Quartic functional equations. (English) Zbl 1072.39024

Lee, Sang Han; Im, Sung Mo; Hwang, In Sung

J. Math. Anal. Appl. 307, No. 2, 387-394 (2005).

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.

Reviewer: János Aczél (Waterloo/Ontario)

MSC:

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

Keywords:

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

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