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Zbl 1072.39024
Lee, Sang Han; Im, Sung Mo; Hwang, In Sung
Quartic functional equations. (English)
[J] J. Math. Anal. Appl. 307, No. 2, 387-394 (2005). ISSN 0022-247X

In analogy to the ``quadratic functional equation" $$f(x+y)+f(x-y)=2f(x)+2f(y),$$ that is, $_s\Delta^2_y f(x)=2f(y),$ the authors call $$f(2x+y)-4f(x+y)+6f(y)-4f(x-y)+f(2x-y)=4! f(x)$$ (rather than $_s\Delta^4_y f(x):= f(x+2y)-4f(x+y)+6f(x)-4f(x-y)+f(x-2y)=4! f(y)$) ``quartic functional equation". They offer its general solution from the real vector space into a real vector space (using solutions of the quadratic equation and four pages of calculations including up to 18-line equations) and a stability theorem for functions from a real normed linear space into a real Banach space.
[János Aczél (Waterloo/Ontario)]

MSC 2000:: *39B52 Functional equations for functions with more general domains
39B42 Matrix and operator functional equations
39B82 Stability, separation, extension, and related topics
46B20 Geometry and structure of normed spaces
46B25 Classical Banach spaces in the general theory of normed spaces

Keywords: stability; real vector spaces; real normed linear spaces; real Banach spaces; quadratic functional equation; quartic functional equation; general solution

Cited in: Zbl pre06112884 Zbl 1117.39020

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