Advertisement

Journal of Global Optimization

, Volume 59, Issue 1, pp 165–171 | Cite as

On the stability of the linear functional equation in a single variable on complete metric groups

  • Soon-Mo Jung
  • Dorian Popa
  • Michael Th. Rassias
Article

Abstract

In this paper we obtain a result on Hyers–Ulam stability of the linear functional equation in a single variable \(f(\varphi (x)) = g(x) \cdot f(x)\) on a complete metric group.

Keywords

Optimization Stability Functional equation Complete metric group Inequalities Banach spaces Operator mapping Euler–Mascheroni constant 

Mathematics Subject Classification

33B15 11B34 41A30 39B22 

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1965)Google Scholar
  2. 2.
    Agarwal, R.P., Xu, B., Zhang, W.: Stability of functional equations in single variable. J. Math. Anal. Appl. 288, 852–869 (2003)CrossRefGoogle Scholar
  3. 3.
    Brydak, J.: On the stability of the functional equation \(\varphi [f(x)]=g(x)\varphi (x)+F(x)\). Proc. Am. Math. Soc. 26, 455–460 (1970)Google Scholar
  4. 4.
    Brzdek, J., Brillouët-Bellout, N., Ciepliński, K.: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. (2012) Art ID. 716936Google Scholar
  5. 5.
    Brzdek, J., Popa, D., Xu, B.: The Hyers-Ulam stability of linear equations of higher orders. Acta Math. Hung. 120, 1–8 (2008)CrossRefGoogle Scholar
  6. 6.
    Brzdek, J., Jung, S.-M.: A note on stability of an operator equation of the second order. Abstr. Appl. Anal. (2011). Art. ID 602713, 15 ppGoogle Scholar
  7. 7.
    Brzdek, J., Popa, D., Xu, B.: On approximate solutions of the linear functional equation of higher order. J. Math. Anal. Appl. 373, 680–689 (2011)CrossRefGoogle Scholar
  8. 8.
    Brzdek, J., Popa, D., Xu, B.: Note on nonstability of the linear functional equation of higher order. Comput. Math. Appl. 62, 2648–2657 (2011)CrossRefGoogle Scholar
  9. 9.
    Brzdek, J., Popa, D., Xu, B.: Selections of set-valued maps satisfying a linear inclusion in a single variable. Nonlinear Anal. 74, 324–330 (2011)CrossRefGoogle Scholar
  10. 10.
    Castillo, E., Ruiz-Cobo, M.R.: Functional Equations and Modelling in Science and Engineering. Marcel Dekker, New York (1992)Google Scholar
  11. 11.
    Cesaro, L.: Optimization Theory and Applications. Problems with Ordinary Differential Equations. Springer, New York (1983)Google Scholar
  12. 12.
    Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge (2002)CrossRefGoogle Scholar
  13. 13.
    Jung, S.-M.: On the modified Hyers–Ulam–Rassias stability of the functional equation for gamma function. Mathematica (Cluj) 39(62), 235–239 (1997)Google Scholar
  14. 14.
    Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optimization and Its Applications, vol. 48. Springer, New York (2011)Google Scholar
  15. 15.
    Kuczma, M.: Functional Equations in a Single Variable. Państwowe Wydawnictwo Naukowe, Warszawa (1968)Google Scholar
  16. 16.
    Pardalos, P.M., Coleman, T.F. (eds.): In: Lectures on Global Optimization. Fields Institute Communications. American Mathematical Society (2009)Google Scholar
  17. 17.
    Polya, G., Szegö, G.: Aufgaben und Lehrsätze aus der Analysis I. Julius Springer, Berlin (1925)CrossRefGoogle Scholar
  18. 18.
    Rockafellar, T.T.: Convex Analysis. Princeton University Press, Princeton (1972)Google Scholar
  19. 19.
    Trif, T.: On the stability of a general gamma-type functional equation. Publ. Math. Debrecen 60, 47–61 (2002)Google Scholar
  20. 20.
    Zwillinger, D.: Standard Mathematical Tables and Formulae, 31st edn. Chapman & Hall/CRC, Boca Raton (2003)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Soon-Mo Jung
    • 1
  • Dorian Popa
    • 2
  • Michael Th. Rassias
    • 3
  1. 1.Mathematics Section, College of Science and TechnologyHongik UniversitySejongRepublic of Korea
  2. 2.Department of MathematicsTechnical University of Cluj-NapocaCluj-NapocaRomania
  3. 3.Department of MathematicsETH-ZürichZurichSwitzerland

Personalised recommendations