Journal of Applied Mathematics

Stability of Jensen-Type Functional Equations on Restricted Domains in a Group and Their Asymptotic Behaviors

Jae-Young Chung, Dohan Kim, and John Michael Rassias

Full-text: Open access

Abstract

We consider the Hyers-Ulam stability problems for the Jensen-type functional equations in general restricted domains. The main purpose of this paper is to find the restricted domains for which the functional inequality satisfied in those domains extends to the inequality for whole domain. As consequences of the results we obtain asymptotic behavior of the equations.

Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 691981, 12 pages.

Dates
First available in Project Euclid: 15 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.jam/1331817619

Digital Object Identifier
doi:10.1155/2012/691981

Mathematical Reviews number (MathSciNet)
MR2846449

Zentralblatt MATH identifier
1235.39019

Citation

Chung, Jae-Young; Kim, Dohan; Rassias, John Michael. Stability of Jensen-Type Functional Equations on Restricted Domains in a Group and Their Asymptotic Behaviors. J. Appl. Math. 2012 (2012), Article ID 691981, 12 pages. doi:10.1155/2012/691981. https://projecteuclid.org/euclid.jam/1331817619


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References

  • S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, NY, USA, 1960.
  • D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222–224, 1941.
  • T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.
  • D. G. Bourgin, “Classes of transformations and bordering transformations,” Bulletin of the American Mathematical Society, vol. 57, pp. 223–237, 1951.
  • D. G. Bourgin, “Multiplicative transformations,” Proceedings of the National Academy of Sciences of the United States of America, vol. 36, pp. 564–570, 1950.
  • T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
  • G. L. Forti, “The stability of homomorphisms and amenability, with applications to functional equations,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 57, pp. 215–226, 1987.
  • D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser Boston, Boston, Mass, USA, 1998.
  • S.-M. Jung, “Hyers-Ulam-Rassias stability of Jensen's equation and its application,” Proceedings of the American Mathematical Society, vol. 126, no. 11, pp. 3137–3143, 1998.
  • K.-W. Jun and H.-M. Kim, “Stability problem for Jensen-type functional equations of cubic mappings,” Acta Mathematica Sinica, vol. 22, no. 6, pp. 1781–1788, 2006.
  • G. H. Kim and Y. H. Lee, “Boundedness of approximate trigonometric functional equations,” Appled Mathematics Letters, vol. 31, pp. 439–443, 2009.
  • C. G. Park, “Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras,” Bulletin des Sciences Mathématiques, vol. 132, no. 2, pp. 87–96, 2008.
  • J. M. Rassias and M. J. Rassias, “On the Ulam stability of Jensen and Jensen type mappings on restricted domains,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 516–524, 2003.
  • J. M. Rassias, “On the Ulam stability of mixed type mappings on restricted domains,” Journal of Mathematical Analysis and Applications, vol. 276, no. 2, pp. 747–762, 2002.
  • J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Journal of Functional Analysis, vol. 46, no. 1, pp. 126–130, 1982.
  • F. Skof, “Sull'approssimazione delle applicazioni localmente $\delta -$additive,” Atti della Accademia delle Scienze di Torino Classe di Scienze Fisiche, Matematiche e Naturali, vol. 117, pp. 377–389, 1983.
  • F. Skof, “Local properties and approximation of operators,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113–129, 1983.
  • B. Batko, “Stability of an alternative functional equation,” Journal of Mathematical Analysis and Applica-tions, vol. 339, no. 1, pp. 303–311, 2008.
  • B. Batko, “On approximation of approximate solutions of Dhombres' equation,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 424–432, 2008.
  • J. Brzdęk, “On stability of a family of functional equations,” Acta Mathematica Hungarica, vol. 128, no. 1-2, pp. 139–149, 2010.
  • J. Brzdęk, “On the quotient stability of a family of functional equations,” Nonlinear Analysis, vol. 71, no. 10, pp. 4396–4404, 2009.
  • J. Brzdęk, “On a method of proving the Hyers-Ulam stability of functional equations on restricted do-mains,” The Australian Journal of Mathematical Analysis and Applications, vol. 6, pp. 1–10, 2009.
  • J. Brzdęk and J. Sikorska, “A conditional exponential functional equation and its stability,” Nonlinear Analysis, vol. 72, no. 6, pp. 2923–2934, 2010.
  • S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48, Springer, New York, NY, USA, 2011.
  • J. Sikorska, “Exponential functional equation on spheres,” Applied Mathematics Letters, vol. 23, no. 2, pp. 156–160, 2010.
  • J. Sikorska, “On two conditional Pexider functional equations and their stabilities,” Nonlinear Analysis, vol. 70, no. 7, pp. 2673–2684, 2009.
  • J. Sikorska, “On a Pexiderized conditional exponential functional equation,” Acta Mathematica Hungarica, vol. 125, no. 3, pp. 287–299, 2009.
  • J. Chung, “Stability of functional equations on restricted domains in a group and their asymptotic behaviors,” Computers & Mathematics with Applications, vol. 60, no. 9, pp. 2653–2665, 2010.