×

zbMATH — the first resource for mathematics

On the Hyers-Ulam stability of functional equations connected with additive and quadratic mappings. (English) Zbl 1101.39017
The author states two lemmas on the stability of the single variable functional equation \(\varphi(2x)=3\varphi(x)+\varphi(-x) \quad (x \in G)\), where \(\varphi\) is a mapping from an abelian group \((G,+)\) into a Banach space \(X\); see G. L. Forti [J. Math. Anal. Appl. 295, No. 1, 127–133 (2004; Zbl 1052.39031)]. Then the author applies the lemmas to investigate the stability of functional equations \[ \begin{aligned}\varphi(x+y)+\varphi(x-y)&= 2\varphi(x)+\varphi(y)+\varphi(-y),\\ \varphi(x+y+z)+\varphi(x)+\varphi(y)+\varphi(z)&= \varphi(x+y)+\varphi(y+z)+\varphi(z+x), \end{aligned} \] which have solutions of the form \(\varphi=a+q\) where \(a\) is an additive mapping and \(q\) is a quadratic one. The obtained results may regard as complementary to those of S.-M. Jung and P. K. Sahoo [Aequationes Math. 64, No. 3, 263–273 (2002; Zbl 1022.39028)] and S.-M. Jung [J. Math. Anal. Appl. 222, No. 1, 126–137 (1998; Zbl 0928.39013)].

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
PDF BibTeX Cite
Full Text: DOI
References:
[1] Aczél, J.; Dhombres, J., Functional equations in several variables, Encyclopedia math. appl., vol. 31, (1989), Cambridge Univ. Press Cambridge · Zbl 0685.39006
[2] Borelli, C.; Forti, G.L., On a general hyers – ulam stability result, Int. J. math. math. sci., 18, 229-236, (1995) · Zbl 0826.39009
[3] Cholewa, P.W., Remarks on the stability of functional equations, Aequationes math., 27, 76-86, (1984) · Zbl 0549.39006
[4] Drygas, H., Quasi-inner products and their applications, (), 13-30
[5] Ebanks, B.R.; Kannappan, Pl.; Sahoo, P.K., A common generalization of functional equations characterizing normed and quasi-inner-product spaces, Canad. math. bull., 35, 321-327, (1992) · Zbl 0712.39021
[6] Fechner, W., On functions with the Cauchy difference bounded by a functional, Bull. Polish acad. sci. math., 52, 265-271, (2004) · Zbl 1099.39018
[7] Fechner, W., On functions with the Cauchy difference bounded by a functional, part II, Int. J. math. sci., 2005, 17, 1889-1898, (2005) · Zbl 1159.39012
[8] Forti, G.L., Comments on the core of the direct method for proving hyers – ulam stability of functional equations, J. math. anal. appl., 295, 127-133, (2004) · Zbl 1052.39031
[9] Hyers, D.H., On the stability of the linear functional equation, Proc. natl. acad. sci. USA, 27, 222-224, (1941) · Zbl 0061.26403
[10] Hyers, D.H.; Isac, G.; Rassias, Th.M., Stability of functional equations in several variables, (1998), Birkhäuser Boston Boston · Zbl 0894.39012
[11] Jung, S.-M., On the hyers – ulam stability of the functional equations that have the quadratic property, J. math. anal. appl., 222, 126-137, (1998) · Zbl 0928.39013
[12] Jung, S.-M.; Sahoo, P.K., Stability of a functional equation of drygas, Aequationes math., 64, 263-273, (2002) · Zbl 1022.39028
[13] Kannappan, Pl., Quadratic functional equation and inner product spaces, Results math., 27, 368-372, (1995) · Zbl 0836.39006
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.