×

Solution and stability of a cubic functional equation. (English) Zbl 1201.39024

Similar to the method used by H.-Y Chu and D.-S. Kang [J. Math. Anal. Appl. 325, No. 1, 595–607 (2007; Zbl 1106.39025)] and A. Najati and Ch. Park [Acta Math. Sin., Engl. Ser. 24, No. 12, 1953–1964 (2008; Zbl 1159.39014)], the authors obtain the general solution and investigate the stability of the following cubic functional equation
\[ f(x+ny)+f(x-ny)+f(nx)=n^2f(x+y)+n^2f(x-y)+(n^3-2n^2+2)f(x), \]
where \(n\geq 2\) is an integer and \(f\) is a map between real vector spaces. They also investigate the stability of the equation above by using the fixed point method.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ulam, S. M.: A Collection of Mathematical Problems, Interscience Publishers, New York, 1968, p. 63 · Zbl 0184.14802
[2] Hyers, D. H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci., 27, 222–224 (1941) · Zbl 0061.26403
[3] Rassias, Th. M.: On the stability of linear mappings in Banach spaces. Proc. Amer. Math. Soc., 72, 297–300 (1978) · Zbl 0398.47040
[4] Gǎvruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl., 184, 431–436 (1994) · Zbl 0818.46043
[5] Jun, K.-W., Kim, H.-M.: The Hyers-Ulam-Rassias stability of a cubic functional equation. J. Math. Anal. Appl., 274, 867–878 (2002) · Zbl 1021.39014
[6] Chu, H.-Y., Kang, D.-S.: On the stability of an n-dimensional cubic functional equation. J. Math. Anal. Appl., 325, 595–607 (2007) · Zbl 1106.39025
[7] Najati, A., Park, C.: On the stability of a cubic functional equation. Acta Mathematica Sinica, English Series, 24, 1953–1964 (2008) · Zbl 1159.39014
[8] Jun, K.-W., Kim, H.-M., Chang, I.-S.: On the Hyers-Ulam stability of an Euler-Lagrange type cubic functional equation. J. Comput. Anal. Appl., 7, 21–33 (2005) · Zbl 1087.39029
[9] Margolis, B., Dias, J. B.: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Amer. Math. Soc., 126, 305–309 (1968) · Zbl 0157.29904
[10] Rus, I. A.: Principles and Applications of Fixed Point Theory (in Romanian), Ed. Dacia, Cluj-Napoca, 1979
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.