Abstract and Applied Analysis

Generalized Hyers-Ulam Stability of Quadratic Functional Inequality

Hark-Mahn Kim, Kil-Woung Jun, and Eunyoung Son

Full-text: Open access

Abstract

We establish the general solution of the functional inequality f x - y + f y - z + f x - z - 3 f x - 3 f y - 3 f z f x + y + z and then investigate the generalized Hyers-Ulam stability of this inequality in Banach spaces and in non-Archimedean Banach spaces.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 564923, 8 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393448670

Digital Object Identifier
doi:10.1155/2013/564923

Mathematical Reviews number (MathSciNet)
MR3129351

Zentralblatt MATH identifier
1291.39054

Citation

Kim, Hark-Mahn; Jun, Kil-Woung; Son, Eunyoung. Generalized Hyers-Ulam Stability of Quadratic Functional Inequality. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 564923, 8 pages. doi:10.1155/2013/564923. https://projecteuclid.org/euclid.aaa/1393448670


Export citation

References

  • S. M. Ulam, A Collection of Mathematical Problems, Interscience, 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.
  • 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.
  • P. Găvruţa, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994.
  • J. Aczél and J. Dhombres, Functional Equations in Several Variables, vol. 31 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, Mass, USA, 1989.
  • D. H. Hyers and T. M. Rassias, “Approximate homomorphisms,” Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125–153, 1992.
  • F. Skof, “Local properties and approximation of operators,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113–129, 1983.
  • P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76–86, 1984.
  • S. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, pp. 59–64, 1992.
  • N. Brillouët-Belluot, J. Brzd\kek, and K. Ciepliński, “On some recent developments in Ulam's type stability,” Abstract and Applied Analysis, vol. 2012, Article ID 716936, 41 pages, 2012.
  • L. Cǎdariu and V. Radu, “Fixed points and the stability of quadratic functional equations,” Analele Universitǎţii de Vest din Timişoara, vol. 41, no. 1, pp. 25–48, 2003.
  • L. C. Cădariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approch,” Grazer Mathematische Berichte, vol. 346, pp. 43–52, 2004.
  • J. B. Diaz and B. Margolis, “A fixed point theorem of the alternative, for contractions on a generalized complete metric space,” Bulletin of the American Mathematical Society, vol. 74, pp. 305–309, 1968.
  • K. Ciepliński, “Applications of fixed point theorems to the Hyers-Ulam stability of functional equations–-a survey,” Annals of Functional Analysis, vol. 3, no. 1, pp. 151–164, 2012.
  • A. Gilányi, “Eine zur Parallelogrammgleichung äquivalente Ungleichung,” Aequationes Mathematicae, vol. 62, no. 3, pp. 303–309, 2001.
  • J. Rätz, “On inequalities associated with the Jordan-von Neumann functional equation,” Aequationes Mathematicae, vol. 66, no. 1-2, pp. 191–200, 2003.
  • W. Fechner, “Stability of a functional inequality associated with the Jordan-von Neumann functional equation,” Aequationes Mathematicae, vol. 71, no. 1-2, pp. 149–161, 2006.
  • A. Gilányi, “On a problem by K. Nikodem,” Mathematical Inequalities & Applications, vol. 5, no. 4, pp. 707–710, 2002.
  • C. Park, Y. S. Cho, and M.-H. Han, “Functional inequalities associated with Jordan-von Neumann-type additive functional equations,” Journal of Inequalities & Applications, vol. 2007, Article ID 41820, 13 pages, 2007.
  • H.-M. Kim, K.-W. Jun, and E. Son, “Hyers-Ulam stability of Jensen functional inequality in $p$-Banach spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 270954, 16 pages, 2012.
  • S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2011.
  • J.-H. Bae and H.-M. Kim, “On the generalized Hyers-Ulam stability of a quadratic mapping,” Far East Journal of Mathematical Sciences, vol. 3, no. 4, pp. 599–608, 2001.
  • T. M. Rassias, “On characterizations of inner product spaces and generalizations of the H. Bohr inequality,” in Topics in Mathematical Analysis, T. M. Rassias, Ed., vol. 11, pp. 803–819, World Scientific, Singapore, 1989.
  • C. Park, “Fixed points, inner product spaces, and functional equations,” Fixed Point Theory and Applications, vol. 2010, Article ID 713675, 14 pages, 2010.
  • S. Jang, C. Park, and H. A. Kenary, “Fixed points and fuzzy stability of functional equations related to inner product,” Journal of Nonlinear Analysis and Application, vol. 2012, Article ID 00109, 15 pages, 2012.
  • S.-M. Jung and Z.-H. Lee, “A fixed point approach to the stability of quadratic functional equation with involution,” Fixed Point Theory and Applications, vol. 2008, Article ID 732086, 11 pages, 2008.
  • M. S. Moslehian and T. M. Rassias, “Stability of functional equations in non-Archimedean spaces,” Applicable Analysis and Discrete Mathematics, vol. 1, no. 2, pp. 325–334, 2007.
  • Y. J. Cho, C. Park, and R. Saadati, “Functional inequalities in non-Archimedean Banach spaces,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1238–1242, 2010.
  • J. Brzd\kek and K. Ciepliński, “A fixed point theorem and the Hyers-Ulam stability in non-Archimedean spaces,” Journal of Mathematical Analysis and Applications, vol. 400, no. 1, pp. 68–75, 2013.