Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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A MEASURE ZERO STABILITY OF A FUNCTIONAL EQUATION ASSOCIATED WITH INNER PRODUCT SPACE

  • Chun, Jaeyoung (Department of Mathematics Kunsan National University) ;
  • Rassias, John Michael (Pedagogical Department E. E., Section of Mathematics and Informatics National and Capodistrian University of Athens)
  • Received : 2016.04.20
  • Published : 2017.03.01
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Abstract

Let X, Y be real normed vector spaces. We exhibit all the solutions $f:X{\rightarrow}Y$ of the functional equation f(rx + sy) + rsf(x - y) = rf(x) + sf(y) for all $x,y{\in}X$, where r, s are nonzero real numbers satisfying r + s = 1. In particular, if Y is a Banach space, we investigate the Hyers-Ulam stability problem of the equation. We also investigate the Hyers-Ulam stability problem on a restricted domain of the following form ${\Omega}{\cap}\{(x,y){\in}X^2:{\parallel}x{\parallel}+{\parallel}y{\parallel}{\geq}d\}$, where ${\Omega}$ is a rotation of $H{\times}H{\subset}X^2$ and $H^c$ is of the first category. As a consequence, we obtain a measure zero Hyers-Ulam stability of the above equation when $f:\mathbb{R}{\rightarrow}Y$.

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References

  1. J. Aczel, Lectures on Functional Equations and Their Applications, Academic Press, new York, 1966.
  2. J. Aczel and J. Dhombres, Functional Equations in Several Variables, vol. 31 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1989.
  3. 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
  4. B. Batko, Stability of an alternative functional equation, J. Math. Anal. Appl. 339 (2008), no. 1, 303-311. https://doi.org/10.1016/j.jmaa.2007.07.001
  5. 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
  6. J. Brzdek, On the quotient stability of a family of functional equations, Nonlinear Anal. 71 (2009), no. 10, 4396-4404. https://doi.org/10.1016/j.na.2009.02.123
  7. J. Brzdek and J. Sikorska, A conditional exponential functional equation and its stability, Nonlinear Anal. 72 (2010), no. 6, 2929-2934.
  8. J. Chung and J. M. Rassias, Quadratic functional equations in a set of Lebesgue measure zero, J. Math. Anal. Appl. 419 (2014), no. 2, 1065-1075. https://doi.org/10.1016/j.jmaa.2014.05.032
  9. A. Gilanyi, Hyers-Ulam stability of monomial functional equations on a general domain, Proc. Natl. Acad. Sci. USA 96 (1999), no. 19, 10588-10590. https://doi.org/10.1073/pnas.96.19.10588
  10. D. H. Hyers, On the stability of the linear functional equations, Proc. Nat. Acad. Sci. USA 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  11. P. Jordan and J. von Neumann, On inner products in linear metric spaces, Ann. of Math. 36 (1935), no. 3, 719-723. https://doi.org/10.2307/1968653
  12. S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), no. 1, 126-137. https://doi.org/10.1006/jmaa.1998.5916
  13. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analisis, Springer, New-York, 2011.
  14. S.-M. Jung, M. S. Moslehian, and P. K. Sahoo, Stability of a generalized Jensen equation on restricted domains, J. Math. Inequal. 4 (2010), no. 2, 191-206.
  15. P. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), no. 3-4, 368-372. https://doi.org/10.1007/BF03322841
  16. A. Najati and S.-M. Jung, Approximately quadratic mappings on restricted domains, J. Inequal. Appl. 2010 (2010), Article ID 503458, 10 pages.
  17. J. C. Oxtoby, Measure and Category, Springer, New-York, 1980.
  18. J. M. Rassias, On the Ulam stability of mixed type mappings on restricted domains, J. Math. Anal. Appl. 281 (2002), no. 2, 747-762.
  19. J. M. Rassias, Asymptotic behavior of mixed type functional equations, Aust. Math. Anal. Appl. 1 (2004), no. 1, 1-21.
  20. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  21. J. Sikorska, On two conditional Pexider functional equations and their stabilities, Nonlinear Analy. 70 (2009), no. 7, 2673-2684. https://doi.org/10.1016/j.na.2008.03.054
  22. F. Skof, Sull' approssimazione delle applicazioni localmente ${\delta}$-additive, Atti della Accademia delle Scienze di Torino, vol. 117, pp. 377-389, 1983.
  23. S. M. Ulam, Problems in Modern Mathematics. Vol. VI, Science Edn. Wiley, New York, 1940.