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.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1965)
Agarwal, R.P., Xu, B., Zhang, W.: Stability of functional equations in single variable. J. Math. Anal. Appl. 288, 852–869 (2003)
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)
Brzdek, J., Brillouët-Bellout, N., Ciepliński, K.: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. (2012) Art ID. 716936
Brzdek, J., Popa, D., Xu, B.: The Hyers-Ulam stability of linear equations of higher orders. Acta Math. Hung. 120, 1–8 (2008)
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 pp
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)
Brzdek, J., Popa, D., Xu, B.: Note on nonstability of the linear functional equation of higher order. Comput. Math. Appl. 62, 2648–2657 (2011)
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)
Castillo, E., Ruiz-Cobo, M.R.: Functional Equations and Modelling in Science and Engineering. Marcel Dekker, New York (1992)
Cesaro, L.: Optimization Theory and Applications. Problems with Ordinary Differential Equations. Springer, New York (1983)
Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge (2002)
Jung, S.-M.: On the modified Hyers–Ulam–Rassias stability of the functional equation for gamma function. Mathematica (Cluj) 39(62), 235–239 (1997)
Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optimization and Its Applications, vol. 48. Springer, New York (2011)
Kuczma, M.: Functional Equations in a Single Variable. Państwowe Wydawnictwo Naukowe, Warszawa (1968)
Pardalos, P.M., Coleman, T.F. (eds.): In: Lectures on Global Optimization. Fields Institute Communications. American Mathematical Society (2009)
Polya, G., Szegö, G.: Aufgaben und Lehrsätze aus der Analysis I. Julius Springer, Berlin (1925)
Rockafellar, T.T.: Convex Analysis. Princeton University Press, Princeton (1972)
Trif, T.: On the stability of a general gamma-type functional equation. Publ. Math. Debrecen 60, 47–61 (2002)
Zwillinger, D.: Standard Mathematical Tables and Formulae, 31st edn. Chapman & Hall/CRC, Boca Raton (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jung, SM., Popa, D. & Rassias, M.T. On the stability of the linear functional equation in a single variable on complete metric groups. J Glob Optim 59, 165–171 (2014). https://doi.org/10.1007/s10898-013-0083-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-013-0083-9
Keywords
- Optimization
- Stability
- Functional equation
- Complete metric group
- Inequalities
- Banach spaces
- Operator mapping
- Euler–Mascheroni constant