# zbMATH — the first resource for mathematics

Hyperstability of general linear functional equation. (English) Zbl 1341.39013
Let $$X,Y$$ be linear spaces over the real or over the complex field, $$g: X\to Y$$, $$A,a_{ij}\in\mathbb{R}$$ or $$\mathbb{C}$$, $$A_i\not=0$$. The main theorem is about hyperstability of the following functional equation $\sum_{i=1}^mA_ig\left(\sum_{j=1}^na_{ij}x_j\right)+A=0,$ assuming certain side conditions for the control function.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges
Full Text:
##### References:
 [1] Bahyrycz, A.; Piszczek, M., Hyperstability of the Jensen functional equation, Acta Math. Hung., 142, 353-365, (2014) · Zbl 1299.39022 [2] Bahyrycz, A., Brzdęk, J., Piszczek, M., Sikorska, J.: Hyperstability of the Fréchet equation and a characterization of inner product spaces. J. Funct. Spaces Appl. Art. ID 496361. (2013). doi:10.1155/2013/496361 · Zbl 1304.39033 [3] Bahyrycz, A., Olko, J.: On stability of the general linear equation. Aequationes Math. doi:10.1007/s00010-014-0317-z · Zbl 1337.39007 [4] Bourgin, D.G., Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16, 385-397, (1949) · Zbl 0033.37702 [5] Brzdęk, J.; Chudziak, J.; Páles, Zs., A fixed point approach to stability of functional equations, Nonlinear Anal., 74, 6728-6732, (2011) · Zbl 1236.39022 [6] Brzdęk, J., Stability of the equation of the $$p$$-wright affine functions, Aequationes Math., 85, 497-503, (2013) · Zbl 1272.39015 [7] Brzdęk, J., Remarks on hyperstability of the Cauchy functional equation, Aequationes Math., 86, 255-267, (2013) · Zbl 1303.39016 [8] Brzdęk, J., Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hung., 141, 58-67, (2013) · Zbl 1313.39037 [9] Brzdęk, J.: A hyperstability result for the Cauchy equation. Bull. Aust. Math. Soc. (2013). doi:10.1017/S0004972713000683 · Zbl 1299.39022 [10] Brzdęk, J., Ciepliński, K.: Hyperstability and Superstability. Abstr. Appl. Anal. (2013). doi:10.1155/2013/401756 · Zbl 1174.39006 [11] Brzdęk, J.: Remarks on stability of some inhomogeneous functional equations. Aequationes Math. doi:10.1007/s00010-014-0274-6 [12] Cǎdariu, L., Radu, V.: Fixed point and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 2. Art. ID 4 (2003) · Zbl 1149.39024 [13] Daróczy, Z.; Lajkó, K.; Lovas, R.L.; Maksa, G.; Páles, Zs., Functional equations involving means, Acta Math. Hung., 166, 79-87, (2007) · Zbl 1174.39006 [14] Gselmann, E., Hyperstability of a functional equation, Acta Math. Hung., 124, 179-188, (2009) · Zbl 1212.39044 [15] Maksa, G., The stability of the entropy of degree alpha, J. Math. Anal. Appl., 346, 17-21, (2008) · Zbl 1149.39024 [16] Maksa, G.; Páles, Zs., Hyperstability of a class of linear functional equations, Acta Math. Acad. Paedag. Nyìregyháziensis, 17, 107-112, (2001) · Zbl 1004.39022 [17] Piszczek, M., Remark on hyperstability of the general linear equation, Aequationes Math., 88, 163-168, (2014) · Zbl 1304.39033 [18] Piszczek, M., Szczawińska, J.: Hyperstability of the Drygas functional equation. J. Funct. Spaces Appl. Art. ID 912718. (2013) · Zbl 1313.39037 [19] Popa, D.; Raşa, I., The Fréchet functional equation with application to the stability of certain operators, J. Approx. Theory, 164, 138-144, (2012) · Zbl 1238.39008
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.