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Approximation septic and octic mappings in quasi-\(\beta\)-normed spaces. (English) Zbl 1279.39024
The stability problem of functional equations originated from a question of S. M. Ulam [A collection of mathematical problems. New York and London: Interscience Publishers (1960; Zbl 0086.24101)] concerning the stability of group homomorphisms. D. H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222–224 (1941; Zbl 0061.26403)] gave a first affirmative partial answer to the question of Ulam for Banach spaces.
The authors introduce and investigate the septic functional equation \[ \begin{split} f(x+4y) -7 f(x+3y) +21 f(x+2y) -35 f(x+y) + 35 f(x) \\ -21f(x-y) + 7 f(x-2y) - f(x-3y) -5040 f(y) =0\end{split}\tag{1} \] and the octic functional equation \[ \begin{split} f(x+4y) -8 f(x+3y) +28 f(x+2y) -56 f(x+y) + 70 f(x) \\ -56f(x-y) + 28 f(x-2y) - 8f(x-3y) +f(x-4y) -40320 f(y) =0.\end{split}\tag{2} \] Using the direct method, the authors prove the Hyers-Ulam stability of the septic functional equation (1) and the octic functional equation (2) in quasi-\(\beta\)-normed spaces; see [T. Z. Xu et al., J. Inequal. Appl. 2010, Article ID 423231, 23 p. (2010; Zbl 1219.39020)] for the concept of quasi-\(\beta\)-normed spaces.
Reviewer’s remark: There are a lot of mathematical errors, e.g.:
1) p. 1110 (\(-19\), \(-18\)): “\(f(x)=x^^7\) is quintic” should be changed to “septic”; “\(f(x)=x^8\) is septic” should be changed ; \(f(x)=x^8\) – is septic (wrong \(\rightarrow\) should be changed into ‘octic’)
2) p. 1111 (+5): What is the meaning of “symmetric” in the 7-additive symmetric map \(A_7 : X^7 \to Y\)? (The notion is not defined.)
3) p. 1111 (\(-9\) – \(-4\)): “degree at most 6” should be changed to “at most 7”), “\(A^6(x)=A^4(x)=A^2(x)=0\)” should be changed to “\(A^6(x)+A^4(x)+A^2(x)=0\)”.
4) p. 1112 (\(-5\)) – p. 1113 (+1): “degree at most 6” should be changed to “at most 8”, “\(A^7(x)=A^5(x)=A^3(x)=A^1(x)=0\)” should be changed to “\(A^7(x)+A^5(x)+A^3(x)+A^1(x)=0\)”.

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges