Abstract and Applied Analysis

Generalized Stability of Euler-Lagrange Quadratic Functional Equation

Hark-Mahn Kim and Min-Young Kim

Full-text: Open access

Abstract

The main goal of this paper is the investigation of the general solution and the generalized Hyers-Ulam stability theorem of the following Euler-Lagrange type quadratic functional equation f ( a x + b y ) + a f ( x - b y ) = ( a + 1 ) b 2 f ( y ) + a ( a + 1 ) f ( x ) , in ( β , p ) -Banach space, where a , b are fixed rational numbers such that a - 1,0 and b 0 .

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 219435, 16 pages.

Dates
First available in Project Euclid: 7 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1399487773

Digital Object Identifier
doi:10.1155/2012/219435

Mathematical Reviews number (MathSciNet)
MR2959767

Zentralblatt MATH identifier
1246.39026

Citation

Kim, Hark-Mahn; Kim, Min-Young. Generalized Stability of Euler-Lagrange Quadratic Functional Equation. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 219435, 16 pages. doi:10.1155/2012/219435. https://projecteuclid.org/euclid.aaa/1399487773


Export citation

References

  • S. M. Ulam, A Collection of the Mathematical Problems, Interscience Tracts in Pure and Applied Mathe-matics No. 8, 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.
  • 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 Th. 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.
  • C. Borelli and G. L. Forti, “On a general Hyers-Ulam stability result,” International Journal of Mathe-matics and Mathematical Sciences, vol. 18, no. 2, pp. 229–236, 1995.
  • S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002.
  • G. L. Forti, “Hyers-Ulam stability of functional equations in several variables,” Aequationes Mathema-ticae, vol. 50, no. 1-2, pp. 143–190, 1995.
  • D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and their Applications 34, Birkhäuser, Boston, Mass, USA, 1998.
  • S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001.
  • Th. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23–130, 2000.
  • J. M. Rassias and H.-M. Kim, “Generalized Hyers-Ulam stability for general additive functional equa-tions in quasi-$\beta $-normed spaces,” Journal of Mathematical Analysis and Applications, vol. 356, no. 1, pp. 302–309, 2009.
  • 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.
  • J. M. Rassias, “On the stability of the Euler-Lagrange functional equation,” Chinese Journal of Mathe-matics, vol. 20, no. 2, pp. 185–190, 1992.
  • M. E. Gordji and H. Khodaei, “On the generalized Hyers-Ulam-Rassias stability of quadratic func-tional equations,” Abstract and Applied Analysis, vol. 2009, Article ID 923476, 11 pages, 2009.
  • K. Jun, H. Kim, and J. Son, “Generalized Hyers-Ulam stability of a quadratic functional equation,” in Functional Equations in Mathematical Analysis, Th. M. Rassias and J. Brzdek, Eds., chapter 12, pp. 153–164, 2011.
  • K.-W. Jun and H.-M. Kim, “Ulam stability problem for generalized $A$-quadratic mappings,” Journal of Mathematical Analysis and Applications, vol. 305, no. 2, pp. 466–476, 2005.
  • J.-H. Bae and W.-G. Park, “Stability of a cauchy-jensen functional equation in quasi-banach spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 151547, 9 pages, 2010.
  • M. E. Gordji and H. Khodaei, “Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces,” Nonlinear Analysis: Theory, Methods & Applica-tions, vol. 71, no. 11, pp. 5629–5643, 2009.
  • A. Najati and G. Z. Eskandani, “Stability of a mixed additive and cubic functional equation in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 1318–1331, 2008.
  • A. Najati and F. Moradlou, “Stability of a quadratic functional equation in quasi-Banach spaces,” Bulletin of the Korean Mathematical Society, vol. 45, no. 3, pp. 587–600, 2008.
  • T. Z. Xu, J. M. Rassias, M. J. Rassias, and W. X. Xu, “A fixed point approach to the stability of quintic and sextic functional equations in quasi-$\beta $-normed spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 423231, 23 pages, 2010.
  • L. G. Wang and B. Liu, “The Hyers-Ulam stability of a functional equation deriving from quadratic and cubic functions in quasi-$\beta $-normed spaces,” Acta Mathematica Sinica (English Series), vol. 26, no. 12, pp. 2335–2348, 2010.