 49 (2007), 321-332.


• [8] H. G. Dales, M. E. Polyakov , Multi-normed spaces, Dissertationes Math. (Rozprawy Mat.), 488 (2012), 165 pages.


• [9] G. L. Forti , Hyers-Ulam stability of functional equations in several variables, Aequationes Math., 50 (1995), 143-190.


• [10] P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings , J. Math. Anal. Appl., 184 (1994), 431-436.


• [11] D. H. Hyers , On the stability of the linear functional equation , Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222-224.


• [12] D. H. Hyers, G. Isac, Th. M. Rassias , Stability of Functional Equations in Several Variables, Birkhäuser, Basel (1998)


• [13] S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Science, New York (2011)


• [14] Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer Science, New York (2009)


• [15] M. S. Moslehian, Approximately vanishing of topological cohomology groups, J. Math. Anal. Appl., 318 (2006), 758-771.


• [16] M. S. Moslehian , Superstability of higher derivatios in multi-Banach algebras , Tamsui Oxford J. Math. Sci., 24 (2008), 417-427.


• [17] M. S. Moslehian, K. Nikodem, D. Popa, Asymptotic aspect of the quadratic functional equation in multi-normed spaces, J. Math. Anal. Appl., 355 (2009), 717-724.


• [18] A. Najati, C. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the pexiderized Cauchy functional equation, J. Math. Anal. Appl., 335 (2007), 763-778.


• [19] A. Najati, M. B. Moghimi , Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces, J. Math. Anal. Appl., 337 (2008), 399-415.


• [20] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.


• [21] J. M. Rassias, M. J. Rassias, On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. Math. Anal. Appl., 281 (2003), 516-524.


• [22] S. M. Ulam , Problems in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York (1964)


• [23] T. Xu, J. M. Rassias, W. Xu , Intuitionistic fuzzy stability of a general mixed additive-cubic equation, J. Math. Phys., 51 (2010), 21 pages.