Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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A Generalization of the Hyers-Ulam-Rassias Stability of the Pexiderized Quadratic Equations, II

  • Jun, Kil-Woung (Department of Mathematics, Chungnam National University) ;
  • Lee, Yang-Hi (Department of Mathematics Education, Kongju National University of Education)
  • Received : 2005.12.14
  • Published : 2007.03.23
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Abstract

In this paper we prove the Hyers-Ulam-Rassias stability by considering the cases that the approximate remainder ${\varphi}$ is defined by $f(x{\ast}y)+f(x{\ast}y^{-1})-2g(x)-2g(y)={\varphi}(x,y)$, $f(x{\ast}y)+g(x{\ast}y^{-1})-2h(x)-2k(y)={\varphi}(x,y)$, where (G, *) is a group, X is a real or complex Hausdorff topological vector space and f, g, h, k are functions from G into X.

Keywords

References

  1. P. W. Cholewa, Remarks on the stability of functional equations, Aeq. Math. 27(1984), 76-86. https://doi.org/10.1007/BF02192660
  2. S. Czerwik, Functional equations and inequalities in several variables, World Scientific Publ. Co., New Jersey, London, Singapore, Hong Kong, 2002.
  3. S. Czerwik On the stability of the quadratic mapping in the normed space, Abh. Math. Sem. Hamburg, 62(1992), 59-64. https://doi.org/10.1007/BF02941618
  4. S. Czerwik, Stability of functional equations of Ulam-Hyers-Rassias, Hadronic Press, Palm Harbor, 2003.
  5. Z. Gajda, On the stability of additive mappings, Internat. J. Math. and Math. Sci., 14(1991), 431-434. https://doi.org/10.1155/S016117129100056X
  6. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. and Appl., 184(1994), 431-436. https://doi.org/10.1006/jmaa.1994.1211
  7. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A., 27(1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  8. D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, 1998.
  9. W. Jian, Some further generalization of the Hyers-Ulam-Rassias Stability of functional equation, J. Math. Anal. Appl., 263(2001), 406-423. https://doi.org/10.1006/jmaa.2001.7587
  10. K. W. Jun and Y. H. Lee, A Generalization of the Hyers-Ulam-Rassias Stability of the Pexiderized Quadratic equations, J. Math. Anal. Appl., 297(2004), 70-86. https://doi.org/10.1016/j.jmaa.2004.04.009
  11. K. W. Jun and Y. H. Lee, On the Hyers-Ulam-Rassias Stability of a Pexiderized quadratic inequality, Math. Ineq. Appl., 4(2001), 93-118.
  12. K. W. Jun, D. S. Shin and B. D. Kim, On the Hyers-Ulam-Rassias Stability of the Pexider Equation, J. Math. Anal. Appl., 239(1999), 20-29. https://doi.org/10.1006/jmaa.1999.6521
  13. Y. H. Lee and K. W. Jun, A Generalization of the Hyers-Ulam-Rassias Stability of Pexider Equation, J. Math. Anal. Appl., 246(2000), 627-638. https://doi.org/10.1006/jmaa.2000.6832
  14. Y. H. Lee and K. W. Jun, A Note on the Hyers-Ulam-Rassias Stability of Pexider Equation, J. Korean Math. Soc., 37(2000), 111-124.
  15. Y. H. Lee and K. W. Jun, On the Stability of Approximately Additive Mappings, Proc. Amer. Math. Soc., 128(2000), 1361-1369. https://doi.org/10.1090/S0002-9939-99-05156-4
  16. Th. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl., 158(1991), 106-113. https://doi.org/10.1016/0022-247X(91)90270-A
  17. 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
  18. Th. M. Rassias, On the stability of the quadratic functional equation, Mathematica, to appear.
  19. Th. M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. "Babes-Bolyai", to appear.
  20. Th. M. Rassias, Report of the 27th Internat. Symposium on Functional Equations, Aeq. Math., 39(1990), 292-293. Problem 16, $2^{\circ}$, Same report, p. 309.
  21. Th. M. Rassias and P. Semrl, On the behavior of mappings which does not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc., 114(1992), 989-993. https://doi.org/10.1090/S0002-9939-1992-1059634-1
  22. F. Skof, Local properties and approximations of operators, Rend. Sem. Mat. Fis. Milano, 53(1983), 113-129. https://doi.org/10.1007/BF02924890
  23. S. M. Ulam, Problems in Modern Mathematics, Chap. VI, Science eds., Wiley, Newyork, 1960.