an:06165029
Zbl 1279.39024
Xu, Tian Zhou; Rassias, John Michael
Approximation septic and octic mappings in quasi-\(\beta\)-normed spaces
EN
J. Comput. Anal. Appl. 15, No. 6, 1110-1119 (2013).
00316200
2013
j
39B82 39B52
Hyers-Ulam stability; septic functional equation; octic functional equation; quasi-\(\beta\)-normed space
The stability problem of functional equations originated from a question of \textit{S. M. Ulam} [A collection of mathematical problems. New York and London: Interscience Publishers (1960; Zbl 0086.24101)] concerning the stability of group homomorphisms. \textit{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
\[
\begin{multlined} f(x+4y) -7 f(x+3y) +21 f(x+2y) -35 f(x+y) + 35 f(x) \\ -21f(x-y) + 7 f(x-2y) - f(x-3y) -5040 f(y) =0\end{multlined}\tag{1}
\]
and the octic functional equation
\[
\begin{multlined} f(x+4y) -8 f(x+3y) +28 f(x+2y) -56 f(x+y) + 70 f(x) \\ -56f(x-y) + 28 f(x-2y) - 8f(x-3y) +f(x-4y) -40320 f(y) =0.\end{multlined}\tag{2}
\]
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 [\textit{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 ``\(A^7(x)+A^5(x)+A^3(x)+A^1(x)=0\)''.
Choonkil Park (Daejeon)
Zbl 0086.24101; Zbl 0061.26403; Zbl 1219.39020