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Remark on hyperstability of the general linear equation. (English) Zbl 1304.39033
Let \(\mathbb{F}\) and \(\mathbb{K}\) be fields of real or complex numbers. Using a fixed point theorem of J. Brzdȩk et al. [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 17, 6728–6732 (2011; Zbl 1236.39022)], the authors prove the following theorem:
Let \(X\) be a normed space over the field \(\mathbb{F}\), \(Y\) be a Banach space over \(\mathbb{K}\), \(a, b \in \mathbb{F} \setminus \{0\}\), \(A, B \in \mathbb{K}\), \(c \geq 0\), \(p < 0\) and \( g: X \to Y\) satisfy \[ \|g(ax + by) - Ag(x) - Bg(y)\| \leq c( \|x\|^{p} + \|y\|^{p}), \;x, y \in X \setminus \{0\}. \] Then \(g\) satisfies the functional equation \[ g(ax + by) = Ag(x) + Bg(y), \;\;x, y \in X \setminus \{0\}. \] .

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
Full Text: DOI
[1] Aoki, T., On the stability of the linear transformation in Banach spaces, J. Math. Soc. Jpn., 2, 64-66, (1950) · Zbl 0040.35501
[2] Badea, C., The general linear equation in stable, Nonlinear Funct. Anal. Appl., 10, 155-164, (2005) · Zbl 1076.39023
[3] Bourgin, D.G., Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16, 385-397, (1949) · Zbl 0033.37702
[4] Brzdȩk, J.: Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hungar. doi:10.1007/s10474-013-0302-3
[5] Brzdȩk, J.; Chudziak, J.; Páles, Zs., A fixed point approach to stability of functional equations, Nonlinear Anal., 74, 6728-6732, (2011) · Zbl 1236.39022
[6] Brzdȩk, J.; Pietrzyk, A., A note on stability of the general linear equation, Aequationes Math., 75, 267-270, (2008) · Zbl 1149.39018
[7] Gajda, Z., On stability of additive mappings, Int. J. Math. Math. Sci., 14, 431-434, (1991) · Zbl 0739.39013
[8] Gǎvruţa, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mapping, J. Math. Anal. Appl., 184, 431-436, (1994) · Zbl 0818.46043
[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. Birkhäuser, Berlin (1998) · Zbl 0907.39025
[11] Jung S.M.: Hyers-Ulam stability of functional equations in mathematical analysis. Hadronic Press, Palm Harbor (2001) · Zbl 0980.39024
[12] Kuczma M.: An introduction to the theory of functional equation and inequalities. PWN, Warszawa (1985) · Zbl 0555.39004
[13] Maksa, Gy.; Páles, Zs., Hyperstability of a class of linear functional equations, Acta Math. Acad. Paedag. Nyìregyháziensis, 17, 107-112, (2001) · Zbl 1004.39022
[14] Popa, D., Hyers-Ulam-Rassias stability of the general linear equation, Nonlinear Funct. Anal Appl., 7, 581-588, (2002) · Zbl 1031.39021
[15] Rassias, Th.M., On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc., 72, 297-300, (1978) · Zbl 0398.47040
[16] Rassias, Th.M., On a modified Hyers-Ulam sequence, J. Math. Anal. Appl., 158, 106-113, (1991) · Zbl 0746.46038
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