Abstract and Applied Analysis

Stability of a Functional Equation Deriving from Quadratic and Additive Functions in Non-Archimedean Normed Spaces

Abasalt Bodaghi and Sang Og Kim

Full-text: Open access

Abstract

We obtain the general solution of the generalized mixed additive and quadratic functional equation f x + m y + f x m y = 2 f x 2 m 2 f y + m 2 f 2 y , m is even; f x + y + f x y 2 m 2 1 f y + m 2 1 f 2 y , m is odd, for a positive integer m . We establish the Hyers-Ulam stability for these functional equations in non-Archimedean normed spaces when m is an even positive integer or m = 3 .

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 198018, 10 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393448673

Digital Object Identifier
doi:10.1155/2013/198018

Mathematical Reviews number (MathSciNet)
MR3121484

Zentralblatt MATH identifier
1291.39039

Citation

Bodaghi, Abasalt; Kim, Sang Og. Stability of a Functional Equation Deriving from Quadratic and Additive Functions in Non-Archimedean Normed Spaces. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 198018, 10 pages. doi:10.1155/2013/198018. https://projecteuclid.org/euclid.aaa/1393448673


Export citation

References

  • S. M. Ulam, Problems in Modern Mathematics, Science Editions, chapter 6, John Wiley & Sons, New York, NY, USA, 1940.
  • 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.
  • 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.
  • St. 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.
  • J. R. Lee, J. S. An, and C. Park, “On the stability of quadratic functional equations,” Abstract and Applied Analysis, vol. 2008, Article ID 628178, 8 pages, 2008.
  • M. E. Gordji and A. Bodaghi, “On the Hyers-Ulam-Rassias stability problem for quadratic functional equations,” East Journal on Approximations, vol. 16, no. 2, pp. 123–130, 2010.
  • L. Cădariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,” in Iteration theory (ECIT '02), vol. 346 of Grazer Mathematische Berichte, pp. 43–52, Karl-Franzens-Universität Graz, Graz, Austria, 2004.
  • L. Cǎdariu and V. Radu, “Fixed points and the stability of quadratic functional equations,” Analele Universităţii de Vest din Timişoara, Seria Matematică-Informatică, vol. 41, no. 1, pp. 25–48, 2003.
  • 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.
  • 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.
  • 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.
  • G. Z. Eskandani, H. Vaezi, and Y. N. Dehghan, “Stability of a mixed additive and quadratic functional equation in non-Archimedean Banach modules,” Taiwanese Journal of Mathematics, vol. 14, no. 4, pp. 1309–1324, 2010.
  • A. Najati and M. B. Moghimi, “Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 399–415, 2008.
  • S. S. Jin and Y. H. Lee, “On the stability of the functional equation deriving from quadratic and additive function in random normed spaces via fixed point method,” Journal of the Chungcheong Mathematical Society, vol. 25, no. 1, pp. 51–63, 2012.
  • 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.
  • K. Hensel, “Uber eine neue Begrndung der Theorie der algebraischen Zahlen,” Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 6, no. 3, pp. 83–88, 1897.
  • L. M. Arriola and W. A. Beyer, “Stability of the Cauchy functional equation over $p$-adic fields,” Real Analysis Exchange, vol. 31, no. 1, pp. 125–132, 2005.
  • J. Brzd\kek and K. Ciepliński, “A fixed point approach to the stability of functional equations in non-Archimedean metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 18, pp. 6861–6867, 2011.
  • 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.
  • J. Schwaiger, “Functional equations for homogeneous polynomials arising from multilinear mappings and their stability,” Annales Mathematicae Silesianae, no. 8, pp. 157–171, 1994.
  • T. Z. Xu, “Stability of multi-Jensen mappings in non-Archimedean normed spaces,” Journal of Mathematical Physics, vol. 53, no. 2, Article ID 023507, 9 pages, 2012.
  • A. Bodaghi, I. A. Alias, and M. H. Ghahramani, “Ulam stability of a quartic functional equation,” Abstract and Applied Analysis, vol. 2012, Article ID 232630, 9 pages, 2012.
  • J. Brzd\kek, “Stability of the equation of the p-Wright affine functions,” Aequationes Mathematicae, vol. 85, no. 3, pp. 497–503, 2013.
  • G. Maksa and Z. Páles, “Hyperstability of a class of linear functional equations,” Acta Mathematica, vol. 17, no. 2, pp. 107–112, 2001.
  • M. Piszczek and J. Szczawińska, “Hyperstability of the drygas functional equation,” Journal of Function Spaces and Applications, vol. 2013, Article ID 912718, 4 pages, 2013.