Hyers-Ulam--Rassias stability of Pexiderized Cauchy functional equation in 2-Banach spaces

Volume 5, Issue 6, pp 459--465 http://dx.doi.org/10.22436/jnsa.005.06.06

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2033 Downloads
• 3041 Views

Authors

G. Zamani Eskandani - Faculty of Mathematical Science, University of Tabriz, Tabriz, Iran. P. Gavruta - Department of Mathematics, University Politehnica of Timisoara, Piata Victoriei No. 2, 300006 Timisoara, Romania.

Abstract

In this paper, we investigate stability of the Pexiderized Cauchy functional equation in 2-Banach spaces and pose an open problem.

Share and Cite

Keywords

• Linear 2-normed space
• Generalized Hyers-Ulam stability
• Pexiderized Cauchy functional equation .

MSC

•  46BXX
•  39B72
•  47Jxx

References

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


• [2] J. Aczél, J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, (1989)


• [3] Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, Colloq. Publ., vol. 48, Amer. Math. Soc., Providence, RI (2000)


• [4] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, New Jersey, London, Singapore, Hong Kong (2002)


• [5] G. Z. Eskandani, On the Hyers-Ulam-Rassias stability of an additive functional equation in quasi-Banach spaces, J. Math. Anal. Appl. , 345 (2008), 405-409.


• [6] G. Z. Eskandani, P. Gavruta, J. M. Rassias, R. Zarghami, Generalized Hyers-Ulam stability for a general mixed functional equation in quasi-\(\beta\)-normed spaces, Mediterr. J. Math. , 8 (2011), 331-348.


• [7] G. Z. Eskandani, H. Vaezi, Y. N. Dehghan, Stability of mixed additive and quadratic functional equation in non-Archimedean Banach modules, Taiwanese J. Math. , 14 (2010), 1309-1324.


• [8] S. Ghler, 2-metrische Rume und ihre topologische Struktur, Math. Nachr., 26 (1963), 115-148.


• [9] S. Ghler, Lineare 2-normierte Rumen, Math. Nachr. , 28 (1964), 1-43.


• [10] L. Găvruţa, P. Găvruţa, G. Z. Eskandani, Hyers-Ulam stability of frames in Hilbert spaces , Bul. Stiint. Univ. Politeh. Timis. Ser. Mat. Fiz., 55(69), 2 (2010), 60-77.


• [11] P. Găvruţa, On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings, J. Math. Anal. Appl. , 261 (2001), 543-553.


• [12] P. Găvruţa, An answer to question of John M. Rassias concerning the stability of Cauchy equation, Advanced in Equation and Inequality, (1999), 67-71.


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


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


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


• [16] S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathimatical Analysis, Hadronic Press, Palm Harbor (2001)


• [17] S. M. Jung, Asymptotic properties of isometries, J. Math. Anal. Appl. , 276 (2002), 642-653.


• [18] Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math., 27 (1995), 368-372.


• [19] A. Najati, G. Z. Eskandani, Stability of a mixed additive and cubic functional equation in quasi-Banach spaces, J. Math. Anal. Appl. , 342 (2008), 1318-1331.


• [20] C. Park, Homomorphisms between Poisson \(JC^*\)-algebras, Bull. Braz. Math. Soc. , 36 (2005), 79-97.


• [21] W. G. Park, Approximate additive mappings in 2-Banach spaces and related topics, J. Math. Anal. Appl. , 376 (2011), 193-202.


• [22] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Journal of Functional Analysis, 46 (1982), 126-130


• [23] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bulletin des Sciences Mathematiques, 108 (1984), 445-446


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


• [25] Th. M. Rassias , On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. , 158 (1991), 106-113.


• [26] Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. , 251 (2000), 264-284.


• [27] Th. M. Rassias , On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (1) (2000), 23-130.


• [28] Th. M. Rassias, P. Šemrl, On the behaviour of mappings which do not satisfy HyersUlam stability, Proc. Amer. Math. Soc. , 114 (1992), 989-993.


• [29] Th. M. Rassias, J. Tabor, What is left of Hyers-Ulam stability?, J. Natural Geometry, 1 (1992), 65-69.


• [30] K. Ravi, M. Arunkumar, J. M. Rassias, Ulam stability for the orthogonally general Euler-Lagrange type functional equation, Intern. J. Math. Stat., 3 (A08) (2008), 36-46.


• [31] S. Rolewicz, Metric Linear Spaces, PWN-Polish Sci. Publ., Warszawa, Reidel, Dordrecht (1984)


• [32] R. Saadati, Y. J. Cho, J. Vahidi, The stability of the quartic functional equation in various spaces, Comput. Math. Appl. , doi:10.1016/j.camwa.2010.07.034. (2010)


• [33] R. Saadati, S. M. Vaezpour, Y. Cho, A note on the On the stability of cubic mappings and quadratic mappings in random normed spaces, J. Inequal. Appl., Article ID 214530. (2009)


• [34] S. M. Ulam, A Collection of the Mathematical Problems , Interscience Publ. New York, (1960), 431-436.