Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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SOLUTION AND STABILITY OF MIXED TYPE FUNCTIONAL EQUATIONS

  • Jun, Kil-Woung (Department of Mathematics, Chungnam National University) ;
  • Jung, Il-Sook (Department of Mathematics, Chungnam National University) ;
  • Kim, Hark-Mahn (Department of Mathematics, Chungnam National University)
  • Received : 2009.10.10
  • Accepted : 2009.11.11
  • Published : 2009.12.30
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Abstract

In this paper we establish the general solution of the following functional equation with mixed type of quadratic and additive mappings f(mx+y)+f(mx-y)+2f(x)=f(x+y)+f(x-y)+2f(mx), where $m{\geq}2$ is a positive integer, and then investigate the generalized Hyers-Ulam stability of this equation in quasi-Banach spaces.

Keywords

Acknowledgement

Supported by : Chungnam National University

References

  1. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66. https://doi.org/10.2969/jmsj/00210064
  2. Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Vol. 1, Colloq. Publ. 48, Amer. Math. Soc. Providence, 2000.
  3. C. Borelli and G.L. Forti, On a general Hyers-Ulam stability result, Internat. J. Math. Math. Sci. 18 (1995), 229-236. https://doi.org/10.1155/S0161171295000287
  4. D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223-237. https://doi.org/10.1090/S0002-9904-1951-09511-7
  5. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64.
  6. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434.
  7. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  8. A. Najati and M. Moghimi, Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces, J. Math. Anal. Appl. 337 (2008), no. 1, 399–415.
  9. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  10. Th. M. Rassias and P. Semrl, On the behaviour of mappings which do not satisfy Hyers–lam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993.
  11. S. Rolewicz, Metric Linear Spaces, Reidel and Dordrecht, and PWN-Polish Sci. Publ. 1984.
  12. F. Skof, Sull'approssimazione delle applicazioni localmente $\delta$-additive, Atti Accad. Sci. Torino Cl Sci. Fis. Mat. Natur. 117 (1983), 377-389.
  13. S. M. Ulam, A collection of the mathematical problems, Interscience Publ. New York, 1960.